asramshk,

wo deve lifers aree lifeobabiles. Thn prod parfitne

mes a

near.

2001; Vamvatsikos and Cornell 2004. Although analytical meth- The generation of probabilistic performance-based seismic de-

odologies for computing response histories have been availablefor decades through the pioneering work of Newmark 1959, andimprovements in computational power since then have been sig-nificant, using second-order inelastic time-history analysis as abasis for design has yet to see widespread application in designpractice for the following reasons: Understanding a systems response is cumbersome because a

large amount of information is produced and its interpretationrequires sound technical knowledge and extensive experience.

Reliable ground motion input records have to be available withvarying attributes, locations, and probabilities of exceedance.

sign model codes with subsequent verification using computa-tional simulation and experimental research has evolved intopractical model codes for performance-based design SAC2000.

Researchers Deierlein and Moehle 2004; Hamburger 2004have begun detailed discussions and planning of future proba-bilistic performance-based design methodologies.

A better understanding of the processes of earthquake groundmotion generation and propagation, and seismic hazard analy-sis along with a growing number of instruments is making alarger number of ground motion time series available to struc-tural engineers.

Developments in soft computing applications in structural en-gineering has helped to automate the design process by intel-ligently performing the time consuming trial-and-error designcycles on the computer.Performance-based design optimization is a combination of

state-of-the-art performance-based seismic engineering and evo-lutionary computation into an automated design environmentwhere design optimization is implicitly built into the process. Al-though there are significant discoveries left to be made, especiallyin the areas of structural behavior, it is now safe to say that theanalytical techniques and computational power have facilitated amovement of design concern from life safety to minimization ofeconomic and human loss during a structures service life.

Structural design, by nature, is a trial-and-error process. Thecomplexities of geometry necessitated by modern architectural

1Visiting Associate Professor, Univ. of WisconsinMadison; AssociateProfessor, Marquette Univ., Milwaukee, WI 53233. E-mail: chris.foley@marquette.edu

2Emison Professor of Civil Engineering, Dept. of Civil Engineering,Univ. of Memphis, Memphis, TN 38152. E-mail: spezeshk@memphis.edu

3Senior Seismic Research Engineer, John A. Martin and Associates,Inc., Los Angeles, CA 90015. E-mail: arzhang@johnmartin.com

Note. Associate Editor: Donald W. White. Discussion open untilNovember 1, 2007. Separate discussions must be submitted for individualpapers. To extend the closing date by one month, a written request mustbe filed with the ASCE Managing Editor. The manuscript for this paperwas submitted for review and possible publication on August 26, 2005;approved on April 19, 2006. This paper is part of the Journal of Struc-tural Engineering, Vol. 133, No. 6, June 1, 2007. ASCE, ISSN 0733-9445/2007/6-757766/$25.00.

JOURNAL OF STRUCTURAL ENGINEERING ASCE / JUNE 2007 / 757Probabilistic Performance-BMoment-Resisting F

Christopher M. Foley, M.ASCE1; Shahram Peze

Abstract: Significant progress has been made in the preceding tquickly migrating from prescriptive procedures intended to preserto quantify risk associated with designs. Therefore, all stakeholdeand risk leading to structural designs that not only reliably preservfrequent ground motions and thereby minimize life-cycle costs. Prtive design methods and full reliability-based design methodologicode performance-based design methodology and casts this desigstory and multistory structural steel frameworks with fully anevolutionary genetic algorithm with radial fitness and balancedapplications of the automated design algorithm to single-story fraand beam-to-column moment capacity ratios.

DOI: 10.1061/ASCE0733-94452007133:6757

CE Database subject headings: Seismic design; NonliOptimization; Algorithms; Probabilistic methods; Inelastic action

Introduction

Performance evaluation of nonlinear dynamic systems is a com-plex process Challa and Hall 1994; Aschheim and Black 2000;Deierlein and Mehanny 2000; Chopra 2001; Chopra and Goeled Optimal Design of Steeles. I: Formulation

M.ASCE2; and Arzhang Alimoradi, A.M.ASCE3

cades in the area of seismic engineering. Design codes are verysafety to reliability-based design with less prescription intendedgiven the opportunity to speak a common language probability

safety after rare ground motions, but minimize damage after moreistic performance-based design is in between traditional prescrip-e present paper provides an overview of a state-of-the-art model-cedure into multiple-objective optimization problems for single-tially restrained connections. A methodology for applying anss functions is discussed in detail. A companion paper providesnd multistory frames with a variety of connection characteristics

analysis; Connections, semi-rigid; Evolutionary computation;

