Shape Optimization of Shear-type Hysteretic Steel Damper for Building Frames using FEM-Analysis and Heuristic Approach

Transcription

1 The Seventh Chna-Japan-Korea Jont Symposum on Optmzaton of Structural and Mechancal Systems Huangshan, June, 18-21, 2012, Chna Shape Optmzaton of Shear-type Hysteretc Steel Damper for Buldng Frames usng FEM-Analyss and Heurstc Approach Hdekazu Watanabe 1*, Makoto Ohsak 1, Junk Nozoe 1 and Satosh Yasuda 2 1 Dept. of Archtecture, Hroshma Unversty, Kagamyama 1-4-1, Hgash-Hroshma, , Japan * {hdekazu-watanabe, ohsak, m124850}@hroshma-u.ac.jp 2 Technology Center, Tase Corporaton, Nase-cho 344-1, Totsuka-ku, Yokohama, , Japan Abstract An optmzaton approach s presented for mprovng energy dsspaton capacty of a passve control devce under cyclc deformaton. A general purpose fnte element software package s combned wth a heurstc optmzaton algorthm. It s demonstrated n the numercal examples that the performance of a shear-type hysteretc steel damper can be successfully mproved by optmzng the geometry of the devce and the thcknesses of the stffeners. Keywords: Shape Optmzaton, Tabu Search, Energy Dsspaton Capacty, Shear-Type Steel Damper 1. Introducton Recently, shear-type hysteretc steel damper s extensvely used for passve control of sesmc responses of buldng frames. The damper s located between the upper and lower beams of a story, and conssts of a low-yeld-pont steel panel that dsspates sesmc energy under forced nterstory. The panel s stffened by several stffeners to prevent local bucklng that leads to premature fracture before a suffcent amount of energy s dsspated. However, n the exstng shear-type hysteretc steel dampers, the locatons and thcknesses of stffeners are emprcally or ntutvely decded by the engneers, and the performance of the damper s confrmed by a physcal test under statc cyclc loadng. The frst author demonstrated through a seres of studes that the performances of structural parts ncludng beams and braces can be successfully mproved through heurstc optmzaton algorthms combned wth hgh-precson fnte element analyss [1,2]. In ths study, we optmze the locatons and thcknesses of the stffeners as well as the aspect rato of the shear-type steel damper. The elasto-plastc responses under statc cyclc loadng are evaluated usng a general-purpose fnte element analyss software package called ABAQUS [3]. The consttutve relaton of the low-yeld-pont steel s defned usng a nonlnear sotropc-knematc hardenng model. The parameters of the consttutve model are dentfed from the exstng materal test results under unaxal cyclc loadng. The panel, flanges, and stffeners are dscretzed nto shell elements. Snce the optmzaton problem s hghly nonlnear, we dscretze the varables nto nteger values and formulate the problem as a combnatoral optmzaton problem. A heurstc optmzaton algorthm called tabu search [4] (hereafter referred to as TS), whch s based on a sngle-pont local search, s used for obtanng approxmate optmal solutons wthn a small number of functon evaluatons. For ths purpose, a tool s developed usng a scrpt language called Python for automatcally exchangng the data between the ABAQUS and the optmzaton program. It s demonstrated n the numercal examples that the performance of a shear-type hysteretc steel damper can be successfully mproved by optmzng the thcknesses and locatons of the stffeners, as well as the aspect rato of panel. By utlzng ths tool, the cost and the tme perod for development of passve control devces can be drastcally reduced and the performances of the devces can be sgnfcantly mproved. 2. Optmzaton algorthm 2.1 Outlne of the TS A heurstc optmzaton algorthm called TS, whch s classfed as a sngle-pont local search [4,5], s used to obtan approxmate optmal solutons wthn a small number of functon evaluatons. In contrast to a populaton-based approach such as genetc algorthm, TS has sngle soluton at each step of local search. Therefore, TS s more sutable

