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Computer-automated synthesis of building frameworks

DONALD E. GRIEKSON A N D GORDON E. CAMERON Solid Mechnr~ic.~ Divisiot~. Del~cirrtrrerzr of' Civil Etzgir~eer-irzg, Ur~iver-sit\) of Wrirerloo. Wrtrer-loo, Otit., Crrr~rrrl(i N 2 L 3C/

Rcccivcd February 6, 1984 Rcviscd manuscript accepted June 14. 1984

Thc paper surveys recent work carried out at the University of Waterloo concerning the computer-automated synthcsis of minimum-weight planar stcel frameworks. For any number of applicd loading schemes, the dcsign of the structurc may be constrained by a varicty of service and (or) ultimate 'limit-states' performance criteria concerning stresses, displaccmcnts. and (or) plastic collapse loads. Fabrication conditions can be imposcd to ensurc member-continuity and structure-symmetry requirements. Member sizes can be taken as either continuous or discrete variables to the synthcsis proccss, depending on whether custom or commercial-standard scctions arc available for the, dcsign. In the latter case, the structure may be automatically sized using the standard sections specified by a variety of different stcel dcsign codes (to datc, Canadian Institute of Steel Construction (CISC) and American Institute of Stecl Construction (AISC) scctions havc bccn invcstigatcd).

The computer-based design mcthod is iterative in nature and is remarkably effective and efficient. Commencing with an arbitrarily chosen initial 'trial' design, the synthcsis process determines the 'minimum-weight' structurc satisfying all of thc imposcd performancc and fabrication conditions without further designer intervention. The number of iterations required to achieve the optimal dcsign is generally small and almost totally independent of the complexity of the structurc.

Three different designs of a building framework are presented to illustrate the scope of the mcthod: limit-states dcsign using CISC sections; working-stress dcsign using AlSC scctions: and dcsign under elastic and plastic pcrformancc criteria using custom sections.

Key w0rri.s: steel. frameworks, standard scctions, custom sections. synthcsis, minimum weight, limit states, computers, automated.

Cet article donne un appcry sur I'optimalisation par ordinatcur de portiqucs plans en acicr. travail realist i I'Univcrsitk de Waterloo. Pour tout nombrc dc miscs en charge. Ic calcul de la structurc peut tcnir comptc dcs critkres dc rupturc Clastique et (OU) plastique conccrnant Ics contraintcs, dCplaccmcnts ct (ou) charges ultimcs. Dcs conditions peuvent etrc imposCcs lors dc la fabrication afin d'obtenir dcs klCmcnts continus et une structurc symCtriquc. Lcs dimensions dcs ClCmcnts pcuvcnt Ctrc constantes ou variables compte tcnu dc I'cxistence des scctions usucllcs. En dcrnier lieu. Ic calcul peut utiliser Ics sections standards spCcifiCcs par differentcs normes de calcul ( h datc, Ics sections dc catalogues CISC ("Canadian Institute of Stccl Construction") et AlSC ("American Institute of Stecl Construction") ont CtC utilisCes).

La mCthode de calcul utilisec par I'ordinateur est itCrativc ct particulikrcmcnt cfficacc ct rapidc. A partir d'un calcul approximatif, la mCthode de synthksc determine la structurc la plus ICgkre satisfaisant B toutes les cxigcnccs concernant la rksistance et la fabrication sans autres interventions du calculatcur. Lc nombrc nbcessaire d'itkration pour obtenir la solution optimale cst gCnCralemcnt petit ct Ic plus souvent indtpcndant de la complcxitC de la structurc.

Lcs calculs dc la structure de trois bitimcnts sont donnCs en cxcrnplc afin d'illustrcr la thCoric du calcul aux Ctats limites en utilisant les scctions du CISC; la thkorie du calcul dcs contraintes en utilisant Ics scctions dc I'AISC; Ics critkrcs dc rupturc des thCories Clastique ct plastique en utilisant des sections pcrsonncllcs.

Mots c1P.s: acier, structures, scctions usuclles, scctions pcrsonncllcs, synthksc. poids minimal. Ctats limitcs, ordinatcurs, automatisation.