Current design procedures involve significant trial and errorand, therefore, will be time consuming and computationallyexpensive if economy of design is pursued.One might argue that headway in overcoming these challenges

has been made through three significant developments:

needs and the stakeholders desire to understand system behavior consider structural components. Nonstructural damage althoughextremely important is not considered.under different loading scenarios challenges a structural engineer

to efficiently generalize the experience of previous designs intonew projects. An adaptive methodology that can be automated isanalogous to the engineers experience-based trial-and-errorprocedures. Genetic algorithms GAs and evolutionary algo-rithms EAs, which operate using computer simulation of naturalevolution, have been applied successfully in recent years to struc-tural design involving complex objectives and constraints with alarge number of design variables Camp et al. 1996; Cheng et al.1999; Pezeshk et al. 2000; Foley 2001; Foley and Schinler 2001;Grierson and Khajehpour 2001; Khajehpour and Grierson 2001;Pezeshk and Camp 2001; Schinler 2001; Schinler and Foley2001; Cheng 2002; Foley and Schinler 2003; Alimoradi 2004.The success of GAs and EAs is in no small part due to theirpower of adaptation and learning.

State-of-the-art research is developing practical, yet accurate,computational tools for simulating nonlinear dynamic responseOpenSees 2001. There also exists a methodology for reliability-based design and evaluation of real structures in test-bed pro-grams MAE 2005; PEER 2005. However, to date, there appearsto be a gap in implementation of new analytical technologies intorobust computational design tools to automate the process ofperformance-based design. This studys objective is to develop apractical and automated computational tool for implementation ofprobabilistic performance-based design of steel moment-resistingframe systems through the use of evolutionary computation andadvanced computational methodologies. It should be noted thatdesign objectives that involve societal consequences are veryimportant considerations for performance-based seismic engineer-ing. However, these objectives are outside the scope of thepresent effort.

Probabilistic Performance-Based Design

Minimization of seismic risk for an individual structure can beachieved by optimizing performance under various seismic haz-ard scenarios. At the same time, efficient use of material andresources necessitates minimization of initial construction costand expected repair costs over the structures service life. Forsteel moment-resisting frames considered in this study, SAC2000 presents analytical methods of quantifying structural per-formance at two levels of seismic hazard: Rare events with lessthan 2% probability of exceedance in 50 years 2/50 events; andmore frequent seismic events with 50% probability of exceedancein 50 years 50/50 events. The present section will outline theimportant fundamentals of the probabilistic design methodologyupon which the present effort is based. Further details can befound in SAC 2000.

Performance ObjectivesA performance objective consists of two components: A statedmaximum level of expected damage also called a performancelevel; and a level of seismic hazard. In a two level methodologySAC 2000, performance objectives can be described as: 2% probability of performance inferior to collapse prevention

in 50 years; and 50% probability of performance inferior to immediate occu-

pancy in 50 years.At this point, it should be emphasized that damage states only758 / JOURNAL OF STRUCTURAL ENGINEERING ASCE / JUNE 2007Collapse prevention CP is a postearthquake damage state inwhich significant degradation in stiffness and strength of thelateral force resisting system has occurred. The structure is onthe verge of total or partial collapse. Immediate occupancy IOis defined as a postevent damage state for which repair is un-necessary. The basic vertical and lateral force resisting systemsof the building retain nearly all of their pre-earthquake strengthand stiffness SAC 2000. CP level design principally controlsthe loss of life and the number of casualties, while IO level de-sign is targeted to control the functionality of the constructedfacility and the economic loss after an earthquake. This designmethod, at its core, is similar to procedures proposed to minimizedeaths, dollars, and downtime Deierlein and Moehle 2004;Hamburger 2004.

As a first step, association of a limit state with a level ofseismic hazard in a stated exposure period e.g., service life ismade. This is done by the designer in collaboration with theowner or other stakeholder considering minimum requirements inthe design provisions. In the course of performance-based design,different levels of damage and functionality can be agreed upon interms of the performance objectives. Within each performanceobjective, various performance levels are often associated withcosts of construction and expected repair. Development of amethodology for creating decision-making curves to elucidatetrade-offs between seismic performance and the cost of construc-tion and expected repair is the focus of the present effort.

Uncertainty in Meeting PerformanceSAC 2000 presents a performance-based design procedure in areliability format. The basic procedure consists of an evaluationof a set of demand-to-capacity ratios at given objective levels forvarious structural response parameters of interest. These responseparameters could be quantities such as interstory drift angle orcolumn compression force. Factors are introduced in the defini-tion of the demand-to-capacity ratios to model uncertainty in re-sponse during seismic events, uncertainty in the analysis beingable to reliably predict response, and uncertainty in the capacity.The demand-to-capacity ratios are used to evaluate confidencelevels associated with the probability of the structure experienc-ing performance worse than the design performance objectiveduring an exposure period.