2 than a genetc algorthm for structural optmzaton problems that demand large computatonal cost for response evaluaton. TS bascally moves to the best neghborhood soluton even f t does not mprove the current soluton. A tabu lst s used to prevent an unfavorable phenomenon called cyclng, n whch several solutons are selected alternatvely. Let J = (J 1,.., J m ) denote the vector of m desgn varables for a combnatoral optmzaton problem to maxmze the objectve functon F(J). The basc algorthm of TS s summarzed as follows: Step 1 Randomly generate a seed soluton J = (J 1,.., J m ) and ntalze the tabu lst T as T = {J}. Evaluate the objectve functon and ntalze the ncumbent optmal objectve value as F opt = F(J). Step 2 Generate a set of q neghborhood solutons J = (J 1,.., J m), (j=1,, q) from J, and evaluate the objectve functon of each soluton. Step 3 Among the solutons n the set {J N1,, J Nq }, select the best one that has the maxmum value of objectve functon, and s not ncluded n the lst T. Assgn the best soluton as the new seed soluton J. Step 4 Update the ncumbent optmal objectve functon as F opt = F(J) when the value s mproved. Step 5 Add the seed soluton J to the lst T f the sze of tabu lst s less than the prescrbed lmt. Step 6 Output F opt and the correspondng optmal soluton, f the number of teratons reaches the specfed value; otherwse, go to Step 2. In Step 2, the neghborhood soluton J1 J m J (,, ) s as defned follows usng unform random numbers r: r : J J 1 (1a) r : r : J J J (1b) J 1 (1c) 2.2 Formulaton of optmzaton algorthm The TS has been developed for optmzaton problems wth nteger varables. Therefore, the desgn varables are dscretzed nto nteger values. Real varables X 1,, X m are defned by nteger varables J 1,, J m wth the specfed standard value X 0 and ncrement ΔX as X X0 JX ( 1,, m) (2) Therefore, all responses of analyss are functons of J. The objectve functon s the total dsspated energy E p. We maxmze E p under a constrant g() J g0, whch s defned for each optmzaton problem. Let s denote the number of samplng values that J can take. The optmzaton problem s formulated as follows: Maxmze F(J) = E p (J) (3a) subject to g() J g0 (3b) J 1,2,, s, ( 1,, m) (3c) 2.3 Optmzaton Process Fg. 1 shows the data flow between TS and FE-analyss usng ABAQUS Ver [3]. The pre-process and post-process are carred out usng the Python scrpt. The computatons of functons and the process of TS are coded usng Fortran.

3 Tabu search (1) Compute objectve functon. (2) Update varables. Post-process (Python Scrpt) Extract responses: dsspated energy, stresses and strans FE-model generaton (Python Scrpt) (1) Generate parts of flange, web, etc. (2) Defne sectons and materal parameters. (3) Combne parts to assemble set. (4) Assgn boundary and loadng condtons. (5) Generate meshes. Output fle: job.odb Analyss usng ABAQUS/Standard Input fle: job.np Fgure 1. Optmzaton algorthm usng TS and ABAQUS. 3. Shear-type hysteretc steel damper 3.1 Standard Model The expermental specmen of the shear-type hysteretc steel damper s shown n Fg. 2. The sze of the specmen s 2/3 of the real sze. The specmen s a stud-type vscoelastc damper, whch s extensvely used as passve control devce of sesmc responses of buldng frames. The materal of center panel of the devce, as shown n hatched area n Fg. 2, s a low-yeld-pont steel. When the devce s attached between the beams, the panel yelds due to the nterstory shear force pror to the bracket of the devce and other members of a frame. Ths way, the panel zone can absorb the earthquake energy effcently wthout damagng other parts. The bucklng restranng stffeners are assgned to prevent premature out-of-plane bucklng of the panel. The stffeners are located longtudnally n the front sde, and laterally n the rear sde of the panel. The materal of the panel, flanges, and bucklng restranng stffeners are LY100 (low-yeld-pont steel), SM490A, and SS400, respectvely. The values of Young's modulus E, Posson's rato ν, yeld stress σ y, and tensle strength σ u obtaned by unaxal tests are lsted n Table1. Fgure 2. An expermental specmen; left: front sde, rght: rear sde) A fnte element model called the standard model s generated from the expermental specmen n Fg.2. The fnte-element mesh s generated usng python scrpt. A quadrlateral thck shell element called S4R s used, and the nomnal sze of mesh s 40 mm for automatc mesh generaton.

