Design optimization of reinforced concrete structures

Transcription

1 Computers and Concrete, Vol. 3, No. 5 (2006) Desgn optmzaton of renforced concrete structures Andres Guerra and Panos D. Kouss Colorado School of Mnes, Dvson of Engneerng, 1500 Illnos St, Golden, CO , USA (Receved Aprl 18, 2006, Accepted August 25, 2006) Abstract. A novel formulaton amng to acheve optmal desgn of renforced concrete (RC) structures s presented here. Optmal szng and renforcng for beam and column members n mult-bay and multstory RC structures ncorporates optmal stffness correlaton among all structural members and results n cost savngs over typcal-practce desgn solutons. A Nonlnear Programmng algorthm searches for a mnmum cost soluton that satsfes ACI 2005 code requrements for axal and flexural loads. Materal and labor costs for formng and placng concrete and steel are ncorporated as a functon of member sze usng RS Means 2005 cost data. Successful mplementaton demonstrates the abltes and performance of MATLAB s (The Mathworks, Inc.) Sequental Quadratc Programmng algorthm for the desgn optmzaton of RC structures. A number of examples are presented that demonstrate the ablty of ths formulaton to acheve optmal desgns. Keywords: sequental quadratc programmng; cost savngs; renforced concrete; optmal stffness dstrbuton; optmal member szng; RS means; nonlnear programmng; desgn optmzaton. 1. Introducton Ths paper presents a novel optmzaton approach for the desgn of renforced concrete (RC) structures. Optmal szng and renforcng for beam and column members n mult-bay and multstory RC structures ncorporates optmal stffness correlaton among structural members and results n cost savngs over typcal state-of-the-practce desgn solutons. The desgn procedures for RC structures that are typcally adapted n practce begn by assumng ntal stffness for the structural skeleton elements. Ths s necessary to calculate the nternal forces of a statcally ndetermnate structure. The fnal member dmensons are then desgned to resst the nternal forces that are the result of the assumed stffness dstrbuton. Ths creates a stuaton where the nternal forces used for desgn may be nconsstent wth the nternal forces that correspond to the fnal desgn dmensons. The redstrbuton of forces n statcally ndetermnate structures at ncpent falure, however, results n the structural performance that s consstent wth the desgn strength of each member. Although ths common practce typcally produces safe structural desgns, t ncludes an nconsstency between the elastc tendences and the ultmate strength of the structure. In some cases ths can cause unsafe structural performance under overloads (e.g. earthquakes) as well as unwanted crackng under normal buldng operatons when factored desgn loads are close to servce loads, (e.g. dead load domnated structures). Ths nconsstency also mples that such desgns are unnecessarly expensve as they do not optmze the structural resstance and often result n Graduate Student, E-mal: [email protected] Assocate Professor, Correspondng Author, E-mal: [email protected]

2 314 Andres Guerra and Panos D. Kouss members wth dmensons and renforcement decded by mnmum code requrements rather than ultmate strength of allowable deflectons. Because of ts sgnfcance n the ndustry, optmzaton of concrete structures has been the subject of multple earler studes. Whereas an exhaustve lterature revew on the subject s outsde the scope of ths paper, some notable optmzaton studes are brefly noted here. For example, Ballng and Yao (1997), and Moharram and Grerson (1993) employed nonlnear programmng (NLP) technques for RC frames that search for contnuous-valued solutons for beam, column, and shear wall members, whch at the end are rounded to realstc magntudes. In more recent studes, Lee and Ahn (2003), and Camp, et al. (2003) mplemented Genetc Algorthms (GA) that search for dscrete-valued solutons of beam and column members n RC frames. The search for dscretevalued solutons n GA s dffcult because of the large number of combnatons of possble member dmensons n the desgn of RC structures. The dffcultes n NLP technques arse from the need to round contnuous-valued solutons to constructble solutons. Also, NLP technques can be computatonally expensve for large models. In general, most studes on optmzaton of RC structures, whether based on dscrete- or contnuous-valued searches, have found success wth small RC structures usng reduced structural models and rather smple cost functons. Issues such as the dependence of materal and labor costs on member szes have been mostly gnored. Also, n an effort to reduce the sze of the problems, smplfyng assumptons about the number of dstnct member szes have often been made based on past practces. Whle economcal solutons n RC structures typcally requre desgns where groups of structural elements wth smlar functonalty have smlar dmensons, the optmal characterstcs and populaton of these groups should be determned usng optmzaton technques rather than predefned restrctons. These ssues are addressed, although not exhaustvely, n ths paper, by ncorporatng more realstc costs and relaxed restrctons on member geometres. Ths study mplements an algorthm that s capable of producng cost-optmum desgns of RC structures based on realstc cost data for materals, formng, and labor, whle, at the same tme, meetng all ACI code and desgn performance requrements. The optmzaton formulaton of the RC structure s developed so that t can be solved usng commercal mathematcal software such as MATLAB by Mathworks, Inc. More specfcally, a sequental quadratc programmng (SQP) algorthm s employed, whch searches for contnuous valued optmal solutons, whch are rounded to dscrete, constructble desgn values. Whereas the algorthm s nherently based on contnuous varables, dscrete adaptatons relatng the wdth and renforcement of each element are mposed durng the search. Ths optmzaton formulaton s demonstrated wth the use of desgn examples that study the stffness dstrbuton effects on optmal span lengths of portal frames, optmal number of supports for a gven span, and optmal szng n mult-story structures. RS Means Concrete and Masonry Cost data (2005) are ncorporated to capture realstc, member sze dependent costs. 2. Optmzaton 2.1. RC structure optmzaton The goal of optmzaton s to fnd the best soluton among a set of canddate solutons usng effcent quanttatve methods. In ths framework, decson varables represent the quanttes to be

