O p t i m u m D e s i g n o f S t e e l P i p e R a c k sSURESH C. ARYA, EDWARD G. FENG AND GEORGE PINCUSThe availabil ity of user-oriented computer programs has hadgreat impact on engineering analysis and design of structuralsystems. This computational tool can be useful not only toobtain fast and accurate values of forces and displacementsin structural systems, but its usefulness has been extended tothe economical (minimum weight) design of structural steelmembers and frames. The design phase can be made veryefficient by following an optimization process when suitablecomputer programs are available. The optimization processmay include a number of design constraints, such as: (a)attainment of overall minimum weight, (b) limiting theselection process to a given range of member sizes, and (c)limiting displacements to certain tolerable values.Current engineering practice often util izes the capabilityof the computer for the analysis phase only, and considers thesubsequent design process a professional art to be left to thediscretion of the "Design Engineer." There are reasons forthis distinction between analysis and design. For example,the application of a design code requires experience andjudgement, the selection of member sizes should be practical,and there is usually a match among the selected membersizes. However, some computer programs available today areso versatile that they not only include code design criteria asbuilt-in algorithms, but in addition the input procedure isrelatively simple and easy to learn and apply. In fact, manyengineers are able to understand the computer code in theirfirst encounter with a coded problem even if lacking in priorcomputer experience. One user-designed computer programwhich has achieved this capability is the MIT-developedIntegrated Civil Engineering System, Structural DesignLanguage (ICES, STRUDLII).1,2This program is available tothe professional and is marketed by many commercialcomputer service bureaus.This paper describes the use of the STRUDL II computerprogram for the optimum design of a steel pipe rack structure commonly used in petrochemical plants. TheSuresh C. Arya is Principal Design Engineer, CE Lummus Company, H o u s t o n , T e x . Edward G. Feng is Senior Design Engineer, CE Lummus Company,H o u s t o n , T e x . George Pincus is Professor and Chairman, Department of CivilEngineering, University of Houston, Houston, Tex.procedure leads to substantial weight (and cost) savings inmost situations. Considering the tonnage of steel used inpetrochemical plant pipe racks, the savings potential may betruly remarkable.

STRUCTURAL SYSTEM—PIPE RACKA two-dimensional frame representing a pipe rack structureis shown in Fig. 1. Even though the example presented here istwo-dimensional, most designers will consider three-dimensional action due to forces acting perpendicular to theplane of the pipe rack (temperature expansion andcontraction or longitudinal wind and/or earthquake). Forsimplicity, the example considers plane frame action. Manytypical racks will not include all the support conditions shown in this example and will, therefore, be much simpler todesign. The frame includes three levels of beams, and eachbeam has a cantilever overhang at one end. One column isextended upwards to function as a "tee" support for pipes at ahigher elevation. The lower level beam has its ends fixed(moment connection) to the columns. The intermediate levelbeam has one end hinged and the other end is allowed to slidehorizontally to provide for possible expansion (connection tothe column is through slotted holes). The upper level beam iscontinuous over the column and becomes a cantileveroverhang on one end, and is allowed to slide at the other end.The bases of the columns are considered fixed. The frame hasa 22-ft span and an overall height of 32 ft. The frames arespaced at 25 ft and are connected to adjacent frames byhorizontal longitudinal beams at each cross-beam level.These frames are restrained in the longitudinal direction byvertical cross-bracing which is usually placed at a spacing of about 100 ft or at each fourth bay. Only the in-plane designof the pipe rack is considered here, although three-dimensional action can be included with minimal additionaleffort.The loads that act on the frame

are shown in Fig. 2.Three independent loadings are considered for design purposes.

Fig. 1. Typical pipe rack steel frame(computer model input )

The loadings corresponding to permanent loads,wind, and temperature changes are shown in Figs. 2a through2c. Temperature change is only considered on the lower levelbeam, since the upper level beams are free to expand orcontract longitudinally. Design loading combinations areindicated in Figs. 3a through 3d. Out-of-plane loadings, suchas longitudinal expansion and contraction forces andlongitudinal wind/earthquake loads, could be considered asadditional loading conditions. Initial sizes for the membersare selected through manual analytical procedures or thesizes might be assumed. These member sizes are then used asinitial trial sizes in the design optimization process and areindicated in Fig. 1. The selection of member sizes can also bearbitrary, as noted above. Regardless of what initial sizes areused, subsequent program iterations will converge to the samefinal design selection. However, the exact number of iterations will vary, depending on how close the initial trialsizes are to the final computer-designed sizes.F ig . 2 . Bas i c load ing cond i t ions

F ig . 2 . Bas i c load ing cond i t ions

DESIGN PROCEDUREAnalysis and design optimization are accomplished by thecomputer program during the following steps (see Fig. 4 forprogram flow chart):1. Analysis of the frame for the given loadingconditions, and combining the results for the designphase.2. Trial members are checked against the provisions of the AISC Specification for strength adequacy for allloading conditions.3. Member sizes are selected by the computer programalgorithm without applying any constraint condition.The AISC Specification is used.4. The member sizes selected in step 3 are used toperform a new stiffness analysis, and again membersizes are selected by the computer programalgorithm. This step serves the purpose of optimization without applying any constraintcondition, i.e., the difference of weights between step3 and step 4 will be negligible.5.