Similar to the practiced method of load and resistance factordesign ACI 2005; AISC 2005, the procedure outlined in SAC2000 is referred to as demand and resistance factor design. Thetheoretical basis and the assumptions made in the derivation ofdesign equations are presented in the literature SAC 2000; Cor-nell et al. 2002; Yun et al. 2002. The overall process is brieflyoutlined for completeness. The confidence parameter is used toquantify confidence in meeting performance objectives. This pa-rameter is evaluated using computed demand and capacity for avariety of response parameters. The general expression for theconfidence parameter is SAC 2000

=aDC

1

where Dmedian demand for a defined response quantity e.g.,column compression force from a structural analysis; Cmediancapacity for the same response quantity; demand variabilityfactor; aanalysis uncertainty factor; and capacity reduction

Table 1. Parameters for Confidence-Level Determination SAC 2000

inear a

drift evfactor. Eq. 1 is evaluated for each structural response parameterconsidered at each performance level defined.

The confidence parameter is used to evaluate confidence levelsin attaining the stated performance objectives. The confidenceparameter is used in the following expression to determine confi-dence levels SAC 2000

= exp bUTKx 12kUT 2where bcoefficient that relates incremental change in demand toan incremental change in ground motion b=1.0 implies a linearrelationship; UTuncertainty measure equal to the vector sumof the logarithmic standard deviation of the variation of demandand capacity from uncertainty; kslope of the hazard curve innatural log-log coordinates; and Kxstandard Gaussian variateassociated with the probability of x not being exceeded as a func-tion of the number of standard deviations above or below themean. The parameters used in the present study for the partiallyrestrained PR and fully restrained FR frames considered aregiven in Table 1.

Components of Design Procedure

The design procedure outlined in SAC 2000 is relatively newand a concise outline of how the procedure is employed in thepresent study is beneficial for the reader to fully appreciate thecomputational demands, the simplified design objectives, andthe simplified constraints that can be formulated using the newmethodology. The approach used in the present optimization for-mulations to address these issues is highlighted in the following.

Evaluation of Demand

For each performance objective, evaluation of demand consists ofselecting input earthquake characteristics e.g., design spectrum,accelerationdisplacement spectrum, or acceleration time histo-ries and appropriate structural analysis procedures that can beused to estimate the response parameters of interest see the nextsection. In the present study, acceleration time histories and in-elastic time-history analysis are used as the basis for demandevaluation.

Fully restrained frames

Immediate occupancyperformance

Collapse preventionperformance

UT,drift=0.20b UT,drift=0.30UT,CCF=0.0225+2a UT,CCF=0.0225+2Cdrift=0.02 Cdrift=0.10CCCF from AISC 2005 CCCF from AISC 2005=1.0b =0.90=1.5 =1.3a=1.02 a=1.0a=e

1.42a a=e1.42

acoefficient of variation of the axial load values from the suite of nonlassigned using yield surfaces.bValues given assume global interstory drift evaluation. Local interstoryJOUResponse Parameters

SAC 2000 uses three response parameters for assessing perfor-mance of steel moment-resisting frames MRFs: interstory driftangle; column axial compression force; and column splice ten-sion force. Interstory drift has long been used as a measure oflateral stability of a structural framing system and for assessmentof serviceability. In prescriptive design specifications or codes,drift limits are set in anticipation that these limits will minimizestructural damage and preserve the stability of the structure dur-ing a ground motion event. These limits are also established topreserve structural stability for postevent recovery and/or rescueoperations. Interstory drift is closely related to plastic rotationdemand in the MRF girders, columns, connections, and P in-stability. As a result, interstory drift is a reasonably good measureof local and global damage to the structure. Interstory drift is usedas a response parameter in the present study.

Column axial compression is also a very useful response pa-rameter for assessing the performance of MRFs. The axial com-pression in a column that resides within a MRF can be used toassess force-controlled behavior within the structural system. Forexample, force-controlled behavior may be inelastic buckling of acolumn and this may lead to inferior frame performance. Thepresent study utilizes column compression force in the assessmentof structural performance.

Column axial splice tension is also a highly important re-sponse measure. It is most critical in situations where columnsplices exist within the structure. These splices may not havesuitable tension capacity and, therefore, they may limit structureperformance during ground motion events. The present study con-siders one- and three-story steel frames and, therefore, it is as-sumed that splices do not exist in the frameworks considered. Asa result, column tension forces are not considered as responseparameters.

Analysis Type

Four different types of structural analysis are allowed in SAC2000 for the design of moment-resisting frames: Linear staticprocedure; linear dynamic procedure; nonlinear static procedure;and nonlinear dynamic procedure NDP. They encompass vary-ing degrees of computational cost and analytical complexity andnaturally result in differing levels of accuracy.