3 Desgn optmzaton of renforced concrete structures 315 Fg. 1 Renforced concrete cross secton and resstve forces determned, and a set of decson varable values consttutes a canddate soluton. An objectve functon, whch s ether maxmzed or mnmzed, expresses the goal, or performance crteron, n terms of the decson varables. The set of allowable solutons, and hence, the objectve functon value, s restrcted by constrants that govern the system. Consder a two dmensonal renforced concrete frame wth members of length L. Each member has a rectangular cross secton wth wdth b and depth h, whch s renforced wth compressve and tensle steel renforcng bars, As 1, and As 2, respectvely (Fg. 1). The set of b, h, As 1,, and As 2, consttute the decson varables. The overall cost attrbuted to concrete materals, renforcng steel, formwork, and labor s the objectve functon. The ACI code requrements for safety and servceablty, as well as other performance requrements set by the owner, consttute the constrants. The formulaton of the problem and the assocated notaton follow: Indces: : RC structural member; beam or column. m : steel renforcng bar szes. Sets: Columns : set of all members that are columns. Beams : set of all members that are beams. Sym : set of pars of column members that are horzontally symmetrcally located on the same story level. Horz : same as Columns, but actvated only when the structure s subjected to horzontal loadng. et : set of member types; ether Columns or Beams. Parameters: C conc, mat l = $/m 3 - Materal Cost of Concrete C steel (et) = 2420 $/metrc ton for beam members and 2340 $/metrc ton for column members L - Length of member, meters (typcally 4 to 10 meters) d' = 7 cm = Concrete Cover to the centrod of the compressve steel - same as the cover to the centrod of thee tensle steel. f c = 28 MPa - Concrete Compressve Strength β 1 = Reducton Factor f c = 28 MPa Ec = 24,900 MPa - Concrete Modulus of Elastcty

4 316 Andres Guerra and Panos D. Kouss Es = 200,000 MPa - Renforcng Bar Modulus of Elastcty f y = 420 MPa - Steel Yeld Stress f s = Stress n Tensle Steel f y bar_numberng = Metrc equvalent bar szes = [#13, #16, #19, #22, #25] bar_dam m = Rebar dameters for m = 1:5..e., [12.7, 15.9, 19.1, 22.2, 25.4] mm bar_area m = Rebar areas for m = 1:5..e., [129, 199, 284, 387, 510] mm 2 c ρ mn = Mnmum rato of steel to concrete cross - sectonal area n all column members = Maxmum rato of steel to concrete cross - sectonal area n all column members c ρ max b ρ mn = Mnmum rato of steel to concrete cross - sectonal area n all beam members Decson Varables: Prmary Varables: b - wdth of member (cm) h - depth of member (cm) As 1, - Compressve steel area of member (cm 2 ) As 2, - Tensle steel area of member (cm 2 ) Auxlary Varables: p - Permeter of member, 2*(b + h ) for columns, and (b + 2*h ) for beams C formng (b, h ) - Cost of forms n placce ($/SMCA) as a functon of cross-sectonal area as descrbed n Fg. 2.1 C conc, place (b, h ) - Cost of placng concrete ($/m 3 ) as a functon of corss-sectonal area as descrbed n Fg. 2.2 P u - Factored Internal Axal Force of member determned va FEA (kn) M u - Factored Internal Moment Force of member determned va FEA (kn m) c - Dstance from most compressve concrete fber to the neutral axs for member (cm) - Locaton of the plastc centrod of member (cm) from the most compressve fber. x Formulaton: subject to: ( C): mn n = 1 p L C formng ( b, h ) + ( b h As 1, As 2, ) L ( C conc, mat'l + C conc, place ( b, h )) + As 1, + As 2, ( ) L C steel ( et) b b h = Columns h = ( j) Sym b j = ( j) Sym h j = ( j) Sym As 1, As 1, j = ( j) Sym As 2, As 2, j = Horz As 1, As 2, (1) (2) (3) (4) (5) (6) (7)

