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Cost optimization of a structural roof system Russell, Alan David 1969

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COST OPTIMIZATION OF A STRUCTURAL ROOF SYSTEM by ALAN DAVID RUSSELL B.A.Sc, University of B r i t i s h Columbia, 1967 A THESIS SUBMITTED IN PARTIAL.FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in the Department of C i v i l Engineering We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA AUGUST, 1969 In presenting this thesis in part ial fulfilment of the requirements for an advanced degree at the University of Br i t ish Columbia, I agree that the Library shall make it freely available for~reference and Study. I 'further agree that permission for extensive copying of this thesis for scholarly/purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thes.is for f inancial gain shall not be allowed without my written permission. Department of P - V (IZ*— VUI&L* The University of Brit ish Colj^bia Vancouver 8, Canada -1-ABSTRACT The object of this investigation is to develop a procedure for the optimization of a roof system to y i e l d minimum cost. The roof system is composed of p a r a l l e l chord, steel trusses, with purlins and decking. The variables considered are geometry, topology, and member siz e . The optimal values o£ the geometry and topology variables are determined by the application of Box's Complex Method. A f u l l y stressed design c r i t e r i o n i s used i n selecting the member sizes. It i s shown that the optimization scheme always converges to a low cost region, although not necessarily to the global optimum. A cost model i s developed that includes member-related costs and weight - related costs. Parameter studies demonstrate the importance of selecting the proper ratios of fixed and . weight-related costs. - i i -TABLE OF CONTENTS PAGE CHAPTER 1 - OUTLINE OF DESIGN PROBLEM 1 Introduction 1 Br i e f Resume of Past Work 2 The Design-Decision Process 6 Organization of Presentation 10 CHAPTER 2 - THE DESIGN PROBLEM STUDIED 12 Problem Statement 12 Structural Parameters to be Chosen by Designer 12 Variables to be Provided by Automated Scheme 13 Objective Function 13 Cost Model 15 Development of Automated Approach 16 CHAPTER 5 - THE MATHEMATICAL PROGRAMMING PROBLEM 21 Cl a s s i c a l Optimization Problem 21 Linear Programming Problem 23 Non-Linear Programming Problem 24 Solution of the Non-Linear Programming Problem 25 The Problem of Structural Optimization 28 Formulation of the Thesis Problem as a Mathematical Programming Problem 31 Investigation of Related Work 33 CHAPTER 4 - SOLUTION OF OPTIMIZATION PROBLEM 3 7 Minimization Algorithm (Box's Complex Method) 37 Program Discussion 40 - i i i -PAGE Modifications to Box's Method Implemented in the Program 40 Presentation of Results 41 V a l i d i t y of the Fully Stressed Design 52 CHAPTER 5 - PARAMETER STUDIES 5 5 Results of Optimization 55 Observations from the Exhaustive Search 68 The Importance of a Wall Cladding Cost and a Topology Cost 70 Influence of Gravity Load 75 S e n s i t i v i t y and Uniqueness of the Best Solution 75 Discussion of Cost Parameters and Cost Model Form 80 CHAPTER 6 - THE EFFECT OF VARIABLE PANEL SPACING 84 Variable Panel Spacing 84 Discussion of Results 93 CHAPTER 7 - CONCLUSIONS 102 Possible Future Extensions 102 BIBLIOGRAPHY 10 4 APPENDIX A - Flowchart of Optimization Scheme 106 APPENDIX B - Sample Computer Output for Typical Optimization Run 108 - i v -LIST OF TABLES TABLE PAGE 4- 1 Groups of Variables Treated by Algorithm 39 4-2 Input Data 44 4-3 Results of Optimization 46 4-4 I n i t i a l and Final Complexes for Run 6 • 47 5-1 Cost Parameters 56 5-2 Input Variables for "High Costs" Optimization Runs 76 5-3 Optimization Run 1 77 5-4 Optimization Run 2 77 5-5 Optimization Run 3 78 5-6 Optimization Run 4 78 5-7 Optimization Run 5 79 5-8 Optimization Run 6 79 6- 1 Solution to Equations (6-4) and (6-7) For 8 Panel Truss 91 6- 2 Pinned Chord - Even Spacing 96 6- 3 Pinned Chord - Variable Spacing 96 6- 4 Continuous Chord - Even Spacing 96 6- 5 Continuous Chord - Variable Spacing 97 6- 6 Continuous Chord - Variable Spacing 97 6- 7 Stress Factors for Continuous Chord and Even Spacing 97 6- 8 Stress Factors for Continuous Chord and Variable Spacing 98 6-9 Stress Factors for Continuous Chord and Variable Spacing 98 - v-TABLE PAGE 6-10 Panel Lengths i n Inches f o r Table 6-5 98 6-11 Panel Lengths i n Inches f o r Table 6-6 99 - v i -LIST OF FIGURES FIGURE PAGE 1-1 Direct Search 5 1-2 Semi-Direct Search 5 1-3 Indirect Search 5 1-4 Design Process Block Diagram 7 1- 5 Design-Analysis-Optimization Block Diagram 10 2- 1 Cost and Weight Model of Roof System 16 2- 2 Program Modules 18 3- 1 Linear Programming Problem 25 3-2 Non-Linear Programming Problem 25 3-3 Local and Global Optima 25 3-4 Non-Linear Objective Function 25 3- 5 Design Space for Two Bar Truss 31 4- 1 to 4-5 Comparison of Optimization Results with Exhaustive Search 48, 49 5- 1 (a to d) Exhaustive Search for Minimum Cost Configuration ("Low Costs") 60 5-2(a to d) Exhaustive Search for Minimum Weight Configuration ("Low Costs") 61 5-3 Minimum Cost D/S Ratio for Different Topologies ("Low Costs") 62 5-4 Web Weight/Total Weight vs D/S Ratio for Different Topologies ("Low Costs") 62 5-5 Minimum Material Cost D/S Ratio for Different Topologies ("Low Costs") 62 - v i i -FIGURE PAGE 5-6 Material Cost/Total Cost vs D/S Ratio for Different Topologies ("Low Costs") 62 5-7(a to d) Exhaustive Search for Minimum Cost Configuration ("High Costs") 63, 64 5-8(a to d) Exhaustive Search for Minimum Weight Configuration ("High Costs") 65, 66 5-9 D/S Ratio for Minimum Total Cost for Different Topologies ("High Costs") 67 5-10 Web Weight/Total Weight vs D/S Ratio for Different Topologies ("High Costs") 67 5-11 D/S Ratio for Minimum Material Cost for Different Topologies ("High Costs") 67 5-12 Material Cost/Total Cost vs D/S Ratio for Different Topologies ("High Costs") 67 5-13(a, b) Study of the Effect of a Wall Cladding Cost on the Best D/S Ratio ("Low Costs") 72 5-14(a, b) Study of the Effect of a Topology Cost on the Best Configuration ("Low Costs") 72 5-15(a to d) Study of the Effect of a Wall Cladding Cost on the Best D/S Ratio ("High Costs") 73 5-16(a to d) Study of the Effect of a Topology Cost on the Best Configuration ("High Costs") 74 5-17(a) Dist r i b u t i o n of Fixed Costs with Truss Spacing ("Low Costs") 82 - v i i i -FIGURE PAGE 5- 17(b) D i s t r i b u t i o n of Fixed Costs with Truss Spacing ("High Costs") 82 6- 1 Panel Spacing Configuration 86 6-2 Continuous Beam Idealization . 89 6-3(a) Solution of Equations (6-4) for 8 Panel Truss 92 6-3(b) Solution of Equations (6-7) for 8 Panel Truss 92 - i x-NOMENCLATURE area of cross section 0.6Fy + mCQ C4 + C5 Cfi NCORD + C_NSPLT.CE o 7 C8 Cg + C 1 Q mL/r cost per pound of steel for j t h member t o t a l cost per square foot 20 for Fy<50 web material cost per pound compression chord material cost per pound tension chord material cost per pound cost of preparation per web member cost of web joint s per web member cost of preparation per chord member cost per spl i c e for chord member cost per square foot of wall cladding roof cost per square foot based on truss spacing roof weight per square foot based on truss spacing roof cost per square foot based on p u r l i n spacing roof weight per square foot based on p u r l i n spacing 535/ \/Fy-13 depth of truss depth to span r a t i o - x i -NPAN = number of panels NPOINT = number of points in complex (k) NSPLICE = number of chord splices P.L. = panel length PLBS = p u r l i n lower bound spacing RATIO = depth to span r a t i o (D/S) r = radius of gyration r^ = s t i f f n e s s c o e f f i c i e n t r^ = pseudo random deviate RNB = random number base S = truss span = ax i a l force i n i t h panel SPACE = truss spacing SPAN = truss span TOL = maximum difference between worst and best point in complex TLBS = lower bound on truss spacing TUBS = upper bound on truss spacing W = material cost of truss per square foot WBC = weight of tension chord material i n pounds WN = number of web members WTC = weight of compression chord material in pounds WWEB = weight of web material in pounds w = distributed load x* = optimum vector (x*,....x*) x-, = number of panels -x-DLBR = D/S lower bound DUBR = D/S upper bound F . = allowable a x i a l compressive stress in i t h member Ai £ a i = axial compressive stress i n i t h member F = nxm matrix of generalized forces ?B = allowable bending stress £ b i = bending stress i n i t h member f(x) = objective function FY =? y i e l d point of web members FYC = y i e l d point of compression chord FYT = y i e l d point of tension chord F y = y i e l d point gi(x) = constraint function 4j « = lower bound on constraint function GUj(x) = upper bound on constraint function k = number of points i n complex k i = i t h panel spacing c o e f f i c i e n t K = s t i f f n e s s matrix (nxn) K l = weight cost c o e f f i c i e n t K 2 = 2x 1A 1+A 1+A 2 K3 = 2A 3x 2 L = truss span m = (6.77+0.079F )/(C -C ) v y p o M i = moment i n i t h member NCORD = number of chord pieces NIT = maximum number of points i n space to be evaluate - x n -= depth to span r a t i o X j = truss spacing x^ = p u r l i n spacing X t-...x L = panel spacing c o e f f i c i e n t s x T -, . . . x = member s i z e s L+1 n Z = s e c t i o n modulus z = o b j e c t i v e f u n c t i o n ( f ( x ) ) a = over r e f l e c t i o n f a c t o r 8 = r e l a t i o n between s e c t i o n modulus and area A = nxm matrix of gen e r a l i z e d displacements y = wL/ 2 £j = length of j t h member p = weight per cubic volume ACKNOWLEDGMENTS The author wishes to express his sincere thanks to Professor Samuel L. Lipson, his supervisor, for the guidance and helpful advice during the course of this study. Special thanks are due to Dr. L.G. Mitten of the Faculty of Commerce at The University of B r i t i s h Columbia (formerly of Northwestern Un i v e r s i t y ) , for his suggestion to use Box's Complex Method, and for his time spent reading the o r i g i n a l draft. To my wife, E l f i e , thank you for your help with the grammar and drawings. CHAPTER 1 OUTLINE OF DESIGN PROBLEM Introduction During the l a s t decade greater emphasis has been placed on the development of r a t i o n a l design for a large variety of struc-tures using automated design-analysis schemes. Heretofore, automated methods had been applied to the analysis of only those st r u c t u r a l configurations with given member properties. The impetus for automated design has come from the a i r c r a f t industry where the most common governing c r i t e r i o n for optimal design i s weight, subject to the s a t i s f a c t i o n of certain functional requirements and relevant constraints. Research i n the design f i e l d is now being a c t i v e l y pursued and applied to a great variety of c i v i l engineering structures. Almost invariably, the measure of a good c i v i l engineering structure is i t s cost and not i t s weight. The design of the structure is characterized by a set of parameters which specify the configuration of the structure and the properties of the s t r u c t u r a l members. Site conditions, code requirements, aesthetic considerations and other relevant factors determine the selection of the structure parameters. The design of the structure, s a t i s f y i n g the above considerations and y i e l d i n g minimum cost, can be obtained only i f i t i s possible to state the requirements e x p l i c i t l y i n mathematical form. The purpose of t h i s investigation is to examine a simple, common, engineering structure. The structure selected i s a roof s t r u c t u r a l system which employs a truss as the main st r u c t u r a l element, with purlins and/or decking. The parameters necessary to describe the system are stated and divided into two cate-gories; parameters set by the designer, and those which can be determined by a n a l y t i c a l means. The f e a s i b i l i t y of automating the analytic decisions i s demonstrated, and a technique i s devised whereby a good solution can always be obtained, although i t should be noted that the solution may not necessarily be the global nor even the l o c a l minimum. The optimal solution i s one for which a lower cost cannot be obtained s a t i s f y i n g the govern-ing constraints, which means that the optimal solution is the global minimum of the objective function. B r i e f Resume of Past Work Discussion w i l l be limited to work which has used a discrete element s t r u c t u r a l i d e a l i z a t i o n . According to G e l l a t l y et a l (1), and Melosh and Luik (2), this work can be examined under the three following headings: " ( i ) Processes which are primarily concerned with the optimization of individual components for a single loading case. ( i i ) Studies of optimization of r e a l i s t i c structures but with conditions and methods which are r e s t r i c t e d to very special conditions. ( i i i ) Analyses which recognize the importance of factors such as m u l t i p l i c i t y of loading and which d i r e c t l y or i n d i r e c t l y advance the state-of-the-art." The types of search procedures u t i l i z e d i n optimal design can be c l a s s i f i e d as d i r e c t , semi-direct and i n d i r e c t . The -3-direct method generally encompasses categories (i) and ( i i ) above. B a s i c a l l y , this procedure consists of e x p l i c i t l y r e l a t i n g each of the st r u c t u r a l elements to the l i m i t i n g conditions (constraints). I f the design variables can be selected to simul-taneously s a t i s f y the upper and lower l i m i t s on the variables, then the corresponding value of the objective function is assumed to be the optimal solution. A p i c t o r i a l representation of the direct search procedure i s presented i n figure (1-1). The semi-direct and i n d i r e c t search procedures are used i n category ( i i i ) and are represented in figures (1-2) and (1-3). Category ( i i i ) may be further divided into three sections. (1) Implementation of existing design procedures (semi-direct  search). This method involves the use of building codes, engineering experience, and i t e r a t i v e analysis and design of the structure. Engineering experience is used to determine the configuration of the structure and the type of members, etc., to be used, based on previous experience with structures performing the same func-t i o n a l requirements. A design c r i t e r i o n i s substituted for the objective function and an i t e r a t i v e solution technique i s used. The substitution of a design c r i t e r i o n involves a t a c i t assump-tion of the behavior c h a r a c t e r i s t i c of the optimum structure, such as the optimum structure being f u l l y stressed. The v a l i d i t y of this type of approach is dependent upon the degree of correla-tion between the design c r i t e r i o n and the objective function. Under this category is a paper by Gray (3) which is closely -4-related in scope to the present study. Gray examined the mini-mum weight solution of a Pratt roof truss system considering depth to span r a t i o , number of panels, and truss spacing, as variables. Due to the small computer size at that time (1958), several simplifying assumptions were made which limited the general usefulness of his approach. (2) Structural synthesis (indirect search method). One of the f i r s t and most concise formulations of the problem of optimal s t r u c t u r a l design was presented by Schmit (4). He defined the s t r u c t u r a l synthesis problem as: "Given a set of load conditions and side conditions for a structure, f i n d , by systematic means, a struc-ture which w i l l support the loads, s a t i s f y the side conditions and maximize the merit c r i t e r i o n by which the structure is to be evaluated." A l i s t of notable contributions to this f i e l d i s presented in a recent paper by Schmit (5). The s t r u c t u r a l synthesis approach employs a mathematical programming technique dependent on the type of the objective function and constraints being considered. B a s i c a l l y , the search procedure depends on an i n i t i a l feasible point and a distance and di r e c t i o n of t r a v e l based on the c r i t e r i o n of minimizing the objective function. (3) Application of knowledge p a r t i c u l a r to the s t r u c t u r a l  f i e l d in developing algorithms for optimum design (semi-direct  plus i n d i r e c t search procedure). Examples include G e l l a t l y , Gallagher and Luberacki (1), -6-The Design-Decision Process The most general problem which could be solved would be f i r s t to specify the functional requirements of the structure ( i . e . , loads, aesthetic requirements, etc.) and the constraints (codes, s i t e conditions); then to examine a l l types of structures which s a t i s f y the requirements and constraints, in order to determine the best structure for a given c r i t e r i o n (cost, weight, s t i f f n e s s , e t c . ) . Such a problem would be almost impossible to solve, and would be computationally so expensive as to make any savings in structure costs i n s i g n i f i c a n t compared to the cost of achieving them. The design process as viewed by Schmit (5) is presented in figure (1-4). The function of this process is to enable the designer to formulate e x p l i c i t l y those areas where engineering judgment i s required, and those areas where an a n a l y t i c a l deter-mination of structure parameters can be implemented. Engineering judgment is used to reduce the general problem to a tractable one. It applies past experience with structures that were designed to s a t i s f y the functional requirements, and chooses the type of struc-ture to be used, based on a non-quantifiable type of optimization. At present, only a small number of the parameters necessary to define a complete structure can be determined a n a l y t i c a l l y by an optimization routine. The input parameters w i l l be termed external parameters for t h i s study, and the variables determined a n a l y t i c a l l y by the optimization routine w i l l be c a l l e d i n t e r n a l variables. The blocks on figure (1-4) are described below and l a b e l l e d i n t e r n a l or external, and the decisions relevant to this study are underlined. -7-NEEDS ESTABLISH CRITERIA GENERATE CONCEPTS DEVELOP MODELS ANALYSIS EVALUATE QUANTITATIVE LY EVALUATE QUALITATIVELY CONSTRUCT AND/ OR TEST OPTIMIZE CHANGE MODEL CHANGE CONCEPT CHANGE CRITERION FIG. 1-4 DESIGN PROCESS BLOCK DIAGRAM NEEDS Needs state the functional purpose of the structure. The function considered herein is a large warehouse type structure or any building where the roof structure employs trusses, with purlins and/or decking (external). CRITERIA The c r i t e r i o n i s stated in the objective function (merit func-t i o n , cost function). The objective function provides the basis of choice between alternate feasible designs. A feasible design is one which s a t i s f i e s both the functional requirements and the constraints. The objective function may be formulated on the basis of cost, weight, s t i f f n e s s , etc., (external). CONCEPTS Concepts embody the experience, judgment, and c r e a t i v i t y of the engineer. Concepts are divided into two i n t e r r e l a t e d sections. (a) Form and type of structure - examples include beams, domes, trusses. The type of structure used i s a bottom chord  bearing, p a r a l l e l chord, truss. (external) (b) Levels of choice based on (a). (i) Material choice - wood, concrete, s t e e l (external). ( i i ) Framing - rigid,, pinned, semi-rigid (external). ( i i i ) Behavior - e l a s t i c analysis, p l a s t i c analysis (external) (iv) Topology) Configuration - number of panels, depth Geometry) to span r a t i o , panel  spacing (internal) . (v) Type of member cross section - WF, tubes, tees, double angles (external). (vi) Member longitudinal properties - haunches, tapered, constant (external). DEVELOP MODELS A lin e a r e l a s t i c analysis i s used, (external) ANALYSIS The s t i f f n e s s matrix of the structure i s formed and the solution of KA=F is c a r r i e d out (K= structure s t i f f n e s s matrix, nxn, A represents an nxm matrix of generalized displacements, and F is an nxm matrix of generalized forces corresponding to A where n is the number of degrees of freedom and m is the number of load' cases.) (internal) - 9 -EVALUATE QUANTITATIVELY The ac c e p t a b i l i t y of the design in terms of the behavior con-s t r a i n t s i s determined. This step includes the redesign of the structure to s a t i s f y the constraints, (internal) OPTIMIZE The objective function for the current design i s evaluated i n this step. Based on the value of the objective function, a direc-tion and distance of tr a v e l is chosen to provide a new set of structure variables to y i e l d an improved value of the objective function, (internal) The other blocks i n figure (1-4) are self-explanatory and w i l l not be elaborated upon here. The dotted portion of figure (1-4) represents the portion of the design process automated for this study. A more detailed examination of the automated portion of the design process is shown i n figure (1-5). The following r e l a t i o n between analysis and design, as stated by Klotz (6) is used. "Analysis i s considered to be the determination of the forces acting in a member in a str u c t u r a l framework subjected to a s p e c i f i e d set of internal and/or external loads. Design i s considered to be the selection of a member's cross sectional geometry and material properties to r e s i s t the member forces i n accordance with the applicable stress and defle c t i o n c r i t e r i a . The v a l i d i t y of a design involves determining i f an assumed member s a t i s f i e s the stress and defle c t i o n s p e c i f i c a t i o n s for the forces determined by analysis, which was in turn based on such a member e x i s t i n g i n the framework." -10-This process, coupled with the objective function evalua-t i o n , i s presented i n figure (1-5). OUTPUT BEST DESIGN STOP STOP SET TOPOLOGY $ GEOMETRY DETERMINE LOADS ASSUME MEMBER CROSS SECTIONAL PROPERTIES ANALYSIS - DETERMINE FORCES DETERMINE MEMBER PROPERTIES I COMPARE MEMBER PROPERTIES EVALUATE OBJECTIVE FUNCTION TEST STOP CRITERION OPTIMIZE NOT SAT. FIG. 1-5 DESIGN-ANALYSIS-OPTIMIZATION BLOCK DIAGRAM Organization of Presentation A detailed examination of the system studied i s presented in Chapter 2. The variables considered are stated, and the object-ive function and cost model are presented. The modular nature of the program developed i s also outlined. In Chapter 3, the general mathematical programming problem i s presented. A b r i e f d i s c u s s i o n of o p t i m i z a t i o n techniques i s i n c l u d e d to c l a r i f y the problem. The problem i s formulated as a mathematical programming problem, and the d e t a i l e d reasons are given as to why no a l l - i n c l u s i v e a l g o r i t h m e x i s t s f o r i t s s o l u t i o n . The problem i s then r e s t a t e d so a s o l u t i o n may be obtained. Chapter 4 presents the a l g o r i t h m used to a r r i v e at the best s o l u t i o n , although no c l a i m i s made t h a t i t represents the optimal s o l u t i o n . An examination of the r e s u l t s i s a l s o i n c l u d e d . Data obtained from an exhaustive search procedure are presented i n Chapter 5. Parameter s t u d i e s are presented along with a d e t e r m i n a t i o n of the r e l a t i v e s e n s i t i v i t i e s of the v a r i a b l e s and the uniqueness of the optimum. A d i s c u s s i o n of the c o s t model i s a l s o i n c l u d e d . Chapter 6 examines the f e a s i b i l i t y o f developing a sub-o p t i m i z a t i o n scheme f o r the p a n e l s p a c i n g c o e f f i c i e n t s . Results of. the study are p r o v i d e d . F i n a l l y , Chapter 7 presents the c o n c l u s i o n s , recommenda-t i o n s f o r f u t u r e changes i n the computer program developed, and p o s s i b l e extensions of the study f o r f u t u r e i n v e s t i g a t i o n . -12-CHAPTER 2 THE DESIGN PROBLEM STUDIED Problem Statement The problem selected i s the designing of a truss to y i e l d the minimum t o t a l cost of the entire s t r u c t u r a l roof system. The roof system is composed of p a r a l l e l chord s t e e l trusses with the roof being applied to the upper chord. The types of member to be used are Tee sections for the chords and Double Angle sections for the webs. A l l j o i n t s are welded. The load cases to be con-sidered are uniform l i v e load on the entire span and uniform l i v e load on h a l f the span, plus dead load for a l l cases. The web configurations that may be used are the Pratt system or the crossed diagonal system. The roof may be applied to the truss at the panel points or anywhere between the panel points by pur-l i n s , or d i r e c t l y to the top chord, the only r e s t r i c t i o n being that the p u r l i n placement must be symmetrical about the center of the truss. For a l l cases, continuity of the chords must be con-sidered. The truss used i s bottom chord bearing. The design of the truss is to be made in accordance with CSA Spec. S16-1965. The chords of the truss are to be of constant area for the entire span. The example chosen for the problem used a span of 120'-0", and the magnitude of the l i v e load was as s p e c i f i e d for snow load i n the National Building Code. Structural parameters to be chosen by the designer (external). (1) Material properties -13-FY - y i e l d point of web members FYC - y i e l d point of compression chord FYT - y i e l d point of tension chord (2) Web configuration - Pratt or crossed diagonal. (3) Roof system (i) Panel point loading ( i i ) P u r l i n loading anywhere on the top chord ( i i i ) Roof deck applied d i r e c t l y to the top chord (4) Number of chord pieces and chord s p l i c e s . (5) Values of cost parameters. (6) Roof model parameters. Variables to be provided by automated scheme ( i n t e r n a l ) . (1) A l l member sizes . (2) Depth to span ra t i o of truss. (3) Number of panels of truss. (4) Panel spacing c o e f f i c i e n t s (optional). (5) Truss spacing. (6) P u r l i n spacing i f considered as a variable. Objective Function The cost figure considered i s the cost C per square foot, C represents the cost of the s t r u c t u r a l items of the roof system, consisting of the truss, purlins ( i f any), deck, and wall cladding above the bottom chord. C is computed as f C = cost/ft2= 1 , SPAN-SPACE C1.WWEB+C2.WTC + C3-WBC + (C 4 + C5)-WN + C6-NCORD + CyNSPLICE + 2•Cg•RATIO• SPAN-SPACE.+ (C 9 + C1Q)-SPAN-SPACE --14-. The c o s t parameters C\ are d e f i n e d as f o l l o w s : = web m a t e r i a l c o s t p e r pound = compression c h o r d m a t e r i a l c o s t p e r pound C - = t e n s i o n c h o r d m a t e r i a l c o s t p e r pound = c o s t o f p r e p a r a t i o n p e r web member C- = c o s t o f web j o i n t s p e r web member ( c o s t o f w e l d i n g b o t h ends t o the chords) Cg = c o s t o f p r e p a r a t i o n p e r cho r d member C- = c o s t p e r s p l i c e f o r cho r d member C„ - c o s t p e r square f o o t o f w a l l c l a d d i n g Cg = r o o f c o s t p e r square f o o t based on t r u s s s p a c i n g (When p u r l i n s are used, Cg r e p r e s e n t s the c o s t o f p u r l i n s -when the deck i s a p p l i e d d i r e c t l y t o the c h o r d , Cg r e p r e s e n t s the deck c o s t . ) C-^Q = r o o f c o s t , p e r square f o o t , based on p u r l i n s p a c i n g o r p a n e l p o i n t d i s t a n c e . When the p a n e l s p a c i n g i s a v a r i a b l e , the r o o f c o s t C-^ g i s assumed t o be governed by the l a r g e s t p a n e l l e n g t h i f p u r l i n s are not used between the p a n e l p o i n t s . The r e m a i n i n g terms i n the o b j e c t i v e f u n c t i o n are d e f i n e d as f o l l o w s : WWEB = weight o f web m a t e r i a l i n pounds WTC = weight o f compression c h o r d m a t e r i a l i n pounds WBC = weight o f t e n s i o n c h o r d m a t e r i a l i n pounds NPAN = number of p a n e l s WN = number o f web members = 2(NPAN) + 1 f o r P r a t t web c o n f i g u r a t i o n = 3(NPAN) + 1 f o r c r o s s e d d i a g o n a l system -15-NCORD number o£ chord pieces NSPLICE number of chord splices RATIO depth to span r a t i o of truss SPACE truss spacing SPAN truss span Cost Model An attempt was made to provide a r e a l i s t i c means of estab-l i s h i n g the cost of the roof s t r u c t u r a l system. It i s doubtful, however, that every fabricator would actually break his costs down in this fashion. In this study, labour and fa b r i c a t i o n charges were s p e c i f i e d independently from material costs. This was done to examine whether or not a topology cost would e f f e c t a change i n the optimum configuration. The topology cost was represented by C^ and Cg, the costs of preparing and j o i n i n g , respectively, the web member. The breakdown in material costs, C^, , and C^ was incor-porated to r e f l e c t the d i f f e r e n t member types (Tees and Double Angles) and to allow a var i a t i o n in the strength properties. Cg and sp e c i f i e d the costs associated with s p l i t t i n g the WF sections to produce Tees, and for s p l i c i n g the Tees. C^, (2^, Cg and Cj represented the fixed costs for a given topology, and the i r e f f e c t on the optimum structure i s examined i n Chapter 4 and Chapter 5. Cg, the cost of wall cladding per square foot, w i l l have an eff e c t on the best depth to span r a t i o . The ^impact of Cg w i l l depend on the r e l a t i v e magnitude of the material and wall cladding costs. Rather than include a design routine for the purlins and -17-an optimization scheme to give the best solution, are exceed-ingly costly to develop, and generally are expensive to run. In order for a program to be economically p r a c t i c a l , i t must at least be competitive with existing design costs, or the savings gained by the application of such a program must outweigh the execution costs. Unfortunately, i t is d i f f i c u l t to estimate the cost of development and the e f f i c i e n c y of a method and, more important, the magnitude of the savings gained over current design techniques. The present program, written in Fortran IV and run on an IBM 360/67, was developed to determine the f e a s i b i l i t y of an automated design approach purporting to y i e l d the optimal^design of a str u c t u r a l system. The largest part of the time spent i n running a s t r u c t u r a l optimization program i s i n the analysis and design phase. Every time the objective function is evaluated, a complete design must be made. Hence i t i s necessary to minimize the number of evaluations of the objective function, and to make the design-analysis cycle as e f f i c i e n t as possible. In order to incorporate v e r s a t i l i t y into the program, i t was written in modular form. Each module performs a s p e c i f i c function, a l l modules being interdependent for values used in the i r execution, but the execution of the module being inde-pendent of and separable from, the rest of the program. For example, i f a di f f e r e n t web configuration i s to be used, a new CONFIG module is substituted and no other changes i n the program are necessary. Figure (2-2) depicts the modules used. An explanation of the function of each module follows: 9 -18-CONFIG BOX ROBOT GEOM LOAD PSPACE SOLTN MODULE FUNCTIONS STRUCT BAND SYMM DESIGN PRICE WEB CHORD TEE FIG. 2-2 PROGRAM MODULES BOX contains the optimization algorithm which is Box's Complex Method. This module, controls the input parameters, and determines the variables to be used in the design and analysis routines. ROBOT performs a l l the functions necessary to achieve a design specified by the variables and parameters transferred from BOX. A l l subsequent modules are controlled by ROBOT. ROBOT performs the necessary bookkeeping to link the modules, and provides for the output of a l l the design information. CONFIG specifies the topology of the structure. It provides the coding information for the members in order to fac i l i ta te the building of the structure stiffness matrix. -19-GEOM takes the data from CONFIG and ROBOT and computes the structure coordinates and the geometry matrices for the load routine. GEOM also c a l l s PSPACE and SOLTN. PSPACE contains the panel spacing algorithm developed i n Chapter 6. SOLTN solves the equations generated i n PSPACE. LOAD builds the load matrix for the load cases. The present load cases include uniform load on f u l l span, uniform load on f u l l span and on half span, uniform load on f u l l span and a single point load at .any panel point, and uniform load placed on an integer number of panels, s t a r t i n g from 1 to NPAN. In a l l cases, dead load i s included and i s always updated to account for change in structure weight. These load routines are f u l l y auto-mated and include the cases where purlins are placed randomly on the top chord. External control i s provided, such that any combination of load cases may be obtained or any pattern of loading achieved. STRUCT builds the member s t i f f n e s s matrices and the structure s t i f f n e s s matrix. A plane frame s t r u c t u r a l element is used. A li n e a r e l a s t i c analysis i s employed to determine the member forces. BAND solves the equation K A= F, using the Choleski scheme. SYMM chooses the governing forces for the web members. -20-DESIGN provides the member cross-sectional properties s a t i s -fying the code requirements. Only h a l f of the web members are designed, since only a symmetrical truss is considered. WEB The properties of the undesigned web members are set equal to t h e i r symmetrical counterparts. CHORD equates the chord size i n each panel with the maximum chord size (constant area constraint). The allowable bending and a x i a l stress for each panel are then found. PRICE evaluates the objective function. TEE contains a table of Tee and Double Angle section prop-e r t i e s , extracted from the AISC handbook. A constant area c r i t e r i o n for the chords was used for two reasons. F i r s t l y , i t provides a f a b r i c a t i o n s i m p l i f i c a t i o n and hence a cost saving. Secondly, i n order to determine the number of d i f f e r e n t chord sizes which should be used, an optimization scheme would have to be devised in order to determine the optimal balance between chord material saving and additional s p l i c e cost. Based on computational experience, i t was found that the deflec-t i o n constraint was never exceeded. For that reason, the present program does not include a redesign scheme to l i m i t d eflection. -21-CHAPTER 3 THE MATHEMATICAL PROGRAMMING PROBLEM C l a s s i c a l optimization problem An optimization problem i s one for which i t i s desired to maximize or minimize a function of one or more variables, subject to certain r e s t r i c t i o n s or constraints, on the variables them-selves, and on functions of the variables. In order to formulate the optimization problem i t i s necessary to know, accurately and quantitatively, how the system variables i n t e r a c t ; to have a measure of system effectiveness (objective function) express-i b l e in terms of the variables; and to estab l i s h a decision apparatus whereby the best solution i s chosen based on the objective function. A general form of the optimization problem i s stated as: determine a vector x*= (x-, * ,. . . x *) which maximizes or minimizes Equation (3-1) represents the objective function, equations (3-2) represent constraint conditions on x, and equations (3-3) represent non-negativity conditions on x. The f i r s t solution of PROBLEM A was by the calculus and imposed four r e s t r i c t i o n s on equations (3-1) to (3-3), namely, no in e q u a l i t i e s appeared in equations (3-2), no non-negativity or discreteness r e s t r i c -tions were permitted on the variables, m<n, and the functions e-fx,,...x ) and f(x,,...x ) were continuous and possessed p a r t i a l derivatives at least through second order. A problem s a t i s f y i n g the above r e s t r i c t i o n s i s termed a c l a s s i c a l optimiza-tion problem. Generally, however, some or a l l of the above r e s t r i c t i o n s are viol a t e d and other techniques must be employed to render a solution to PROBLEM A. PROBLEM A constitutes the general statement of the mathema-t i c a l programming problem. To f a c i l i t a t e a meaningful discussion of the non-linear programming problem, the following d e f i n i t i o n s and theorem are presented along with a b r i e f examination of the properties of the li n e a r programming problem. De f i n i t i o n A set of points T i s a convex set i f every convex combination of points in T i s also in T. That i s , for every X,0<_X<1, and any two points x^, x 2 e T, {Xx-^  + (1-X)x 2> e T. Def i n i t i o n A function f is a convex function of x i n a nonempty convex set S i f for every two points x^ e S and x 2 e S, and every X, where 0<X<1, f U x j + (1-X)x 2>£ X f ^ ) + (1-X ) f ( x 2 ) . D e f i n i t i o n A function g(x) i s concave i n a convex set S i f -g(x) is a convex function i n S. Def i n i t i o n The convex programming problem i s written as follows: minimize z = f(x) subject to g^(x)>_0, i=l,...m where f(x) is a convex function and each g^(x) i s a concave function. Theorem (Local-Global Convexity Property) Every l o c a l minimum x* of the convex programming problem is a global minimum (7). -23-Linear Programming Problem Find x* such that the constraints n g i(x 1,...x n) = E a i-x- {<,=,>} b i , i=l,m j = 1 x j 1 °» J = 1> n PROBLEM B are s a t i s f i e d and the objective function n z = f (x-, , . . .x ) = E c-x. j=l J J / is maximized or minimized, where a--, b. and c. are known 13 1 1 constants. Note that no other r e s t r i c t i o n s may be s p e c i f i e d on the variables. For example, i f i t i s required that a l l or some of the Xj_ be integers, PROBLEM B would then become a non-linear Four important properties of the l i n e a r programming problem are: (a) The set of feasible solutions which s a t i s f y the constraints and the non-negativity r e s t r i c t i o n s i s a convex set which has a f i n i t e number of corners c a l l e d extreme points. (b) The set of a l l n-tuples (x 1...x n) which y i e l d a s p e c i f i e d value of the objective function i s a hyperplane. The hyper-planes corresponding to d i f f e r e n t values of the objective function are p a r a l l e l . (c) A l o c a l maximum or minimum i s also the global maximum or minimum. (d) If the optimal value of the objective function is bounded, at least one of the extreme points of the convex set of feasible solutions w i l l be an optimal solution. The l i n e a r programming problem is r e a d i l y solved by the use of the simplex method, which is an exceedingly e f f i c i e n t algorithm. problem. -24-Non-linear Programming Problem I£ any of the functions are non-linear, or i f integer constraints apply to any of the variables of PROBLEM A, then PROBLEM A is termed a non-linear mathematical programming problem. Terms such as convex, concave, separable, quadratic, and factorable apply to special cases of PROBLEM A. Some or a l l of the properties which characterize a l i n e a r programming problem may be vi o l a t e d in the non-linear problem. If the set of feasible solutions is convex, the optimal solution may be at an i n t e r i o r point of the feasible set, at an extreme point, or tangent