Caution should be exercised in selecting the method of analy-

Partially restrained frames

Immediate occupancyperformance

Collapse preventionperformance

UT.drift=0.20 UT,drift=0.35UT,CCF=0.0225+2 UT,CCF=0.0225+2

Cdrift=0.01 Cdrift=0.10CCCF from AISC 2005 CCCF from AISC 2005

=1.0 =0.85=1.4 =1.4a=1.02 a=1.03a=e

1.42 a=e1.42

nalyses. It should be noted that column compression force capacities are

aluation was omitted in the present study.RNAL OF STRUCTURAL ENGINEERING ASCE / JUNE 2007 / 759

sis as some methods may not be applicable to specific structuralsystems e.g., regular frames, torsion-sensitive buildings. How-

are used as input to the analytical model to compute the medianresponse of the structure for the performance levels associatedever, if appropriate engineering judgment is exercised, NDP canbe applied to all systems. It commonly results in more accurateresponse quantities than other methods, because all modes ofvibration, geometric and material nonlinearity, and second-ordereffects can be captured in the analysis. The additional computa-tional cost associated with carrying out this level of analysis issignificant. Furthermore, selecting an appropriate suite of groundmotions for carrying out the nonlinear dynamic analysis must bedone with due diligence. The reason for this is that design spectrainclude a wide range of ground motion characteristics for a build-ing site. A single acceleration time history or limited suites ofground motions is thought to be incapable of providing reliabledesigns for buildings. As a result, care must be exercised in se-lecting ground motion acceleration time histories for a given siteand recurrence intervals.

In this study, nonlinear second-order time-history analysis isused as the analytical basis for the design automation. Todayscomputational advancements justify employment of such so-phisticated analytical engines in the design optimization and/orautomation to fully leverage state-of-the-art computation andstate-of-the-art design methodologies.

Ground MotionsThe input ground motion can be represented in different ways,with the simplest form being a time history that is scaled to user-defined peak ground accelerations, and more precise descriptionsbeing a design spectrum compatible strong motion time history.The intensity level associated with the ground motion used instructural analysis should be consistent with the performance ob-jectives of design.

The most accurate estimate of response is normally attainedthrough time-history analysis using large sets of input groundmotion records whose median spectral acceleration falls as closeas possible to a target design spectrum that described the hazardlevel associated with the design performance objectives at thesite. In this study, a suite of seven ground motion accelerationtime histories for IO performance and a second suite of sevenhistories for CP performance are utilized for determining responseand evaluating performance.

Strong ground motion records representing 2 and 50% prob-abilities of exceedance in 50 years for the city of Los Angeleswere chosen from the records developed in the SAC steel projectSomerville et al. 1997. A newly developed GA-based groundmotion search and scaling program Naeim et al. 2004 could alsobe used to perform the task of ground motion selection and opti-mal scaling.

The pool of ground motion records used to represent targetdesign spectra of the National Earthquake Hazard Reduction Pro-gram Site Category D ATC 1997a,b with deaggregation of haz-ards of M 6.757.5 at closest distance of 220 km, and M 57 at515 km, for 2/50 and 50/50, respectively. The horizontal com-ponents of the acceleration time histories are provided in strike-normal and strike-parallel components. The 2/50 earthquakerecords are chosen from the near-fault recordings or simulationswith virtually no scaling, and the simulated time histories are formagnitude 7.1 events on the Elysian Park fault a blind thrustfault with shallowest depth of 10 km. The 50/50 events are fromcrustal earthquakes on soil category D. The near-fault recordscover a balance of faulting mechanisms such as strike slip, ob-lique, and dip slip Somerville et al. 1997. These time histories760 / JOURNAL OF STRUCTURAL ENGINEERING ASCE / JUNE 2007with the record probabilities. Fig. 1 illustrates the ground motionrecords used for the response simulation in the present study.

Evaluation of CapacityCapacity is the maximum force or displacement that a structuralmember, substructure, or structural system can support or undergoat a defined limit state. Quantifying uncertainty in meeting per-formance objectives through a mechanism such as Eq. 1 re-quires that capacities be defined. Each of the response parametersused in the performance evaluation have associated capacities thatrequire definition.

FEMA-350 SAC 2000 defines interstory drift angle capaci-ties at two levels: Global and local. The global interstory driftcapacity is thought to be a very good parameter for assessing astructures ability to resist P instability and collapse SAC2000. However, large interstory drifts can also place large de-mands on the connections within the structure, and interstory driftcapacities defined with consideration of local connection responseare also needed. The present study utilizes global interstory driftangle capacities. Local interstory drift angle capacities are omit-ted from consideration.