5 Desgn optmzaton of renforced concrete structures 317 h 0.85 b h f c --- As 1, f y d As 2, f y ( h d ) x 2 = b h f c As 1, f y + As 2, f y h c d = f y Es P u 0.8 φ 0.85f c ( b h ( As 1, + As 2, )) + f y ( As 1, + As 2, ) As 1, As 2, b h h 5 b b bar_dam M ( As 2 bar_area M ) 2 + As 2 ((, bar_area M ) 2 1) max( 1.0, 1.0 bar_dam M ) (8) (9) (10) (11) (12) (13) (14) b ρ mn, As , b h As 2, b h Beams (15) As 2, b h f ---- s As1, Beams f y b h (16) c ρ mn, As , b h As 2, b h Columns (17) As , b h As 2, b h c ρ max, Columns M u φ a o a 1 P u φ a 2 ( P u φ) 2 a 3 ( P u φ) 3 a 4 ( P u φ) 4 a 5 ( P u φ) 5 0 (18) (19) b 16 cm, h 16 cm, As 1, 258 mm 2, As 2, 258 mm 2, b 500 cm, h 500 mm, As 1, 130, 000 mm 2, As 2, 130, 000 mm 2 (20) The objectve functon C, n Eq. (1), descrbes the cost of a renforced concrete structure and ncludes, n order of appearance, forms n place cost, concrete materals cost, concrete placement and vbratng ncludng labor and equpment cost, and renforcement n place usng A615 Grade 60 steel ncludng accessores and labor cost. The costs of formng and placng concrete are a functon of the cross-sectonal dmensons b and h of the structural elements. These costs are detaled n Table 1 and n Fgs. 2, 3, and 4. As shown n Fgs. 2 through 4, lnear nterpolaton between ponts s used to calculate cost of formng and the cost of placng concrete. Note that RS Means provdes only the dscrete ponts. The assumpton of lnear nterpolaton between these ponts s made by the authors due to lack of better estmates. The constrants n Eqs. (2) through (7) defne relatve geometres for members n one of the specfed sets: Columns, Sym, and Horz. Eq. (8) defnes the locaton of the plastc centrod of element as a functon of the decson varables. Eq. (9), defnes the locaton of the neutral axs. Eq.

6 318 Andres Guerra and Panos D. Kouss Table 1 RS Means 2005 concrete cost data all data n englsh unts from means concrete & masonry cost data Copyrght Reed Constructon Data, Kngston, MA ; All rghts reserved. Product Descrpton Total Cost Incl. Unts Overhead and Proft REINFORCING IN PLACE A615 Grade 60, ncludng access. Labor Beams and Grders, #3 to # (2200) $/metrc ton ($/ton) Columns, #3 to # (2125) $/metrc ton ($/ton) CONCRETE READY MIX Normal weght 4000 ps (92.5) $/m 3 ($/Yd 3 ) PLACING CONCRETE and Vbratng, ncludng labor and equpment. Beams, elevated, small beams, pumped (61.0) $/m 3 ($/Yd 3 ) (small =< 929 cm2 (144 n 2 )) Beams, elevated, large beams, pumped (40.5) $/m 3 ($/Yd 3 ) (large =>929 cm2 (144 n 2 )) Columns, square or round, 30.5 cm (12") thck, pumped 79.8 (61.0) $/m 3 ($/Yd 3 ) 45.7 cm (18") thck, pumped 53.0 (40.5) $/m 3 ($/Yd 3 ) 70.0 cm (24") thck, pumped 51.7 (39.5) $/m 3 ($/Yd 3 ) 91.4 cm (36") thck, pumped 34.0 (26.0) $/m 3 ($/Yd 3 ) FORMS IN PLACE, BEAMS AND GIRDERS Interor beam, job-bult plywood, 30.5 cm (12") wde, 1 use 41.0 (12.5) $/SMCA* ($/SFCA) 70.0 cm (24") wde, 1 use 35.8 (10.9) $/SMCA ($/SFCA) Job-bult plywood, cm (8" 8") columns, 1 use 41.0 (12.5) $/SMCA ($/SFCA) cm (12" 12") columns, 1 use 37.1 (11.3) $/SMCA ($/SFCA) cm (16" 16") columns, 1 use 36.3 (11.05) $/SMCA ($/SFCA) cm (24" 24") columns, 1 use 36.7 (11.2) $/SMCA ($/SFCA) cm (36" 36") columns, 1 use 34.3 (10.45) $/SMCA ($/SFCA) *Square Meter Contact Area and Square Foot Contact Area (10) ensures that the appled factored axal load P u s less than φp n for the mnmum requred eccentrcty, as defned by ACI Eq. (10-2). Eq. (11) mantans that the tensle steel area s greater than the compressve steel area. The ntent of ths restrcton s to facltate the algorthmc search. Eqs. (12) and (13) are problem specfc restrctons related to the wdth, b, and depth, h, of all members. Whereas these restrctons are common practce n low sesmcty areas, they are by no means general requrements for all constructon. Whle Eq. (12) ensures that the wdth s less than the depth, Eq. (13) prevents the creaton of large shear walls and mantans mostly frame acton for the desgn convenence of ths study. Eq. (14) ensures that the tensle steel can be placed n element wth approprate spacng and concrete cover as specfed by ACI In Eq. (14), the subscrpt M on bar_dam and bar_area corresponds to the dscrete bar area that s closest to and not less than the contnuous value of As 2,. The constrants lsted n Eqs. (15) through (18) ensure that the amount of renforcng steel s between code specfed mnmum and maxmum values. And fnally, Eq. (19) ensures that the appled axal and bendng forces of element, P u and M u, determned wth a Fnte Element Analyss (FEA), are wthn the bounds of the factored P-M nteracton dagram whch s