to a constraint condition. The.problem of l o c a l optima which a r e n o t the g l o b a l optimum c a u s e s considerable d i f f i c u l t y . O n l y i n the c a s e o f t h e c o n v e x programming problem is there any assurance that a l o c a l optimum i s the global optimum. When the set of feasible solutions does not form a convex set or the objective function i s non-linear, then the solution to the problem becomes exceedingly d i f f i c u l t . P i c t o r i a l representations for the solution spaces for a line a r programming problem (convex li n e a r solution space) 1, a non-linear problem with a convex solution space (non-linear constraints), a non-convex solution space, and a convex solution space with non-linear objective function, are shown in figures (3-1) to (3-4). CONSTRAINTS -<——— DECREASING VALUE OF OBJECTIVE FUNCTION ISOVALUE CONTOUR OF OBJECTIVE FUNCTION NON-LINEAR CONSTRAINTS OPTIMUM LINEAR OBJECTIVE Y \FUNCTION FIG. 3-1 LINEAR PROGRAMMING PROBLEM FIG. 3-2 NON-LINEAR PROGRAMMING PROBLEM NON-CONVEX SOLUTION SPACE \ "°LOCAL ^ OPTIMUM \\ \GLQ£AL OPTIMUM FIG. 3-3 LOCAL AND GLOBAL OPTIMA ISOVALUE CONTOURS ,OF NON-LINEAR OBJECTIVE FUNCTION CONVEX 'SOLUTION SPACE -GLOBAL OPT/MUM LOCAL OPTIMUM FIG. 3-4 NON-LINEAR OBJECTIVE FUNCTION Solution of the Non-linear Programming Problem At present there are three basic approaches to the solution of the non-linear problem. Depending on the form of the objec-tive function and constraints, an i t e r a t i v e scheme using l i n e a r programming as the solution algorithm, can be applied (Hadley (8), -26-Cornell et a l (9)). Application of this technique, however, does not necessarily lead to the global optimum. Another group of solution algorithms u t i l i z e s some type of gradient approach. The t h i r d method is the exhaustive search method which involves the evaluation of a l l possible combinations of the variables. The exhaustive search method i s only used when no other algorithm exists for the solution of the problem. B a s i c a l l y , a l l optimi-zation schemes exploit the structure of the problem i n order to arrive at a solution algorithm, and hence, no general a l l -inclusive solution technique exists for the non-linear program-ming problem. Four basic steps are required for a search procedure: (1) An i n i t i a l feasible point, (2) A d i r e c t i o n of t r a v e l , (3) A distance of t r a v e l , and (4) A termination c r i t e r i o n for the search. The following outline provides a b r i e f resume of the decisions to be made and possible ways of making them for the solution of the non-linear problem. Detailed explanations of algorithms employed in solving the non-linear problem may be found in references (8), (10) , and (11). General problem Minimize f(x) where x = (x....x )e R n 1 n General solution technique Select an i n i t i a l point x Q For i = 0, 1..., point = (x|, x i , . . ^ 1 ) is known. Select a d i r e c t i o n s 1 = ( s ^ , . . . s * ) . Select a distance aj_ (scalar) . Then, a new point i s given by -27-i+1 i . i - -, ' " i + l i ^ i x = x + a.s or, m general, xn = x, + a.s. 1 ' k k i k for k = 1,.. Check the termination c r i t e r i o n . Choice of d i r e c t i o n , s 1 Based on derivatives (i) F i r s t order methods - steepest ascent or descent (gradient methods). ( i i ) Second order methods. In the v i c i n i t y of the minimum x*, the f i r s t p a r t i a l s vanish and the Taylor series expansion gives (to second order terms) f(x) = f(x*) + i(x-x*) TH f(x*) (x-x*) where H £(x*) is the matrix of second p a r t i a l s ' of f (Hessian of f) evaluated at x*. Methods to be employed include Fletcher-Powell, conjugate gradient, Powell's method. Choice of distance a^ Choose a to make g(a) = f ( x 1 + ^ ) = f ( x 1 + as 1) a minimum. Set the derivative equal to zero. da k+1 8x, i+l da . k+1 3x, i + l K k k or, in vector form, vffx 1*"*")^s 1 = 0 An alternative method for choosing a^ would be to f i t a quadratic or cubic to f ( x 1 + "^) and minimize the resulting f i t t e d curve. Termination c r i t e r i a (1) Based on change i n slope. < e (i) max, k 3x, - 2 8 -n 2 ( i i ) E ( 3 f ) 2 < e k = l 3x, k ( 2 ) B a s e d o n c h a n g e i n f u n c t i o n v a l u e i i + l f ( x ) - f ( x ) < e ( 3 ) B a s e d o n c h a n g e i n x max, I x , 1 + ^ - x.11 < e k k k' T h e P r o b l e m o f S t r u c t u r a l O p t i m i z a t i o n T h e s t r u c t u r a l o p t i m i z a t i o n p r o b l e m may b e s t a t e d a s : c h o o s e t h a t s e t o f t o p o l o g i c a l , g e o m e t r i c a l a n d m ember v a r i a b l e s w h i c h w i l l m i n i m i z e t h e s t r u c t u r e m e r i t f u n c t i o n s u b j e c t t o c o d e r e q u i r e m e n t s , c o n t i n u i t y r e q u i r e m e n t s , s i t e c o n d i t i o n s , a e s t h e t i c c o n s i d e r a t i o n s a n d o t h e r s p e c i f i e d b e h a v i o u r a l r e q u i r e -m e n t s . L e t t h e d e s i g n v a r i a b l e s b e c o n t a i n e d i n t h e v e c t o r x . T h e d e s i g n v a r i a b l e s a r e s u b j e c t t o t w o t y p e s o f c o n s t r a i n t s , n a m e l y , s i d e c o n s t r a i n t s a n d b e h a v i o u r c o n s t r a i n t s . S i d e c o n -s t r a i n t s a r e t h o s e t h a t s p e c i f y u p p e r a n d l o w e r b o u n d s o n e a c h o f t h e d e s i g n v a r i a b l e s , w h e r e a s b e h a v i o u r c o n s t r a i n t s a r e f u n c t i o n s o f t h e d e s i g n v a r i a b l e s . L e t X^ a n d X ^ be t h e l o w e r a n d u p p e r l i m i t s o n t h e d e s i g n v a r i a b l e s . L e t g^.. ( x ) b e t h e i t h b e h a v i o u r f u n c t i o n f o r t h e j t h l o a d c o n d i t i o n . F u r t h e r , l e t G^j a n d G^j b e t h e l o w e r a n d u p p e r l i m i t s r e s p e c t i v e l y o n t h e b e h a v i o u r f u n c t i o n g . . ( x ) . I n g e n e r a l , G ^ - a n d GV. w i l l d e p e n d o n t h e v a l u e s o f t h e d e s i g n v a r i a b l e s ( e . g . a l l o w a b l e s t r e s s i n c o m p r e s s i o n m e m b e r ) . H e n c e , t h e b o u n d s o n t h e b e h a v i o u r a l c o n -s t r a i n t s c a n o n l y b e s t a t e d i m p l i c i t l y . A design x, which s a t i s f i e s both the side constraints and the behaviour constraints, i s classed as a feasible design point in the solution space. The choice between alternate acceptable (feasible) designs i s accomplished by-means of the objective function f ( x ) . Using the above d e f i n i -tions, the problem of structural optimization has been stated in matrix form by Schmit (5) as: "Given the preassigned parameters and the design load system (generally includes a m u l t i p l i c i t y of d i s t i n c t load conditions) as well as the design limitations X , X^' G^(x), a minimum value." The preceding d e f i n i t i o n constitutes the formal state-ment of the s t r u c t u r a l synthesis problem and i s invariably a non-linear programming problem. Much work has been done for the case where f(x) is a l i n e a r function and gj, j ( x) a r e non-line a r functions. This has usually meant that only member sizes have been considered, with the configuration and material of the structure treated as constant. Even in this case, there is no guarantee that any optimum found is a global optimum. A simple design space for a two bar truss i s shown in figure (3-5). Note that a l l the functions are considered to be l i n e a r and that X^ i s not active. L Find x* subject to X L and G (x*) < g.-(x*) _ U < x* £ X G U(x*) such that f(x* < ) takes on -30-•''''/ \ J CO. i » x i \ / S2' x2 d 2 ^ S = - P ^ i n a ^ + S 2 1 c o s B 2 S.. equals the ax i a l force in the i t h member for the j t h load case. < S 1 2 = P 2 s i n a 2 + S 2 2 c o s 3 2 cosg. cosB. S21 = P i ^ c o s a i + sina^tanB^) S 2 2 = P2(cosot2 " sina 2tanB 1) sinB 2 + cosB^anB-^ sin B 2 + cosB 2tanB^ f(x) = dp(x^ + x2 ) (minimum weight) sing^ sinB 2 X L = {0,0} X U = {x^, x^} QT = {-a, -0} G u = {a, a} L _U (compression considered positive) Let B7 = 60°, 8 7 = 45°, a, = 20°, a- = 45' Then, S = .436P , S = .793P , S 1 2 = 1.032P2, S 2 2 268P, Further, l e t P =P =P=20 kips and l e t o =20 k s i Also, define f'(x) = f(x) = 1.15x, + 1.41x "dp - 1 2 The design space is as shown below. 1.5 |2 NyC* SIZING UPPER LIMIT 1.0-. 5- • i x LOWERNJ i ^ 1 LIMIT > REGION OF ACCEPTABLE DESIGNS ! DUE TO P n i x X l SIZING UPPER LIMIT \ l i 1 —^ ... SIZING LOWER LIMIT \ .. | V i ^ P ^ l L 0 W E R LIMIT DUE TO P, TO P. MsOVALUE CONTOUR OF OBJECTIVE FUNCTION (f'(x)) f x ^ S I Z I N G LOWER LIMIT 2 \ x 0 LOWER LIMIT" DUE TO P, .5 1.0 1.5 2.0 FIG. 3-5 DESIGN SPACE FOR TWO BAR TRUSS Formulation of the thesis problem as a mathematical program- ming problem. Let x^ = number of panels j x^ = depth to span ratio X j = truss spacing x^ = p u r l i n spacing x ,...x = panel spacing c o e f f i c i e n t s x ,..x = member sizes L+l' n TOPOLOGY GEOMETRY -32-Minimize f(x,,...x ) 1 n S x 3 S P^.x.c. + 2x A + A + A o + 2A,Sx„x 7 + A.Sx, j=L+l J J J 1 1 1 2 3 2 3 4 3 where S = span p = weight per cubic volume c. = cost per pound of s t e e l ; one of C-, C 7 or C?. H. = length of j t h member. \ = C4 + C5 A 2 = CgNCORD + CyNSPLICE A 3 = C g A4 = C9 + C10 = § C x 3 ' x4^ (C^ is defined i n Chapter 2) subject to 6 £ x^ £ 16, x^ an even integer DLBR < x 2 £ DUBR TLBS < x^ < TUBS PLBS < x„ < S/x, — 4 — 1 x^ , ...x^ > 0 x T,'.,.x > minimum area in member table L+l' n — E x. - 0.5 = 0 j = 5 3 (DLBR, DUBR, TLBS, TUBS, and PLBS are defined i n Chapter 4.) x L + l , ' " ' x n m u s t s a t i s f y a l l code requirements and are chosen from a discrete member spectrum. The problem as stated, i s a non-linear, non-convex math-ematical programming problem for which no a l l - i n c l u s i v e optimization algorithm exists. -33-Investigation of related work The cost optimization of topology and geometry for struct-u r a l frames has been investigated by Soosaar (13). Soosaar considered the cost to be composed of two major components: the fixed component which represents the cost of including a member, and the weight related costs. His work treated the case of rect-angular multi-storey frames and i s summarized as follows (12): " . . . I t is assumed, and l a t e r demonstrated, that for the purposes of configurational optimization, a determinate or si m i l a r s i m p l i f i e d treatment i s s u f f i c i e n t as well as necessary to model the actual complex e l a s t i c behaviour of the frame. The determinate portal and cantilever methods are employed for wind behaviour; gravity loads are treated by a combination of , i n f l e c t i o n - p o i n t assumption and a one or two-cycle moment d i s t r i b u t i o n . Since the structure is then "determinate", the optimum member-level solution is f u l l y stressed. A continuous spectrum of member sizes is assumed as well as functional relationships among the member properties. Member selection is by design sub-optimization based on the AISC s p e c i f i c a t i o n . These above assumptions result in major s i m p l i f i c a t i o n s in the formulation i n which the cost objective function contains only the topological (member existence) and geometrical (span length) variables, and in which the constraints are l i n e a r . A solution to the mathematical programming problem i s obtained by a two-stage process using C l a s s i c a l optimization to determine the topology l e v e l solution, followed by a quadratic program-ming algorithm to determine the geometry...." -34-Two important conclusions stated by Soosaar regarding the optimum configuration are: "(1) The optimum configuration of the frame is nearly wholly dependent on the r e l a t i v e proportions of the weight costs and fixed costs. High fixed member costs lead to fewer columns. (2) Unlike the objective functions i n most s t r u c t u r a l optimi-zation problems, the cost vs. topology curves for frames are far from f l a t when fixed costs are included; data obtained from steel fabricators indicate that the fixed costs are high. A minimum weight optimization ( i . e . , zero fixed costs) can result in a structure actually more costly by as much as 70%." It i s noted that Soosaar treated the storey height and frame spacing as input parameters. His approach to the con-fig u r a t i o n problem was not used for this study because of the following reasons: (1) For a m u l t i p l i c i t y of load cases, the e x p l i c i t formulation of member forces i n terms of geometric variables i s not p r a c t i c a l . (2) Soosaar demonstrated the hierarchy of variables to be topo-logy, geometry, and member sizes. Based on this d i v i s i o n he was able to show that the optimization could proceed on these three levels sequentially without a l t e r i n g the previous optimization level (e.g., the geometry optimization did not affe c t the optimal topology). The d i v i s i o n , however, of the configuration problem into a two-level solution procedure i s not v a l i d when the geometrical variables, truss depth and spacing, are considered. (The d i v i s i o n i s v a l i d when the geometry l e v e l consists only of the panel spacing.) The effect on the objective function, of a change in truss spacing or depth to span r a t i o , i s of much greater magnitude than a change i n topology (see Chapter 5). This implies a di f f e r e n t hierarchy of variables than assumed by Soosaar. (3) The formulation of the member sizes i n terms of geometrical variables, using a f u l l y stressed design c r i t e r i o n , i s not possible i f constraints are put on member sizes (e.g., chords of constant area) . In almost a l l work to date, the problem of member l e v e l optimization has been considered using a continuous member spec-trum, and assuming functional relationships between the member properties. G e l l a t l y et a l (1), G e l l a t l y (14), and Cornell et al (9), have made notable recent contributions to thi s field;and have also included geometry as a variable. In a l l cases, weight has formed the basis of the objective function. The problem of a discrete member spectrum has been treated by Toakley (15). As of yet, no e f f i c i e n t algorithm exists for the solution of.large integer programming problems, and further, considerable d i f f i c u l t y has been encountered i n assuring convergence of the algorithm used. In order for an a l l - i n c l u s i v e algorithm to be applicable to the problem at hand, i t would have to be capable of handling a mixed integer-continuous programming problem with a non-linear objective function, subject to non-linear constraints. At present, no such algorithm e x i s t s , and even i f the problem i s subdivided based on a hierarchy of variables, e f f i c i e n t algorithms providing the optimal solution for each subdivision do not e x i s t . -36-The problem investigated herein was divided as follows i n order to achieve a solution, although not an optimal one. (1) The variables x-p.-.x^ are considered i n a separate optimization scheme. (x^,...x-^ were also included but l a t e r deleted.) (Chapter 4) (2) X J - , . . X L are generated in a sub-optimization scheme. (Chapter 6) (3) A f u l l y stressed design for x T -,..x was performed, subject to code requirements and constant area constraints. A f u l l y stressed design i s considered to be one where every member i s stressed to the f u l l allowable l i m i t under at least one load condition. The constraint of constant area for the chords meant that only one section would govern the design of the chords. The algorithm developed for panel spacing in-Chapter 6 used this fact to t ry and stress the compression chord in each panel to the f u l l allowable l i m i t , to determine whether such a procedure would lead to a reduction i n the-chord weight. -37-CHAPTER 4 SOLUTION OF OPTIMIZATION PROBLEM Minimization algorithm (Box's Complex Method) The algorithm used to determine the optimal values of x* is examined, and the v a l i d i t y of the f u l l y stressed design as the minimum weight design for x*, i s discussed. The algorithm employed i n the program was developed by M.J. Box (16). The method, c a l l e d the Complex Method, was devised for finding the maximum or minimum of a general non-li n e a r function of several variables within a constrained region. This constrained optimum is characterized by some of the variables lying at the l i m i t s of t h e i r permissible ranges. Box points out that i f the problem studied does not have a con-strained optimum, then a method with no provision for bounding the variables w i l l produce the same r e s u l t . Although the variables used i n this study do not, in general, l i e at the extremities of t h e i r ranges, Box's method i s of such a general nature that i t can be adapted to the present problem. Box's Complex Method searches for the maximum value of a function f(x-j_,...x ) subject to m constraints of the form g^ < X j , < hj,, k=l,m where x n +-^,..x m are functions of x •]_,,. x n, and the lower and upper constraints and h^ are either con-stants or functions of x^,...x . (For a minimization problem, -f is maximized.) Assume that the set of feasible solutions is not a n u l l set. Then, an i n i t i a l point x° = {x£...x^} which s a t i s f i e s the m constraints, must ex i s t . For the search procedure, k > n+l points form the complex, of which one i s the -38-i n i t i a l point. The k-1 points used with the i n i t i a l feasible point to form the s t a r t i n g complex, are obtained independently by the use of pseudo random numbers, x^ being computed as x^ = g^ + r^(h^ - g^) where r^ is a pseudo random deviate rectangularly d i s t r i b u t e d over the i n t e r v a l (0,1). According to Box, a point so selected must s a t i s f y the e x p l i c i t con-stra i n t s but not the i m p l i c i t constraints. (The i m p l i c i t con-st r a i n t s are x n +-^. . «x .) For the program developed, an i m p l i c i t constraint existed only for the panel spacing c o e f f i c i e n t s , and i t was always s a t i s f i e d before evaluation of the objective function. Following the generation of the k-1 points, the objective function is evaluated at each point. The point corresponding to least function value, x^, is replaced by a new point, x^. x^ is located a>l times as far from the centroid of the remaining points (x1^ excluded) as the r e f l e c t i o n of the worst point i n the centroid. i s c o l l i n e a r with the rejected point and the centroid of the retained points. Note that the centroid of the objective function cannot be used, as the value of the corres-ponding variables cannot be determined. If the new point is no improvement over the worst point, a is halved. This procedure is repeated u n t i l a constraint i s v i o l a t e d or no improvement in the objective function can be made. If a t r i a l vector x does not s a t i s f y an e x p l i c i t constraint on a variable x^, the variable i s set equal to a value e .inside the appropriate l i m i t . "The use of over-reflection by a factor a>l, tends to cause -39-a continual enlargement of the complex and thus to compensate for the moves halfway toward the centroid. Furthermore, i t enables rapid progress to be made when the i n i t i a l point is remote from the optimum. It is also an aid towards maintaining the f u l l dimensionality of the complex. So, too, i s the use of k>n+l points, since with k=n+l points only, the complex is l i a b l e to collapse into a subspace. In p a r t i c u l a r , i t tends to f l a t t e n i t s e l f against the f i r s t located constraint and thus be unable to move along an additional constraint when a corner is reached." (16) The values of a and k suggested by Box are 1.3 and 2n, respectively, with the algorithm not being prejudiced adversely by using alternative values. The effect of varying a and, k is studied in the results presented. The