Column compression force CCF capacities are another veryimportant contributor to the process of quantifying the confidencelevels in meeting performance objectives. The present study usesyield surfaces with critical parameters that define the yield surfaceassigned using design specifications AISC 2005. Fig. 2 illus-trates the yield surfaces for beam elements and beamcolumnelements used in the present study. Pyttensile yield capacityof the member assuming the gross cross-sectional area. Mp+ andMp

positive and negative plastic moment capacities for the baresteel member. Finally, the axial compression capacity in the ab-sence of bending moment, Pncminor, is taken as the out of planeinelastic buckling capacity of the member. This point on the in-teraction diagram and the kink is connected conservatively witha straight line.

Column splice tensile capacities are also integral to the SAC2000 methodology. As discussed earlier, the present study islimited to buildings that are three stories and less and the inclu-sion of column splices in these structures was omitted.

Collapse of a building structure is associated with the MRFsinability to support gravity loading during or after an event. Thiscan also be defined as a system capacity consideration, although itis not considered directly in the SAC 2000 procedure. Collapsehas been described as a ratcheting over of a framework resultingin progressive P excursions Gupta and Krawinkler 2000;Ibarra and Krawinkler 2004. Collapse can be described as asimple loss of lateral stiffness in combination with the framebeing left in a significantly displaced configuration at the comple-tion of the ground motion event. Deierlein and Mehanny 2000introduced an objective measure called the stability index in orderto assess the tendency for collapse. The incremental dynamicanalysis technique Vamvatsikos and Cornell 2004; Kunnath2005 is another approach that will explicitly allow the designerto assess the tendency for collapse during or immediately follow-ing severe ground motion events.

Assessing collapse or near collapse via interstory drift is arelatively complex procedure and a standard methodology is lack-ing in this regard. Monitoring the numerical algorithm executionfor appearance of negative definite or singular stiffness matriceswas done, but it should be pointed out that the damping levels

theassumed in computing structure response 5% in the presentstudy may maintain a numerically stable structure during theground motion event. A structure was deemed unstable if, duringcomputed response, the numerical algorithm indicated an ill-conditioned system of equations. As methods for improving as-sessment of near collapse improve, these methodologies can berelatively easily included in the methodology proposed.

Confidence in Meeting PerformanceThe essence of probabilistic performance-based design is to in-clude uncertainty in a direct fashion. The SAC 2000 procedureincorporates uncertainty through definition of confidence levelson meeting performance objectives. Median structural demandparameters obtained from a set of response history analyses isused to find a ratio of probabilistic demand to capacity calledthe confidence parameterEq. 1. The ratio is calculated forall response parameters considered in the design. As outlined ear-lier, the present study considers column compression force andglobal interstory drift angle. Eqs. 1 and 2 are used to defineconfidence levels in meeting performance by considering globalinterstory drift and column compression force independently. Thelower confidence level then becomes the controlling level for as-sessing a designs expected performance. Table 1 contains theparameters needed to utilize Eqs. 1 and 2.

Confidence levels are computed by considering responseparameters and then computing the corresponding confidence pa-

Fig. 1. SAC ground motion acceleration time histories used in1 in./ s2=0.0254 m/s2.JOUrameters. Assuming all SAC 2000 response parameters areconsidered, confidence parameters for each performance objectivewould be computed as

IO = maxdriftGlobal_IO,driftLocal_IO,CCFIO ,spliceIO 3

CP = maxdriftGlobal_CP,driftLocal_CP,CCFCP ,spliceCP 4where is defined in Eq. 1. The subscript drift stands for inter-story drift angle and CCF represents the maximum mediancolumn compression force demand-to-capacity ratio obtained

Fig. 2. Yield surfaces for beam and beamcolumn members: abeam member with no axial force bendingmoment interaction; bbeamcolumn member with axial-force bendingmoment-interaction

present study Somerville et al. 1997. It should be noted thatRNAL OF STRUCTURAL ENGINEERING ASCE / JUNE 2007 / 761

simply stated as: Minimize the initial cost of construction; maxi-mize the confidence level on the probability of damage not toconsidering the entire response history for all column members inthe framework. The resulting IO and CP confidence parametersare then used in conjunction with Eq. 2 to compute confidencelevels associated with meeting each of these performance objec-tives. Based upon earlier discussions regarding response param-eters, the confidence parameters for the present study are basedupon the following controlling conditions:

IO = maxdriftGlobal_IO,CCFIO 5

CP = maxdriftGlobal_CP,CCFCP 6It is worth noting that column compression force demand

confidence parameter calculation appears to be inconsistent withuse of interaction yield surfaces in inelastic analysis. The reasonfor this is that the interaction surface can replace use of designspecification equations. This is often called advanced analysisClarke et al. 1992; White 1992, 1993; White and Nukala 1997;Bridge et al. 1998; Foley 2001; Foley and Schinler 2003. Designfor CCF is based on the evaluation of Eq. 1. Here, demand is thelargest median column compression force found for all membersin the frame obtained from nonlinear second-order time-historyanalysis of the structure that is subject to a set of input groundmotion time histories. Capacity is given in FEMA-350, whichbasically follows the AISC specifications. Uncertainty and capac-ity reduction factors are shown in Table 1.