7 Desgn optmzaton of renforced concrete structures 319 Fg. 2 Cost of FORMS IN PLACE for columns Fg. 3 Cost of FORMS IN PLACE for beams modeled as a splne nterpolaton of fve strategcally selected (M n, P n ) pars. The lower and upper bounds desgnate the range of permssble values for the decson varables. The lower bounds on the wdth and depth are formulated from the code requred mnmum amount of steel and the mnmum cover and spacng. Upper bounds decrease the range of feasble solutons by excludng excessvely large members Optmzaton technque for RC structures Varous optmzaton algorthms can be used dependng on the mathematcal structure of the problem. MathWork s MATLAB s used to apply an SQP optmzaton algorthm to the descrbed problem through MATLAB s ntrnsc functon fmncon, whch s desgned to solve problems of the form: Fnd a mnmum of a constraned nonlnear multvarable functon, f(x), subject to

8 320 Andres Guerra and Panos D. Kouss Fg. 4 Cost of PLACING CONCRETE and vbratng, ncludng labor and equpment Fg. 5 Optmzaton routne flow chart g( x) = 0; h( x) 0; Ib x ub; where x are the decson varables, g(x) and h(x) are constrant functons, f(x) s a nonlnear objectve functon that returns a scalar (cost), and Ib and ub are the lower and upper bounds on the decson varables. All varables n the optmzaton model must be contnuous. The SQP method approxmates the problem as a quadratc functon wth lnear constrants wthn each teraton, n order to determne the search drecton and dstance to travel (Edgar and Hmmelblau 1998).

9 Desgn optmzaton of renforced concrete structures 321 The flow chart n Fg. 5 demonstrates the entre optmzaton procedure from generatng ntal decson varable values, x o, to selectng the best locally optmal soluton from a set of optmal solutons found by varyng x o. Intal decson varable values are found by solvng the descrbed optmzaton formulaton for each ndvdual element subjected to nternal forces of an assumed stffness dstrbuton. At least ten dfferent assumed stffness dstrbutons are utlzed; each leads to a local optmal soluton. Comparson of all local optmal solutons, not all of whch are dfferent, provdes a reasonable estmaton of the global optmum soluton. Whereas the ntal decson s based on an element-by-element optmzaton approach, the fnal optmzaton (Eqs. 1-20) s global and allows all element dmensons to vary smultaneously and ndependently n order to acheve the optmal soluton. As such, the fnal desgn s acheved at an optmal nternal stffness confguraton. Ths corresponds to the nternal force dstrbuton that ultmately results n the most economcal desgn. 3. Desgn requrements 3.1. Cross-secton resstve strength Consder a concrete cross secton renforced as shown n Fg. 1, subjected to axal loadng and bendng about the z-axs. The resstve forces of the RC cross-secton nclude the compressve strength of concrete and the compressve and tensle forces of steel, and are calculated n terms of the desgn varables (b, h, As 1,, As 2, ), the locaton of the neutral axs c, and the concrete and steel materal propertes. It s assumed that concrete crushes n compresson at ε c =0.003 and that the strans assocated wth axal loadng and bendng very lnearly along the depth of the crosssecton. The bendng resstve capacty M n for a gven compressve load P n s calculated teratvely by assumng ε c =0.003 at the most compressve fber of the cross-secton, and by varyng c untl force equlbrum s acheved. The strength reducton factor s calculated based on Fg. 6 Interacton dagram at falure state

10 322 Andres Guerra and Panos D. Kouss the stran n the tensle steel. At ths state, the resultng moment s evaluated, and the par (M n, P n ) at falure s obtaned. The locus of all (M n, P n ) falure pars s known as the M-P nteracton dagram for a member (Fg. 6) M-P nteracton dagram Safety of any element requres that the factored pars of appled bendng moment and axal compresson ( M u φ, P u φ) fall wthn the M-P nteracton dagram. The strength reducton factor, φ, s evaluated based on the stran of the most tensle renforcement and s 0.65 for tensle stran less than 0.002, 0.9 for tensle strans greater than 0.005, and s lnearly nterpolated between 0.65 and 0.9 for strans between and 0.005, as defned n ACI , Secton 9.3. Fnally, an axal compresson cutoff for the cases of small eccentrcty was placed equal to P u φ = 0.8[ 0.85f c ( A g A st ) + f y A st ] as per ACI Eq. (10-2). Mathematcally, f FM ( n, P n ) = 0 s a functon that descrbes the nteracton dagram, safety requres that FM ( u φ, P u φ) 0 for all members. For a gven cross-secton, the nteracton dagram s typcally obtaned pont-wse by fndng numerous combnatons (M n, P n ) that descrbe falure. For the purpose of ths study, the nteracton dagram s modeled as a cubc splne based on fve ponts (Fg. 6), three of whch are the balance falure pont ( M nb, P nb ), the pont of zero moment, and the pont that corresponds to a neutral axs locaton at the level of the compressve steel axs. The remanng two ponts are located ether above or below the balance falure pont ( M nb, P nb ) dependng on whether the appled axal compresson load s greater or smaller than P nb, respectvely. Fg. 6 shows the three fxed ponts as sold crcles and the two condtonal ponts whch are located below or above the balance falure pont as open crcles and open trangles, respectvely. It s assumed that the desgn for shear loads does not alter the optmal desgn decson varables b, h, As 1,, and As 2,. Ths assumpton s typcally acceptable for long slender elements where the combnatons of flexural and axal loads commonly control the element dmensons. It s also assumed that the optmal soluton s not senstve to connecton detalng. For structures n Sesmc Desgn Category A, B, and C as classfed n the ASCE 7 Standard (SEI/ASCE 7-98) ths assumpton s acceptable Roundng the contnuous soluton Concrete desgn s ultmately a dscrete desgn problem, where typcal element dmensons are multples of 50 mm and steel renforcement conssts of a fnte number of commonly avalable renforcng bars. Roundng b and h to dscrete values s ncorporated through the use of a secondary optmzaton process that fnds the optmal renforcng steel amounts for fxed b and h. Varous combnatons of roundng b and h ether up or down to 5 cm multples are examned to fnd the dscrete soluton wth the lowest total cost. Selectng a dscrete number and sze of longtudnal renforcng steel from contnuous-valued solutons s accomplshed by fndng the dscrete number and sze that s closest to and not less than the contnuous-valued optmal soluton. Ths s mplemented nto the optmzaton model so that the mnmum wdth b mn that s requred to ft the selected renforcng steel becomes a lower bound that ensures proper cover and spacng of the longtudnal renforcng steel. Ths process begns by calculatng the dscrete number of bars that are necessary to provde at least the steel area of the contnuous soluton. The mnmum wdth requred for proper cover and spacng for each set of