type of optimum ( l o c a l or global) derived from the algorithm can be checked by i n i t i a t i n g the search from a d i f f e r -ent s t a r t i n g point and from a di f f e r e n t random number base. It is of interest to note that Box assumed that the set of feasible solutions formed a convex set. * TOPOLOGY x l D/S RATIO x2 TRUSS SP • x3 TOPOLOGY x l D/S RATIO x2 TRUSS SP , x3 PURLIN SP.x 4 TOPOLOGY x l D/S RATIO X2 TRUSS SP , x3 PANEL SP.x 5,x L TOPOLOGY X l D/S RATIO X2 TRUSS SP • x3 PURLIN SP.x 4 PANEL SP.x 5,x L TABLE 4-1 GROUPS OF VARIABLES TREATED BY ALGORITHM •40-Program Discussion The groups of variables which the optimization algorithm was set up to handle are summarized i n Table (4-1). The para-meters input to the algorithm are l i s t e d and described below. NPOINT Number of points i n the complex (k) NIT Maximum number of points in the space to be considered (includes i n i t i a l complex). NIT was incorporated to l i m i t the computational cost, and hence i t acts as a termination c r i t e r i o n . TOL TOL i s the maximum percentage difference between the worst point and the best point of the complex. (Termination c r i t e r i o n ) o v e r - r e f l e c t i o n factor lower and upper bounds respectively on the truss spacing. lower and upper bounds respectively on the truss depth to span r a t i o . PLBS lower bound on p u r l i n spacing. The upper bound, PUBS, was considered to be the panel length. RNB random number base. The topology constraints were b u i l t into the program. ALPHA ( a) TLBS,TUBS DLBR,DUBR Modifications to Box's Method implemented in the Program When more than one e x p l i c i t constraint was v i o l a t e d , a was halved, and the objective function of this new point was evalu-ated only i f the e x p l i c i t constraints were s a t i s f i e d by the new point. I f the constraints were s t i l l not s a t i s f i e d , the -41-centroid was used as the new point, as i t must always be a feas-ib l e point. If the objective function of a new point was worse than the previous worst point of the complex, a was again halved. If no improvement was made, the centroid was used. If s t i l l no improvement resulted, a point midway between the best point of the complex and the centroid was used. If no improvement result-ed, the search was terminated. For the case when the region surrounding the optimum was very f l a t , these modifications man-aged to keep the search in progress. For the examples run, the region surrounding the optimum was found to be very f l a t , and hence, many alternate designs having the same cost were possible. Because a non-optimal topology solution could contain a geometry optimum, adjacent topologies were examined. Although this had the e f f e c t of making the search more expensive, i t greatly aided the convergence of the complex to the area of the optimum. A flow chart of the optimization process is displayed in Appendix A. Presentation of Results A number of computer runs were made to study the e f f e c t of varying the number of points in the complex, the value of a, the s t a r t i n g point for the optimization, and the random number base. For this purpose the cost parameters for the model outlined in Chapter 2 are as follows: C 1 = C 2 = C 3 = $0.08 C 4 + C 5 = $13.00 C 6 = $15.00 C_ = $5.00 C g = $0.20 -42-TRUSS SPACING IN FEET 0 - 5 5 - 10 10 - 15 15 - 20 20 - 25 C g ($/FT 2) C 9S(#/FT 2) .06 . 50 . 10 1.00 . 18 1.60 .24 2.10 .32 4.00 PURL] IN SPACING IN FEET 0 - 6 6 - 10 10 - 17 17 - 24 24 - 29 C 1 0 ($/FT 2) C 1 QSC#/FT 2) . 25 3.00 .50 6.00 . 75 9.00 1.00 12.00 1. 50 18.00 For run 17, C g was set at $2.00. For run 18, + was set at $40.00. Except for runs 15 and'16 where the l i v e load was 60 psf, the l i v e load used was 40 psf. For run 19 the upper l i m i t on the truss spacing was 40 feet and the cost associated with truss spacing was set equal to zero. A36 steel was used for a l l designs. Table (4-2) l i s t s the input data for each of the runs made, and Table (4-3) l i s t s the r e s u l t s . Appendix B contains a t y p i c a l computer output. A comparison of the algorithm answer and the results from an exhaustive search procedure is shown in figures (4-1) to (4-5). For the panel point loading case, the second best point of the f i n a l complex was used in figure (4-2), since the best result from the exhaustive search was employed as an i n i t i a l point and could not be improved upon. Table (4-4) l i s t s the i n i t i a l and f i n a l complex for run 6. < (a) k = 1.5n ot = 1.5 (runs 1, 2, 5) In a l l cases, a good i n i t i a l point was used. For run 3, -43-the best point of run 2 was used as the i n i t i a l point. Because the complex was small, the worst points were removed quickly and convergence to a low cost, feasible region, resulted. It was noted i n run 1, however, that the complex became locked i n a subspace and no improvement could be made. The major advan-tage of a small complex i s i t s replacement by a better complex with the minimum number of objective function evaluations. The major disadvantage of a small complex is that a uniform d i s t r i -bution over the solution space cannot be obtained with a small number of points for the i n i t i a l complex, and hence the tendency to get locked i n a subspace i s more prevalent. (b) k = 2n a = 1.3 (runs 4, 5, 6, 9) • The same i n i t i a l point was used for a l l runs except 6. The only v a r i a t i o n permitted was i n the random number base. The use of a larger complex allowed a more uniform d i s t r i b u t i o n of s t a r t -ing points. The i n i t i a l point of run 6 was purposely chosen to be a poor s t a r t i n g point i n order to judge the bias created by a good i n i t i a l point. Apparently, the effect of the choice of the i n i t i a l point does not appreciably a l t e r the f i n a l r e s u l t . •(c) k 2.5n a = 1.3 (runs 11 B, 12) The e f f e c t of a large complex is to make the convergence of the entire complex to a low cost region very slow. It was noted, that while in parts (a) and (b) above, the maximum percentage difference between the best and worst point of the f i n a l complex was of the order of £ to 3 per cent, the difference for k = 2.5n was approximately between 3 to 6 per cent. -44-(d) Ef f e c t of g (runs 7,8, 9, 10) The i n i t i a l point and the random number base were held con-stant for a l l runs. Except for run 7, the choice of a does not appear to be c r i t i c a l . For run 7, the complex became locked in a subspace, and while small improvements could be made, the small value of a precluded the opportunity to enlarge the region of the complex s u f f i c i e n t l y to allow movement of the complex from the subspace. RUN NPOINT NIT a RNB D/S RATIO TRUSS SP. PURLIN SP. DLBR DUBR TLBS TUBS PLBS PUBS 1 6 40 1.3 .45 .07 .14 12. 25. 4. P.L. 2 6 20 1.3 .45 .07 . 12 15. 25. 4. 4. 3 6 40 1.3 . 75 .07 . 12 15. 25. 4. P.L. 4 8 30 1.3 .90 .07 .12 15. 25. 4. P.L. 5 8 40 1.3 . 10 .07 . 12 15. 25. 4. P.L. 6 8 30 1.3 . 45 .07 .12 15. 25. 4. P.L. 7 8 40 1.1 .45 .07 .12 15 . 25. 4. P.L. 8 8 40 1.2 . 45 .07 .12 15. 25 . 4. P.L. 9 8 30 1.3 .45 .07 .12 15. 25 , 4. P.L. 10 8 30 1.4 .45 .07 . 12 15. 25 . 4. P.L. 11 10 30 1.3 . 45 .07 .12 15. 25. 4. P.L. 12 10 30 1.3 . 80 .07 . 12 15 . 25. 4. P.L. 13 8 30 1.3 .45 .07 .12 15. 25 . P.L. 14 8 30 1.3 .20 .07 . 12 15. 25. P.L. 15 8 30 1.3 .45 .07 . 12 15. 25. 4. P.L. 16 8 30 1.3 .10 .07 . 12 15. 25. 4. P.L. 17 8 30 1.3 .60 .07 .12 15. 25 . 4. P.L. 18 8 30 1.3 . 70 .07 .12 15 . 25 . 4. P.L. 19 8 30 1.3 . 80 .07 . 12 15. 40. 4. P.L. NOTE: For a l l runs, TOL equalled 0.005, and the lower and upper bounds on the number of panels were 6 and 16, respectively. P.L. designates panel point loading. TABLE 4-2 INPUT DATA -45-(e) Runs 13 to 19 Runs 13 to 19 investigate trusses subjected to panel point loading, the ef f e c t of increased load, and changes in the parameter values of the cost model. (See figures (4-2), (4-3), (4-4), (4-5).) From figures (4-1) to (4-5), i t i s noted that the corres-pondence of the best algorithm point to the best exhaustive search point i s not always exact. Two reasons are put forward to account for this discrepancy. F i r s t l y , the exhaustive search procedure considered only a very limited number of points, and therefore i t i s possible that a better answer e x i s t s , although any improvement i n the best point would not be greater than about 1 per cent. Secondly, the truss spacing variable in the optimization routine was treated as a continuous variable. In f a c t , i t should have been treated as a discrete variable because of the assumed form of the cost model. The following study v e r i f i e s the l a t t e r conclusion. . I N I T I A L P 0 I N T S F I N A L P O I N T S SEARCH TERM-* x2 A A ft INATION RUN X1 x? x3 x4 WEIGHT £(x) x l x3 x4 WEIGHT f(x ) j. J X CRITERION 1 10 . 100 15. 6. 9231. 1.0708 10 . 1000 15. 6. 9231. 1.0708 NO IMPROVEMENT 2 10 . 100 15. 4. 9034. 1.0621 8 ,0839 19. 15 4. 11375. 1.0463 NIT 3 8 .084 19. 2 4. 11376. 1.0449 8 . 0834 19. 35 4. 11389. 1.0410 NO IMPROVEMENT 4 10 .100 15. 6. 9231. 1.0708 10 .0950 15. 4 .615 8986. 1.0508 NO IMPROVEMENT 5 10 . 100 15. 6. 9231. 1.0708 8 .0829 19. 38 4. 11379. 1.0397 NIT 6 12 . 120 15. 10. 9233. 1.3578 8 .0932 19. 89 4. 286 11533. 1.0357 NIT 7 10 . 100 15. 6. 9231. 1.0708 10 .097 19. 01 5. 1.0565 TIME TERM 8 10 .100 15. 6. 9231. 1.0708 8 .0992 19. 95 4. 11615. 1.0394 TOL 9 10 . 100 15. 6. 9231. 1.0708 10 .0972 19. 79 4. 11029. 1.0448 NIT 10 10 . 100 15. 6. 9231. 1.0708 12 . 098 19. 85 4. 10358. 1.0430 NO IMPROVEMENT 11 10 . 100 15. 6. 9231. 1.0708 10 .0801 15. 5 . 9175. 1.0604 NO IMPROVEMENT 12 12 . 120 15. 6. 9725 . 1.1297 10 .0971 19. 44 4.615 11017. 1.0537 NIT 13 10 .100 15. P.L. 9379. 1.5774 12 .0964 15. P.L. 8608. 1.3206 NO IMPROVEMENT 14 12 .100 20. P.L. 10866. 1.3068 12 .1000 20. P.L. 10866. 1.3068 NIT 15 10 . 100 20. 6. 14826. 1.1671 10 .0962 18. 49 4. 13195. 1.1587 NO IMPROVEMENT 16 12 . 120 15. 10. 11565. 1.4614 10 .1063 17. 82 4.615 13110. 1.1832 NIT 17 10 .100 20. 6. 11302 . 1.4096 10 .0700 18. 47 4.286 11569. 1.3425 NIT 18 12 . 100 20. 6. 10994. 1.3423 8 . 0894 19. 71 4.615 12657. 1.2708 NO IMPROVEMENT 19 12 . 120 15. 6. 9725. 0. 9497 10 .1102 4. 6. 19467. 0.6700 NIT TABLE 4-3 RESULTS OF OPTIMIZATION -47-I N I r r I A L C 0 M I ' L E X x l x2 x3 x4 WEIGHT f(x) 12 .120 15. 10. 9233. 1.3578 14 .086 18. 50 7.5 11637. 1.3949 6 .075 18.95 20 . 15629. 1.9249 8 . 105 17.07 10. 13243. 1.4416 16 .098 22.43 6. 12628. 1.1698 16 .084 22.63 5. 12976. 1.1698 12 .073 16.87 4. 29 10092. 1.1130 6 .078 19.42 15. 20520. 1.8284 F I N A L C : 0 M P I , E X x l X2 X3 X4 WEIGHT f(x) 8 .0879 19.34 4. 11462. 1.0456 10 . 1078 19. 79 4.615 11319. 1.0588 12 . 0985 19.12 4. 286 10147. 1.0553 8 .0961 18.62 4.615 11343. 1.0648 8 .0809 17.95 4. 11007. 1. 0661 8 .0969 19.59 4.615 11623. 1.0481 8* .0932* 19.89* 4.286* 11533. * 1.0357* 12 . 0990 19.81 4. 10396. 1.0456 * denotes optimum TABLE 4-4 INITIAL AND FINAL COMPLEXES FOR RUN 6 -48-£ 1.10 ^ 1.00 \ to 8 T" 08 .10 D/S RATIO 12 Exhaustive Search - 8 panel Exhaustive Search - 10 panel Exhaustive Search for 4-0" p u r l i n spacing 20-0" truss spacing © Indicates algorithm result Live load = 40 psf FIG. 4-1 COMPARISON OF OPTIMIZATION RESULTS WITH EXHAUSTIVE SEARCH .08 .] .0 ' . ] L214-12. Exhaustive Search - 12 panel Exhaustive Search - 14 panel Exhaustive Search for 20'-0" truss spacing Panel point loading only Live load = 40 psf (Second best point of Complex used for Run 14) D/S RATIO FIG. 4-2 COMPARISON OF OPTIMIZATION RESULTS WITH EXHAUSTIVE SEARCH \ \ y s * ^ t .c 8 .1 L0 .] 12 Exhaustive Search - 10 panel Exhaustive Search - 12 panel Exhaustive Search for 4'-0" p u r l i n spacing 20-0" truss spacing Live load = 60 psf D/S RATIO FIG. 4-3 COMPARISON OF OPTIMIZATION RESULTS WITH EXHAUSTIVE SEARCH -49-1. 50 •p t/) o 1.40 EH 1.30 A 12. '1 y // 8 * * X * r / •V s s .08 .] .0 .] L2 Exhaustive Search - 8 panel Exhaustive Search - 10 panel Exhaustive Search - 12 panel Exhaustive Search for 410" p u r l i n spacing 2010" truss spacing Live load = 40 psf Wall cladding cost $2.00/ft 2 D/S RATIO FIG. 4-4 COMPARISON OF OPTIMIZATION RESULTS WITH EXHAUSTIVE SEARCH 30 rt 1.20 IO s ""I a .08 . ] L0 . ] L2Exhaustive Search - 8 panel Exhaustive Search - 10 panel Exhaustive Search for 4-0" p u r l i n spacing 20'-0" truss spacing Live load=40 psf Topology cost $40./web member D/S RATIO FIG. 4-5 COMPARISON OF OPTIMIZATION RESULTS WITH EXHAUSTIVE SEARCH -50-The objective function may be written as f(x) = I (K W(x x..x n,S,F) + 2x 1A 1 + A 1 + A 2 + 2A 3x 2x 3S Sx 3 + A 4x 3S) = W*(x ..xn,S,F) + 2x 1A 1 + A± + A 2 + 2A 3x 2 + A 4 (4-1). where W*(x ..x^,S,F) represents the material cost of the truss per square foot, S is the span of the truss, F represents the magnitude of the loading, x,..x are defined in Chapter 3. I n • Let K 2 = 2x^A^+ A-^  + A 2 equal the topology cost plus the chord labour cost. Hence, represents the fixed cost per truss and is independent of the area supported by the truss. Let = wall cladding cost = 2A 3x 2- Also, A^ = C g(x 3) + C-^ g (x^) Hence, f becomes f(x) = W*(x 1..x n,S,F) + K 2 + K 3 + C g(x 3) + C 1 0 ( x 4 ) (4-2) x^S Consider S and x^ to be constant. The change of f with respect to x^ is given as 3f = 3W* . K + 3C (X_) (4-3) -^zr- 2 » 3 3 x 3 3 x 3 v 2 c 3 X x 3 b 3 Consider a region of the cost function,x 3 £ x 3 £ x 3, where Cg is constant. Then, M_ = ™1 - K 2 C 4 _ 4 ) 3x_ 3x x T T > 1 4 4 J 3 J x-zS -51-and the value of the objective function at a point x^ in the int e r v a l is given by, f = f + / 3 9f 3x • . (4-5) ° x| 3xJ 3 Two p o s s i b i l i t i e s arise for the given region. (1) If the slope 3f i s of constant sign in the region 3x 3 considered, then f should be evaluated at one of the end points, depending on the sign of 3f . 9 x 3 (2) If 3f changes sign in the i n t e r v a l , values of f between 3x 3 the end points should be examined. From computational experience, 3W* was found to be negative, 3x 3 and hence, f(x) should be evaluated at. the upper l i m i t of con-stant roof cost with respect to truss spacing. An a l t e r a t i o n in the program r e f l e c t i n g the above conclusion was implemented for a second set of cost parameters in Chapter 5. The effect of evaluating the objective function between the li m i t s of a region of constant C^ cost is p a r t i c u l a r l y pronounced for runs 15, 16, and 17. If the objective function had been evaluated at the upper l i m i t of the constant cost region for run 16, approximately 3 to 4 cents per square foot would have been saved, (based on slope of 3f ), thereby making the algorithm answer comparable 3x 3 to the best exhaustive search answer. Note in Table (4-2) that approximately 50 per cent of the runs were terminated by the i t e r a t i o n counter NIT which was employed to control the computation cost. Probably the best way to use the program would be to make an i n i t i a l run with wide constraints and then make a second run, tightening the constraint l i m i t s , and changing the random number base. -52-V a l i d i t y of the f u l l y stressed design A claim is made, that given x^...x^, a unique design results from the f u l l y stressed design procedure, and this design corres-ponds to the minimum weight design. A f u l l y stressed design is considered to be a design where, for a m u l t i p l i c i t y of loading conditions, the governing section of each member i s stressed to the f u l l allowable stress under at least one loading condition. A minimum weight design, on the other hand, is an arrangement of s t r u c t u r a l elements where a l l the design requirements such as stresses, deflections, and geometric constraints are s a t i s f i e d and the t o t a l s t r u c t u r a l weight is minimized. The minimum weight design problem is formalized by stating i t as a mathema-t i c a l programming problem. The r e l a t i o n of the minimum weight design to the f u l l y stressed design is discussed by Razani (17) , who considered the minimum weight design of trusses not subject to buckling, and Soosaar (13), who extended Razani's work to include a combined stress state. The c r i t e r i o n proposed uses the Kuhn Tucker optimality conditions and indicates whether or not the f u l l y stressed design i s a l o c a l optimum, but does not guarantee that the l o c a l optimum i s the global optimum in a non-convex space. The method developed by Razani i s not used herein. Two basic questions regarding the f u l l y stressed design must be answered. F i r s t : Does the method of f u l l y stressed design converge to a unique answer regardless of the i n i t i a l s t a r t i n g point? Second: Is the f u l l y stressed design the minimum weight design? The design of the members of the trusses considered i n this study was always governed by stress l i m i t a t i o n s . At no time was the deflection constraint active. Note that i n the case of active deflection constraints, the minimum weight design i s not a f u l l y stressed design. For a s t a t i c a l l y determinate structure, the f u l l y stressed design is always unique since the member forces are independent of the member properties. Razani demonstrated that the mini-mum weight problem for a determinate structure is a l i n e a r programming problem, and as discussed i n Chapter 3, any l o c a l optimum is also a global optimum, and further, the optimum must occur at one of the extreme points. Because a f u l l y stressed design occurs at an extreme point, the minimum weight design corresponds to the f u l l y stressed design. The uniqueness of the f u l l y stressed design for this study for a given truss configuration was determined by a compu-ta t i o n a l method. The i n i t i a l design was started at d i f f e r e n t extreme points, and i n a l l cases, converged to the same design. If the structure can be reduced to a determinate structure, then the argument r e l a t i n g the f u l l y stressed design to the minimum weight design may be employed. Assume the geometry and topology of the truss is fixed. Consider the truss web members. Neglecting the shear carried by the continuous chords, the web member forces are determined by the laws of s t a t i c s and hence the forces are independent of the cross-sectional properties of the web members. (Single diagonal system considered.) Applying the constraint of constant area to the chords i n effect reduces -54-them to determinate members. The d i s t r i b u t i o n of axial force due to gravity loads is independent of the chord cross-sectional properties. For constant panel spacing, the bending s t i f f n e s s I/L of the compression chord of each panel i s uniquely set, and hence the moment pattern is established. Therefore, the same section of the chord w i l l always govern the design of the. chord. Based on the argument that the force d i s t r i b u t i o n i s independent of the member properties, the f u l l y stressed design is the minimum weight design. In Chapter 6 i t is demonstrated that the preceding argu-ment does not hold i f the panel lengths are allowed to vary during the design cycle. The study in Chapter 6 attempts.