Optmzaton of Damage Index n RC Structures Usng Genetc Algorthm Gh. Ghodrat Amr 1, B. Moheb 2 and N. Maddah 3 1 Professor, Center of Excellence for Fundamental Studes n Structural Engneerng, College of Cvl Engneerng, Iran Unversty of Scence & Technology, Narmak, Tehran 16846, Iran 2 PhD Student, College of Cvl Engneerng, Iran Unversty of Scence & Technology, Tehran, Iran 3 MSc Graduate, College of Cvl Engneerng, Iran Unversty of Scence & Technology, Tehran, Iran Emal: ghodrat@ust.ac.r, moheb@ust.ac.r ABSTRACT : The am of ths research s to use Genetc Algorthm n optmal desgn of renforced concrete frames. Durng desgn of structures, dfferent parameters lke stress n members, deflecton of members, nter- drfts and etc. are consdered. In ths research damage ndex as a desgn parameter has been used and wth mnmzng damage ndex as a constrant, the total weght of structure has been mnmzed. The Park model has been appled as a damage estmatng model. Also the computer program, IDARC, has been used for evaluatng ftness functon wth dfferent sze of beams and columns and dfferent steel areas. KEYWORDS: Optmzaton, Damage Index, RC Structures, Genetc Algorthm 1. INTRODUCTION In performance assessment and desgn verfcaton of buldng structures, approxmate nonlnear statc procedures (NSPs) are becomng commonplace n engneerng practce to estmate sesmc demands. In fact, some sesmc codes have begun to nclude them to ad n performance assessment of structural systems. Although sesmc demands are best estmated usng nonlnear tme-h (NTH) analyses, NSPs are frequently used n ordnary engneerng applcatons to avod the ntrnsc complexty and addtonal computatonal effort requred by the former. As a result, smplfed NSPs recommended n ATC-40 (1996) and FEMA-356 (2000) have become popular. These procedures are based on monotoncally ncreasng predefned load patterns untl some target dsplacement s acheved (Kalkan, E. and Kunnath, S. K, 2006). There are some crterons for assessment of structures behavor n the target dsplacement. One of these approaches s dscussed n FEMA-356 (2000). Ths method s based on evaluaton of plastc rotaton of plastc hnges n structural elements. Ths method s a prmtve procedure n assessment of damage parameter n structures. Some researchers have proposed more complcated procedures for assessment of damage parameter. Park et al. (Park et al., 1985) suggested a damage ndex whch s based on deformaton and energy concepts n structural elements. For reducng damage ndex n structures we have to change dmensons of structural elements and amount of used materal. Ths wll nduce the ncrease of constructon cost. In ths paper t s tred to smultaneously reduce the amount of used materal n the structure and holdng damage ndex as a constant value or reduce t. For ths purpose two renforced concrete frames whch have 4 and 8 stores were selected. For solvng ths optmzaton problem genetc algorthm have been appled. The objectve functon n the GA, as mentoned before, s an ndex of structure weght, amount of rebar n the beams and columns and overall damage ndex of structure. After mplementaton of ths algorthm the weght of structure and damage ndex have been decreased. By usng ths method the cost of constructon s thoroughly reduced and performance of structure s mproved. 2. DAMAGE INDEX One of the most popular damage models whch s used n research studes s Park & Ang damage model (Park et al., 1985), ths model was ncorporated n IDARC program (1996) snce the orgnal release of the program.
The Park & Ang damage ndex for a structural element s defned as: δ m β DI P& I = + deh (2.1) δ δ P u u y Where δ m s the maxmum experenced deformaton; δ u s the ultmate deformaton of the element; Py s the yeld strength of the element; de h s the hysteretc energy absorbed by the element durng the response h; and β s a model constant parameter. The Park & Ang damage model (Park et al., 1985) accounts for damage due to maxmum nelastc excursons, as well as damage due to the h of deformatons. Both components of damage are lnearly combned. Three damage ndces are computed usng ths damage model: 1. Element damage ndex: column, beams or shear wall elements. 2. Story damage ndex: vertcal and horzontal components and total damage. 3. Overall buldng damage. Drect applcaton of the damage model to a structural element, a, or to the overall buldng requres the determnaton of the correspondng overall element,, or buldng ultmate deformatons. Snce the nelastc behavor s confned to plastc zones near the ends of some members, the relaton between element, or top deformatons, wth the local plastc rotatons s dffcult to establsh. For the element end secton damage the followng modfcatons were appled. DI θ θ + β E m r = h (2.2) θ u θ r M yθu Where θ m s the maxmum rotaton attaned durng the loadng h; θ u s the ultmate rotaton capacty of the secton; θ r s the recoverable rotaton when unloadng; M y s the yeld moment; and E h s the dsspated energy n the secton. The element damage s then selected as the bggest damage ndex of the end sectons. The two addtonal ndces: and overall damage ndces are computed usng weghtng factors based on dsspated hysteretc energy at component and levels respectvely: DI = ( λ ) component ( DI ) component (2.3) (λ ) component E = E component DI overall = ( λ ) ( DI) (2.4) (λ ) E = E Where λ are the energy weghtng factors; and E are the total absorbed energy by the component or. Table 1 presents the calbrated damage ndex wth the degree of observed damage n the structure.