Size of members which are continuous are madeequal for practical reasons; for example, a column isa continuous member of uniform size.6. Constraint conditions regarding a given range of member dimensions are applied, and a new set of member sizes is obtained using the AISCSpecification for all design loading conditions.Subsequently, step 5 is repeated to arrive at practicalmember sizes.7. Using the new member sizes selected in step 6, theframe is reanalyzed for all loading conditions. Usingthese new design forces, the member sizes are againselected such that the limiting deflection constraintcondition (which is applied at specific controllingpoints of the frame) is satisfied. This deflectionconstraint condition is in addition to the previousconstraint condition listed under step 6. Step 5 isagain repeated to arrive at practical member sizes.8. The member sizes obtained from the previous stepare used once more to reanalyze the frame and as aF ig . 3 . Load ing combina t i ons86ENGINEERING JOURNAL / AMERICAN INSTITUTE OF STEEL CONSTRUCTION© 2003 by American Institute of Steel Construction, Inc. All rights reserved. This publication or any part thereof must not be reproduced in any form without the written permission of the publisher.final AISC Specification check. This step assuresthat the member forces obtained from the lateststiffness analysis have not changed significantly,which would require a further change in members i zes .9. Finally, the member force, the reactions at thesupports, and the deflection of the frame aretabulated for subsequent design. These results arenormally required for the design of the connections,including the column base details and thefoundations, and also to further assure that therequired deflection limitations have been achieved.Summarizing the previous nine steps for the optimum designprocess:Steps 1-5 accomplish minimum weight design.Step 6 accomplishes minimum weight design with aconstraint condition on desirable dimensions (width anddepths) of members.Step 7 accomplishes minimum weight design withconstraint conditions on member dimensionsand limitingdeflection criteria.Steps 8-9 document the results of steps 1-2.COMPUTER MODELINGComputer modeling is a technique of frame idealization suchthat computer coding may be conveniently applied. Thefollowing steps are followed in coding a steel pipe rack forSTRUDL-II computer processing:1. The frame is described by a single line diagramshowing the proper end conditions, as given in Fig. 1.2. The location of the joints is selected. The joints arelocated at: (a) the support points, (b) the free end of members, (c) the intersection of beams and columns,and (d) the mid-points of beams. Joints may also beFig. 4. Computer program input/output flow chart.(Note: Broken lines indicate optional flow path)87THIRD QUARTER / 1979© 2003 by American Institute of Steel Construction, Inc. All rights reserved. This publication or any part thereof must not be reproduced in any form without the written permission of the publisher.established at any other points where knowledge of displacements and forces is desired. Any joint(usually the left bottom support) is selected as theorigin andX (horizontal) andY (vertical) global axesare indicated as shown in Fig. 1. The globalZ -ax i s i sperpendicular to the frame. The coordinates of eachjoint from the origin are then computed. Joints arenumbered in ascending order such that the differencein numbering between joints at the two ends of amember is as