11 Desgn optmzaton of renforced concrete structures 323 renforcng steel bars s calculated. Next, the combned cost of each bar set and the correspondng mnmum requred wdth s evaluated. The requred wdth that s assocated wth the lowest combned cost s used as the mnmum wdth requrement. It s noted that when a dscrete soluton requres more than fve renforcng bars, the mnmum requred wdth s calculated based on bundles of two bars placed sde-by-sde. Also, for beam members, the compresson and tensle renforcements are desgned to use the same sze bars for constructon convenence, as long as the strength requrements are stll met. Fnally, strrups consst of #13 renforcng bars. 4. Desgn examples 4.1. Optmzaton desgn examples Three examples of optmal desgn for mult-story and mult-bay renforced concrete frames are presented here to demonstrate the method. The frst example studes the optmal desgn of a onestory portal frame wth varyng span length. The second example studes the optmal desgn of a mult-bay one-story frame wth varyng number of bays but wth a constant over-all grder length of 24 meters. The thrd example creates optmal desgns of mult-story, sngle-bay RC structures wth and wthout horzontal sesmc forces. Desgns based on standard approaches are created to compare optmal and typcal-practce desgns. Whle the same cost functon used n the optmzaton formulaton s mplemented to calculate the Typcal Desgn Costs, the desgn dmensons for the Typcal Desgn Costs are based on a smplfed state-of-the-practce desgn method. Ths method ntally assumes that all columns wll be 25 cm by 25 cm and that the wdth of all beams s 25 cm. Addtonally, for beams the depth s equal to the wdth tmes a factor of one-thrd the beam length n meters (McCormac 2001). At ths pont an nternal stffness dstrbuton has been defned and nternal forces can be calculated. The amount of renforcement s then desgned to meet strength and code requrements for each member. The wdth of each beam member, or wdth and depth of each column member, s ncreased n ncrements of 5 cm f the assumed sze of the member s not suffcent to hold the needed steel to mantan strength requrements. Note that the nternal forces are based on the ntally assumed stffness dstrbuton and not on the desgn dmensons. For all examples presented here, the length of columns s four meters, the compressve strength of concrete s 28 MPa, the yeld stress of steel s 420 MPa, the cover of compressve and tensle renforcement s 7 centmeters, the unt weght of steel renforced concrete s 24 kn/m 3, and the assocated materals and constructon costs are lsted n Table Loadng condtons All frames examned here are loaded by ther self weght w G, an addtonal gravty dead load, w D = 30 kn/m, and a gravty lve load, w L = 30 kn/m. Sesmc horzontal forces, wherever they are appled, are determned usng the ASCE 7 (SEI/ ASCE7-98) equvalent lateral load procedure for a structure n Denver, Colorado that s classfed as a substantal publc hazard due to occupancy and use, and s founded on ste class C sol (SEI/ ASCE7-98). A base shear force V s calculated usng gravty loads equal to 1.0(w D )+0.25w L, and s

12 324 Andres Guerra and Panos D. Kouss Table 2 Calculated equvalent lateral loads for mult-story, one-bay structures Lateral load at: Number of Stores: Frst floor Second Floor Thrd Floor Fourth Floor Ffth Floor (kn) (kn) (kn) (kn) (kn) One Story Two Story Three Story Four Story Fve Story then dstrbuted approprately to each frame story. Table 2 detals the equvalent sesmc horzontal forces for the mult-story desgn examples. The frames that are subjected to gravty loads only are desgned for factored loads of 1.2 w G +1.2 w D w L = 1.2 w G + 84 kn/m. The frames that are subjected to gravty and sesmc loads are desgned for the worse combnaton of the followng factored load combnatons (ACI ): 1.2 w G w D w L = 1.2 w G +84 kn/m 1.2 w G w D w L E = 1.2 w G + 66 kn/m E 0.9 w G w D E = 0.9 w G + 27 kn/m E Fg. 7 Portal frame Table 3 Optmal portal frame costs for varous span lengths Span Length (L) Typcal Optmal Desgn Desgn Costs Cost Cost Savngs Over Typ. Desgn Cost Optmal Cost per Foot of Structure Optmal Cost per Foot of Beam meters Dollars Dollars Percent Dollars Dollars