Table 1 Interpretaton of overall damage ndex (1996) Degree of Physcal Appearance Damage State of Buldng Damage Index Collapse Partal or total collapse of buldng > 1.0 Loss of buldng Severe Extensve crashng of concrete; 0.4-1.0 Beyond repar dsclosure of buckled renforcement Moderate Extensve large cracks; spallng of < 0.4 Reparable concrete n weaker elements Mnor Mnor cracks; partal crushng of concrete n columns Slght Sporadc occurrence of crackng 3. OPTIMIZATION APPROACH As t mentoned before for solvng the optmzaton problem, genetc algorthm has been used. Genetc algorthms developed by Holland (Holland J.H., 1975) that combne problem solvng algorthms wth the prncples of evoluton demonstrate excellent operatons n combnatoral optmzaton that have a fnte soluton. Potental solutons are represented as ndvduals, combnatons as chromosomes, and then they are evolved gradually through the genetc operatons such as selecton, crossover, and mutaton to fnd optmal solutons. Also, a ftness functon, whch orgnates from an objectve functon of the problem, evaluates each ndvdual. The searchng schematcs of a smple genetc algorthm can be generalzed to the followng steps: Step 1. Organze ntal populaton P(0) of solutons, whch ncludes M number ndvduals as the ntal generaton, g=0; Step 2. Evaluate ftness of all ndvduals n P(0); Step 3. Fnsh operatons when the Stop Crteron Satsfed, otherwse proceed to Step 4; Step 4. Select the more ft ndvduals based on ftness from P(g) and transform them to new ndvduals, called offsprng. (Lee, C.K. and Km, S.K., 2007) In ths problem ndvduals consst of dmenson of beams and columns of frame and the amount of rebar area n each of them. For columns one rebar ndex as the overall rebar area, and for beams two steel ndces as top and bottom rebar area of beam have been consdered. Columns secton s square and beams secton s rectangle. Therefore there are two parameters for each beam dmenson and one parameter for each column dmenson. Consequently each ndvdual has N bt, whch N s defned below: N = (NBEAMS + NCOLUMNS) (AsBt + DmBt) Where: N: Number of bts n each ndvdual. NBeams: Number of beams n structure. NColumns: Number of columns n structure. AsBt: Number of bts whch used as each rebar steel ndex (n ths research AsBt=12). DmBt: Number of bts whch used as each dmenson ndex (n ths research DmBt=9). For example N for consdered 4 buldng s equal to 840. The objectve functon s defned as a functon of concrete and rears volume, and overall structural damage ndex. For defnng ths objectve functon, weghted sum method has been used. By usng penalty method, constrants (e.g. maxmum and mnmum area of rebar n beams) were appled. The ftness functon s defned below: F= (Wd Sumd) + (Was Sumas) + (Wc Sumc) + (Aspen) Where:
Wd: Weght of damage ndex (n ths research Wd=20). Sumd: Damage ndex. Was: Weght of rebar volume ndex (n ths research Was=20). Sumas: Rebar volume ndex. Wc: Weght of concrete volume ndex (n ths research Wc=100). Sumc: Concrete volume ndex. Aspen: Penalty ndex for controllng range of rebar area. Each populaton has 26 ndvduals. Crossover and mutaton rato were consdered equal to 0.8 and 0.01 respectvely. 4. MODELING Two renforced concrete frames whch have four and eght stores were selected. These frames were desgned for regon of hgh sesmcty n accordance wth 2800 code (2005) wth consderng ACI318-05 (2005) provsons for ntermedate moment resstng frames. The sol of the ste s assumed to be sol type II accordng to 2800 code (2005) (smlar to sol type C n FEMA 356), a regular buldng n plan was consdered and an nteror frame was selected. Buldng plan and selected frame are shown n Fgure 1. Fgure 1 Buldng plan and selected frame Concrete compressve strength and yeld stress of steel rebar are supposed to be 24 Mpa and 400 Mpa respectvely. The dead and lve load of stores are assumed to be 700 kg/m 2 and 200 kg/m 2 respectvely, consequently the contrbuton of the selected frame s 2800 kg/m for dead and 800 kg/m for lve load. Table 2 Dmensons of renforced concrete elements sectons Secton Name Depth Wdth BEAM30X30 30 30 BEAM35X30 35 30 COL35X35 35 35 COL40X40 40 40 COL30X30 30 30 COL45X45 45 45