small as possible. For pipe racks, thejoint numbering scheme is not as critical as in astructure with many joints and, therefore, may bearb i t ra ry . 3. A member always exists between two joints. Thesemembers are numbered in sequential order and areindicated within a circle in Fig. 1 in order to createdifferentiation between joint and member numbers.4. The loading conditions are shown in Fig. 2. Loadswhich may act independently are indicated onseparate sketches, e.g., as shown in Figs. 2a through2c. Combined loadings formed from the factored sumof independent loadings are shown in Figs. 3athrough 3d.INPUT TO COMPUTER PROGRAM—EXAMPLEInput data is taken from Figs. 1 through 3. The data isentered in groups in certain sequential logical order. Thisprocedure is described below using the example problem.A typical pipe rack frame which is used in thepetrochemical industry is shown in Fig. 1, and the basic anddesign loadings are shown in Figs. 2 and 3, respectively. Aconstraint condition on deflection is selected, i.e., themaximum vertical deflection in the beams at joints 4, 9, and15 should not exceed 1.0 in. and the lateral displacement of the frame at joint 3 is limited to 0.75 in. The input data iscoded on 80 column sheets, and a free format procedure maybe followed, i.e., data may be coded in any column and extraspaces between data are ignored. Each command statement orline of data should occupy a single line. There are nine setsof data, A through I, which are required to code and solvethis problem, and these are illustrated in Tables 1-9. Thesedata sets must be preceded by a command STRUDL, whichactivates the computer program stored in the system. Thecommand statements and the input data which are keypunched on computer cards are shown on the left side of Tables 1 through 9 in capital letters. Explanation for eachcommand is given on the rightTable 1. Coordinates of Frame GeometryI n p u t D a t a S e t ' A ' E x p l a n a t i o n U N I T F E E T I n d i c a t e s l e n g t h u n i t s T Y P E P L A N E F R A M E I n d i c a t e s s t r u c t u r e t y p e JO INT COORDINATES 120.22.0.0.S U P P O R T S U P P O R T Desc r ibes the geomet ry o f the f rame and po in ts where des ign in fo rmat ion i s des i red . 3 0 . 1 8 . 4 1 1 . 1 8 . 5 2 2 . 1 8 . 6 2 7 . 5 1 8 7 – 4 . 0 2 2 . 8 0 . 2 2 . 9 1 1 . 2 2 . 1 0 2 2 . 2 2 . 1 1 – 5 . 5 2 6 . 1 2 0 . 2 6 . 1 3 1 1 . 2 6 . 1 4 2 2 . 2 6 . 1 5 2 2 . 3 2 . Y - coord ina te (Z -coord ina te au tomat i ca l l y se t to ze ro )X -coord ina te Jo in t Number88ENGINEERING JOURNAL / AMERICAN INSTITUTE OF STEEL CONSTRUCTION© 2003 by American Institute of Steel Construction, Inc. All rights reserved. This publication or any part thereof must not be reproduced in any form without the written permission of the publisher.Table 2. Connectivity of Structure TopologyI n p u t D a t a S e t ' B ' E x p l a n a t i o n M E M B E R I N C I D E N C E S I n d i c a t e s c o n n e c t i o n o f m e m b e r be tween j o in t s 1 1 32 3 83 8 1 2 4 2 55 5 1 0 6 1 0 1 4 7 1 4 1 5 8 3 49 4 51 0 5 6 1 1 7 8 1 2 8 9 1 3 9 1 0 1 4 1 1 1 2 1 5 1 2 1 3 1 6 1 3 1 4 End jo in t number (End jo in t i s a lways fu r ther away f rom o r ig in than s ta r t j o i n t )S ta r t j o i n t numberMember numberTable 3. Member Releases, Properties and Elastic Constants

I n p u t D a t a S e t ' C ' E x p l a n a t i o n M E M B E R R E L E A S E S R e l e a s e t h e e n d s o f m e m b e r t o m o d e l t h e d e s i r e d b o u n d a r y cond i t i ons . 3 1 3 1 6 E N D M O M E N T Z M a k e s t h e f a r e n d o f m e m b e r s 3 , 1 3 a n d 1 6 h i n g e d . 1 2 S T A R T M O M E N T Z M a k e s t h e n e a r e n d o f m e m b e r 1 2 h i n g e d . 1 3 1 6 E N D F O R C E X M a k e s t h e f a r e n d o f m e m b e r s 1 3 a n d 1 6 f r e e t o m o v e hor i zonta l l y ( s l o t t ed ) . U N I T K I P S I N C H E S M E M B E R P R O P E R T I E S T h e s e c t i o n p r o p e r t i e s o f m e m b e r s f o l l o w . 1 T O 7 T A B L E ' S T E E L W ' ' W 8 X 2 8 ' S t r u c t u r a l s h a p e s a s s u m e d f o r m e m b e r s a s n o t e d , e . g . , 1 t h r o u g h 7 a r e W 8 X 2 8 . 8 9 1 2 1 3 1 4 1 5 1 6 T A B L E ' S T E E L W ' ' W 1 0 X 2 9 ' 1 0 1 1 T A B L E ' S T E E L W ' ' W 8 X 1 5 ' C O N S T A N T S E 2 9 0 0 0 . A L L M a t e r i a l m o d u l u s o f e l a s t i c i t y . C T E . 0 0 0 0 0 6 5 A L L C o e ffi c i e n t o f t h e r m a l e x p a n s i o n . B E T A 0 . A L L O r i e n t a t i o n o f m e m b e r s e c t i o n , i . e . , l o c a l m e m b e r s t r o n g ax i s i s pa ra l le l t o the g loba l Z -ax i s . U N I T S I N P O U N D S C O N S T A N T D E N S I T Y . 2 8 4 A L L W e i g h t d e n s i t y o f m a t e r i a l .89THIRD QUARTER / 1979© 2003 by American Institute of Steel Construction, Inc. All rights reserved. This publication or any part thereof must not be reproduced in any form without the written permission of the publisher.Table 4. Basic Loadings and CombinationsI n p u t D a t a S e t ' D ' E x p l a n a t i o n U N I T S K I P S F E E T F A H R E N H E I T L O A D I N G ' D E D + L I V E ' ' P E R M A N E N T L O A D S ( D E A D + M A X L I V E ) ' D e s c r i b e s l o a d i n g c o n d i t i o n d u e t o p e r m a n e n t l o a d s . D E A D L O A D W I T H M O M E N T S Y - 1 . 0 M e m b e r d e a d l o a d i n c l u d i n g fi x e d e n d m o m e n t s . J O I N T L O A D S 1 3 F O R C E Y -3 . C o n c e n t r a t e d l o a d s a n d m o m e n t s a c t i n g a t j o i n t s , e . g . , a fo rce o f 3 k ips a t j o i n t 13 ac t ing down . 1 5 F O R C E Y - 2 . M O M E N T Z 4 . M E M B E R L O A D S L o a d s a c t i n g o v e r t h e l e n g t h o f t h e m e m b e r . 8 9 F O R Y U N I F O R M W - . 7 5 U n i f o r m l o a d o v e r f u l l l e n g t h o f m e m b e r s 8 a n d 9 o f i n tens i ty 0 .75 k ips / f t . 1 1 F O R Y U N I F O R M W - . 4 L A 0 . L B 3 . 0 U n i f o r m l o a d o v e r p a r t i a l l e n g t h i . e . , f r o m s t a r t t o 3 f e e t f rom s ta r t . 1 4 F O R Y L I N E A R W A - . 2 W B 0 . 6 L A 0 . L B 4 . 5 L i n e a r l y v a r y i n g l o a d o v e r p a r t i a l l e n g t h w i t h i n i t i a l i n tens i ty . 2 k ips / f t . a t s t a r t to . 6 k ips / f t . 4 .5 f t . f rom s ta r tac t i ng downward (nega t i ve y -d i rec t i on ) . 1 0 F O R Y C O N C E N T R A T E D P - 2 . 0 L 4 . 0 C o n c e n t r a t e d l o a d a p p l i e d a t a d i s t a n c e f r o m s t a r t , e . g . , 2 k ips on member 10 , a t 4 f t . f rom s ta r t a c t i ng down(nega t i ve y -d i rec t i on ) . 1 2 1 3 F O R Y C O N