[Traduit par la rcvuc]

Can. J. Civ. Ens. 11. 863-874 (1984)

Introduction The paper addresses the task faced by designers of

structures for which limit-states criteria must be satis- fied at one or more distinct loading levels. For example, the specified limit states may concern acceptable elastic displacements under service loads, acceptable elastic stresses under factored service loads, and adequate postelastic strength reserve of the structure under ulti- mate loads. Ideally, while satisfying the various per- formance criteria, the most economical design of the structure is sought.

The conventional approach to such design is to sepa-

rately proportion the structure to satisfy one set of per- formance criteria (e.g., stress limits), and to then mod- ify the structure to satisfy the one or more other sets of criteria that are of concern to the design (e.g., displace- ment limits, failure limits, etc.). A drawback of this approach, however, is that decisions taken at any one time to satisfy some of the performance criteria are usually made in the absence of explicit information as to their consequences for the other criteria. As such, they may result in the violation of criteria that were otherwise satisfied at a previous design stage. More- over, such an approach makes it difficult to have explic-

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8 64 CAN. 1. CIV. ENG. VOL. l I . 1981

CONTINUOUS VARIATION c.1 (custom sections] ciated with it. First, as it is generally impossible to uniformly scale the member sizes to achieve a standard- section design, stress redistributions will occur that may result in the violation of one or more of the ~erformance criteria. Second, there is no guarantee that the scaled

b l ~ ~ ~ ~ - 1 ~ 1 ;::~;;,R:ISCRETE design defines the minimum-weight standard-section structure.

lcustorn sections 1 Further studies by Lee (1983) and by Grierson and Lee (1983, 1984) extended the design &hod such that the sizes of the cross sections for the members of planar

( c l 1 TL [ ~ ; ; ; ~ ~ - O I S C R E T E frameworks can be taken as discrete variables to the is londord s e c t i o n s ~ design. This capability allows the synthesis technique

to be applied for two different design situations, de- FIG. 1. Different types of member-sizing variables for pending on the nature of the prevailing limit-states cri-

design. teria. First, for the case in which both elastic and plastic performance criteria must be satisfied, the design cross

it concern for design economy. At best, a somewhat section for each member is automatically selected from cumbersome trial-and-error process is required to among a specified set of regular-discrete sections (spe- achieve a reasonably efficient design. cifically, a set of discrete cross sections of different

An initial study by Grierson and Schmit, Jr. (1982) sizes but of the same shape; see Fig. I b ) . Second, for considered thin-walled structures comprised of bar, the case in which elastic stress and (or) displacement membrane, and (or) shear-panel elements under uni- axial stress, and developed a computer-automated synthesis capability whereby a minimum-weight struc- ture is found while satisfying all specified limit-states criteria simultaneously. For first-order behaviour (small-deformation theory) and proportional static loading, the limit states may involve ;ny combination of performance criteria concerning acceptable elastic stresses and displacements under service loads and ade- quate postelast~c strength reserve of the structure under ultimate loads. As well, fabrication conditions can be imposed to ensure member-continuity and structure- symmetry requirements. Commencing with an arbi- trarily chosen initial 'trial' design, the synthesis process determines the 'minimum-weight' structure without further designer intervention.

Further studies by Chiu (1982) and by Grierson and Chiu (1982, 1984) extended the design method to planar frameworks comprised of beam and column members under combined axial and bending stresses. These studies took simultaneous account of service and ultimate performance criteria, but considered the sizes of the cross sections for the members of the structure as continuous variables to the synthesis process (see Fig. 1, where the different types of member-sizing vari- ables considered herein are graphically illustrated). In theory, this design approach tacitly assumes the availability of custom-fabricated sections that have the exact size, stiffness, and strength properties required for the members of the minimum-weight structure.

criteria alone are to be satisfied, the design cross section for each member is automatically selected from among available commercial-standard sections (which do not obey a constant-shape rule as their size varies; see Fig. I L . ) . In the latter regard, the initial applications of the design method considered the wide-flange, tee, and double-angle steel sections specified by the Canadian Institute of Steel Construction (CISC 1980).

The strategy employed by the synthesis technique to classify the data bank of standard steel sections is quite general and independent of the units of measurement adopted for the design (Grierson and Lee 1984). This implies, then, that the design method may be directly applied for a variety of different steel codes (Canadian, American, British, etc.). To date, an ongoing study by Cameron (1984) has extended the synthesis capability to allow computer-automated design accounting for the standard steel sections specified by both ClSC (1980) and the American lnstitute of Steel Construction (AISC 1980).