A typcal vew of selected frames for analyss s shown n Fgure 2. The heght of each s supposed equal to 3 meters. Fgure 2 Typcal vews of selected frames 5. ANALYSIS METHOD For evaluatng the ftness of each ndvdual an ActveX control has been wrtten by Delph program (1998). Ths control gets an ndvdual as a strng and gves ftness of t as a real number. The Genetc algorthm has been appled by GA Toolbox n MATLAB program (2005). All nonlnear analyses have been performed by IDARC program (1996). Optmzaton has been done by followng steps and s shown n Fgure 3: Step 1: Generatng populaton n MATLAB program (2005). Step 2: Sendng each ndvdual as a strng to ActveX control. Step 3: Decodng the strng n the ActveX control and generatng the nput fle of IDARC program (1996) wth respect to dmensons and steel areas whch are obtaned from decoded ndvdual. Step 4: Runnng IDARC program (1996) by ActveX control automatcally and generatng output fle. Step 5: Readng overall damage ndex of structure from output fle and evaluatng ftness wth respect to damage ndex and concrete and rebar s volume. Step 6: Sendng ftness of ndvdual to MATLAB program (2005) as a real number. Step 7: checkng stop crterons n MATLAB (2005) and applyng reproductons (selecton, mutaton and crossover). Step 8: proceedng to step 2, f the stop crtera s not satsfed. As mentoned before, push over analyss has been performed as a nonlnear analyss. The load pattern assumed to be trangular as mentoned n FEMA 356 (2000). All structures were pushed to reach the dsplacement equal to sx percent of structure heght. 5.1. ActveX Control ActveX control s the software, based on Mcrosoft's Component Object Model (COM) (1998). It lets a program to add Its functonalty by callng that components and t wll be appear as normal parts of the program. Ths type of control can be wrtten n many wndows programmng languages (e.g. Vsual basc,
Delph) and can be used n each language. In ths paper the ActveX control has been wrtten n Delph programmng language (1998) and has been used n MATLAB software. Start Generate Populaton wth Random Strng n MATLAB Sendng Strng to Wrtten ActveX Control Decodng the Strng and Generatng IDARC Input Fle by ActveX Control Runnng IDARC and Creatng Output Fle Readng overall Damage and Evaluatng Ftness Sendng Ftness to MATLAB Stop Crtera s satsfed Yes Stop No Applyng Reproducton: Selecton Crossover Mutaton Fgure 3 Typcal flowchart of analyss method 6. ANALYSIS RESULTS By method whch descrbed n the last secton, two RC frames have been analyzed and Results are presented here. Genetc algorthm convergence graph are shown n Fgures 4 and 5. Fgure 4 Genetc algorthm convergence graph for 4 frame
Fgure 5 Genetc algorthm convergence graph for 8 frame As t s shown n the last fgures, the algorthm converged after performng some generaton steps. Comparson between the orgnal desgn and the results of ths algorthm s shown n table 3. Frame Number Table 3 Comparson between orgnal desgn and analyss results Orgnal desgn Analyss Results Concrete Volume Rebar steel volume Overall Damage Index Concrete Volume Rebar steel volume Overall Damage Index 4-12.33 0.196 0.246 7.33 0.172 0.144 8-27.6 0.696 0.258 32.13 0.549 0.224 7. CONCLUSION By usng ths method the cost of constructon wll be decreased. The damage ndex can predct behavor of structure more accurate and can be used as a parameter to measure performance of structures. Wth respect to analyss results, n the 4 frame the volume of concrete and rebar steel decreased 41 and 13 percents respectvely. Also overall damage ndex reduced. In the 8 frame the volume of concrete and rebar steel decreased 14 and 22 percents respectvely. Also overall damage ndex reduced. Therefore ths method can be used to optmal analyss of structures. REFERENCES ACI 318M-05, Amercan Concrete Insttute ACI Commttee 318. (2005). Buldng Code Requrements for Structural Concrete and Commentary. ATC40, Appled Technology Councl. (1996). Sesmc Evaluaton and Retroft of Concrete Buldngs. Volume 1, ATC-40 Report, Redwood Cty, Calforna. FEMA356, Federal Emergency Management Agency. (2000). Pre-standard and Commentary for the Sesmc Rehabltaton of Buldngs. Washngton, D.C. Holland J.H., Adaptaton n Natural and Artfcal Systems, Unversty of Mchgan Press, Ann Arbor, 1975. Kalkan, E. and Kunnath, S. K. (2006). Assessment of Current Nonlnear Statc Procedures for Sesmc Evaluaton of Buldngs. Journal of Engneerng Structures. Lee, C.K. and Km, S.K. (2007). GA-based Algorthm for Selectng Optmal Repar and Rehabltaton Methods for Renforced Concrete RC Brdge Decks. Journal of Automaton n Constructon, 16, 153-164. Marco Cantu, Tm Gooch, John F. Lam (1998), Delph Developer's Handbook, Sybex.
MATLAB, The Language of Techncal Computng. (2005). MathWorks, Inc. USA. Park, Y.J. and Ang, A.H.S., (1985). Mechanstc Sesmc Damage Model for Renforced Concrete. Journal of Structural Engneerng, 111:4. Standard No.2800, Iranan Code of Practce for Sesmc Resstant desgn of Buldngs. (2005). Buldng and Housng Research Center. Tehran. Iran. Valles, R. E., Renhorn, A. M., Kunnath, S. K., L, C. and Madan, A. (1996). IDARC 2D Verson 4.0: A Program for the Inelastc Damage Analyss of Buldngs, Techncal Report NCEER-96-0010