P - 5 . 0 L 5 . 5 1 6 F O R Y C O N P - 4 . 0 L 5 . 0 1 6 F O R Y C O N P - 2 . 0 L 9 . 0 L O A D I N G ' W I N D ' ' W I N D L O A D L A T E R A L D I R E C T I O N ' D e s c r i b e s a n o t h e r l o a d i n g c o n d i t i o n d u e t o w i n d . J O I N T L O A D S L o a d s a p p l i e d a t t h e j o i n t s . 1 5 F O R × 0 . 6 A f o r c e o f 0 . 6 k i p s i s a c t i n g i n h o r i z o n t a l d i r e c t i o n . o n j o i n t 1 5 . 3 8 1 2 F O R × 2 . 1 L O A D I N G ' T E M P * 5 0 F ' ' T E M P V A R I A T I O N R I S E 5 0 F ' D e s c r i b e s a n o t h e r l o a d i n g c o n d i t i o n d u e t o t e m p e r a t u r e . M E M B E R 8 9 T E M P A X I A L 5 0 A x i a l t e m p e r a t u r e c h a n g e o f 5 0 d e g r e e s F a p p l i e d t o b e a m s 8 a n d 9 . L O A D I N G C O M B ' A ' ' P E R M A N E N T L O A D S + W I N D ' - C o m b i n e s t w o b a s i c l o a d i n g c o n d i t i o n s f o r d e s i g n o f member , i . e . , 0 .75 × 'DED+L IVE ' p lus 0 .75 × 'WIND ' . ( the end dash i nd i ca tes cont inued on nex t ca rd ) .C O M B I N E ' D E D + L I V E ' 0 . 7 5 ' W I N D ' . 0 7 5 L O A D I N G C O M B ' B ' ' P E R M A N E N T L O A D S + T E M P R I S E + 5 0 F ' - C o m b i n e s t h e b a s i c l o a d i n g c o n d i t i o n w i t h t e m p e r a t u r e expans ion o f 50 degrees F . C O M B I N E ' D E D + L I V E ' 1 . 0 ' T E M P * 5 0 F ' 1 . 0 L O A D I N G C O M B ' C ' ' P E R M A N E N T L O A D S + T E M P F A L L – 5 0 F ' - C o m b i n e s t h e b a s i c l o a d i n g c o n d i t i o n w i t h t e m p e r a t u r e cont rac t i on o f 50 deg rees F .COMBINE 'DED+L IVE ' 1 .0 ' TEMP*50F ' -1 .0 L O A D I N G C O M B ' A A ' ' F U L L P E R M A N E N T L O A D S + W I N D ' - C o m b i n e s t w o b a s i c l o a d i n g c o n d i t i o n s f o r l i m i t i n g deflec t ion cons t ra in t cond i t i on , i . e . , 1 .0 × 'DED+LIVE ' p l u s 1 . 0 × ' W I N D ' . C O M B I N E ' D E D + L I V E ' 1 . 0 ' W I N D ' 1 . 090ENGINEERING JOURNAL / AMERICAN INSTITUTE OF STEEL CONSTRUCTION© 2003 by American Institute of Steel Construction, Inc. All rights reserved. This publication or any part thereof must not be reproduced in any form without the written permission of the publisher.Table 5. Stiffness Analysis and Output ResultsI n p u t D a t a S e t ' E ' E x p l a n a t i o n P R I N T D A T A A L L P r i n t s o u t i n p u t d a t a . S T I F F N E S S A N A L Y S I S T h i s c o m m a n d p e r f o r m s s t r u c t u r a l a n a l y s i s o f t h e f r a m e . PLOT PLANE S t ruc tu re geomet ry i s p r in ted out a s a check on the i n p u t . U N I T S K I P S I N C H E S O U T P U T B Y J O I N T L IST D ISPLACEMENTS ALL Resu l t s a re p r i n ted out fo r deflec t i ons and ro ta t i ons o f j o i n t s . O U T P U T B Y L O A D S L IST FORCES REACT IONS ALL Resu l t s a re p r i n ted ou t fo r f o rces , moments a t j o i n t s , and reac t ions a t suppor t s . O U T P U T B Y M E M B E R S S E C T I O N F R A C T I O N N S 2 0 . 1 . L IST FORCE ENVELOPE ALL MEMBERS Max imum fo rces and moments i n each member a repr in ted out . Cons iders a l l l oad ing cond i t i ons , and twosec t i ons on each member , the s ta r t and the end .L IST STRESS ENVELOPE ALL MEMBERS Max imum s t resses in each member i s p r i n ted ou ts im i l a r to above .side of each table. Each table contains the followinginformation (capital letters are actual commands):Data Set ' A ' and 'B ' (Tab les 1 and 2)—Information istaken from Fig. 1 and includes a description of the framegeometry and topology. Note that for three-dimensionalaction, TYPE PLANE FRAME, would become TYPESPACE FRAME and three sets of coordinates (x , y , z ,)would be listed for each joint.Data Set 'C ' (