Optimized Design Problem Statements

One of the attractions to performance-based design optimizationis that the design statements become very direct and greatly sim-plified from a stakeholders perspective. For instance, a multi-objective reliability-based optimization problem can be relatively

Fig. 3. Connection models used in the present study: a nonlinearmomentrotation response without gap; b nonlinear momentrotation response with gap762 / JOURNAL OF STRUCTURAL ENGINEERING ASCE / JUNE 2007exceed CP performance during the structures service life e.g., 50years; and maximize the confidence that damage will not beworse than that associated with IO performance during the struc-tures service life. These types of design formulations can bemore appealing to owners of buildings.

The present study considered both portal frame design andmultistory frame design. It was decided to formulate two opti-mization problem statements. The portal frame analysis wouldlikely not be as sensitive to the column compression force re-sponse parameter because the axial loading in the columns islikely to change very little during the ground motion events.However, in the case of multistory frames, the column compres-sion forces during the events may change significantly and, there-fore, the column compression force response parameter maybecome more important in assessing confidence in meeting per-formance. The present section outlines the optimization problemstatements considered.

The first-optimization problem statement considered has beenshown to be applicable to the portal frame considered in thepresent study Alimoradi 2004. Therefore, the probabilisticperformance-based design optimization problem statement forportal frames with partially or fully restrained connections can bestated as

Minimize:

Z1 = Cinitial

i=1

Nmem

iAiLi Vcolumns + Vbeams 7

Z2 =1

qCP8

Z3 =1

qIO9

Subject to:

Mpcol 1.2

j=1

Nb

Mpbm full-strength connections 10

Mpcol 1.2

j=1

Nb

Mcycn partial-strength connections 11

qIO qlimitIO 12

qCP qlimitCP 13

where Cinitialinitial construction cost; imaterial weight den-sity for member i; Aicross-sectional area of member i;Lilength of member i; Vcolumnsvolume of the columns in theframe; Vbeamsmaterial volume of the beams in the frame;qIOconfidence level associated with meeting IO performance;qCPconfidence level in meeting CP performance; Mpcolplasticmoment capacity of the column at a given beam-to-column joint;Mp

bmplastic moment capacity of beam j at the same beam-to-column joint; Mcycnyield moment capacity of connection j at thebeam-to-column joint; and Nbnumber of beams framing into thebeam-to-column joint considered. Minimization of is equivalentto maximization of q.

Two nonlinear connection characteristics are considered inthe present study. Both are shown in Fig. 3. The first is a bilinear

construction objectives and constraint functions as they develop.Other objectives and constraints for steel framed structures thatconnection with initial stiffness, Kc, hardening stiffness ratio,

, and yield moment capacity, Mcy. In this connection, unloadingis assumed to take place with the initial stiffness and there isno limit to the connections rotation capacity. The second connec-tion is also bilinear. However, hysteretic pinching is modeled inthis connection. The hardening stiffness and the initial elasticstiffness are used to define the gap behavior associated with thisconnection.

Naturally, a given design that has a lower demand to capacityratio provides higher level of confidence on the performance levelfor which the ratio of demand to capacity is obtained. Throughoutthis study, the minimization of is always analogous to maximi-zation of the level of confidence. The choice of either variable is,however, up to the user. If communicating level of confidencewith the stakeholder is preferred, the final minimal ratio of de-mand to capacity could be easily transformed to a level of confi-dence following Eq. 2.

The SAC 2000 procedure includes minimum confidence lev-els for assessing performance. If one assumes global behaviorlimited by interstory drift as the controlling response parameter,as done in the present study, the SAC 2000 methodology re-quires at least 50% confidence in attaining IO performance and90% confidence in meeting CP performance objectives. Thepresent optimization problem includes maximization of confi-dence for both performance levels and, therefore, a lower-boundlimit was used as shown in Eqs. 12 and 13.

The second optimization problem formulation consideredsought to ensure economy of designs by establishing a mecha-nism in which the confidence level based upon the global inter-story drift response parameter is high and is as close as possibleto the confidence level computed, using the column compressionforce response parameter Alimoradi 2004. This formulation willbe termed balanced design. An optimization problem formulatedin this manner can be mathematically stated as

Minimize:

Z1 = Vcolumns + Vbeams 14

Z2 =qCCF

CP qISD

CP 2

qCP15

Z3 =qCCF

IO qISD

IO 2

qIO16

Subject to:qISD

CP= qo

CB, qISD

IO= qo

IO 17

in which qoCP and qoIOtarget levels of confidence for design.