-to stress the compression chord to the f u l l allowable l i m i t at more than one section by maintaining the constant area c r i t e r i o n but varying the panel lengths. It is shown that f u l l y stressing the compression chord at least at one section in every panel can lead to a heavier design than stressing the chord to the allowable l i m i t in only one panel. -55-CHAPTER 5 PARAMETER STUDIES Results of optimization Extensive results are presented in order to: (i) Investi-gate the effect of variations in the cost parameters; ( i i ) To determine the s e n s i t i v i t y of the objective function to changes in the structure variables; and, ( i i i ) To examine the uniqueness of the optimum. The effect of varying the magnitude of the gravity load is also b r i e f l y discussed. A second set of cost parameters was used, in addition to that in Chapter 4. Both sets are l i s t e d in Table (5-1), and are designated as "low costs" and "high costs". In the f i r s t set (used previously) the labor costs for preparing and welding a l l members are substantially lower than in the second. S i m i l a r l y , the wall cladding cost (Cg) is much lower in the f i r s t set. The differences in a l l other items are f a i r l y small. The t o t a l cost per square foot, C, i s given as: C = 1 (C -WWEB + C-WTC + C-'WBC + (C4+C-)-WN + Cg-NCORD SPAN.SPACE 1 + Cy-NSPLICE + 2Cg-RATIO-SPAN-SPACE + (Cq+C )•SPAN'SPACE) (See Chapter 2 for defini t i o n s ) A span of 120'-0" was used throughout. A l l members were de-signed on the basis of 36 k. s . i s t e e l . A l i v e load of 40 psf was used, except when the effect of the magnitude of l i v e load was examined. Only even panel spacing was considered. A very coarse grid of structure variables was evaluated to obtain the exhaus--56-t i v e search r e s u l t s . The exhaustive search g r i d s were made as f o l l o w s : ITEM LOW COSTS" "HIGH COSTS" C l $ 0 .08 $ 0 08 c 2=c 3 $ 0 .08 $ 0 09 C4 + C5 $13 . 00 $31 00 C6 $15 . 00 $15 00 C7 $ 5 .00 $40 00 C8 $ 0 . 20 $ 1 50 NCORD 4 ( NSPLICE 2 L \ c 9 0--5 f t $0 .06 0--17ft $0 .17 5--10 . 10 17--24 .25 10--15 . 18 24--31 .34 15--20 .24 31--38 . 36 20--25 . 32 38--45 .42 CnS 0--5 f t 50 lbs 0--17ft 566 lbs 5--10 1. 00 17--24 830 10--15 1. 60 24--31 1 141 15--20 2 . 10 31--38 1 503 20--25 4. 00 38--45 1 742 C10 0--6 f t $0 .25 0--7 f t $0 . 40 6--10 . 50 7--8 .50 10--17 . 75 8--11 .5 . 70 17-•24 1 .00 11 5- 12 5 . 80 24--29 1 . 50 12 5- 20 1 .00 C 1 0 S 0--6 f t 3. 00 lbs 0--7 f t 2 020 lbs 6--10 6. 00 7--8 2 400 10--17 9. 00 8--11 . 5 2 500 17--24 12 . 00 11 5- 12. 5 3 000 24--29 18. 00 12 5- 20 3 664 TABLE (5-1) COST PARAMETERS -57-VARIABLE LOWER BOUND UPPER BOUND INCREMENT TRUSS SPACING 15.00 25.00 2.5 D/S RATIO 0.08 0.12 .02 "LOW NUMBER OF PANELS 8 16 2 COSTS" PURLIN SPACING 4. 4. TRUSS SPACING 17.00 45.00 3 . 5 D/S RATIO 0.07 0.11 .01 "HIGH NUMBER OF PANELS 8 14 2 COSTS" PURLIN SPACING 6. 6. For the "low costs" , depth to span ratios of 0.07, 0. 09, and 0.11 were evaluated for the 20 ' -0" truss spacing for a l l topologies. For the "high costs", the roof deck was based on a wood system. Although the cost figures C P and C ^ ^ are not exact, the ratios between the steps of the roof model are of the correct magnitude. The numerical values of C to C 0 are not intended to 1 o represent actual, current, fa b r i c a t i o n charges. Results for the exhaustive search are presented i n figures (5-1) to (5-6). The "high costs" were based on a roof system consisting of steel j o i s t s for purlins ( l i g h t weight j o i s t s to 31'-0" ($0.30/#), long span j o i s t s for truss spacings greater than 31'-0" ($0.24/#)), and steel Q decking. Two ste e l fabricators were con-sulted regarding costs, and average values of the cost parameters provided were assumed. Results of the exhaustive search are presented in figures (5-7) to (5-12). It was found that the most economical trusses resulted when the p u r l i n spacing was within the lowest constant cost region for C - ^ Q . For panel point loading, the chords of the truss are design-ed for ax i a l load plus bending due to the secondary moments. When purlins are placed between panel points, the compression -58-chord must be designed for ax i a l load plus primary bending moments caused by member loading and secondary moments. Hence for a given topology and dead plus l i v e load, the panel point loaded truss results in less weight for the truss, and there-fore less cost. This cost saving, however, is offset by the increased dead load due to the longer span of the decking'neces -sary for panel point loading, and by the increased cost of the deck because of increased weight. For the cost models employed, the increased cost due to heavier deck type was always substan-t i a l l y greater than the material cost saving of panel point loading over member loading. "Secondary moments" as used above,' and in the balance of the thesis, refers to the bending moments in the top chord occasioned by the fact that the chord i s continuous and the truss deflects under load. A comparison of the minimum cost solution and the minimum weight solution is presented in figures (5-1) and (5-2) for the "low costs" and figures (5-7) and (5-8) for the "high costs". The d i s c o n t i n u i t i e s in the t o t a l cost vs truss spacing curves are the result of two factors. F i r s t l y , the form of the cost model Cg, and, secondly, the form of the dead weight model CgS.: The l a t t e r changes the material cost. Hence, a discontinuity should be shown in the minimum weight curves, but in order to show this e f f e c t , i t would be necessary to evaluate, for adjoining constant cost steps, the upper l i m i t of the lower cost region and the lower l i m i t of the higher cost region. This was not done, and only a design for the upper l i m i t of the lower cost region was performed. -59-The exhaustive search figures i l l u s t r a t e the conclusion of Chapter 4, namely, that the objective function should be evaluated only at the upper l i m i t of a constant Cg cost region. EXHAUSTIVE SEARCH FOR MINIMUM COST CONFIGURATION ("LOW COSTS") 1.20 1.10 1.00 1.20 1.10 1. 00 8 PANEL SEARCH FOR VARIOUS D/£ : - - / 2-u i •• •08 •IO io \ \ 4' • 0" PI IRLIN \ SPAC1 :NG 15 20 25 TRUSS SPACING IN FEET FIG. 5-l(a) 12 PANEL SEARCH FOR VARIOUS — - D/S RATIOS' T 5 7 T ) , r-TRUSS SPACING IN FEET FIG. 5-l(c) CO O U < C H H CL( H CO o u < H O H 1.20 1.10 1.00 1.20 1.10 1.00 10 PANEL SEARCH FOR VARIOUS D/S RATIOS 15 20 25 TRUSS SPACING IN FEET FIG. 5-l(b) 14 PANEL SEARCH FOR VARIOUS D/S = •« D/S RATIOS T 5 ZD TRUSS SPACING IN 25 FEET FIG. 5-l(d) O EXHAUSTIVE SEARCH FOR MINIMUM WEIGHT CONFIGURATION ("LOW COSTS") 0.45 0.40 0 .35 D/S = . 0 8 I? 8 PAN EL .10 \ / \ \ / 4' -•0" Pl JRLIN SPACI NG 15 20 25 TRUSS SPACING IN FEET FIG. 5-2(a) CM EH H in O u < i—i w < 0.45 0 . 40 0 . 35 i o p; D/S = . 12 . 0 8 . I O J N 4' -0" PI JRLIN > SPAC] [NG 15 20 TRUSS SPACING IN FEET FIG. 5-2 Cb) 25 0 . 45 0.40 0 .35 12 PA NEL D/S = . 1 2 . 0 8 . 1 0 4' -C " PUP, .LIN S PACIN G 0.45 15 20 25 TRUSS SPACING IN FEET FIG. 5-2(c) EH V-i H Ul c u H I < I—i Pi M E H < 0. 40 0 . 35 -D/S _ .12 14 VP iNFT in . IU . 0 8 4' -( )" PUF ILIN ^ [G 15 20 TRUSS SPACING IN FEET FIG. 5-2(d) 25 TRUSS SPACING 20'-0" PURLIN SPACING 4*-0' MINIMUM COST D/S RATIO FOR DIFFERENT TOPOLOGIES 1. 20 1.10 1.00 PANELS 50 .10 D/S RATIO FIG. 5-3 MINIMUM MATERIAL COST D/S RATIO FOR DIFFERENT TOPOLOGIES 40 £ PANELS 30 08 .10 D/S RATIO FIG. 5-5 H i—I m < H O H U l—I W CO W CO O C J < H O H H CO O u t-H •< v—l PH W H LIVE LOAD 40 PSF ("LOW COSTS") WEB WEIGHT/TOTAL WEIGHT VS D/S PATIO FOR DIFFERENT TOPOLOGIES 50 30 10 14 12 10 8 08 .10 D/S RATIO 12 50 FIG. 5-4 - MATERIAL COST/TOTAL COST VS D/S RATIO FOR DIFFERENT TOPOLOGIES 40 30 PANELS 1NJ PANELS . 10 D/S RATIO FIG. 5-6 EXHAUSTIVE SEARCH FOR MINIMUM COST CONFIGURATION ("HIGH COSTS") °/Sv (RESULTS PLOTTED FOR DIFFERENT D/S RATIOS) 3Tj 413 4 T TRUSS SPACING IN FEET FIG. 5-7(a) D/S 25 30 35 TRUSS SPACING IN FEET TO rr FIG. 5-7(b) EXHAUSTIVE SEARCH FOR MINIMUM COST CONFIGURATION ("HIGH COSTS") 1. 80 1. 70 1,60 •09 N •07 \ \ \ \ \ \ \ \ \ \ v > \ \ \ \ \ \ \ \„\ \\ V \ s \ \ \\ \ .\ s:—_ \\ \\ ^ , \_ -~X^ 12 P A.NEL TRUSS SPACING IN FEET FIG. 5-7(c) 1. 80 1. 70 1.60 •09 \ > v \ •07 v \ \ \ \ \ \ X \ \ V \ \ \ \ VH N. \ i \ \ \ \ \ N \ \ . \ S \ \ \ \ s N N 14 P A.NEL 15 20 2 5 30 35 40 45 TRUSS' SPACING IN FEET FIG. 5-7(d) ON EXHAUSTIVE SEARCH FOR MINIMUM WEIGHT CONFIGURATION ("HIGH COSTS") (RESULTS PLOTTED FOR DIFFERENT D/S RATIOS) EXHAUSTIVE SEARCH FOR MINIMUM WEIGHT CONFIGURATION ("HIGH COSTS") (RESULTS PLOTTED FOR DIFFERENT D/S RATIOS) 12 PANEL 15 20 25 30 TRUSS SPACING IN FEET 35 40 ;FIG. 5-8(c) 45 D/S .07-.11 -"""" . — ^ 14 P \NEL 15 20 25 30 35 TRUSS SPACING IN FEET FIG. 40 5-8(d) 45 M A T E R I A L C O S T / F T T O T A L C O S T / F T O On < tn t-1 o > a O n3 CO a CO o oo o cn cn • o CO an H i CO r-> // / f J / / / / fy i // / '/ / 1 i/ 1 i/ 1/ » / /' / /' / /' / // i / f f 1 ff 1 /' II' w ON S PAN ELS RAT MAT n tn O 73 r—1 Tl > O f 73 a cn —,» a o CO O r-H CO oo H n tn H 73 O i—i tn H O > H t-1 H cn h-' O o O •"d CO n O H r—1 tr1 o < • Cl CO 1—1 On tn a i CO h-> -CO O H-i r—1 on on C A ) on cs o i > tn CO 2s s-m H w o tn n 1—4 o len a i—i H n O n H m > 73 tn 2; H tn H cn O X) O tr* < 0 CO cn tn CO CO -19--68-Observations from the Exhaustive Search A "Low Costs" (1) For constant p u r l i n spacing, the t o t a l cost C i s most sen-s i t i v e to a va r i a t i o n in truss spacing. (2) The maximum va r i a t i o n in results was 23.2 per cent for a l l the depth to span r a t i o s , topologies and truss spacings con-sidered. (3) The best truss spacing was always 20'-0" for minimum t o t a l cost. (4) Although the configuration for minimum weight was not sought, results from the exhaustive search indicate that the truss spacing for minimum weight occurs outside of the region considered, and is substantially higher than the spacing for the minimum cost solution, (5) The t o t a l cost is not affected greatly by a va r i a t i o n i n the D/S r a t i o . The t o t a l v a r i a t i o n i n cost, in the range .07 < D/S < ,12, is about 7 per cent of the minimum value in the case of the 8 panel truss, and about 3 per cent in the case of the 10 panel truss. (6) The maximum difference in t o t a l cost caused by a va r i a t i o n in topology occurred for a depth to span ra t i o of 0.12 and was 8.48 per cent. (7) For minimum weight, a l l designs are very close together except for the 8 panel truss. (8) No s i g n i f i c a n t difference is observed between the minimum t o t a l cost depth to span ratio and the minimum weight depth to span r a t i o , for a given truss spacing. This s i m i l a r i t y arises from the fact that a very low wall cladding cost was used ($0.20). -69-B "High Costs" (1) For the exhaustive search (6'-0" p u r l i n spacing), the most expensive truss had a t o t a l cost of $1.9620 per square foot and the minimum was $1.5755 per square foot. This represents a t o t a l v a r i a t i o n of 24.5 per cent. (2) Of the 180 complete designs made for the exhaustive search, 145 were within 10 per cent of the minimum cost of $1.5755 per square foot. This indicates that for a constant p u r l i n spacing, there are many alternate feasible designs with substantial variations in configuration. (3) Two d i s t i n c t regions were found to exist for minimum cost based on truss spacing. These were the region surrounding the 24 ,foot spacing and that surrounding the 38 foot spacing. The 38 foot spacing, however, was the better of the two. (4) For the minimum search, except in the case of the 8 panel topology, the minimum weight configuration occurred for a.truss spacing between 38 and 45 feet, with the slope of the curve in this region being almost zero. (5) For a truss spacing of 38'-0" and p u r l i n spacing of 6'-0", the following conclusions can be made, based on figures (5-9) to (5-12). (a) A r e a l i s t i c wall cladding cost sub s t a n t i a l l y separates the minimum t o t a l cost depth to span ra t i o from the minimum weight depth to span r a t i o . The minimum cost D/S r a t i o was 0.07 except in the case of the 14 panel topology, while the minimum weight D/S ra t i o occurred at a value outside the range of the exhaustive search. (b) The maximum spread i n cost for a l l topologies and depth -70-to span r a t i o was 7.05 per cent. (c) From figure (5-12) i t i s seen that minimizing the ra t i o of material cost to t o t a l cost does not lead to minimum cost. The Importance of a Wall Cladding Cost and a Topology Cost (1) Wall Cladding Cost Figures (5-13(a)) and (b) depict the impact of a wall clad-ding cost on diff e r e n t topologies for the "low costs", and figures (5-15(a)) to (c) represent the same for the "high costs". By increasing the depth to span r a t i o of a truss, the ax i a l force i n the chords decreases and hence the chord area required decreases, whereas the web weight increases. Therefore there i s , at some stage, a depth to span ra t i o y i e l d i n g minimum weight. This minimum weight depth to span ra t i o is very dependent on the magnitude of the load carried. Associated with the height of a truss is the area to be covered with wall cladding. As this area increases, so does the cost of covering i t . By including a wall cladding cost, a balance i s achieved between high walls, minimizing truss weight, and low walls, minimizing wall cost, hence y i e l d i n g an optimal cost solution. When no cost i s associated with the truss height (minimum weight case for D/S r a t i o ) , the maximum effect of the D/S r a t i o on the t o t a l cost is 8.5 per cent for the "low costs", and 6.2 per cent for the "high costs", within the range of D/S r a t i o considered. For r e a l i s t i c wall costs ($1.20 to $2.00), the effect of the cladding cost is to lower the optimum D/S r a t i o . The. maximum variation in t o t a l cost was 14.4 per cent and 7 per cent for the -71-"low costs" and "high costs", respectively. For small variations (0.01) from the optimum D/S r a t i o , the cost v a r i a t i o n from the optimal value i s approximately 2 to 5 per cent. (2) Topology Cost The topology cost (represented by C^+C^) influences the str u c t u r a l configuration i n the following two ways: (i) It represents a fixed cost item and hence, d i r e c t l y affects the best truss spacing. ( i i ) For a given truss spacing and depth to span r a t i o , varying the topology affects the truss weight. The greater the number of panels, the shorter the panel length and hence the less the reduction in allowable stress for the compression chord. Also, the primary moments, due to transverse loading on the com-pression chord decrease with an increasing number of panels. Therefore, the compression chord weight tends to decrease with increasing number of panels. Only the impact of part ( i i ) is considered herein. Figures (5-14(a)) and (b) and figures (5-16(a)) to (c) i l l u s t r a t e the effe c t of a topology cost on the design with low and high costs, respectively. The topology cost i s considered to be the cost of adding a web member to the truss. For zero topology cost, the maximum va r i a t i o n in cost was 3.5 per cent for the "low costs" and 3.8 per cent for the "high costs", demonstrating that the topology vs cost curve is f l a t for zero topology cost. For a high topology cost ($48.00), the curves are not f l a t , and the maximum var i a t i o n for the "low" and "high costs" are 12.9 and 6.7 per cent, respectively. For topologies adjacent to the optimal topology, the cost variation i s less than 5 per cent. STUDY OF THE EFFECT OF A WALL CLADDING COST ON THE BEST D/S RATIO ("LOW COSTS") .08 .10 .12 D/S RATIO FIG. 5-13(a) STUDY OF THE EFFECT OF A TOPOLOGY COST , 801 I I | |  8 10 12 14 16 NUMBER OF PANELS FIG. 5-14(a) D/S RATIO FIG. 5-13(b) ON THE BEST CONFIGURATION ("LOW COSTS") . 80 8 10 12 14 16 NUMBER OF PANELS FIG. 5-14(b) STUDY OF THE EFFECT OF A WALL CLADDING COST ON THE .08 .10 ,12 "' " . 0 8 .10 .12 D/S RATIO FIG. 5-15(c) D/S RATIO FIG. 5-15(d) TRUSS SPACING 38*-0" PURLIN SPACING 6'-0" LIVE LOAD 40 PSF STUDY OF THE EFFECT OF A TOPOLOGY COST ON THE  BEST CONFIGURATION ("HIGH COSTS") 1. 80 D/S=0.07 10 12 14 NUMBER OF PANELS FIG. 5-16(a) S=0.09 8 10 12 14 NUMBER OF PANELS FIG. 5-16(c) TRUSS SPACING 38'-0" CN] H H CO O U < H O H H E-i tn o C J I-J >< E-i O H 1. 60 1.80 3=0.08 10 12 14 NUMBER OF PANELS FIG. 5-l6(b) 5=0.10 PURLIN SPACING 6' - 0" 8 10 12 14 NUMBER OF PANELS FIG. 5-16(d) LIVE LOAD 40 PSF -75-Influence of Gravity Load An examination of the influence of gravity load on the structure variables was made with loads of 20, 40, 60, and 80 psf. The "low" cost model was employed, along with a truss spacing of 20'-0" and a p u r l i n spacing of 4'-0". Varying the magnitude of the gravity load results in the following con-clusions : (1) For l i g h t loadings, the design is i n e f f i c i e n t as minimum member areas govern. For compression members, the small member sizes r e s u l t i n low allowable stresses, and therefore, a low depth to span ra t i o governs. (2) The best depth to span r a t i o and number of panels increases with increasing load. (3) In general, the minimum weight geometry and topology do not correspond with the minimum cost geometry and topology. (4) The r a t i o of material cost to t o t a l cost increases substan-t i a l l y with increasing gravity load. S e n s i t i v i t y and Uniqueness of the Best Solution The s e n s i t i v i t y of the best solution to variations i n the structure variables, is d i r e c t l y dependent on the ratios of the cost parameters. For high fixed costs and system costs (chord costs, topology cost, wall and roof costs), the hierarchy of structure variables is p u r l i n spacing, truss spacing, depth to span ratio and number of panels. For low topology cost, the best cost answer is not sensitive to small variations in the number of nanels. With the truss spacing and p u r l i n spacing held con-stant, many possible solutions exist which come within 5 per cent -76-of one another for small changes in depth to span r a t i o and topology. Results obtained from Box's Complex Method indicate that the region surrounding the optimum is very f l a t , and many possible alternate good designs exist for the solution of the problem. Results obtained from the optimization runs for the Vhigh costs" are presented in Tables (5-3) to (5-8). The i n i t i a l and f i n a l complexes are shown to demonstrate the many alternate, good feasible designs possible. Note that for the 6 runs made, the maximum deviation from the best answer of 1.5749 is only: 1.155 per cent. As stated in Chapter 4, only the values at the upper l i m i t s of the constant Cg region were considered, instead of treating the truss spacing as a continuous variable. Table (5-2) defines the input variables. RUN NPOINT NIT ALPHA RNB D/S RATIO TRUSS SP. PURLIN SP. DLBR DUBR TLBS TUBS PLBS PUBS 1 8 30 1.3 .05 .06 . 11 17. 45. 6. P.L. 2 8 30 1.3 . 45 . 06 .12 17. 45. 6. P.L. 3 8 30 1.3 .90 . 06 .11 17. 45. 6. P.L. 4 10 30 1.3 .05 .06 . 11 17. 45. 6. P.L. 5- 10 30 1.3 . 45 .06 .12 17. 45. 6. P.L. 6 10 30 1.3 .90 . 06 .11 17. 45. 6. P.L. Note: C 9S(1) = 16.99 feet. TABLE 5-2 INPUT VARIABLES FOR "HIGH COSTS" OPTIMIZATION RUNS I N I T I A L C 0 M P L E X x l x2 x 3 x 4 WEIGHT f(x) 10 . 100 24. 6. 11885. 1.6211 14 .060 24. 5.455 15333. 1.6989 14 .096 38. 5.455 18279. 1.6470 16 .090 31. 6.667 15544. 1.7151 10 .073 31. 12. 17256. 2.0094 14 . 061 45. 6.667 28625. 1.6855 8 . 098 45. 5.455 22403. 1.6211 16 . 068 24. 6.667 14168. 1.7266 F I N A L C 0 M P L E X x l x2 x 3 x4 WEIGHT f(x) 8 .0721 45. 5. 25451. 1. 5957 10 . 0753 38. 5. 21201. 1.5929 8 . 0750 45. 5.4545 24938. 1.5957 8 .0747 45. 5.4545 24927. 1.5945 8 .0862 38. 5. 20797. 1.5897 8 .0716 45. 5. 25376. 1. 5928' 8* .0744* 38. * 5.4545* 22168. * 1.5821* 8 . 0750 45. 5.4545 24938. 1.5957 denotes optimum TABLE 5-3 OPTIMIZATION RUN 1 I N I T I A L C 0 M P L E X X l x2 x 3 x4 WEIGHT f(x) 10 . 100 24. 6. 11885. 1.6211 14 .079 31. 7.5 16948. 1.7841 6 .066 31. 20. 23862. 2.2796 8 .103 24. 10. 14900. 1.9794 16 . 093 38. 6 .667 18977. 1.6796 16 .077 45. 6. 24292. 1.6813 12 . 063 24. 5 . 15017. 1.6553 6 .070 31. 15. 26259. 2.3487 F I N A I C 0 M P L E X x l x2 x 3 x4 WEIGHT f( x ) 8 .0896 45. 6.667 23862. 1.6206 8 .0909 38. 6.667 21190. 1.6112 12 .0866 38. 6.667 19180. 1.6126 12 .0843 38. 6.667 19930. 1.6209 8 .0912 38. 6.667 21141. 1.6112 10 .0905 38. 6.667 19812. 1.6097 10* . 0917* 38. * 6.* 18659.* 1. 5906* 8 .0919 38. 6.667 20444. 1.5996 TABLE 5-4 OPTIMIZATION RUN 2 I N I T I A L C 0 M P L E X x l x2 x 3 x4 WEIGHT f (x) 10 . 100 24. 