Table 3)—Information on member releases(program assumes all member ends fully fixed, unlessmodified by this command) and member properties aretaken from Fig. 1. The parameters under the commandCONSTANTS are for steel material. The value of BETAis taken as zero, since the web of the sections is in theplane of paper (members are oriented such that bendingis about their strong axis). BETA becomes 90 degreeswhen the web of a member is perpendicular to the planeXY.Data Set 'D ' (Table 4)—Information for loadingconditions is as shown on Fig. 2. Three loadingcombinations 'A', 'B', 'C' are performed following currentdesign practice, and combination 'AA' is formulated tocheck for the limiting deflection criteria. Information onloading combinations is obtained from Fig. 3.Data Set ' E ' (Table 5)—This data set includes commandsfor the listing of the stiffness analysis results. The resultsinclude the forces in the members, displacements of thefree joints, and reactions at the supports for all basic andcombined loading conditions. Maximum stress enveloperesults are also printed out for design purposes.Data Set ' F ' (Table 6 )—This data set defines theparameters that are required for the design of steelmembers. Effective and unbraced lengths for columns aredefined, but beam and cantilever members are consideredto be totally braced along their compression flange, sincethey are attached to pipes perpendicular to their lengths.A code check on the assumed member sizes is performedfor strength adequacy. Subsequently, member sizes areselected by the computer program for the forcesgenerated by the stiffness analysis of data set 'E'.Another cycle of stiffness analysis is performed using thenew sizes as input for member properties. Then, resultsof the stiffness analysis are used to select new membersizes.The preceding steps accomplish minimum weightoptimization using the computer program withoutapplying any constraint condition.91THIRD QUARTER / 1979© 2003 by American Institute of Steel Construction, Inc. All rights reserved. This publication or any part thereof must not be reproduced in any form without the written permission of the publisher.Table 6. Prescribed Design Parameters, AISC Specification Check andComputer DesignI n p u t D a t a S e t ' F -1 ' E x p l a n a t i o n L O A D I N G L I S T ' A ' ' B ' ' C ' ' D E D + L I V E ' D e s i g n l o a d i n g c o n d i t i o n s a r e c o n s i d e r e d . P A R A M E T E R S D e s i g n p a r a m e t e r s a r e p r e s c r i b e d . ' C O D E ' ' S P 6 9 ' A L L A I S C - C o d e i s u s e d . ' U N L C F ' 6 6 . 0 8 9 1 2 1 3 1 5 1 6 U n b r a c e d l e n g t h o f m e m b e r c o m p r e s s i o n fl a n g e , i .e . , 66 i nches fo r members 8 , 9 , 12 , 13 , 15 and 16 . ' F L Y D ' 3 6 . A L L Y i e l d s t r e n g t h o f m a t e r i a l . ' S E C N D A R Y ' 1 A L L I n d i c a t e s t h a t a l l m e m b e r s a r e p r i m a r y m e m b e r s . ' P R I D T A ' 1