4 A constant dstrbuted load that s equvalent to a compressve axal force of 710 kn s frst appled at the top plate pror to the forced horzontal cyclc deformaton. The loadng program of the statc shear force of fve cycles s controlled by the peak drft angle of 1/60 rad. There have been many crtera for ductle fracture of steel materals. However, we use the equvalent plastc stran as a measure of damage of the materal, because most of the crtera are based on the equvalent plastc stran. The results of standard model are shown n Table 2, where E p, ε max, R max, and M max denote the total dsspated energy, the maxmum equvalent plastc stran among all elements, the maxmum horzontal reacton force, and the maxmum reacton moment, respectvely. Table 1. Materal propertes E (kn/mm 2 ) ν σy (N/mm 2 ) σu (N/mm 2 ) LY SM490A (PL-19) SM490A (PL-16) SS Table 2. Analyss Results Ep(kN m) εmax Rmax(kN) Mmax(kN m) Standard Model Fg. 3 shows dstrbuton of equvalent plastc stran of the standard model at fnal step. It can be observed from ths fgure that the equvalent plastc stran has larger values manly n the red regon around the center of the shear panel. Fgure 3. Dstrbuton of equvalent plastc stran of the standard model at fnal step; left: front sde, rght: rear sde. 4. Optmzaton of the shear-type hysteretc steel damper In ths secton, we demonstrate n the numercal examples that the performance of a shear-type hysteretc steel damper can be successfully mproved by optmzaton. The locatons and thcknesses of the bucklng restranng stffeners are optmzed. The aspect rato of the panel s next optmzed. Fnally, all varables n the frst and second problems are optmzed. The objectve functon to be maxmzed s the total dsspated energy E p when the maxmum value of equvalent plastc stran among all elements reaches ε max of the standard model shown n Table Optmzaton of locatons and thcknesses of bucklng restranng stffeners In ths optmzaton problem, the vector of desgn varables conssts of the locatons of the bucklng restranng stffeners (J 1, J 2 ) and the thcknesses of the stffeners (J 3, J 4 ). In order to preserve the symmetry, the numbers of ndependent varables for locaton and thckness are one, respectvely, for the stffeners n front and rear. The locaton X

5 of the stffeners at each sde s defned by J 1 and J 2 as X JX J 1,2,,7, ( 1, 2) (4) where X1 and X2 are 1/18 of the wdth and heght of the panel, respectvely. The ndependent varables for thcknesses T 1 and T 2 of the stffeners n two sded are defned by S 1 and S 2 as T0.002 ST, S 1,,8, ( 1, 2) (5) where T1 and T2 are m. For the TS, the number of neghborhood solutons s q = 4, and the number of steps n = 15. Optmzaton s carred out from three dfferent random ntal solutons to nvestgate dependence of the optmal soluton on the ntal soluton. The maxmum value was obtaned for E p n two trals; therefore, ths soluton s regarded as the optmal soluton. The optmal values of nteger varables are (J 1, J 2, S 1, S 2 ) = (3, 1, 7, 6). The soluton s hereafter called the optmal soluton S. As the result of optmzaton the total dsspated energy E p was mproved by 19.1 %from the standard model. The dstrbuton of equvalent plastc stran of the optmal soluton S at fnal step s shown n Fg. 4. In ths optmal soluton, the lateral bucklng restranng stffeners n the rear sde are shfted to the center and the thcknesses of stffeners n both sdes are ncreased from the standard model. It s seen from Fgs. 3 and 4 that the equvalent plastc stran has large value n wder regon as a result of optmzaton so that the large deformaton around the center of the standard model s restraned by movng the lateral stffeners n the rear sde to the center. Ths way, the energy dsspaton property can be mproved through ths optmzaton. In ths model, the effect of optmzaton s summarzed as follows: 1. More energy can be dsspated to ncrease the area of large plastc deformaton. 2. The maxmum equvalent plastc stran can be reduced by ncreasng the stffnesses of the stffeners; hence, the number of cycles before reachng the specfed bound of maxmum stran can be ncreases. Note that a too much ncrease of the stffness n the panel leads to a large deformaton of brackets, and accordngly, small deformaton of the panel. Therefore, the most approprate thcknesses of the bucklng restranng stffeners have been selected by optmzaton. Fgure 4. Dstrbuton of equvalent plastc stran of Model S at fnal step; left: front sde, rght: rear sde. 4.2 Optmzaton of aspect rato of the panel We next optmze the aspect rato of the panel. In order to preserve symmetry property, the ndependent varables are the panel wdth H 1 and the panel heght H 2, whch are defned by two nteger varables K 1 and K 2 as H1 K17 H1 K1 1,2,,5 (6a) H K 7 H K 1,2,,8 (6b) where H1 and H2 are 1/10 of the wdth and heght, respectvely, of the panel of the standard model. The enumeraton method s used here, because the number of varables s 2, and the small numbers 5 and 8 are assgned for samplng values of the varables. The constrants are added to consder the nfluence of the desgn modfcaton of the panel on the responses of buldng frames;.e., the maxmum horzontal reacton force R max and the maxmum reacton moment M max should be less than the values of the standard model as shown n Table 2. As the result of enumeraton, the maxmum value of E p s correspondng to the optmal soluton (K 1, K 2 ) = (4, 7). Ths soluton