6. 11885. 1.6211 14 . 104 24. 5.455 12547. 1.7359 16 .061 31. 6.667 19078. 1.7165 16 .094 31. 5.455 15618. 1.7284 8 . 093 31. 8. 571 18030. 1.9527 8 .108 45. 8.571 23005 . 1.9602 12 .068 24. 7.5 14540. 1.7539 8 .076 38. 12. 23131. 2.0070 F I N A L C 0 M P L E X x l x2 x 3 x4 WEIGHT f (x) 8 .0960 38. 6.667 20249. 1.6077 8* .0815* 38. * 6.667* 21665.* 1. 5931* 8 .0950 38. 6.667 20215. 1.6039 8 .0930 38. 6.667 20923. 1.6123 8 .0945 38. 6.667 20552 . 1. 6093 8 .0949 38. 6.667 20211. 1.6035 10 .0934 38. 6.667 19571. 1.6136 8 .0950 38. 6.667 20215. 1.6039 TABLE 5-5 OPTIMIZATION RUN 3 I N I T I A L C 0 I \ P L E X x l x2 / x 3 x 4 WEIGHT £(x) 10 .100 .24. 6. 11885. 1.6211 14 .096 38. 7.5 18813. 1.7571 14 .090 31. 7.5 16020. 1. 7943 16 .073 31. 6. 17686. 1.7185 10 .061 45. 7.5 27818. 1.7262 14 .098 45. 7.5 22526. 1.7894 8 .068 24. 10. 16553. 1.9300 16 .064 24. 7.5 14394. 1.8242 6 .098 45. 12 . 27359. 2.0797 8 .085 24. 12 . 15669 . 2.0544 F I N A L C 0 M P I , E X x l x2 x 3 x 4 WEIGHT f( x ) 10 .1000 24. 6. 11885. 1.6211 10 .0600 38. 5.4545 24223. 1.6073 10 .0794 45 . 6. 23124. 1.6012 10 .0600 38. 6. 24223. 1.6073 10 .0711 45. 6.667 25383. 1.6148 10 .0600 38. 5.4545 24223. 1.6073 8* .0773* 45.* 6.667* 24335.* 1. 5923* 8 .0736 31. 5.4545 18951. 1.6190 8 .0831 45. 6. 24068. 1.6051 8 .0802 45. 6.667 24377. 1.6016 TABLE 5-6 OPTIMIZATION RUN 4 I N I T I A L C O M P L E X x l x2 x 3 x4 WEIGHT 10 .100 24. 6. 11885. 1.6211 14 .066 31. 7.5 18341. 1.7809 6 . 103 24. 20. 14613. 2.2271 8 .093 38. 10. 21254. 1.9189 16 .077 45. 6.667 24292. 1.6813 16 . 063 24. 6.667 14892. 1.7364 12 .070 31. 5.455 17737. 1.6444 6 . 104 45. 20. 22940. 2 .2233 8 .093 38. 12. 22349. 2.0411 10 .078 • 24. 10. 15177. 1.9620 F I N A L C 0 M P L E X x l x2 x 3 x 4 WEIGHT £(x) 8 . 0970 38. 5.4545 20362. 1.6129 10 . 0801 45. 5. 23260. 1.6056 8 .0878 38. 6.667 21581. 1.6097 8 .0812 38. 5. 20978. 1. 5786 8 .0810 38. 5.4545 21693. 1.5922 10 .0740 38. 6. 20811. 1.5813 8 . 0828 45. 6.667 24070. 1.6042 8 . 0747 38. 6. 22175. 1.5829 8 .0777 38. 5.4545 22008. 1.5886 8 .0884 38. 6.667 21418. 1.6083 I N I T I A L C O M P L E X 10 14 16 16 8 8 12 8 16 6 10 10 8 8 8 10 . 100 .061 .094 .093 .108 .068 .076 .067 .067 .080 24. 31. 31. 31. 45 . 24. 38. 45. 31. 31. 6. 7.5 5.455 7.5 8.571 12 . 10. 15. 6.667 20. WEIGHT 11885, 19738, 15618, 15827 23005, 18208 19837 28761, 17763 20949 F I N A L C O M P L E X 0968 0871 0699 0687 0686 0833 0716 0885 0929 0 79 7 x. 38. 38. 38. 38. 38. 45. 45. 38. 38. 45. 6. 5.4545 6. 5. 6.667 6.667 6.667 6. 5. 6.66 7 WEIGHT 18105, 20040, 22493, 22979, 23455, 24031, 26664, 19229, 20497, 24475, f ( x ) 1.6211 1. 7993 1.7284 1.8295 1.9602 2.0826 1.8960 2.2340 1.7040 2.2498 f ( x ) 5944 6044 5749 5810 5900 6050 1.6142 1.5925 1.6033 1.6019 TABLE 5-7 OPTIMIZATION RUN 5 TABLE 5-8 OPTIMIZATION RUN 6 -80-D i s c u s s i o n o f C o s t P a r a m e t e r s and C o s t Model Form A model w h i c h p u r p o r t s t o r e p r e s e n t t h e c o s t o f a s t r u c t u r a l s y s t e m must i n c l u d e c o s t o f m a t e r i a l s , d e s i g n , d e t a i l i n g , f a b r i c a -t i o n , t r a n s p o r t a t i o n , and e r e c t i o n . Some o f t h e s e c o s t s c a n be r e p r e s e n t e d by a lump sum w h i c h i s a l m o s t i n d e p e n d e n t o f t h e s i z e and shape o f t h e t r u s s u s e d . T h i s f i g u r e c o u l d i n c l u d e d e s i g n and d e t a i l i n g c o s t s , as w e l l as some " f i x e d " c o s t s a s s o c i a t e d w i t h f a b r i c a t i o n and e r e c t i o n . S i n c e , however, s u c h c o s t s do n o t a f f e c t t h e o p t i m i z i n g p r o c e d u r e , t h e y have n o t been i n c l u d e d . The c o s t p a r a m e t e r s i n c l u d e d i n t h e model employed a r e examined below. C^, , C_ These p a r a m e t e r s r e p r e s e n t t h e m a t e r i a l c o s t . The m a t e r i a l c o s t s a r i s e f r o m two d i s t i n c t a r e a s , w h i c h a r e , t h e c o s t o f t h e m a t e r i a l d e l i v e r e d f r o m t h e m i l l t o t h e f a b r i c a t o r , and u n i t c o s t s w h i c h a r i s e f r o m f a b r i c a t i o n and e r e c t i o n . Examples o f t h e l a t t e r c o s t s a r e p a i n t i n g , and t r a n s p o r t a t i o n f r o m t h e f a b r i c a t o r t o t h e s i t e . C^, Cr-, Cg, C_ T h e s e p a r a m e t e r s a r e i n t e n d e d t o r e p r e s e n t l a b o r c o s t o f f a b r i c a t i o n . These i n c l u d e t h e l a b o r c o s t s i n v o l v e d i n c u t t i n g , p u n c h i n g , f i t t i n g , and w e l d i n g , and a r e r a t h e r i n s e n s i -t i v e t o w e i g h t , w i t h i n l i m i t s . O t h e r c o s t s r e l a t e d t o i n d i v i d u a l members as opposed t o w e i g h t r e l a t e d c o s t s , may be i n c l u d e d . In t h e model u s e d , no v a r i a t i o n i s c o n s i d e r e d i n t h e s e p a r a -m e t e r s . I n f a c t , however, t h e s e v a r i a b l e s a r e d e p e n d e n t on t h e magnitude o f t h e f o r c e s c a r r i e d ( f u n c t i o n o f g e o m e t r y , t o p o l o g y , and code r e q u i r e m e n t s ) , and on t h e member s i z e . G e n e r a l l y , t h e v a r i a t i o n i n C^...C- w i t h r e s p e c t t o geomet r y , t o p o l o g y , and f o r c e l e v e l , i s r e l a t i v e l y s m a l l , and an a v e r a g e v a l u e may be -81-used without s i g n i f i c a n t error. Depending on the magnitude of the parameters and C,_, the cost vs topology curve may be f l a t , or have a d e f i n i t e slope, thereby establishing alternate topologies as the optimal solution, or a unique topology as the optimum. From information extracted from f a b r i c a t o r s , i t was noted that a large v a r i a t i o n existed for C^...C . The importance of choosing the correct values for these parameters i s shown i n figures (5-18(a)) and (b) for the low and high costs. The curves depict the effect of the topology cost and chord costs d i s t r i -buted over the area supported by one truss. Coupled with the effect of decreasing fixed cost per unit area with increased truss spacing, is the decrease of weight with increased spacing.-This effect i s not included in the curves. The cost model steps of Cg are shown above the curves. When the combined decrease in cost, due to fixed cost and weight related costs, is less than the increase i n roof cost due to a new constant cost region, the best truss spacing is defined. Cg No v a r i a t i o n i n the wall cost, with respect to truss spacing, is possible with the representation used. For reason-able truss spacings, a constant cost parameter is s u f f i c i e n t . The v a r i a t i o n of wall cladding cost with spacing is very depend-ent on the type of material used. It is conceivable that for large truss spacings a variation i n Cg, with respect to spacing, would have to be considered in order to permit a v a l i d cost representation. THE DISTRIBUTION OF FIXED COSTS WITH TRUSS SPACING RESULTS PLOTTED FOR DIFFERENT TOPOLOGIES TRUSS SPACING IN FEET TRUSS SPACING IN FEET FIG. 5-17(a) "LOW COSTS' FIG. 5-17 (b) "HIGH COSTS - 8 3 -Roo£ Cost C G and C C ^ Q , the cost o£ the decking, is represented exactly by the model used. C ^, the p u r l i n cost, is a function of truss spacing and p u r l i n spacing. Only that v a r i a t i o n related to truss spacing is modelled. The effect of p u r l i n spacing has been ignored, and an average value used. -84-CHAPTER 6 THE EFFECT OF VARIABLE PANEL SPACING Variable Panel Spacing The design of the chords i s always governed by the applica-tion of a uniform load to the entire span. The two chords constitute 40 to 80 per cent of the truss weight, while 55 to 75 per cent of the chord weight is comprised of the compression chord. In the discussion of previous chapters, a f u l l y stressed design of the chords was defined as a design where at least one cross section of the chord was stressed to the f u l l allowable l i m i t . Using this d e f i n i t i o n , and varying the panel spacing, there i s possible an i n f i n i t e number of f u l l y stressed designs. There i s , however, one unique design which f u l l y stresses to the allowable l i m i t at least one cross section of the compression chord i n each panel. Employing the c r i t e r i o n of constant chord area, the configuration which stresses each panel to the allow-able l i m i t is sought. The assumption i m p l i c i t i n this method is that the minimum compression chord weight i s obtained by stress-ing each panel to the l i m i t . The algorithm developed herein may be c l a s s i f i e d as a sub-optimization scheme applied to a sub-ordinate element, namely, the compression chord, with the merit c r i t e r i o n being weight. The v a l i d i t y of the substitution of a dif f e r e n t c r i t e r i o n for the selection of the best truss is questionable and must be rigorously examined. Note that the algorithm developed i s dependent on the fact that the allowable compression stress is a variable, governed by the panel lengths. The cases examined are: A pin connected truss with panel point loading: A continuous chord truss with panel point loading: A continuous chord truss with transverse loading on the compression chord. For panel point loading, the roof type is governed by the longest panel length. A detailed develop-ment i s presented for the case of an eight panel truss, and results are included for d i f f e r e n t topologies. -86-Cases ( i ) and ( i i ) P i n n e d and c o n t i n u o u s c h o r d s t u d i e s w i t h  p a n e l p o i n t l o a d i n g . The a l g o r i t h m was deve l o p e d f o r the p i n n e d c h o r d s t u d y and was a l s o employed when the chords were t r e a t e d as c o n t i n u o u s members, w i t h p a n e l p o i n t l o a d i n g . The r e s u l t s i n the l a t t e r case were a f f e c t e d , however, by the e x i s t e n c e o f the secondary moments. Y=wL FIGURE 6-1 PANEL SPACING CONFIGURATION The a x i a l f o r c e s S i n the compression c h o r d are d e f i n e d as: w L 2 ( k + k + k ) 2 2d w L 2 ( k + k j 2 2d 4 w L 2 k 2 (6-1) 2d 4 - 8 7 -T h e f o r m u l a f o r t h e a l l o w a b l e s t r e s s i n t h e i n t e r m e d i a t e r a n g e i s a s s u m e d t o a p p l y . F ^ , t h e a l l o w a b l e c o m p r e s s i v e s t r e s s i s g i v e n b y F A = 0.6F - mO^L - C ) ( 6 - 2 ) r w h e r e F = y i e l d s t r e s s i n k . s . i . m = 6.77 + 0.0 79F„ C = 535 C p - C o V P / ( F y - 1 3 ) C Q = 20 w h e n F y <_ 50 r = r a d i u s o f g y r a t i o n F ^ may b e w r i t t e n a s f o l l o w s : F A = A * - B * k i w h e r e A* = 0.6F.. + mC = c o n s t a n t ( 6 - 3 ) y o B* = mL r F o r a d e s i g n t o s a t i s f y t h e c o d e r e q u i r e m e n t s , f - / F . < 1, w h e r e f . i s t h e a c t u a l a x i a l s t r e s s i n t h e i t h member. T h e a i f u l l y s t r e s s e d d e s i g n r e q u i r e s t h a t t h e i n e q u a l i t y b e r e m o v e d a n d f - / F . . = 1. H e n c e , t h e a r e a A c a n b e w r i t t e n a s A = S - / F . - . a i A i * i A l I f a c o n t i n u o u s member s p e c t r u m w a s u s e d , a n d a f u n c t i o n a l r e l a -t i o n s h i p b e t w e e n t h e r a d i u s o f g y r a t i o n r a n d a r e a A a s s u m e d , i t w o u l d a l w a y s b e p o s s i b l e t o make ^-aj_/^^ ~ 1- When a d i s c r e t e m ember s p e c t r u m i s u s e d , f„ n-/F.- c a n o n l y a p p r o x i m a t e t h e r i g h t -a 1 y\ l h a n d s i d e . A p p l y i n g t h e c o n s t r a i n t o f c o n s t a n t a r e a f o r t h e c o m p r e s s i o n c h o r d a n d l e t t i n g N e q u a l t h e n u m b e r o f p a n e l s i n -88-h a l f the truss, the following set of non-linear simultaneous algebraic equations r e s u l t s . N _ k^ - 0.5 = 0 (constraint equation) i = l (6-4) S i/(A* - B*k.) - S i + 1 / ( A * - B*k. + 1) - 0 An i t e r a t i v e scheme must be employed to solve equations (6-4), and, moreover, a span slenderness r a t i o L/r must be assumed. Note that, for panel point loading, r i s the smaller of r x and r . Hence i t w i l l be necessary to solve the equa-tions for each design made. ( i i i ) Continuous chord with uniform transverse load In this case the compression chord is subject to large primary bending moments, and these must be formulated in an e x p l i c i t form in terms of the panel spacing c o e f f i c i e n t s . In order to obtain these moments the chord was ide a l i z e d as a continuous beam on r i g i d supports subjected to a uniform transverse load of intensity w, and a two pass moment d i s t r i -bution was performed. Advantage was taken of the fact that the area and the moment of i n e r t i a of the chord are constant for the entire span. No attempt was made to model the secondary moments a r i s i n g from deflection of the truss. The moment expressions are stated i n equations (6-5). -89-w#/ft . k i _ L _ k 2L "7 k-L k,L. FIGURE 6-2 CONTINUOUS BEAM IDEALIZATION M. M, M. M, = wL2_(1.5k2 + r ^ k ^ - l . S k 2 ) - 0. ( k 2 - k 2 ) ) = ^ L i ^ l + r 3 ( k 3 _ k 2 ) + 0.5r 2(k 2-1.5k 2) + 0. 5 r 3 (r_ (k 2-k 3) 12 r 2(k2 - 1.5k 2))) wL 2(k 2 + r f k ^ - k 2 ) + 0.5r ( l - r _ ) ( k 2 - k 2 ) ) wL 2(k 2 +0.5r 6(k 2-k 2) - 0.5r 4r 6(k' 2-k 2)) 12 (6-5) r - 3k 2/ C3k 2 +4k 1) r = 1 - r 2 1 r 3 - V^VV r = 1-r 4 3 r 5 = k 4 / ( k 3 + k 4 ) r 6 = ^ 5 The a x i a l force expressions were assumed to be the same as for the pin connected case. The code requirement for a combined stress case i s presented in the form of an interaction formula, £ a i / F A i + £ b i / F B 1 !• where f = actual a x i a l stress = S-/A ai 1 F„. = allowable axial stress = A* - B*k. Ai 1 f, . = actual bending stress = M/Z bi F B = allowable bending stress = constant -90-A simple functional relationship of the form Z = 3A was assumed between the section modulus Z and area A. For a f u l l y stressed design, ai Ai b i B Applying the stress d e f i n i t i o n s , the in t e r a c t i o n formula can be written as: S i/(A* - B*k i) + Mj/BF B = A (6-6) Applying the constraint of constant chord area, the following set of equations was formed. N E k. - 0.5 = 0 (constraint equation) i=l 1 (6-7) S i/(A*-B*k i) - S i + 1 / ( A * - B*k i + 1) + (M.-M j + 1)/BF B =0 Note that the governing member moments in equations (6-7) can arise from three locations i n the member. These are the largest of the two negative end moments, or the maximum posit i v e moment between ends. Corresponding to the sign of the moments, d i f f e r -ent 3 values are applicable. Since the compression chord i s continuously supported out of plane, r corresponds to r . The solutions for equations (6-4) and (6-7) are summarized in Table (6-1) and figures (6-3) for an 8 panel truss. They were obtained by a generalized Newton-Raphson scheme. The truss span was 120'-0", the truss spacing 20'-0", and the l i v e load 40 psf. The same cost model as in chapter 4 and 5 was used. For the case o:f transverse load, the deck was applied d i r e c t l y to the chord. Results are presented i n Tables (6-2) to (6-6) for the following cases. -91-TABLE CHORD FIXITY PANEL SPACING LOAD TYPE 6-6-6-6-6-6 PINNED PINNED CONTINUOUS CONTINUOUS CONTINUOUS EVEN VARIABLE EVEN VARIABLE VARIABLE PANEL POINT PANEL POINT CONTINUOUS CONTINUOUS CONTINUOUS L/r h k 3 300 . 2441 . 1170 .0750 .638 400 .2233 .1207 .0832 . 728 500 .2073 . 1229 .0896 .0801 600 .1945 .1243 .0949 . 0863 700 .1839 . 1252 . 0993 .0916 800 .1749 . 1258 .1031 .0962 900 . 1671 .1261 .1064 .1004 1000 . 1603 . 1263 .1094 . 1041 1100 . 1542 .1263 . 1120 . 1075 1200 . 1488 . 1263 .1144 .1106 1300 . 1440 .1261 . 1165 . 1134 1400 . 139 5 . 1260 .1185 .1160 SOLUTION TO EQUATIONS (6 -4) FOR 8 PANEL TRUSS (INDEPENDENT OF D/S) L/r k l k2 h k 4 400 .1611 .0564 .1554 .1271 500 .1592 . 0681 .1467 . 1260 600 . 1574 .0789 .1397 . 1240 700 .1551 .0879 .1345 . 1224 800 .1528 . 0957 . 1305 . 1210 900 .1504 .1025 .1274 .1198 1000 . 1479 . 1083 .1249 .1188 1100 . 1455 . 1132 .1231 . 1182 1200 .1430 . 1171 . 1219 .1180 1300 . 1404 .1203 . 1212 . 1181 SOLUTION TO EQUATIONS (6-7) FOR 8 PANEL TRUSS (D/S 0.10) TABLE 6-1 -93-Discussion of Results In examining and interpreting the results , the following facts must be borne in mind. (1) The chords carry load most ef f ic ient ly when the depth of the truss is large. This minimizes the chord area. (2) The allowable stress in the compression chord is highest when the area of the chord is large. (3) Webs carry the shear most eff ic ient ly when they are short and the angle between the horizontal and the web member is large. (4) For panel point loading, the shear carried by the webs increases with an increasing number of-panels. (5) The truss stiffness is proportional to the square of the depth. Hence, the deeper the truss, the smaller the second-ary moments, and the more accurate is the representation of the compression chord as a continuous beam on r ig id supports. (6) The maximum slenderness ratio for tension members can govern the size of the diagonals. (7) The allowable bending stress and axial stress are constants for the tension chord. The smaller the length,of the center panel, the larger the force in the tension chord.. (8) The axial force in the compression chord for the center panel is a constant, independent of the panel spacing. In the following discussion, a l l comparisons refer to the  even panel spacing case. (1) Pin connected case - the same load for even and variable spacing. -94-A l l members are subjected to a uni a x i a l state of stress. No deck dead load was applied, so that both the even and the variable panel spacing cases would be subject to the same loads. From Tables (6-2) and (6-3) i t is seen that: (i) The bottom chord weight remained the same in almost a l l cases. , ( i i ) No s i g n i f i c a n t gain or loss was made i n the web weight. ( i i i ) In almost a l l designs, a saving was made in the top chord weight, which was of s u f f i c i e n t magnitude to offset any increase in web weight or bottom chord weight. Although.not i d e n t i c a l l y equal to 1.0, the stress factors for the compression chord for each panel were almost equal to one another, and as close to 1.0 as allowed by the use of a discrete member spectrum. (iv) The weight-saving ranged from -2.14 per cent to 6.95 per cent with the average saving between approximately 3 and 4 per cent. . (v) The d i s t r i b u t i o n of panel c o e f f i c i e n t s is parabolic for constant L/r. (vi) The roof deck cost is governed by the longest panel length. Hence, the cost of the variable panel spacing is ad-versely affected by the form of the cost model. Several cases were run with the eff e c t of roof dead load being included. For these cases, the truss weight for the v a r i -able panel spacing was substantially higher than for the even panel case, r e f l e c t i n g the change i n deck weight due to the long end panel. - 9 5 -Continuous chord case - secondary moments includ e d . The above d i s t r i b u t i o n of panel spacing c o e f f i c i e n t s was not v a l i d . To s a t i s f y code requirements, the chord was designed on the ba s i s of an i n t e r a c t i o n formula. Because of the length of the end pan e l , a reduction i n allowable bending s t r e s s had to be made. For long panel lengths, the moment term dominated the i n t e r a c t i o n formula because of the low allowable bending s t r e s s . Hence no weight saving was a f f e c t e d , and f u r t h e r , the t o t a l cost was s u b s t a n t i a l l y higher due to the long end panel length governing the deck cost. Because the d i s t r i b u t i o n of the panel spacing c o e f f i c i e n t s i s p a r a b o l i c , f o r the p i n connected t r u s s , a t e s t was made using a p a r a b o l i c d i s t r i b u t i o n of the form y = a Q + a-^x2, w i t h , the r a t i o of the outside panel to center panel set at 1.5. A small weight saving was made i n se v e r a l i n s t a n c e s , and the roof cost was-maintained at the same l e v e l as f o r the even panel case. ; ( 2 ) Continuous chord w i t h transverse load. In order to minimize the number of approximations made, a uniform load was used even though the main concern was w i t h a p u r l i n loading. This meant that only two approximations re-mained, i . e . , that a two pass moment d i s t r i b u t i o n was s u f f i c i e n t and that the secondary moments could be neglected. The v a l i d i t y of the two pass moment d i s t r i b u t i o n f o r large variations-; i n panel spacing was not r i g o r o u s l y examined. I t was found that the secondary moments had a s i g n i f i c a n t e f f e c t on the attempt to achieve f u l l y s t r e s s e d sections i n a l l panels, by the use of the above equations. -96-L/r NPAN D/S TR WT WEB WT TC WT BC WT USED ACTUAL COST/FTz MAT C/FT2 8 . 