A L L N o d i a g n o s t i c o u t p u t i s d e s i r e d . ' K Z ' 1 . 2 M E M 1 4 E ff e c t i v e l e n g t h f a c t o r f o r c o l u m n s i n l o c a l Z - a x i s , i . e . , the p lane o f the f rame. ' K Z ' 2 . 0 M E M 2 3 5 6 7 ' K Y ' 0 . 8 0 M E M 3 6 7 E ff e c t i v e l e n g t h f a c t o r f o r c o l u m n s i n l o c a l Y - a x i s , i . e . , pe rpend i cu la r to the f rame. ' K Y ' 0 . 6 5 M E M 1 2 4 5 ' L Z ' 2 4 0 . 0 M E M 1 4 U n b r a c e d l e n g t h a g a i n s t b u c k l i n g i n l o c a l Z -a x i s , i . e . , i n the p lane o f the f rame. ' L Z ' 9 6 . 0 M E M 7 Inpu t Data Se t ' F -2 ' C H E C K C O D E F O R M E M B E R S 1 t o 1 6 A I S C - C o d e c h e c k i s p e r f o r m e d o n a l l m e m b e r s . Inpu t Data Set ' F -3 ' S E C T I O N F R N S 2 0 . 1 . M E M 1 t o 1 6 S e c t i o n s a r e s e l e c t e d f o r m e m b e r d e s i g n c o n s i d e r i n g moments and ax ia l f o rces a t bo th ends . S E L E C T M E M 1 t o 1 6 W I T H ' C O M B I N E D ' A I S C - C o d e a x i a l a n d b e n d i n g f o r c e s i n t e r a c t i v e fo rmu lae (ax ia l l oad and moment ) a re used .Data Set 'G ' ( Tab le 7 )—The constraint condition on memberdimensions are given in this table. A stiffness analysis isperformed on the member sizes selected in the previous step.Member sizes are reselected for the new member forces,taking into account the constraint conditions. Segments of continuous members may be selected by the computerprogram with different sizes. Thus, segments common to asingle or continuous member are made equal, based on thelargest required moment of inertia for any one segment, sinceflexural stresses control the design process.Data Set 'H ' ( Tab le 8 )—The commands in this data set applyan additional constraint condition on maximum permissibledeflections at certain joints (in the vertical direction at mid-span of the beams and in the horizontal direction at theintersection of the first-level beam and column). First, thestiffness analysis is performed on the frame using themember sizes obtained in the preceding table. Redesign of those members which are affected by the constraint condition is performed by applying the constraint condition of limitingdeflection. Continuous member sizes are rationalized andsection properties are printed.Data Set ' I ' ( Tab le 9 )—The commands in this tablereanalyze the frame for all the loading conditions, a codecheck of the members is performed, and a printout of the finaldeflections at the joints, the support reactions, and themember properties is produced.92ENGINEERING JOURNAL / AMERICAN INSTITUTE OF STEEL CONSTRUCTION© 2003 by American Institute of Steel Construction, Inc. All rights reserved. This publication or any part thereof must not be reproduced in any form without the written permission of the publisher.Table 7. Rationalization of Sizes and Constraint Condition of Member SizesI n p u t D a t a S e t ' G - 1 ' E x p l a n a t i o n T A K E M E M 1 T O 3 A S L A R ' I Z ' o f M E M 1 t o 3 S a m e s e c t i o n b a s e d o n m a x i m u m m o m e n t o f i n e r t i a i s used on ent i re leng th o f each co lumn. T A K E M E M 4 T O 7 A S L A R ' I Z ' O F M E M 4 T O 7 T A K E M E M 1 4 1 5 1 6 A S L A R ' I Z ' o f M E M 1 4 1 5 1 6 S a m e s e c t i o n b a s e d o n m a x i m u m m o m e n t o f i n e r t i a i s used on ent i re leng th o f each beam. T A K E M E M 1 2 1 3 A S L A R ' I Z ' O F M E M 1 2 1 3 T A K E M E M 8 9 A S L A R ' I Z ' O F M E M 8 9 T A K E M E M 1 0 1 1 A S L A R ' I Z ' O F M E M 1 0 1 1 S a m e s e c t i o n i s u s e d f o r n o n c o n t i n u o u s c a n t i l e v e r s based on max imum moment o f i ne r t i a . P R I N T M E M B E R P R O P A L L