13 Desgn optmzaton of renforced concrete structures 325 Fg. 8 Optmal costs for varyng span lengths n a one-story structure Fg. 9 Normalzed optmal cost for varyng span lengths n a one-story structure One-story portal frame wth varyng span length Consder a sngle-story portal frame subjected to gravty factored lve and dead loads, w U, totalng 84 kn/m. The structural model s shown n Fg. 7. The heght of the structure s 4 meters, and the span, L, vares n order to study the effect of the span length on the optmal soluton. Comparson of the costs at the specfed span lengths s presented n Table 3, whle a graphcal presentaton of the same data s shown n Fg. 8. The normalzed cost per foot s presented n Fg. 9 to demonstrate the pure cost burden per foot assocated wth the larger span lengths. It should be noted that every pont n the cost calculaton of Fgs. 8 and 9, wth the excepton of the typcal desgn ponts, represents an optmal soluton for the specfc structure. Although the cost ncreases smoothly as a functon of span length, the assocated element cross-sectonal dmensons do not follow an equally smooth change pattern. For spans of length up to 12 meters optmal solutons result n small columns and a large grder. The grders under such desgn develop moment dagrams that have small negatve endmoments, large postve md-span moments, and behave almost as smply supported beams. For span lengths of 14 m or larger, the pattern changes abruptly to one where columns become

14 326 Andres Guerra and Panos D. Kouss Fg. 10 Grder moments at optmal soluton values Table 4 Portal frame, L = 12 meters, cost = $8408 t = 3.4 mnutes b h As 1 As 2 P u M u φ Element Number cm cm Comp. Tens. kn kn-m #25 6# #25 6# #25 9# Table 5 Portal frame, L = 14 meters, cost = $10702 t = 1.7 mnutes b h As 1 As 2 P u M u φ Element Number cm cm Comp. Tens. kn kn-m #25 13# #25 13# #25 9# comparable n sze to the grder, and are assocated to bendng moment dstrbutons where the grders have equal negatve and postve moment magntudes. Fg. 10 llustrates the grder moment dagrams at the optmal soluton values for the 12 and 14 meter span lengths. The characterstcs and performance of the one-story portal frame for the 12 and 14 meter span lengths are presented n Tables 4 and 5. P u and M u are the crtcal nternal forces of each element at the optmal soluton and t s the total computaton tme n mnutes usng a Pentum GHz laptop wth a Wndows XP Pro operatng system. Optmal desgns result n cost savngs of just under 1% for the 8-meter span to 17% for the 24- meter span (See Fg. 8 and Table 3). It s very nterestng to note that for certan span lengths the typcal desgn costs are relatvely close to the optmal desgn costs. Ths demonstrates the regons where the typcal practce assumptons result n effcent structures and where they do not. A beam length of eght meters for a portal frame wth four meter long columns appears to be the most effcent structure for the typcal practce assumptons as t results n a cost closest to the optmal cost Increasng the number of bays n a constant 24-meter span In ths example, a seres of mult-bay one story frames are desgned wth a total span of 24 meters (Fg. 11). The frames range from a one bay structure supported by two columns (bay length of 24.0 m) to a twenty-four bay structure supported by 25 equally spaced columns (bay length of 1.0 m). A factored gravty load of 1.2 w G + 84 kn/m s placed on the entre 24-meter grder. The cost of each

15 Desgn optmzaton of renforced concrete structures 327 Table 6 Optmal cost for ncreasng number of bays n a 24 meter span Number of Bays Typcal Desgn Cost ($) Optmal Desgn Cost ($) Cost Savngs (%) Fg. 11 Frame of length 24 m wth n bays typcal and optmal desgn s presented n Table 6 and s plotted aganst the number of bays n Fg. 12. Ths example covers the entre range of ( M u φ, P u φ) combnatons n the nteracton dagram. The nner grders carry vrtually no axal load (tenson controlled), whle the outer grders are loaded at hgh eccentrcty (tenson controlled or transton). The nner columns carry ther loads wth very small eccentrcty and are controlled by the code cap on axal compresson for columns of small eccentrctes. Fnally, the outer columns are compresson controlled wth a sgnfcant bendng moment component. Note n Fg. 12, the lnear relatonshp between cost values for frames wth 21, 22, and 24 bays. These correspond to solutons where all members are controlled by mnmum code requrements for member dmensons and renforcng steel: 20 cm by 20 cm members wth a total of 4#13 renforcng steel bars. For 20 bays or less, mnmum code requrements gradually stop controllng the problem, startng wth the outer beams. The lowest cost corresponds to seven spans of 3.4 meters each. It s noted agan that every desgn presented n Fg. 12 s optmal for the specfc geometry,.e., span length.