The relatively simplistic format of the optimization problemsoutlined above hides the complexity of the analysis needed toallow such simplistic statements. One should also note that thereare two constraints that fall outside the realm of understanding forowners: The strong column weak beam SCWB criteria. Theremaining expressions are posed in the language of owners. Thisis the beauty of probabilistic performance-based design.

It is recognized that minimizing initial construction cost on thebasis of structure volume weight is overly simplistic for steelframe structures. However, the methodology being proposed ismost certainly capable of easily including more complex initialJOUinclude constructibility have been proposed and successfullyimplemented in optimized steel frame design problems Liu et al.2006.

The evaluation of confidence levels of the structure in attain-ing IO and CP seismic performance will require specific point-in-time loading at the time of the events. The following load com-binations are used to evaluate seismic demands:

1.0D + 0.25L + 1.0E2/50 18

1.0D + 0.25L + 1.0E50/50 19

where E2/50 and E50/50 represent earthquake effects for 2/50 and50/50 ground motions, respectively.

Evolutionary Genetic AlgorithmThe multiple objective optimization problem considered in thepresent study is highly complex when one considers that the ana-lytical modeling for demand prediction requires inelastic time-history analysis to accurately quantify the stated objectives andconstraint violations. Therefore, classical gradient-based optimi-zation algorithms will be difficult if not impossible to utilize.However, the GA a class of EA modeled on survival of the fittesttheory has been successfully applied in many areas of structuralengineering as alluded to previously. The attraction to using a GAfor solving problems like that considered is that gradients of ob-jective functions and constraints need not be included in the al-gorithm for solution. This is especially important when inelastictime-history analysis is considered as well as uncertainty. In fact,it may not be possible to formulate a traditional gradient-basedsolution for the problem currently under consideration. EAs havebeen shown to easily provide optimized and automated designalgorithms for complex inelastic-analysis-based design Foley2001; Foley and Schinler 2003.

A GA is a computational simulation of the natural evolutionaryprocess to solve search and optimization problems. Early formu-lations of adaptable systems on machines go back to the earlystages of computer software and hardware development Levy1992. It has taken a significant length of time for this subject tomature to the point where it is a practical tool. The pioneeringwork by Goldberg 1989, subsequent researchers, and the wide-spread availability of high-speed computers have paved the wayfor many applications of GAs in engineering. The fundamentalsof the GA have been diligently and thoroughly described through-out the literature Goldberg 1989; Mitchell 1997; Haupt andHaupt 1998; Michalewicz 1999; Coley 2001 and details will notbe repeated here.

A GA driver Carroll 2004 is used in the present study as afront end in the automated design algorithm. Various structuraldesigns are represented in the traditional manner using binarystring chromosomes that decode to the structural steel shapes usedas design variables in the problems considered. The parametersused to control the GA and improve its performance are often-problem dependent. Discussion of the two frames in the compan-ion paper Alimoradi et al. 2006 includes the GA parametersused to execute the automated design.RNAL OF STRUCTURAL ENGINEERING ASCE / JUNE 2007 / 763

Populations with higher fitness should have greater dispersionin objective space. In other words, higher fitness should beFitness Formulations

Application of the GA in the search for the optimal combinationof design variables requires that the optimization problem state-ments be recast into unconstrained optimization problems. Sinceseveral optimization problem statements were considered in thepresent study, multiple fitness formulations were also considered.

Consider two competing objectives Z1 and Z2, where it is de-sired to minimize both. If the two objectives compete with oneanother, the candidate designs can form decision-making curvesin objective space. The group of designs that dominate all othersis termed the Pareto front in objective space. If gradient-basedoptimization algorithms are used, this front can be approximatedusing weighting of individual objectives and then solving single-objective problems to find a set of single optimal solution pointson the curve. Other approaches to solving multiple-objectiveoptimization problems are available Balling and Wilson 2001;Deb 2001. Two distinct fitness formulations are used in the GAformulations presented in this study. Each will be discussed insome detail in the following sections.

Radial Fitness FormulationThe first formation of GA fitness is done within the context ofmultiple-objective optimization and Pareto optimal theory. Detailsof the formulation can be found in Alimoradi 2004. For thepresent discussion, we will limit fitness space to two dimensionsas shown in Fig. 4. The cloud of circles in Fig. 4 representscandidate designs that have been generated during the executionof the GA. The hypothetical Pareto front is shown as the solidline. When more than two objectives are considered, these frontsare hyperquadrics in n-dimensional space. These fronts may behyperspheres, hyperellipsoids, or hyperhyperboloids dependingupon the convexity of the relationship among objectives. In allcases, the Pareto front assuming minimization of all objectivescan be obtained by linking individual fitness to that individualsproximity to the origin in objective space. The physical analogyis very simplistic. The origin in objective space representsthe best possible solution to which both objectives are at theirminimumszero.