M E M B E R S T a b l e o f m e m b e r s i z e s i s p r i n t e d o u t . S T E E L T A K E O F F S t e e l w e i g h t i s c a l c u l a t e d f o r t h e c u r r e n t sec t ions . Inpu t Data Set 'G -2 ' M E M B E R C O N S T R A I N T M e m b e r s i z e s c a n b e l i m i t e d t o d e s i r e d r a n g e o f d imens ions . 1 T O 7 C O N S ' Y D ' G E 7 . 5 C o l u m n l e a s t d e p t h i s i n d i c a t e d , e . g . , m e m b e r s 1 th rough 7 shou ld be a t l eas t 7 .5 i nches deep . 1 T O 7 C O N S ' Y D ' L E 1 8 . 0 C o l u m n s m a x i m u m d e p t h b e l e s s o r e q u a l t o 1 8 i n c h e s . 1 T O 7 C O N S ' Z D ' G E 5 . 0 C o l u m n fl a n g e l e a s t w i d t h i s i n d i c a t e d , e . g . , m e m b e r s flange w id th shou ld be equa l o r g rea te r than 5 .0 i nches . 1 T O 7 C O N S ' Z D ' L E 1 0 . 0 C o l u m n ' s m a x i m u m fl a n g e w i d t h b e e q u a l o r l e s s t h a n 10 inches . 8 9 1 2 1 3 1 5 1 6 C O N S ' Y D ' G E 9 . 5 M i n i m u m b e a m d e p t h o f 9 . 5 i n . o r l e s s i s i n d i c a t e d f o r m e m b e r s 8 , 9 , 1 2 , 1 3 , 1 5 a n d 1 6 . 8 9 1 2 1 3 1 5 1 6 C O N S ' Z D ' G E 5 . 0 M a x i m u m b e a m fl a n g e w i d t h o f 5 . 0 i n . o r g r e a t e r i s i n d i c a t e d f o r m e m b e r s 8 , 9 , 1 2 , 1 3 , 1 5 a n d 1 6 .Table 8. Constraint Condition on Limiting DeflectionI n p u t D a t a S e t ' H ' E x p l a n a t i o n L O A D I N G L I S T ' A A ' O n l y l o a d i n g c o m b i n a t i o n f o r w h i c h d e fl e c t i o n cons t ra in t cond i t i on i s p resc r i bed i s cons idered . A D D I T I O N S A d d i t i o n a l d e s i g n p a r a m e t e r a r e d e s c r i b e d . P A R A M E T E R ' D E F L E C T N ' 1 3 2 . 0 8 1 2 1 5 D e fl e c t i o n c o n s t r a i n t i s d e s c r i b e d o n b e a m s , e . g . , a max imum deflec t ion o f L /132 i s des i red , wh i ch i s 1 in.' D E F L E C T N ' 2 8 8 . 0 1 4 D e fl e c t i o n c o n s t r a i n t i s d e s c r i b e d o n c o l u m n s , e . g . , a max imum deflec t ion o f L /288 i s des i red , wh i ch i s 0 . 7 5 i n . ' L O A D I N G ' ' A A ' 1 4 8 1 2 1 5 L o a d i n g c o n d i t i o n i s a l s o s e l e c t e d .93THIRD QUARTER / 1979© 2003 by American Institute of Steel Construction, Inc. All rights reserved. This publication or any part thereof must not be reproduced in any form without the written permission of the publisher.

Table 9. Stiffness Analysis and Output ResultsI n p u t D a t a S e t ' I - 1 ' E x p l a n a t i o n L O A D I N G L I S T A L L A l l b a s i c a n d c o m b i n a t i o n l o a d i n g c o n d i t i o n s are cons ide red . S T I F F N E S S A N A L Y S I S Sta t i c ana ly s i s i s pe r fo rmed on p rev ious lys e lec ted sec t i ons .LOADING L IST 'A ' 'B ' ' C ' 'AA ' 'DED + L IVE 'Des i red l oad ing cond i t i ons a re ac t i va ted . L I S T R E A C T I O N S D I S P A L L R e a c t i o n s a t s u p p o r t s a n d d e fl e c t i o n s a t j o i n t s fo r the des i red l oad ings a re p r i n ted ou t . Input Da ta Se t ' I - 2 ' P R I N T M E M B E R P R O P E R T I E S A L L M E M B E R S F i n a l s e c t i o n p r o p e r t i e s f o r a l l m e m b e r s a r e pr in ted out i n tabu la r fo rm. S T E E L T A K E S t e e l w e i g h t f o r t h e fi n a l d e s i g n e d f r a m e i s ca l cu la ted . F I N I S H