The design method involves an iterative process having the following essential features: ( I ) for a given design (e.g., the initial 'trial' design), sensitivity analysis techniques are employed to approximate the performance constraints as linear functions of the member-sizing variables; (2) optimization techniques are applied to find an improved (i.e., lower weight) design; (3) sensitivity analysis is en~ployed again to update the performance constraints for the next weight optimization; and (4) the process is repeated until

Alternatively, the designer can scale up the continuous- weight convergence occurs hfter a number of design variable values found for the design to arrive at stages. commercial-standard sections for the members. This The design method is remarkably effective and effi- latter approach, however, has several difficulties asso- cient. The use of sensitivity analysis techniques permits

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GRIERSON A N D CAMERON 865

the performance criteria for even the most complicated structure to be directly expressed explicitly in terms of the sizing variables for the design. The use of first-order Taylor's series approximations of the performance criteria to reduce them to linear functions of the sizing variables results in minimal computational effort for the algorithm applied to conduct the weight optimization for each design stage. The number of iterations required to achieve the optimal design is generally small and almost totally independent of the complexity of the structure. The first-order representations of the per- formance criteria are 'exact' at the optimal design stage and, therefore, an 'accurate' final design is rendered.

The essential details of the design method are described first, and then three example designs of a building framework are presented to illustrate the method.

The design problem All loads are static, and the distinct loading levels

associated with the various limit states of concern to the design are proportionally related to each other. The framework is discretized into an assemblage of tz pris- matic members, which may be of a variety of types (wide-flange beams, hollow-box columns, double- angle bracing struts, etc.). The design variable for each member i is its cross section area a,. In its general form, the minimum-weight design problem is

I ,

[ l a ] C w , n , I - I

Subject to:

[ Ib] 6 , s 6 , s 6 , ( j = I . 2. .... (1)

[ I d ] q,,,Sa,,,S&,,, (tn = I , 2, ..., p )

[ l e ] a , E A, ( i = I, 2, .... t ~ )

Equation [I a] defines the weight of the structure (wi is the weight coefficient for member i = material density x member length); [ I b] defines the d constraints on elastic displacements 6, due to service loads (the 'hat' symbol A denotes the specified limited values); [I c ] defines the s constraints on elastic stresses a, due to service loads (or due to factored service loads if yield- stress limit states are of concern to the design); [I clj defines the p ultimate-load constraints on the plastic- collapse load factors a,,,; [I e] specifies that the design cross-section area a , for each member i is to be selected from among a predetermined set of discrete cross- section areas A, = {a , , a? , . . .Ii. If the design is to be conducted for continuously varying rather than discrete

section sizes, [ l el is replaced by the size constraints

[ I f ] c;ri 5 a , 5 i; ( i = I , 2, ,.., t7)

where p, and ii are specified bounds on member cross- section areas.

The lower and upper bounds on displacements and stresses have negative and positive signs, respectively, to account for the two possible senses of response action (e.g., left or right sway, compressive or tensile stress). The compressive-stress bounds serve to guard against the local buckling of members and of the flange and web elements of cross sections. For the synthesis tech- nique to date, and herein, these limiting stress values are a priori calculated using assumed slenderness ratios and thereafter remain constant throughout the design history (Chiu 1982). An ongoing study is introducing the capability to progressively update the compressive- stress-bounds fk to reflect the actual melnber and section properties that prevail at each stage of the design history (Cameron 1984).

The positive-valued lower bounds on plastic-collapse load factors are typically taken to be the same limiting value $,,, = 9 (117 = 1, 2, ..., p ) for all ultimate-load constraints (e.g., 9 = 1.8 requires that the structure be capable of withstanding an 80% overload beyond the service-load level without failure occurring in any plas- tic mechanism mode). The corresponding upper bounds &,,, are somewhat superfluous to the design and may be taken arbitrarily large or omitted altogether (although it is possible to conceive of limiting the strength of a structure in certain failure modes).

If the structure is to be sized accounting for elastic limit states alone, [ 1 dl is omitted from the design for- mulation. Conversely, [ I b] and [ 1 c ] are omitted if only plastic limit states are of concern to the design. In...