16 328 Andres Guerra and Panos D. Kouss Fg. 12 Increasng number of bays n a 24 meter span In all cases wth 20 bays and less, the model s such that beams ncrease n sze to keep the columns as small as possble. Only the two-span and the one-span frames have columns wth crosssectonal dmensons larger than 25 cm. In all mult-span structures, the outermost beams carry larger load and, n most cases, the nner beams are close to mnmum values. Substantal costs savngs over typcal desgn s demonstrated for mult-bay structures. Note n Fg. 12 that after reachng the lowest typcal desgn cost of approxmately $9700 for the sx-bay structure, the costs ncrease lnearly wth each addtonal bay. The lnear ncrease ndcates that mnmum dmensons have been reached for the typcal desgn assumptons. Thus, each addtonal bay ncreases the total cost by the cost of one column and one beam Mult-story desgn examples The desgns presented n ths secton address three groups of mult-story, sngle bay frames. The frst group conssts of frames that have a span length of 4 m, one to sx stores and are subjected to Fg. 13 Mult-story sngle-bay structures

17 Desgn optmzaton of renforced concrete structures 329 gravty loads only. The second group s smlar to the frst group. However, the frames have a span length of 10 m. The thrd group conssts of frames that have a span length of 10 m, one to fve stores, and are subjected to gravty and sesmc loads. Fg. 13 dsplays the one, two, and three story frames subjected to gravty and sesmc forces. Element numberng and loadng for the four-, fve- and sx-story structures follows the same pattern as n the frames presented n Fg. 13. The magntudes of the sesmc forces at each floor are lsted n Table Mult-story - vertcal load only Gven the effects of grder length on the optmal desgn patterns of portal frames, short-span and long-span, (4m and 10 m respectvely) multstory frame desgns are examned here. The optmal costs of mult-story frames subjected to gravty loads only are presented n Fg. 14, where crcular marks represent the long span frames and rectangular marks represent the short span frames. Note that the long-span results are presented n two groups: open crcles for the frames that have an odd number of stores (1, 3, or 5), and sold crcles for the frames that have an even number of stores (2, 4, or 6). No such dstncton s necessary for the short span frames. In general, t can be seen from Fg. 14 that the cost ncreases lnearly wth the number of stores for both the long- and short-span frames. The lnear relaton between the number of stores and cost s almost exact n the case of short-span frames. On the other hand, a closer examnaton of the long-span frames (see data ponts and ther least-square-ft lnes) ndcates that the odd-story frames are slghtly more expensve than the even-story frames. The practcal sgnfcance of ths observaton s not clear. Nevertheless, from the theoretcal standpont, ths s an nterestng, and rather unexpected fndng that merts further analyss and explanaton. Let us consder the n-story frame of Fg. 15. Note that the end-rotatonal tendences of each grder are ressted by one column above and one column below at each end, wth the Fg. 14 Increasng cost of multple story structures subject to vertcal loadng only

18 330 Andres Guerra and Panos D. Kouss Fg. 15 Column stffness tendences for Long-Span multstory frames under gravty loads excepton of the roof grder, where only one column below the grder provdes the rotatonal resstance at each end of the grder. To assst each of the long grders carry ther large moments n a cost-effectve way, columns tend to be stff n order to restrct rotaton and develop suffcent negatve end moment, whch n turn reduces the md-span postve moment. Thus, consderng the n-story frame of Fg. 15, the columns C n underneath the top grder G n, tend to be stff. The bottomends of these columns (C n ) also provde large rotatonal resstance to the next grder G n-1. As a result, the next set of columns C n-1 does not need to be as stff. Ths s ndcated n Fg. 15 by the label soft next to columns C n-1. Snce the bottom-end of these columns do not provde suffcent rotatonal stffness to the next grder G n-2, the next set of columns C n-2 must be stff (see label n Fg. 15). Thus, an alternatng pattern of stff-soft columns s created. In frames of even number of stores, ths pattern results n an equal number of stff and soft sets of columns. On the other hand, n frames of odd number of stores, the same pattern results n one extra story of stff columns. Thus, the odd-story frames are relatvely more expensve than the even-story frames. It should be ponted out however, that a soft column at a lower story tends to be stffer than a soft column at a hgher story, snce t carres larger axal load. Tables 7 and 8 detal the optmal soluton results for the four- and fve-story frame wthout lateral loads. For short-span structures, creaton of stff columns to help dstrbute the moment more effcently n the grder s not cost effectve, gven that grders and columns have a smlar length, whch makes two small columns and one large grder less expensve. Thus n the case of short-span frames, the stff-soft pattern descrbed earler s not effcent, and there s no dstncton between

19 Desgn optmzaton of renforced concrete structures 331 Table 7 Four-story, sngle bay long span structure, vertcal load only t = 17.7 mnutes b h As 1 As 2 P u M u φ Element Number cm cm Comp. Tens. kn kn-m #16 4# #16 4# #22 14# #19 5# #19 5# #25 6# #13 4# #13 4# #22 14# #19 14# #19 14# #19 12# Table 8 Fve-story, sngle bay long span structure, vertcal load only t = 18.5 mnutes b h As 1 As 2 P u M u φ Element Number cm cm Comp. Tens. kn kn-m #25 5# #25 5# #25 7# #16 11# #16 11# #22 13# #22 15# #22 15# #22 12# #19 10# #19 10# #22 11# #19 6# #19 6# #22 6# odd- and even-story frames. The structural tendences descrbed above were also observed n the one-story portal frame dscussed n secton It was found there that smaller span frames favored small end moments (small columns-larger grder), whle the larger span frames were more economcal when larger end moments were developed (large columns-smaller grder). In the one story example of secton the transton between small and large span occurred between 12 m and 14 m. In the multstory frames of ths secton the transton occurred at less than 10 m, due to the dfferent end condtons of the grders.