Any definition of fitness employed in the GA should nothinder the development of enough points on the Pareto front toreasonably represent the relative significance of different objec-tives in decision space. In general, individual fitness should havethe following characteristics: Individuals closer to the origin should have higher fitness; and

Fig. 4. Pareto front approximation in genetic algorithm for multiple-objective optimization764 / JOURNAL OF STRUCTURAL ENGINEERING ASCE / JUNE 2007assigned to populations where greater separation between can-didate designs along the Pareto front exists. This will aid de-signers with higher quality decision curves.

The aforementioned objectives may be translated into a polar co-ordinate system fitness function based upon the distance of solu-tions from the worst-case scenario design. In the two-dimensionalobjective space as shown in Fig. 4, this becomes minimization of1 /R, where the radial distance Ri for any candidate design is

Ri = Zmax1 Zi12 + Zmax2 Zi22 20The terms in Eq. 20 are defined in Fig. 4. The radial distance isnothing more than the magnitude of the position vector connect-ing the worst design, C, to the design considered.

The radial fitness is defined in the present study usingEqs. 713 as its basis. When column compression force isomitted as a response parameter appropriate for portal framesconsidered in this present study, the confidence levels in meetingperformance objectives can be defined using global interstorydrift limits. This correspondence is given below SAC 2000

qlimitCP

= 0.70, dlimitCP = 6.6% SMRF, dlimitCP = 5.9% OMRF

qlimitIO

= 0.70, dlimitIO = 1.5% SMRF, dlimitIO = 0.78% OMRF

SMRF denotes a special moment-resisting framework and OMRFdenotes an ordinary moment-resisting frame. In general, SMRFscontain detailing and members that are thought to be capable ofundergoing larger seismic demands more reliably than OMRFs.

SCWB behavior is also included. The fitness of the jth indi-vidual at the kth generation during the evolution can be written asAlimoradi 2004

Fjk = RjkRmaxdlimitCP

djkCP

dlimitIOdjkIO Mpcol1.2 Mpbm 21

in which

Rjk = dlimitIO djkIO2 + dlimitCP djkCP2 + Wmax Wjk2 22

Rmax = dlimitIO 2 + dlimitCP 2 + Wmax2 23djkIO and djkCPmean interstory drift demands for individual j ingeneration k for IO and CP ground motion records, respectively. Itshould be noted that if partially restrained connections are in-cluded in the frame considered, Eq. 21 should have the plasticmoment capacity of the beams changed to the corresponding endconnection plastic moment capacity.

Balanced Confidence FormulationThe second fitness formulation was developed to balance confi-dence levels for attaining performance objectives using predomi-nant response parameters: global interstory drift and column com-pression force. This formulation stems from the design objectivesposed in Eqs. 1417. In a manner similar to that used for theradial fitness formulation, a second fitness for individual j at gen-eration k in the evolution is written as Alimoradi 2004

Fjk = 16,717Vjk 0 . 012

CPIO

PCPPIO 24

where

CP = 0.01 for ISDCP CCFCP 1 25 ReferencesISDCP CCFCP 2 otherwiseIO = 0.01 for ISDIO CCFIO 1ISDIO CCFIO 2 otherwise 26

PCP =1

1 + ISDCP oISDCP0.05 227

PIO =1

1 + ISDIO oISDIO0.05 228

in which ISDCP and ISDIO confidence parameters for global inter-

story drift at CP and IO performance objectives, respectively;oISD

CP target confidence parameter for global interstory drift atthe CP performance level based upon a user-defined level ofconfidence; oISDIO target confidence parameter for global inter-story drift at the IO performance level based upon a user-definedlevel of confidence; and Vjkmaterial volume of individual j atgeneration k. The value 16,717 is the largest material volumepossible given the section sizes chosen for the set of all designvariables.

Concluding Remarks

A procedure for design of steel frames using a probabilistic-confidence-level-based formulation consistent with the SAC2000 methodology has been presented. This probabilistic meth-odology for design has been cast into several optimal designproblem formats. Both optimal design formats contain multipleobjectives. The first seeks to minimize the volume of structuralmembers, while also seeking to maximize the confidence levelsfor attaining immediate occupancy and collapse prevention per-formance. The second optimization problem formulation makesan attempt at ensuring economy in design by establishing amechanism in which the confidence level based upon the globalinterstory drift response parameter is high and is as close as pos-sible to the confidence level computed using the column compres-sion force response parameter. The formulations presented arecapable of considering partial connection restraint and assessmentof confidence levels for performance are based upon inelastictime-history analysis.

The optimization problems are intended to be solved using aGA and to this end, two fitness function definitions have beenpresented. The first utilizes a radial polar coordinate formulationthat is applicable to multidimensional objective space. The secondis a more traditional penalty function formulation.

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