T e r m i n a t i o n o f p r o g r a m .DISCUSSION OF RESULTSA two-dimensional frame of structural steel, commonlycalled "pipe rack" in the petrochemical industry, wasdesigned using the AISC Specification by manualcomputation, and is then compared with optimum sizesobtained by using the computer program STRUDL-II. Thesizes are selected by the computer program initially withoutany constraint condition, and again using constraint conditions on member sizes and on limiting deflections. Thesizes obtained by the above investigation are shown in Fig.5b. Comparing the sizes of Fig. 5a (manually designed) withFig. 5b (designed by the computer program), it may be notedthat the former uses the same section (w8×28) for bothcolumns, while the latter uses different sections, i.e., w10×21and w14×30 for the left and right side columns, respectively.This occurs because: (1) a horizontal sliding hingeconnection is used between the two upper-level beams andcolumns, and (2) the right side column projects above thetop-level beam. The beams selected by the computer programare lighter but deeper compared to those selected manually.This indicates that the computer program is organized toselect the most economical sections. Comparing the sizesshown in Fig. 5c (designed by the computer program withconstraint condition on member sizes as given in Table 7)with the sizes shown on Fig. 5b (design by the computer withno constraints), it may be noted that, except for the beams atthe two upper levels, all sizes are the same. The two upper-level beam sections w10×21, are reduced in depth (from 12to 10 inches), but became slightly heavier (from 19 plf to 21plf). Thus, having a size constraint condition has a mildeffect on the weight, whereas manual vs. computer design hasa strong effect on weight. The sizes shown in Fig. 5d areobtained by the computer program with an additionalconstraint condition of limiting deflection to a maximumvalue of 1 in. in the vertical direction at mid-span of allbeams and 0.75 in. in the horizontal direction in the columnsat the first-level beam. Thus, the additional design constraintresults in an increase in size of the right side columncompared to the previous design (w14×30 to w14×34), andindicates that the previous computer designed sizes did not,in fact, meet the limiting deflection criteria in the horizontaldirection. Comparing the deflection results (caused by fulldead, live, and wind loading) corresponding to Figs. 5a and5b and which are plotted in Figs. 6 and 7, respectively, itmay be noted that the horizontal deflection at the lower-levelbeam is 1.60 in. for the manual design and 0.72 in. in thecomputer design. Thus, according to the prescribed limit of 0.75 in., the manual design is unacceptable, but thecomputer-designed sizes adequately meet the establishedcriteria. It should be noted that manual design seldom checksfor maximum tolerable deflection, due to the fact that hand orhandbook formulas or procedures do not provide a simpleway of computing deflections. Concerning the verticaldeflection in the beams, the maximum value is 0.64 in. forthe manual design and 0.94 in. for the computer design, whilethe prescribed limit is 1.0 in. This indicates that the computerprogram selects sizes which approach the limitingrequirements, while manually selected sizes areoverdesigned. Similarly, this is also true in the selectionprocess for the cantilevers. A size of w6×8 is selected by thecomputer program and a size of w8×15 is selected by themanual procedure. One fact worth clarifying in these figuresis the difference between respective horizontal deflectionvalues for the right and left side columns at the two upper-level beams. The reason that the right side column does notdeflect as much as the left side column is because the rightcolumn is allowed to slide freely in the horizontal directionwhile the left side column is not free to slide horizontally.94ENGINEERING JOURNAL / AMERICAN INSTITUTE OF STEEL CONSTRUCTION© 2003 by American Institute of Steel Construction, Inc. All rights reserved. This publication or any part thereof must not be reproduced in any form without the written permission of the publisher.

Table 10. Weight Optimization—Comparative ResultsW E I G H T O F S T E E L - P O U N D S M A N U A L D E S I G N D E S I G N B Y C O M P U T E R P R O G R A M D E S C R I P T I O N W I T H O U T C O N S T R A I N T C O N D I T I O N S NOC O N S T R A I N T C O N D I T IO N C O N S T R A I N T O N M E M B E R S IZES C O N S T R A I N T O N M E M B E R S I Z E S A N D D E F L E C T I O N M E M B E R S I Z E S W I T H O U T R A T I O N A L I Z A T I O N -1 s t . C Y C L E = 2 , 4 5 4 2 n d . C Y C L E = 2 , 5 6 9 2 , 8 7 6 3 , 2 3 2 M E M B E R S I Z E S W I T H R A T I O N A L I Z A T I O N 3 , 8 5 3 3 , 1 1 2 3 , 2 1 5 3 , 3 4 3 P E R C E N T A G E S A V I N G S O V E R M A N U A L D E S I G N 0 1 9 . 2 1 6 . 6 1 3 . 2Fig. 5. Design comparisonA weight comparison of the four designs in Fig. 5 isgiven in Table 10. The upper row of steel weights is for thecase when different sizes are selected by the computerprogram for different segments of a single continuous mem-ber. However, the size of a segment which has the largestmoment of inertia should be selected as a common size forthe remaining segments to achieve a practical design and toreduce fabrication costs. The weight of the members after95THIRD QUARTER / 1979© 2003 by American Institute of Steel Construction, Inc. All rights reserved. This publication or any part thereof must not be reproduced in any form without the written permission of the publisher.