20 332 Andres Guerra and Panos D. Kouss Fg. 16 Comparson of optmal and typcal desgn costs n multple story structures subject to vertcal loadng only Fg. 17 Increasng cost of multple story structures subject to horzontal loadng Fg. 16 llustrates the same optmal costs for the long-span structures along wth typcal desgn costs. The typcal desgn assumptons resulted n effcent stffness dstrbutons for the one-story structure. The two-through sx-story structures showed cost savngs of 11.6%, 3.0%, 9.4%, 6.9%, and 6.2%, respectvely Mult-story frames wth lateral loads The long-span mult-story frames of the prevous secton are desgned here for gravty and sesmc loads, as descrbed n Table 2. Followng ACI requrements, both heavy (66 kn/m) and lght (27 kn/m) gravty loads are consdered as calculated n the Secton 4.2, on Loadng Condtons. The ncreased stffness requrements due to the lateral loads elmnate the stff-soft patterns observed n the gravty-only examples of the prevous secton. The cost of each optmzed desgn s presented n Fg. 17, along wth the costs of the gravty only frame desgns to ndcate the cost ncrease due to lateral loadng. It can be seen that the sesmc loads do not cause sgnfcant cost burdens for frames

21 Desgn optmzaton of renforced concrete structures 333 wth three stores or less, but they become farly costly for taller structures. Ths observaton s ste dependent, and can be dfferent for sesmc loads that correspond to a dfferent ste. 5. Conclusons Ths paper presents a novel approach for optmal szng and renforcng mult-bay and mult-story RC structures ncorporatng optmal stffness correlaton among structural members. Ths study ncorporates realstc materals, formng, and labor costs that are based on member dmensons, and mplements a structural model wth dstnct desgn varables for each member. The resultng optmal desgns show costs savngs of up to 23% over a typcal desgn method. Comparson between optmal costs and typcal desgn method costs demonstrates nstances where typcal desgn assumptons resulted n effcent structures and where they dd not. The formulaton, ncludng the structural FEA, the ACI member szng and the cost evaluaton, was programmed n MATLAB (Mathworks, Inc.) and was solved to obtan the mnmum cost desgn usng the SQP algorthm mplemented n MATLAB s ntrnsc optmzaton functon fmncon. A number of farly smple structural optmzaton problems were solved to demonstrate the use of the method to acheve optmal desgns, as well as to dentfy characterstcs of optmal geometrc spacng for these structures. It was found that optmal portal frame desgns follow dfferent patterns for small and large bay lengths. More specfcally t was found that short-span portal frames are optmzed wth grders that are stff compared to the columns, thus ensurng grder smple supported acton, whle long-span portal frames are optmzed wth grders that are approxmately as stff as the columns, thus splttng the overall grder moment to approxmately equal negatve and postve parts. It was also found that grders that are supported by multple supports, as n the case of mult-bay frames, have an optmal span length, below whch the desgn becomes uneconomcal because some members are controlled by code mposed mnmum szes, and above whch the desgn also becomes uneconomcal as the member szes tend to become excessve. Fnally t was found that optmal desgn mult-story frames present smlar characterstcs to onestory portal frames where short-bay desgns are optmal wth grders that are stff compared to the columns, and long-span desgns are optmal wth grders that are approxmately as stff as the columns. For gravty domnated long-span mult-story frames, optmal desgns tend to have alternatng stff and soft columns. It s not however clear that ths pattern exsts n optmal desgns of mult-bay multstory frames, consderng that the nteror columns typcally do not carry sgnfcant moments due to gravty. Fnally, the alternatng stff/soft column pattern was not observed when the horzontal sesmc loads had a sgnfcant nfluence on the desgn. References Amercan Concrete Insttute (ACI)(2002), Commttee 318 Buldng Code Requrements for Structural Concrete (ACI ) and Commentary (ACI 318R-02), Detrot. Ballng, R.J. and Yao, X. (1997), Optmzaton of renforced concrete frames, J. Struct. Eng., ASCE, 123(2), Camp, C.V., Pezeshk, S., and Hansson, H. (2003), Flexural desgn of renforced concrete frames usng a genetc

22 334 Andres Guerra and Panos D. Kouss algorthm, J. Struct. Eng., ASCE, 129(1), Constructon Publshers and Consultants (2005), RS Means Concrete and Masonry Cost Data 23 rd Ed. Reed Constructon Data, Inc. MA. Edgar, T.F., and Hmmelblau, D.M. (1998), Optmzaton of Chemcal Processes, 2 nd ed. New York, McGraw Hll. Lee, C., and Ahn, J. (2003), Flexural desgn of renforced concrete frames by genetc algorthm, J. Struct. Eng., ASCE, 129(6), McCormac, J.C., (2001), Desgn of Renforced Concrete, 5 th ed. New York, John Wley and Sons, Inc. Moharram, H., and Grerson, D.E., (1993), Computer-automated desgn of renforced concrete frameworks, J. Struct. Eng., ASCE, 119(7), The Mathworks Inc. Constraned optmzaton, utor13b.shtml#51152 Accessed Aprl 1, SEI/ASCE 7-98, (2000), Mnmum Desgn Loads for Buldngs and Other Structures, Amercan Socety of Cvl Engneers, Reston VA. CM