6 s hereafter called the optmal soluton W, the soluton acheved 31.1% ncrease of the total dsspated energy E p from the standard model. The dstrbuton of equvalent plastc stran of the optmal soluton W at fnal step s shown n Fg. 5. The optmal soluton W has larger panel wdth and panel heght than those of the standard model. Hence, the ncrease of the panel volume as a result of optmzaton leads to the mprovement of energy dsspaton property. However, f the heght becomes too large, then the average shear angle s reduced, and we cannot have enough plastc deformaton n the panel. Therefore, the heght dd not reach ts upper bound through optmzaton. Fgure 5. Dstrbuton of equvalent plastc stran of Model W at fnal step; left: front sde, rght: rear sde. 4.3 Smultaneous optmzaton of stffeners and aspect rato In ths secton, we optmze the thcknesses and locatons of the bucklng restranng stffeners, as well as the aspect rato of the panel, smultaneously. Hence, the vectors of desgn varables are J, S and K defned n Secs. 4.2 and 4.2. TS s used for optmzaton wth the number of neghborhood solutons q = 6 and the number of steps n = 15. Optmzaton was frst carred out from a randomly generated ntal soluton under constrants on reacton force and moment defned n Sec. 4.3 to obtan the maxmum value for E p by the soluton (J 1, J 2, S 1, S 2, K 1, K 2 ) = (3, 1, 7, 3, 4, 8). Ths soluton was found at the 13th step, whch s nearly the end of the total 15 steps. Therefore, optmzaton of 15 steps was carred out agan from ths soluton to obtan the maxmum value of the E p by the optmal soluton (J 1, J 2, S 1, S 2, K 1, K 2 ) = (3, 1, 8, 5, 4, 8). The soluton s hereafter called the optmal soluton SW, whch acheves 60.5% ncrease of the total dsspated energy E p from the standard model. The dstrbuton of equvalent plastc stran of the optmal soluton SW at fnal step s shown n Fg. 6. Snce S 1 and K 2 of the optmal soluton SW have ther upper-bound values, t s possble to optmze these varables wth larger upper bounds. Fgure 5. Dstrbuton of equvalent plastc stran of Model SW at fnal step; left: front sde, rght: rear sde.

7 5. Dscussons on performance of optmzaton Optmzaton results are summarzed n Table 3. In the optmal soluton S, the maxmum performance s acheved through optmzaton of stffeners usng the panel of the standard model. On the other hand, n the optmal soluton W, the geometry of panel s also mproved. The performance of devce s further mproved by the optmal soluton SW through smultaneous optmzaton of the stffeners and the panel. Table 3. Optmzaton Results Ep(kN m) εmax Rmax(kN) Mmax(kN m) Standard Model Optmal Soluton S Optmal Soluton W Optmal Soluton SW The mportant pont for structural optmzaton problems s to reduce the number of analyses to obtan an approxmate optmal soluton, because structural analyss problems are computatonally expensve. The number of analyses N opt s defned for TS as N qn N (7) opt where N tabu s the number of solutons rejected by the tabu lst. In the optmal soluton S, the total number of N opt for three trals s 173 (the total number of possble combnatons s 3136). In the optmal soluton SW, the total number of N opt for two consecutve optmzaton processes s 178 (the total number of possble combnatons s ). Therefore, t may be concluded from these results that approxmate optmal solutons can be successfully found wth small number of analyses usng TS algorthm. A PC wth Intel Xeon W3680 (3.33 GHz) and 12 GB memory s used for the computatons. Although the PC has sx cores, only sngle core s avalable for computaton usng ABAQUS, and the CPU tme s 439 seconds n the case of standard model. 6. Conclusons Optmzaton has been carred out for a shear-type hysteretc steel damper subjected to statc cyclc deformaton. The objectve functon s the total dsspated before the maxmum equvalent stress reaches the specfed value. The conclusons drawn from ths study are summarzed as follows: 1. Energy dsspaton property successfully mproved by optmzng the shape of the panel, as well as the locatons and the thcknesses of the stffeners. About 60% of the total dsspated energy has been ncreased as a result of optmzaton from the standard model. 2. A heurstc approach called TS can be effectvely used to obtan an approxmate of a computatonally expensve structural optmzaton problem wth practcally acceptable small number of functon evaluatons. 3. A FE-analyss software package ABAQUS can be combned wth an optmzaton algorthm usng the Python scrpt. Acknowledgments Ths work s partally supported by the Kajma Foundaton's Research Grant, and Grant-n-Ad for Scentfc Research (No ) from JSPS, Japan. References 1. P. Pan, M. Ohsak and H. Tagawa, Shape optmzaton of H-beam flange for maxmum plastc energy dsspaton, J. Struct. Eng., ASCE, Vol. 133(8), pp , M. Ohsak and T. Nakajma, Optmzaton of lnk member of eccentrcally braced frames for maxmum energy dsspaton, J. Constructonal Steel Research, Vol. 75, pp , ABAQUS Ver User s Manual, SIMULIA, F. Glover, Tabu search: Part I, ORSE J. Computng, Vol. 1(3), pp , M. Ohsak, Optmzaton of Fnte Dmensonal Structures, CRC Press, tabu