08 10907 2622 5100 3185 720 1.5068 .364 8 . 10 9861 3380 4080 2401 770 1.4800 .329 8 . 12 9808 4287 3480 2042 :z • i—i 765 1.4862 .327 10 .08 10348 2543 4620 3185 u 726 1.5099 .345 1.0 .10 9503 3291 3660 2552 PH 800 1.4897 . .317 10 .12 10039 4776 3180 2082 CO 955 1.5155 .335 12 .08 10005 2740 4080 3185 iz: w > 770 1.2701 .334 12 .10 9612 3580 3480 2552 765 1.2650 .320 12 . 12 9810 4847 2880 2082 765 1.2796 .327 TABLE 6-2 PINNED CHORD - EVEN SPACING NPAN D/S TR WT WEB WT TC WT BC WT L/r USED ACTUAL COST/FT2 MAT C/FT2 8 8 8 10 10 10 12 12 12 .08 .10 .12 .08 .10 .12 .08 . 10 . 12 10344 9488 10032 9636 9257 9340 9866 9302 9 604 2539 3096 4469 2611 3225 4378 2841 3570 4921 4620 3840 3480 3840 3480 2880 3840 3180 2580 3185 2552 2082 3185 2552 2082 3185 2552 2103 726 710 765 710 765 765 710 955 783 726 710 765 710 765 765 710 955 783 1.7380 1.7175 1.7436 1.7361 1.7315 1.7423 1.7654 1.7547 1.7727 .345 .316 .334 .321 .309 .311 .329 .310 .520 TABLE 6-3 PINNED CHORD - VARIABLE SPACING NPAN D/S TR WT WEB WT TC WT BC WT L/r USED ACTUAL COST/FT2 MAT C/FT2 8 8 8 10 10 10 12 12 12 .08 .10 . 12 .08 . 10 .12 .08 .10 . 12 14957 13844 13580 13653 12386 12947 12730 11676 12570 2953 3819 4641 2918 3801 5144 3133 3991 5367 7920 6840 6354 6354 5400 5100 5400 4500 4500 4083 3185 2585 4381 3185 2703 4198 3185 2703 |Z M C J < PH CO & > w 358 389 395 395 379 466 379 454 454 1.6518 1.6227 1.6219 1.6300 1. 5958 1.6225 1.6209 1.5938 1.6316 .498 .461 . 453 . 455 .413 .432 • . 424 . 389 . 419 TABLE 6-4 CONTINUOUS CHORD - EVEN SPACING\ Note: TR WT = tr u s s weight WEB WT = web weight TC WT = top chord weight BC WT =• bottom chord weight MAT C = m a t e r i a l cost NPAN = number of panels -97-L/r COST/FT2 NPAN D/S TR WT WEB WT TC WT BC WT USED ACTUAL MAT C/FT2 8 .08 16519 3138 9000 4381 354 354 1.7039 .550 8 . 10 16099 3914 9000 3185 354 354 1.6979 . 536 8 . 12 15756 5132 7920 2703 358 358 1.6944 . 525 10 .08 15181 3178 7920 4083 358 358 1.6810 .505 10 .10 14524 3899 7440 3185 378 378 1.6671 .484 10 . 12 15534 5029 7920 2585 358 358 1.7087 .518 12 .08 14674 3151 7440 4083 378 378 1.6857 .489 12 .10 12713 4128 5400 3185 379 379 1.6283 . 423 12 . 12 13547 5444 5400 2703 379 379 1.6642 .451 -TABLE 6 -5 CONTINUOUS CHORD - VARIABLE SPACING L/r COST/FT2 NPAN D/S TR WT WEB WT TC WT BC WT USED ACTUAL MAT C/FTZ 8 .08 14961 2958 7920 4083 1500 358 1.6519 .498 8 .10 14424 3799 7440 3185 1500 378 1.6420 .480 8 .12 14036 4611 6840 2585 1500 389 1.6371 .468 10 .08 13628 2893 6354 4381 1500 395 1.6292 .454 10 . 10 12407 3822 5400 3185 1500 379 1.5965 ..414 10 . 12 13047 4944 5400 2703 1500 379 1.6258 . 436 12 .08 12694 ' 3097 5400 4198 1500 379 1.6197 :.42 3 12 . 10 12269 3984 5100 3185 1500 466 1.6135 . 409 12 .12 12497 5294 4500 2703 1500 454 1.6291 .416 TABLE 6-6 CONTINUOUS CHORD - VARIABLE SPACING NPAN D/S *2 U *5 £6 8 .08 .682 . 899 . 839 . 880 8 . 10 . 802 1.016 . 895 1.000 8 .12 . 772 .956 . 830 .933 10 .08 .580 .790 . 822 .907 .872 10 . 10 .576 . 772 . 777 . 855 . 880 10 .12 . 720 .916 .879 .961 1.001 12 .08 . 464 .668 . 722 . 834 .901 .932 12 . 10 .579 . 788 . 842 .957 . .964 .987 12 .12 .541 . 717 . 749 . 845 . 842 . 880 TABLE 6-7 STRESS FACTORS FOR CONTINUOUS CHORD § EVEN SPACING -98-NPAN D/S £ 1 £2 £3 f4 £5 £6 8 .08 .997 .911 .947 .861 8 .10 . 963 . 892 .903 .872 8 .12 .970 .906 .925 . 896 10 .08 .892 . 846 . 802 . 779 . . 785 10 . 10 .992 .931 . 886 . 841 .837 10 . 12 .988 .945 . 799 . 767 . 745 12 .08 .992 .933 .858 . 821 . 746 . 710 12 .10 .977 .941 .912 .889 . 783 . 732 12 .12 1.006 .970 . 889 . 862 . 720 . 718 TABLE 6-8 STRESS FACTORS FOR CONTINUOUS CHORD § VARIABLE SPACING NPAN D/S £ 1 £2 £3 £4 £5 £6 8 .08 . 752 .950 .825 . 842 8 . 10 . 821 .998 . 800 .872 8 . 12 . 847 1.003 . 769 . 851 10 .08 . 754 .928 .774 . 833 -.814 10 . 10 . 758 .918 . 758 . 815 . 796 10 . 12 . 733 . 863 .667 . 715 . 701 12 .08 .696 . 862 . 795 . 866 . 888 . 896 12 . 10 . 836 .971 .813 .872 . 781 . 785 12 . 12 .856 .978 . 792 . 849 . 747 . 763 TABLE 6-9 STRESS FACTORS FOR CONTINUOUS CHORD § VARIABLE SPACING NPAN D/S k 2L k 3 L k 4L k 5 L k 6L 8 .08 236. 55 76. 22 226. 47 180.76 8 . 10 233. 05 73. 67 230 . 04 183.24 8 .12 230. 07 75. 38 229. 54 185.01 10 .08 201. 91 58. 90 189. 22 118.42 151. 55 10 . 10 197. 52 53. 77 194. 15 118.01 156. 55 10 . 12 194. 17 51. 78 196. 17 118.52 159. 36 12 .08 179. 27 43. 52 165. 90 83. 81 133. 13 114. 37 12 . 10 172. 76 49. 28 160. 53 90 . 41 131. 05 115. 97 12 .12 168. 50 45. 53 162. 49 88.99 135 . 40 119. 09 TABLE 6-10 PANEL LENGTHS IN INCHES FOR TABLE 6-5 -99-NPAN D/S k i L k 2 L k.L 4 k 5 L k 6L 8 .08 194. 16 179. 04 174. 67 172. 12 8 .10 193. 90 178. 58 175. 03 172 . 49 8 . 12 193. 64 178. 14 175. 37 172 . 85 10 .08 172. 33 143. 78 141. 02 132. 18 130. 69 10 .10 171. 34 142. 12 141. 83 132. 95 131. 75 10 . 12 170. 39 140. 52 142. 51 133. 58 133. 00 12 .08 157. 79 121. 88 122. 24 109. 08 105. 43 103 . 57 12 . 10 156. 00 119. 14 123. 13 109. 78 106. 99 104. 96 12 . 12 154. 41 116. 73 123. 88 110. 36 108. 45 106. 18 TABLE 6-11 PANEL LENGTHS IN INCHES FOR TABLE 6-6 Tables (6-4) to (6-6), (6-7) to (6-9), and (6-10) to (6-11) i l l u s t r a t e cost and weight f i g u r e s , s t r e s s f a c t o r s , and panel lengths r e s p e c t i v e l y , f o r the runs made. The columns headed L/r i n d i c a t e the span slenderness r a t i o employed to o b t a i n the panel spacing c o e f f i c i e n t s , and the a c t u a l values f o r the chords designed on t h i s b a s i s . Although the loads c a r r i e d by the even and v a r i a b l e panel spaced trusses are the same, the v a r i a b l e panel spacing c o n f i g u r a t i o n r e s u l t e d i n s u b s t a n t i a l l y heavier compression chords. In Table (6-8), the s t r e s s f a c t o r s f o r the compression chord increase from the center outward. This d i s t r i b u t i o n was caused by the secondary moments which were not accounted f o r i n the moment expressions used f o r the c a l c u l a t i o n of panel spacing c o e f f i c i e n t s . The secondary moments have an e f f e c t opposite to that of the primary moments. Considering the e f f e c t of a d i s c r e t e member spectrum and the secondary moments, the v a r i a b l e spacing scheme r e s u l t e d i n a design more c l o s e l y r e l a t e d to a f u l l y s t r e s s e d design than the even panel spacing, but with heavier chords. -100-By using a high span slenderness r a t i o , a d i s t r i b u t i o n s i m i l a r to the d i s t r i b u t i o n due to the true span slenderness r a t i o , was achieved. A span slenderness ratio of 1500 was used to obtain the results presented in Table (6-6). This r a t i o was employed in order to investigate what occurs when a small varia-tion from even panel spacing is permitted, while using a tech-nique which should lead to a f u l l y stressed design. For this study, the compression chord size was not increased greatly, and in some instances remained the same as for the even panel con-f i g u r a t i o n . A reduction in the web weight was occasionally achieved, but no o v e r - a l l net saving was made. In order to determine the form of'the minimum weight solu-t i o n for the compression chord, the problem should be treated as a mathematical programming problem, which could be stated as: Minimize A =: Sm/-(A* - B*l<m) + Mm/3 F g subject to S i/(A* - B*k i) + Mj/BFg < A, i = 1 N, i f m N E k. - 0.5 = 0 i = l x where S^, i = 1....N and are defined in equations (6-1) and (6-5). At least one panel (mth panel) would be stressed to the f u l l allowable l i m i t for the minimum weight solution, but which panel this would be is not known beforehand. Therefore, the problem would have to be solved N times, with m varying from 1 to N. Before such a procedure should be attempted, a method of model-l i n g the secondary moments has to be derived. Although i t i s -101-perhaps possible to fi n d a d i s t r i b u t i o n of panel spacing c o e f f i c i e n t s y i e l d i n g lower structure weight than for even panel spacing, the t o t a l cost saving would l i k e l y be less than about 1 per cent or 2 per cent. Further, the substitution of a p u r l i n loading for a continuous loading, which would be an accurate modelling of the real roof system, invalidates the moment expressions developed, unless the p u r l i n spacing is small (2 to 3 f e e t ) , and the number of panels is small. A more lucrativ e search area would be to abandon the constant area c r i t e r i o n for long spans, and fin d the optimal number of chord size changes to be made, based on weight savings and spl i c e cost. -102-CHAPTER 7 CONCLUSIONS (1) The f e a s i b i l i t y of automating the analysis and design of a s t r u c t u r a l system to y i e l d low cost, has been demonstrated. (2) The use of an optimization scheme employing Box's Complex Method i s more economical than an exhaustive search procedure. (3) No claim is made that Box's Method yields a global mini-mum in the cost space (4) The cost model examined provides a reasonable representa-t i o n of the cost of the s t r u c t u r a l system considered, and allows the inclusion of any costs which are dependent either on the weight, or on the numbers of members in the truss. (5) High labor' costs, associated with numbers of members, lead to larger truss spacing. (6) Small deviations in the s t r u c t u r a l variables from the opti-mal configuration result i n an increase in cost of less than 5 per cent. (7) The minimum weight s t r u c t u r a l configuration i s not a mini-mum cost configuration for the cost model studied. (8) The hierarchy of s t r u c t u r a l variables for the system examined i s p u r l i n spacing, truss spacing, depth to span r a t i o , and number of panels (topology). (9) The optimal structural configuration i s d i r e c t l y dependent on the ratios between the cost parameters. POSSIBLE FUTURE EXTENSIONS (1) A preliminary design routine should be written so that only -103-approximate designs are made u n t i l the optimization algorithm converges to a low cost region. Such a change would substanti-a l l y reduce the design - analysis cycle cost. (2) A greater number of possible web configurations could be included in the program, as well as the design of trusses with sloping chords. (3) The use of tables of members, other than Tees and Double Angles, could be implemented. (4) A design scheme for deflection control could be developed. (5) The str u c t u r a l system could be extended to include the columns, thus allowing the design of a complete bent. -104-BIBLIOGRAPHY 1. G e l l a t l y , R.A. , Gallagher, R.H. , and Luberacki, W.A., "Development o£ a Procedure for Automated Synthesis of Minimum Weight Structures", Defense Documentation Center, Report AD-611-310, Oct. 1964. 2. Johnson, J.R., Melosh, R.J., and Luik, R., "Optimum Structural Design", presented at the 25th Meeting of The Structures and Materials Advisory Group for Aerospace Research and Development, Ottawa, Sept. 1967. 3. Gray, G.M., "A Method of Applying The Electronic Computer to Roof Truss Proportioning", F i r s t Conference on Elect r o n i c Computation, ASCE Structural D i v i s i o n , Kansas City, Nov. 1958, pp.401-415. 4. Schmit, L.A., "Structural Design by Systematic Synthesis", Second Conference on Electronic Computation, ASCE Structural D i v i s i o n , Pittsburgh, 1960, pp.105-132. 5. Schmit L.A., "A Review of Some Structural Synthesis Studies", presented at The International Symposium on The Use of Electronic D i g i t a l Computers in St r u c t u r a l , Newcastle, July, 1966. 6. Klotz, L.H., "On the Application of ICES and Universal Program Software Systems", ASCE C i v i l Eng., Feb. 1969, pp.72 - 77. l'. Fiacco, A.V. and McCormick, G.P., "Nonlinear Programming: Sequential Unconstrained Minimization Techniques"., Wiley (1968). 8. Hadely, G., "Nonlinear and Dynamic Programming", Addison-Wesley (1964). j 9. Cornell, C.A., Reinschmidt, K.F., and Brotchie, J.F., "A Method For The Optimum Design of Structures", . presented at The International Symposium on The Use of Electronic D i g i t a l Computers in Structural Engineering, Newcastle, July, 1966. 10. Gue, R.L.? and Thomas, M.E., "Mathematical Methods in Operations Research", Macmillan (1968). 11. Wilde, D.F. and Beightler, C.S., "Foundations of Optimization", Prentice-Hall (1967). 12 Soosaar, K. and Cornell, C.A., "Optimization of Topology and Geometry of Structural Frames", presented at the ASCE Joint Specialty Conference on Optimization and Nonlinear Problems, Chicago, A p r i l , 1968. -105-13. Soosaar, K. , "Optimization of Topology and Geometry of S t r u c t u r a l Frames", Sc.D. Thesis of M.I.T., May, 1967. 14. G e l l a t l y , R.A., "Development of Procedures f o r Large Scale Automated Minimum Weight Design", AFFDL-TR-66-180, Wright-Patterson AFB, Ohio, Dec. 1966. 15. Toakley, A.R., "Optimum Design Using A v a i l a b l e S e c t i o n s " , Jour. S t r u c t u r a l D i v i s i o n , V o l . 94, No. ST5, May, 1968, pp. 1219-1241. 16- Box, M.J., "A New Method of Constrained O p t i m i z a t i o n and a Comparison w i t h Other Methods", Computer J o u r n a l , V o l . 8, 1965, pp. 42-52. 17. Razani, R., "Behavior of F u l l y Stressed Design of Structures and I t s R e l a t i o n s h i p to Minimum-Weight Design", AIAA J o u r n a l , V o l . 3 / No. 12 , 1965 , pp. 2262-2268. -106-APPENDIX A GENERATE EVALUATE f(x'), i=l,...k M = k FIND WORST VALUE OF f(x') = = f(x J) COMPUTE CENTROID OF REMAINING POINTS 107-t SATISFY INTEGER CONSTRAINT EVALUATE WITH ADJACENt TOPOLOGIES PICK BEST RFS NO. 015589 UNIVERSITY OF B C COMPUTING CENTRE HTSIAN059) JOB START: 13:23:16 06-03-69 $SIGNON ADR T=10.0M P»50 -____JJ-S,T_SlGNON_WAS: 12:36:15 _ 06-03T69___ ._ .. USER "ADR." SIGNED ON AT 13:23:22 ON 06-03-69 • RUN B0X*ROBOT<-PRATT*SU8R»DEC2 1 = TEE EXECUTION BEGINS DESIGN FOR PARALLEL CHORD TRUSSES USING TEE SECTIONS FOR CHORD MEMBERS AND DOUBLE ANGLE SECTIONS FOR WEB MEMBERS. _XiELD_ST.RESS_.FOR_HEB .MEMBERS E0UALS...36.0 KSI. . _ . YIELD STRESS FOR COMPRESSION CHORD EQUALS 36.0 KSI. YIELD STRESS FOR TENSION CHORD EOUALS 36.0 KSI. DEFLECTION CONSTRAINT DELTA/SPAN EQUALS _UP_P.ER._6UUNDS_0N_T.RUSS..SPACING, INCREMENTS.ARE ,AS_F.0LLOWS_i_ 16.99 24.00 31.00 38.00 DEAD LOAD OF ROOF PER SQUARE FOOT FOR EACH INCREMENT IS AS FOLLOWS-. 0.57 0.B3 1.1* 1.50 UPPER BOUNDS ON PURLIN SPACING INCREMENTS ARE AS FOLLOWS-7..J10 __8.00_ _ 11.50. . 12.50 20.,.00_ DEAD LOAD OF ROOF PER SQUARE FOOT FOR EACH INCREMENT IS AS FOLLOHS-2.02 2.40 2.50 3.00 3.66 LIVE LOAOS AND ROOF DEAD LOAO APPLIED ON CHORDS BY PURLINS. TRUSS DEAD LOAD APPLIEO AT PANEL POINTS. _E.V.eN_P_AN6L_SP_ACI.NG_C0NSI0ERED. ._ ... CHORDS OF TRUSS CONSIDERED TO BE CONTINUOUS - 1 0 9 -THE NUMBER OF V A R I A B L E S C O N S I D E R E D FOUALS 4 I N I T I A L PARAMETERS P R I N T E D OUT P A N E L S RAT IO TRUSS S P . P U R L I N S P . PANEL S P A C I N G C O E F F I C I E N T S 1 0 . 0 0 0 0 . 100 2 4 . 0 0 0 6 . 0 0 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 1 4 . 0 0 0 0 . 0 6 6 3 1 . 0 0 0 7 . 5 0 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 6 . 0 0 0 0 . 103 2 4 . 0 0 0 2 0 . 0 0 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 8 . 0 0 0 0 . 0 9 3 3 8 . 0 0 0 1 0 . 0 0 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 . .. 1 6 . 0 0 0 0 . 0 7 7 4 5 . 0 0 0 6 . 6 6 7 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 1 6 . 0 0 0 0 . 0 6 3 2 4 . 0 0 0 6 . 6 6 7 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 1 2 . 0 0 0 0 . 0 7 0 3 1 . 0 0 0 5 . 4 5 5 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 6 . 0 0 0 0 . 1 0 4 4 5 . 0 0 0 2 0 . 0 0 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 a. ooo 0 . 0 9 3 3 9 . 0 0 0 1 2 . 0 0 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 1 0 . 0 0 0 0 . 0 7 S 2 4 . 0 0 0 1 0 . 0 0 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 TOTAL COST PER SQUARE FOOT EOUALS t 1 . 6 2 1 1 TOTAL COST PER SQUARE FOOT EOUALS t 1 . 7 8 0 9 TOTAL COST PER SQUARE FOOT EQUALS » 2 . 2 2 7 1 TOTAL COST PER SQUARE FOOT EQUALS * 1 . 9 1 8 9 TOTAL COST PEP. SQUARE FOOT EOUALS t 1 . 6 8 1 3 TOTAL COST PER SOUARE FOOT EOU A I S t 1 . 7 3 6 4 TOTAL COST PER SQUARE FOOT EQUALS » 1 . 6 4 4 4 TOTAL COST PER SQUARE FOOT EQUALS I 2 . 2 2 3 3 TOTAL COST PER SQUARE FOOT EQUALS t 2 . 0 4 1 1 TOTAL COST PER SQUARE FOOT EQUALS  1 . 9 6 2 0 WORST PO INT PARAMETERS 1 4 6 1 3 . 0 2 6 . 0 0 0 0 . 1 0 3 2 4 . 0 0 0 2 0 . 0 0 0 2 . 2 2 7 CENTRO ID V A R I A B L E V A L U E S 1 8 7 6 2 . 8 7 1 1 . 1 1 1 0 . 0 8 3 3 3 . 3 3 3 9 . 3 6 5 1 . 8 4 5 NEW POINT PARAMETERS BEFORE S A T I S F A C T I O N UF INTEGER CONSTRA INTS 0 . 0 1 7 . 7 5 6 0 . 0 5 7 4 5 . 4 6 7 - 4 . 4 6 0  C O N S T R A I N T E X C E E O E O . KKKK EQUALS 4 NEW P O I N T PARAMETERS BEFORE S A T I S F A C T I O N OF INTEGER C O N S T R A I N T S _ 0 . 0 _ 1 4 . 4 3 3 0 . 0 7 0 3 9 . 4 0 0 2 . 4 5 3 CONSTRA INT E X C E E D E O . KKKK EQUALS 1 TOTAL COST PER SOUARE FOOT EQUALS » 1 . 6 5 3 0 TOTAL COST PER SQUARE FOOT EQUALS t 1 . 6 7 5 4 TOTAL COST PER SQUARE FOOT EQUALS » 1 . 6 4 9 0 WORST P O I N T PARAMETERS . 2 2 9 4 0 . 4 3 . . 6 . 0 0 0 0 . 1 0 4 4 5 . 0 0 0 2 0 . 0 0 0 2 . 2 2 3 C E N T R O I D V A R I A B L E V A L U E S 1 9 1 3 7 . 3 4 1 1 . 7 7 8 0 . 0 7 9 3 3 . 3 3 3 7 . 6 9 9 1 . 7 8 2 NEW PO INT PARAMETERS BEFORE S A T I S F A C T I O N OF INTEGER CONSTRA INTS 0 . 0 1 9 . 2 8 9 0 . 0 4 7 1 8 . 1 6 7 - 8 . 2 9 3 CONSTRA INT E X C E E D E D . KKKK EQUALS 3 _NEW_POINT_ PARAMETERS BEFORE S A T I S F A C T I O N OF INTEGER C O N S T R A I N T S 0 . 0 1 5 . 5 3 3 0 . 0 6 3 2 5 . 7 5 0 - 0 . 2 9 7 CONSTRA INT E X C E E D E D . KKKK EQUALS 1 TOTAL COST PER SQUARE FOOT EQUALS » 1 . 7 1 6 2 TOTAL COST PER SQUARE FOOT EQUALS t 1 . 6 9 5 4 WORST P O I N T PARAMETERS _ _ 2 2 3 4 8 . 6 3 8 . 0 0 0 . . . 0 . 0 9 3 3 8 . 0 0 0 1 2 . 0 0 0 2 . 0 4 1 C E N T R O I D V A R I A B L E V A L U E S 1 8 7 9 8 . 3 2 1 2 . 4 4 4 0 . 0 7 6 3 2 . 5 5 6 6 . 9 2 1 1 . 7 4 3 NEW PO INT PARAMETERS BEFORE S A T I S F A C T I O N OF INTEGER CONSTRA INTS 0 . 0 1 8 . 2 2 2 0 . 0 5 2 2 5 . 4 7 8 0 . 3 1 8 CONSTRA INT E X C E E D E O . KKKK EQUALS 3 _NEW_PJI INT_.PARAMETERS B E F O R E ^ S A T I S F A C T I O N .OF INTEGER .CONSTRAINTS 0 . 0 1 5 . 3 3 3 0 . 0 6 4 2 9 . 0 1 7 3 . 6 1 9 C O N S T R A I N T E X C E E O E O . KKKK EQUALS 1 TOTAL COST PER SQUARE FOOT EQUALS » U I I 2 2 -HQ-T O T A L C O S T P E R S O U A R E F O O T E Q U A L S - I . W O R S T P O I N T P A R A M E T E R S 1 5 1 7 6 . 8 * . 1 0 . 0 0 0 0 . 0 7 8 2 4 . 0 0 0 1 0 . C E N T R O I D V A R I A B L E V A L U E S 1 9 1 7 9 . 3 8 1 2 . 8 8 9 0 . C 7 4 3 3 . 3 3 3 6 , N E W P O I N T P A R A M E T E R S B E F O R E S A T I S F A C T I O N 6 8 2 1 . 0 0 0 1 . 9 6 2 3 6 5 1 . 7 1 2 O F I N T E G E R C O N S T R A I N T S 0 . 0 1 6 . 6 4 4 0 . 0 6 8 4 5 . 4 6 7 C O N S T R A I N T E X C E E D E D . KKKK E Q U A L S 3 N E W P O I N T P A R A M E T E R S B E F O R E S A T I S F A C T I O N 0 . 0 1 4 . 7 6 7 0 . 0 7 1 3 9 . 4 0 0 4 . C O N S T R A I N T E X C E E D E D . KKKK E Q U A L S 1 O F I N T E G E R C O N S T R A I N T S 0 0 3 T O T A L C O S T P E R S Q U A R E F O O T E Q U A L S I 1 . 6 9 0 4 T O T A L C O S T P E R S Q U A R E F O O T E Q U A L S S 1 . 6 3 0 5 H U R S T P O I N T P A R A M E T E R S 2 1 2 5 3 . 8 3 8 . 0 0 0 0 . 0 9 3 3 8 . 0 0 0 1 0 . 0 0 0 1 . 9 1 9 C E N T R O I D V A R I A B L E . V A L U E S 1 9 5 9 3 . 2 5 1 3 . 3 3 3 0 . 0 7 2 3 4 . I l l 5 . 8 1 0 1 : 6 8 0 N E W P O I N T P A R A M E T E R S B E F O R E S A T I S F A C T I O N 0 . 0 2 0 . 2 6 7 0 . 0 4 3 2 9 . 0 5 6 0 . C O N S T R A I N T E X C E E O E O . KKKK E Q U A L S 3 N E W P O I N T P A R A M E T E R S B E F O R E S A T I S F A C T I O N 0 . 0 1 6 . 8 0 0 0 . 0 5 7 3 1 . 5 8 3 3 . C O N S T R A I N T E X C E E D E D . KKKK E Q U A L S 3 O F I N T E G E R C O N S T R A I N T S 3 6 2 O F I N T E G E R C O N S T R A I N T S 0 8 6 C E N T R O I D U S E D A S N E W P O I N T T O T A L C O S T P E R S Q U A R E F O O T E O U A L S « T O T A L C O S T P E R S Q U A R E F O O T E Q U A L S « T O T A L C O S T P E R S Q U A R E F O O T E O U A L S -W O R S T P G I N T P A R A M E T E R S 1 8 3 4 0 . 8 4 1 4 . 0 0 0 0 . 0 6 6 3 1 . 0 0 0 6 3 4 9 6 6 2 0 6 1 5 4 C E N T R O I D V A R I A B L E V A L U E S 1 9 9 4 8 . 9 8 1 3 . 1 1 1 0 . 0 7 2 3 4 . 8 8 9 5 . N E W P C I N T P A R A M E T E R S B E F O R E S A T I S F A C T I O N 0 . 0 1 1 . 9 5 6 0 . 0 8 0 3 9 . 9 4 4 3 . C O N S T R A I N T E X C E E D E O . KKKK E Q U A L S 1 T O T A L C O S T P E R S Q U A R E F O O T E O U A L S - 1 6 4 3 1 . 6 6 2 O F I N T E G E R C O N S T R A I N T S 2 2 9 T O T A L C O S T P E R S Q U A R E F O O T E O U A L S i 1 . 6 5 6 0 T O T A L C O S T P E R S O U A R E F O O T E Q U A L S $ 1 . 6 0 5 6 W O R S T P O I N T P A R A M E T E R S 1 4 8 9 1 . 7 2 1 6 . 0 0 0 0 . 0 6 3 2 4 . 0 0 0 6 . 6 6 7 1 . 7 3 6 C E N T R O I D V A R I A B L E V A L U E S 2 0 8 7 8 . 8 0 1 2 . 4 4 4 0 . 0 7 4 3 7 . 2 2 2 5 . 4 5 8 1 . 6 4 7 N E W P O I N T P A R A M E T E R S B E F O R E S A T I S F A C T I O N 0 . 0 7 . 8 2 2 O . O S B 5 4 . 4 1 1 3 , C O N S T R A I N T E X C E E D E D . KKKK E Q U A L S 2 N E W P O I N T P A R A M E T E R S B E F O R E S A T I S F A C T I O N 0 . 0 1 0 . 1 3 3 0 . 0 8 1 4 5 . 8 1 7 4 . C O N S T R A I N T E X C E E O E D . KKKK F Q U A L S 2 O F I N T E G E R C O N S T R A I N T S 8 8 7 O F I N T E G E R C O N S T R A I N T S 6 7 2 C E N T R O I D U S E D A S N E W P O I N T T O T A L C O S T P E R S Q U A R E F O O T E Q U A L S t T O T A L C O S T P E R S Q U A R E F n O T E Q U A L S S T O T A L C O S T P E R S Q U A R E F O O T E Q U A L S « W O R S T P O I N T P A R A M E T E R S 1 9 2 9 7 . 4 9 1 4 . 0 0 0 0 . 0 6 3 3 1 . 0 0 0 6 1 8 3 6 3 1 7 5 8 1 3 C E N T R O I D V A R I A B L E V A L U E S 2 1 0 4 6 . 9 3 1 2 . 0 0 0 0 . 0 7 5 3 8 . 0 0 0 5 N E W P O I N T P A R A M E T E R S B E F O R E S A T I S F A C T I O N 0 . 0 9 . 4 0 0 0 . 0 9 1 4 7 . 1 0 0 6 C O N S T R A I N T E X C E E D E D . KKKK E Q U A L S 1 T O T A L C O S T P E R S Q U A R E F O O T E Q U A L S > 1, 5 6 9 1 . 6 3 5 O F I N T E G E R C O N S T R A I N T S 3 0 9 TOTAL COST PER SQUARE FOOT EOUALS ( 1 . 6 3 3 6 TOTAL COST PER SQUARE FOOT EOUALS t 1 . 6 2 4 1 WORST P O I N T PARAMETERS 1 8 6 0 6 . 3 4 1 4 . 0 0 0 0 . 0 6 4 3 1 . 0 0 0 5 . 0 0 0 1 . 6 8 2 C E N T R O I D V A R I A B L E VALUES 2 1 6 1 9 . 3 0 1 1 . 3 3 3 0 . 0 7 8 3 9 . 5 5 6 5 . 7 5 4 1 . 6 2 8 .NEW P O I N T PARAMETERS BEFORE S A T I S F A C T I O N OF INTEGER CONSTRA INTS 0 . 0 7 . 8 6 7 0 . 0 9 7 5 0 . 6 7 8 6 . 7 3 5 C O N S T R A I N T E X C E E D E D . _KKKK_E.QUALS 1 ... TOTAL COST PER SQUARE FOOT EQUALS i 1 . 7 1 7 6 TOTAL COST PER SQUARE FOOT EQUALS t 1 . 7 3 1 7 TOTAL COST PER SQUARE FOOT EQUALS i 1 . 7 7 5 7 NEW P O I N T I S NO IMPROVEMENT OVER WORST P O I N T . HALVE A L P H A . NEW P O I N T PARAMETERS BEFORE S A T I S F A C T I O N OF INTEGER C O N S T R A I N T S 2 2 3 9 9 . 1 4 9 . 6 0 0 _ 0 . 0 8 8 4 5 . 1 17 . 6 . 2 4 4 C O N S T R A I N T E X C E E D E D . KKKK EQUALS 1 TOTAL COST PER SQUARE FOOT EQUALS t 1 . 6 2 7 8 TOTAL COST PER SQUARE FOOT EQUALS * 1 . 6 3 8 5 TOTAL COST PER SQUARE FOOT EQUALS $ 1 . 6 1 4 9 WORST PO INT PARAMETERS 2 4 2 9 1 . 7 0 1 6 . 0 0 0 0 . 0 7 7 , 4 5 . 0 0 . 0 6 . 6 6 7 1 . 6 8 1 C E N T R O I D V A R I A B L E V A L U E S 2 1 5 7 2 . 5 7 1 0 . 4 4 4 0 . 0 7 9 3 9 . 5 5 6 5 . 7 5 4 1 . 6 2 1 NEW POINT PARAMETERS BEFORE S A T I S F A C T I O N OF INTEGER CONSTRA INTS 0 . 0 3 . 2 2 2 0 . 0 8 3 3 2 . 4 7 8 4 . 5 6 8 C O N S T R A I N T E X C E E D E D . KKKK EQUALS 2 NEW P O I N T PARAMETERS BEFORE S A T I S F A C T I O N OF INTEGER CONSTRA INTS 0 . 0 6 . 8 3 3 0 . 0 8 1 3 6 . 0 1 7 5 . 1 6 1 TOTAL COST PER SQUARE FOOT EQUALS t 1 . 6 4 9 6 TOTAL COST PER SQUARE FOOT EQUALS t 1 . 5 9 2 2  WORST P O I N T PARAMETERS 2 6 3 1 0 . 6 4 1 2 . 0 0 0 0 . 0 7 0 4 5 . 0 0 0 5 . 0 0 0 1 . 6 4 9 C E N T R O I D V A R I A B L E VALUES . . _ . 2 1 0 5 9 . 4 8 1 0 . 0 0 0 0 . 0 B 1 3 8 . 7 7 8 5 . 8 0 5 1 . 6 1 4 NEW PO INT PARAMETERS BEFORE S A T I S F A C T I O N OF INTEGER C O N S T R A I N T S 0 . 0 7 . 4 0 0 0 . 0 9 5 3 0 . 6 B 9 6 . 8 5 1  TOTAL COST PER SQUARE FOOT EQUALS » 1 . 7 1 9 5 TOTAL COST PER SOUARE FOOT EQUALS J 1 . 7 5 0 0 _rQIAL_COSJ_P.ER_SQUARE FOOT EQUALS . . - 1 . 7 8 3 1 „ NEW P O I N T I S NO IMPROVEMENT OVER WORST P O I N T . HALVE A L P H A . NEW PO INT PARAMETERS BEFORE S A T I S F A C T I O N OF INTEGER CONSTRA INTS 1 6 4 1 3 . 1 5 8 . 7 0 0 0 . 0 8 8 3 4 . 7 3 3 6 . 3 2 8  TOTAL COST PER SQUARE FOOT EQUALS $ 1 . 6 0 9 7 TOTAL COST PER SQUARE FOOT EQUALS S 1 . 6 1 2 3 TOTAL COST PER SQUARE FOOT EQUALS S . 1 .6 .485 WORST P O I N T PARAMETERS 1 7 7 3 7 . 2 0 1 2 . 0 0 0 0 . 0 7 0 3 1 . 0 0 0 5 . 4 5 5 1 . 6 4 4 C E N T R O I D V A R I A B L E V A L U E S :  2 1 * 8 6 . 5 7 9 . 5 5 6 0 . 0 8 3 3 9 . 5 5 6 5 . 9 3 9 1 . 6 1 1 NEW P O I N T PARAMETERS BEFORE S A T I S F A C T I O N OF INTEGER CONSTRA INTS 0 . 0 6 . 3 7 8 0 . 1 0 Q 5 . 0 . 678 . . . . 6 . 5 7 0 C O N S T R A I N T E X C E E D E D . KKKK EQUALS 1 TOTAL COST PER SQUARE FOOT EQUALS t 1 . 6 8 4 1 TOTAL COST PER SQUARE FOOT EQUALS t 1 . 6 3 3 3  WORST PO INT PARAMETERS 2 2 8 5 1 . 5 9 8 . 0 0 0 0 . 1 0 0 4 5 . 0 0 0 6 . 6 6 7 1 . 6 3 3 C E N T R O I D V A R I A B L E V A L U E S . . 2 1 4 8 6 . 5 7 9 . 5 5 6 0 . 0 8 3 3 9 . 5 5 6 5 . 9 3 9 1 . 6 1 1 NEW POINT PARAMETERS BEFORE S A T I S F A C T I O N OF INTEGER C O N S T R A I N T S 0 - 0 1 1 . 5 7 8 0 . 0 6 1 3 2 . 4 7 8 4 . 9 9 4 C O N S T R A I N T E X C E E O E O . K K K K E O U A L S 1 T O T A L C O S T P E R S Q U A R E F O O T E Q U A L S * 1 . 6 2 1 7 T O T A L C O S T P E R S Q U A R E F O O T E Q U A L S i 1 . 6 5 5 9 T O T A L C O S T P E R S Q U A R E F O O T E Q U A L S t 1 . 6 2 1 8 W O R S T P O I N T P A R A M E T E R S 2 4 9 7 8 . 6 4 1 2 . 0 0 0 0 . 0 7 1 4 5 . 0 0 0 5 . 0 0 0 1 . 6 3 0 C E N T R O I D V A R I A B L E V A L U E S 2 1 3 1 8 . 6 9 9 . 5 5 6 0 . 0 8 2 3 8 . 7 7 8 5 . 9 3 9 1 . 6 1 0 . . N E W . . P . 0 I N T . - P A R A M E T E R S B E F O R E S A T I S F A C T I O N O F I N T E G E R C O N S T R A I N T S 0 . 0 6 . 3 7 8 0 . 0 9 5 3 0 . 6 8 9 7 . 1 6 1 T O T A L C O S T P E R S Q U A R E F O O T E O U A L S 1 1 . 7 8 1 4 T O T A L C O S T P E R S O U A R E F O O T E O U A L S t 1 . 7 2 0 9  N E W P O I N T I S N O I M P R O V E M E N T O V E R W O R S T P O I N T . H A L V E A L P H A . N E W P O I N T P A R A M E T E R S B E F O R E S A T I S F A C T I O N O F I N T E G E R C O N S T R A I N T S . 1 6 . 4 3 1 . 8 1 7 . 9 6 7 0 . 0 8 8 3 4 . 7 3 3 6 . 5 5 0 T O T A L C O S T P E R S O U A R E F O O T E Q U A L S t 1 . 6 0 8 3 T O T A L C O S T P E R S Q U A R E F O O T E Q U A L S t 1 . 6 1 4 5 T O T A L C O S T P E R S Q U A R E F O O T E Q U A L S i 1 . 6 4 9 9  W O R S T P O I N T P A R A M E T E R S 2 3 7 5 7 . 6 4 8 . 0 0 0 0 . 0 9 1 4 5 . 0 0 0 6 . 6 6 7 1 . 6 2 4 . C E N T R O I O V A R I A B L E V A L U E S 2 1 0 5 8 . 7 2 9 . 5 5 6 0 . 0 8 1 3 8 . 0 0 0 5 . 9 3 9 1 . 6 0 8 N E W P O I N T P A R A M E T E R S B E F O R E S A T I S F A C T I O N O F I N T E G E R C O N S T R A I N T S 0 . 0 1 1 . 5 7 8 0 . 0 6 8 2 8 . 9 0 0 4 . 9 9 4  C O N S T R A I N T E X C E E D E D . K K K K E Q U A L S 1 T O T A L C O S T P E R S Q U A R E F C O T E Q U A L S * 1 . 6 4 7 0 . . T O T A L C O S T . . P E R . S Q U A R E F O O T E Q U A L S . $ 1 . 6 7 9 3 T O T A L C O S T P E R S Q U A R E F O O T E Q U A L S $ 1 . 6 3 4 5 N E W P O I N T I S N O I M P R O V E M E N T O V E R W O R S T P O I N T . H A L V E A L P H A . N E W P O I N T P A R A M E T E R S B E F O R E S A T I S F A C T I O N O F I N T E G E R C O N S T R A I N T S 1 8 8 9 1 . 0 1 1 0 . 5 6 7 0 . 0 7 5 3 3 . 4 5 0 5 . 4 6 7 T O T A L C O S T P E R S Q U A R E F O O T E Q U A L S t 1 . 5 9 0 1 T O T A L C O S T P E R S Q U A R E F O O T E Q U A L S » 1 . 6 1 3 5 T O T A L C O S T P E R S Q U A R E F C O T E O U A L S » 1 . 5 8 2 9 W O R S T P O I N T P A R A M E T E R S 2 3 4 6 7 . 7 5 1 2 . 0 0 0 0 . 0 6 1 3 8 . 0 0 0 5 . 0 0 0 1 . 6 2 2 C E N T R O I O V A R I A B L E V A L U E S 2 0 9 1 5 . 0 5 9 . 1 1 1 0 . 0 8 3 3 8 . 0 0 0 6 . 0 5 1 1 . 6 0 3 N E W P O I N T P A R A M E T E R S B E F O R E S A T I S F A C T I O N O F I N T E G E R C O N S T R A I N T S 0 . 0 5 . 3 5 6 0 . 1 1 1 3 8 . 0 0 0 7 . 4 1 6 C O N S T R A I N T E X C E E D E D . K K K K F O U A L S 1 T O T A L C O S T P E R S Q U A R E F O O T E Q U A L S S 1 . 8 1 8 3  T O T A L ' C O S T P E R S Q U A R E F O O T E Q U A L S t 1 . 7 5 2 2 N E W P O I N T I S N O I M P R O V E M E N T O V E R W O R S T P O I N T . H A L V F A L P H A . N E W P O I N T P A R A M E T E R S B E F O R E S A T I S F A C T I O N O F I N T E G E R C O N S T R A I N T S 2 1 9 0 5 . 1 8 7 . 2 3 3 0 . 0 9 7 3 8 . 0 0 0 . 6 . 7 3 3 T O T A L C O S T P E R S Q U A R E F O O T E O U A L S i 1 . 7 2 1 8 T O T A L C O S T P E R S O U A R E F O O T E O U A L S S 1 . 7 3 2 8  T O T A L C O S T P E R S Q U A R E F O O T E Q U A L S $ 1 . 7 7 9 0 C E N T R O I O U S E D A S N E W P O I N T T O T A L C O S T P E R S Q U A R E F O O T E Q U A L S i 1 . 6 2 2 4 T O T A L C O S T P E R S Q U A R E F O O T E O U A L S i 1 . 6 3 2 5 T O T A L C O S T P E R S Q U A R E F O O T E Q U A L S I 1 . 6 0 4 2 W O R S T P O I N T P A R A M E T E R S  1 1 8 8 4 . 9 2 1 0 . 0 0 0 0 . 1 0 0 2 4 . 0 0 0 6 . 0 0 0 1 . 6 2 1 C E N T R O I D V A R I A B L E V A L U E S 2 2 2 6 8 . 8 9 8 . B B 9 ' 0 . 0 8 1 4 0 . 3 3 3 6 . 1 2 5 1 . 6 0 2 N E W P O I N T P A R A M E T E R S B E F O R E S A T I S F A C T I O N O F I N T E G E R C O N S T R A I N T S 0 . 0 7 . 4 4 4 C O N S T R A I N T E X C E E O E O , . 0 5 6 6 1 . 5 6 7 K K K K E Q U A L S N E W P O I N T P A R A M E T E R S B E F O R E S A T I S F A C T I O N O F I N T E G E R C O N S T R A I N T S 0 . 0 8 . 1 6 7 0 . 0 6 8 5 0 . 9 5 0 6 . 2 0 6 C O N S T R A I N T E X C F E D E D . K K K K E Q U A L S 1 T O T A L C O S T P E R S Q U A R E F O O T E Q U A L S . « 1 . 6 2 0 2 T O T A L C O S T P E R S Q U A R E F O O T E Q U A L S « 1 . 6 2 1 0 T O T A L C O S T P E R S Q U A R E F O O T E Q U A L S S 1 . 6 5 7 2  W O R S T P O I N T P A R A M E T E R S 2 7 5 9 4 . 1 2 8 . 0 0 0 0 . 0 6 3 4 5 . 0 0 0 6 . 6 6 7 1 . 6 2 0 C E N T R O I D V A R I A B L E V A L U E S 2 2 2 6 8 . 8 9 8 . 8 8 9 0 . 0 8 1 4 0 . 3 3 3 6 . 1 2 5 1 . 6 0 2 N E W P O I N T P A R A M E T E R S B E F O R E S A T I S F A C T I O N O F I N T E G E R C O N S T R A I N T S 0 . 0 1 0 . 0 4 4 0 . 0 9 7 3 4 . 2 6 7 T O T A L C O S T P E R S Q U A R E F O O T E Q U A L S * 1 . 6 1 3 4 T O T A L C O S T P E R S Q U A R E F O O T E O U A L S $ 1 . 6 3 0 4 T O T A L C O S T P E R S Q U A R E F O O T E O U A L S * 1 . 6 1 2 9 W O R S T P O I N T P A R A M E T E R S 2 1 5 4 2 . 3 9 1 2 . 0 0 0 0 . 0 7 2 3 8 . 0 0 0 6 . 0 0 0 C E N T R O I D V A R I A B L E V A L U E S  1 . 6 1 5 2 2 1 3 7 . 7 6 8 . 4 4 4 0 . 0 8 4 4 0 . 3 3 3 6 . 0 6 4 1 . 6 0 1 N E W P O I N T P A R A M E T E R S B E F O R E S A T I S F A C T I O N O F I N T E G E R C O N S T R A I N T S 0 . 0 3 . B 2 2 0 . 1 0 0 4 3 . 3 6 7 6 . 1 4 7 C O N S t R A I N T E X C E E D E D . K K K K E Q U A L S 1 T O T A L C O S T P E R S Q U A R E F O O T E O U A L S « 1 . 6 8 3 9 T O T A L C O S T P E R S Q U A R E F O O T E Q U A L S i 1 . 6 3 3 0  N E W P O I N T I S N O I M P R O V E M E N T O V E R W O R S T P O I N T . H A L V E A L P H A . N E W P O I N T P A R A M E T E R S B E F O R E S A T I S F A C T I O N O F I N T E G E R C O N S T R A I N T S 2 2 8 4 8 . 5 1 6 . 1 3 3 0 . 0 9 2 4 1 . 8 5 0 6 . 1 0 6 T O T A L C O S T P E R S Q U A R E F O O T E Q U A L S t 1 . 6 6 6 7 T O T A L C O S T P E R S Q U A R E F O O T E Q U A L S t 1 . 6 2 5 4 C E N T R O I D U S E D A S N E W P O I N T  T O T A L C O S T P E R S Q U A R E F O O T E Q U A L S t 1 . 6 1 2 9 T O T A L C O S T P E R S O U A R E F O O T E O U A L S - 1 . 6 1 9 4 T O T A L C O S T P E R S O U A R E F O O T E Q U A L S 1 1 . 6 5 6 2 W O R S T P O I N T P A R A M E T E R S 2 3 8 7 1 . 1 8 8 . 0 0 0 0 . 0 8 8 4 5 . 0 0 0 6 . 6 6 7 C E N T R O I D V A R I A B L E V A L U E S  1 . 6 1 5 2 2 1 9 9 . 9 8 8 . 4 4 4 0 . 0 8 3 4 0 . 3 3 3 6 . 0 6 4 1 . 6 0 1 N E W P O I N T P A R A M E T E R S B E F O R E S A T I S F A C T I O N O F I N T E G E R C O N S T R A I N T S 0 . 0 9 . 0 2 2 0 . 0 7 8 3 4 . 2 6 7 5 . 2 8 0 T O T A L C O S T P E R S Q U A R E F O O T E Q U A L S t 1 . 5 9 1 3 T O T A L C 0 S 1 P E R S Q U A R E F O O T E Q U A L S t 1 . 6 0 6 0 T O T A L C O S T P E R S Q U A R E F O O T E O U A L S t 1 . 5 8 8 6  W O R S T P O I N T P A R A M E T E R S 2 4 4 3 1 . 1 2 8 . 0 0 0 0 . 0 8 4 4 5 . 0 0 0 6 . 6 6 7 1 . 6 1 3 C E N T R O I D V A R I A B L E V A L U E S 2 1 9 3 0 . 7 1 8 . 4 4 4 0 . 0 8 3 3 9 . 5 5 6 5 . 9 2 9 1 . 5 9 8 N E W P O I N T P A R A M E T E R S B E F O R E S A T I S F A C T I O N O F I N T E G E R C O N S T R A I N T S 0 . 0 9 . 0 2 2 0 . 0 8 1 3 2 . 4 7 8 4 . 9 7 1  C O N S T R A I N T E X C E E D E D . K K K K E Q U A L S 1 T O T A L C O S T P E R S Q U A R E F O O T E Q U A L S $ 1 . 5 9 7 4 T O T A L C O S T P E R S O U A R E F O O T E Q U A L S « 1 . 6 1 6 0 T O T A L C O S T P E R S Q U A R E F O O T E Q U A L S « 1 . 5 7 8 6 S E A R C H T E R M I N A T E D . M A X I M U M N O . O F I T E R A T I O N S R E A C H E D . -114-D E S I G N D F S T A T I C A L L Y I N D E T F R M I N A T E T R U S S S P A N I N I N C H E S E Q U A L S 1 4 4 0 . 0 D E P T H T O S P A N R A T I O E Q U A L S 0 . 0 B 1 2 N U M B E R O F P A N E L S E Q U A L S 8  N U M B E R O F J O I N T S E Q U A L S 1 8 N U M B E R O F M E M B E R S E Q U A L S 3 3 N U M B E R O F D E G R E E S O F F R E E D O M E Q U A L S 5 1 H A L F B A N D W I D T H E Q U A L S 1 2 P U R L I N S P A C I N G I N I N C H E S E Q U A L S 6 0 . 0 0 T R U S S S P A C I N G I N F E E T E O U A L S 3 8 . 0 0 . R O O F D E A D L O A D I N K I P S P E R F O O T O F S P A N E Q U A L S 0 . 1 3 3 9 I N T E R N A L L O A D R O U T I N E S A R E U S E D . T H E F O L L O W I N G L O A D C A S E S A R E C O N S I D E R E D  U N I F O R M L I V E L O A D O N F U L L S P A N A N D U N I F O R M L I V E L O A D O N H A L F S P A N R O O F D E A D L O A D A N D T R U S S D E A O L O A D A O O E O I N F O R A L L C A S E S T H E M A G N I T U D E O F T H E U N I F O R M L Y D I S T R I B U T E D L O A D I S 1 . 5 2 K I P S P E R F O O T . T H E M A G N I T U D E O F T H E P O I N T L O A D I S . 0 . 0 K I P S . T O T A L N U M B E R O F L O A D C A S E S C O N S I D E R E D I S 2 -115-W E I G H T U S E D F O R R E D E S I G N E O U A L S 2 0 5 0 8 . 6 2 W E I G H T U S E D F O R R E D E S I G N E O U A L S 2 0 5 0 0 . 4 3 W E I G H T U S E D F O R R E D E S I G N E O U A L S 2 0 8 1 4 . 8 4 P A N E L D E P T H T O S P A N R A T I O E O U A L S 0 . 6 4 9 5 M E M B E R I N F O R M A T I O N F O R D E S I G N - C O M P R E S S I O N N E G A T I V E - F O R C F I N K I P S - A R E A S O U A R E I N C H E S D E S I G N O F W E B M E M B E R S M E J 4 B E R N U M B E R M A X . C O M P R E S S I O N L O A D M A X . T E N S I O N L O A D D E S I G N A R E A L E N G T H S T R E S S F A C T O R I D E N T I F I C A T I O N 1 - 1 0 9 . 2 8 1 0 . 0 7 . 0 6 0 1 1 6 . 9 1 1 . 0 0 8 5 . 0 0 3 . 5 0 0 . 4 3 8 D O U B L F A N G L E 3 0 . 0 1 7 9 . 0 5 7 8 . 3 6 0 2 1 4 . 6 3 0 . 9 9 2 5 . 0 0 5 . 0 0 0 . 4 3 8 D O U B L E A N G L F 5 - 9 7 . 7 2 7 0 . 0 7 . 0 0 0 1 1 6 . 9 1 1 . 0 1 8 4 . 0 0 3 . 5 0 0 . 5 0 0 D O U B L E A N G L E 7 0 . 0 1 2 4 . 3 7 8 5 . 7 2 0 2 1 4 . 6 3 1 . 0 0 7 4 . 0 0 4 . 0 0 0 . 3 7 5 D O U B L E A N G l E 9 - 6 7 . 2 6 6 0 . 0 4 . 9 6 0 1 1 6 . 9 1 0 . 9 7 2 4 . 0 0 3 . 0 0 0 . 3 7 5 D O U B L E A N G L E 1 1 0 . 0 7 2 . 8 9 0 3 . 3 8 0 2 1 4 . 6 3 0 . 9 9 8 3 . 5 0 3 . 5 0 0 . 2 5 0 D O U B L E A N G L E 1 3 - 4 0 . 3 8 8 0 . 0 3 . 5 6 0 1 1 6 . 9 1 0 . 9 0 5 3 . 5 0 2 . 5 0 0 . 3 1 3 D O U B L E A N G L E 1 5 - 1 7 . 4 2 5 4 8 . 1 7 9 4 . 6 0 0 2 1 4 . 6 3 0 . 9 8 6 3 . 5 0 3 . 0 0 0 . 3 7 5 D O U B L E A N G L E 1 7 - 2 8 . 3 0 6 0 . 0 3 . 2 4 0 1 1 6 . 9 1 0 . 9 0 7 3 . 0 0 2 . 5 0 0 . 3 1 3 D O U B L E A N G L E 1 9 - 1 7 . 4 2 5 4 8 . 1 7 9 4 . 6 0 0 2 1 4 . 6 3 0 . 9 B 6 3 . 5 0 3 . 0 0 0 . 3 7 5 D O U B L E A N G L E 2 1 _ . . _ - 4 0 . 3 8 8 0 . 0 3 . 5 6 0 1 1 6 . 9 1 0 . 9 0 5 3 . 5 0 2 . 5 0 0 . 3 1 3 D O U B L E A N G L F 2 3 0 . 0 7 2 . 8 9 0 3 . 3 R 0 2 1 4 . 6 3 0 . 9 9 8 3 . 5 0 3 . 5 0 0 . 2 5 0 O O U R L E A N G L E 2 5 - 6 7 . 2 6 6 0 . 0 4 . 9 6 0 1 1 6 . 9 1 0 . 9 7 2 4 . 0 0 3 . 0 0 0 . 3 7 5 D O U B L E A N G L E 2 7 0 . 0 1 2 4 . 3 7 8 5 . 7 2 0 2 1 4 . 6 3 1 . 0 0 7 4 . 0 0 4 . 0 0 0 . 3 7 5 D O U B L E A N G L E 2 9 - 9 7 . 7 2 7 0 . 0 7 . 0 0 0 1 1 6 . 9 1 1 . 0 1 8 4 . 0 0 3 . 5 0 0 . 5 0 0 D O U B L F A N G L F 3 1 0 . 0 1 7 9 . 0 5 7 8 . 3 6 0 2 1 4 . 6 3 0 . 9 9 2 5 . 0 0 5 . 0 0 - 0 . 4 3 8 D O U B L E A N G L E 33 - 1 0 9 . 2 8 1 0 . 0 7 . 0 6 0 1 1 6 . 9 1 1 . O O B 5 . 0 0 3 . 5 0 0 . 4 3 8 D O U B L E A N G L E D E S I G N O F C H O R D M E M B E R S M E M B E R N U M B E R D E S I G N A R E A I N E R T I A L E N G T H S T R E S S F A C T O R I D E N T I F I C A T 1 0 N 2 1 5 . 8 9 3 4 9 . 5 2 1 8 0 . 0 0 0 . 0 1 7 1 5 5 4 . 0 0 T E E 4 2 2 . 5 0 4 4 0 . 7 4 1 8 0 . 0 0 0 . 7 4 4 1 8 8 5 . 0 0 T E E 6 1 5 . 8 9 3 4 9 . 5 2 1 8 0 . 0 0 0 . 4 5 5 1 5 5 4 . 0 0 T E E 8 2 2 . 5 0 4 4 0 . 7 4 1 8 0 . 0 0 1 . 0 0 6 1 8 8 5 . 0 0 T E F 1 0 1 5 . 8 9 3 4 9 . 5 2 1 8 0 . 0 0 0 . 8 0 1 1 5 5 4 . 0 0 T E E 1 2 2 2 . 5 0 4 4 0 . 7 4 1 8 0 . 0 0 1 . 0 0 9 1 8 8 5 . 0 0 T E E 1 4 1 5 . 8 9 3 4 9 . 5 2 1 8 0 . 0 0 0 . 9 7 9 1 5 5 4 . 0 0 T E E 1 6 2 2 . 5 0 4 4 0 . 7 4 1 8 0 . 0 0 0 . 9 7 2 1 8 8 5 . 0 0 T E E . . . 1 8 . . 1 5 . 8 9 3 4 9 . 5 2 1 8 0 . 0 0 0 . 9 7 9 1 5 5 4 . 0 0 T E E 2 0 2 2 . 5 0 4 4 0 . 7 4 1 8 0 . 0 0 0 . 9 7 3 1 8 8 5 . 0 0 T E E 2 2 1 5 . 8 9 3 4 9 . 5 2 1 8 0 . 0 0 0 . 8 0 1 1 5 5 4 . 0 0 T F E 2 4 2 2 . 5 0 4 4 0 . 7 4 1 8 0 . 0 0 1 . 0 0 9 1 8 8 5 . 0 0 T E E 2 6 1 5 . 8 9 3 4 9 . 5 2 1 8 0 . 0 0 0 . 4 5 5 1 5 5 4 . 0 0 T E E 2 8 2 2 . 5 0 4 4 0 . 7 4 1 8 0 . 0 0 1 . 0 0 6 1 8 8 5 . 0 0 T E E 3 0 1 5 . 8 9 3 4 9 . 5 2 1 8 0 . 0 0 0 . 0 1 7 1 5 5 4 . 0 0 T E E 3 2 2 2 . 5 0 4 4 0 . 7 4 1 8 0 . 0 0 0 . 7 4 4 1 8 8 5 . 0 0 T E E C H O R D F O R C E S - A X I A L F O R C E I N K I P S - B E N D I N G M O M E N T I N K I P I N C H E S M E M B E R N U M B E R L O A O C A S E A X I A L F O R C E B M J N L B M J N G 2 1 - 0 . 0 0 4 0 . 0 3 4 . 8 3 4 2 - 0 . 0 0 2 0 . 0 3 2 . 2 9 8 4 1 - 1 5 0 . 1 6 1 0 . 0 - 3 4 8 . 4 0 7 2 - 1 1 5 . 1 3 3 0 . 0 - 3 5 4 . 1 8 8 6 1 1 5 0 . 1 5 9 - 3 4 . 8 3 3 3 4 . 9 2 T . 2 . . 1 1 5 . 1 3 3 - 3 2 . 2 9 6 4 0 . 8 4 7 8 1 - 2 5 4 . 4 6 8 3 4 8 . 4 1 4 - 2 2 7 . 4 5 8 2 - 1 8 3 . 9 9 2 3 5 4 . 1 9 3 - 2 0 0 . 9 9 1 1 0 1 2 5 4 . 4 6 5 - 3 4 . 9 2 4 1 2 2 . 0 8 7 2 1 8 3 . 9 9 1 - 4 0 . 8 4 5 9 2 . 7 2 8 1 2 1 - 3 1 5 . 5 9 5 2 2 7 . 4 7 1 - 7 2 . 0 6 5 2 . . . - 2 1 1 . 5 3 5 2 0 1 . 0 0 0 - 1 8 7 . 4 5 2 1 4 1 3 1 5 . 5 9 3 - 1 2 2 . 0 8 4 8 6 . 0 9 7 2 2 1 1 . 5 3 4 - 9 2 . 7 2 5 5 0 . 2 3 5 1 6 1 - 3 3 7 . 6 9 3 7 2 . 0 7 5 - 1 5 6 . 5 9 4 2 - 1 9 6 . 9 2 1 1 8 7 . 4 6 7 - 7 7 . 2 5 0 1 8 1 3 1 5 . 5 9 5 - 6 6 . 0 9 5 1 2 2 . 0 8 9 _ 2 1 5 6 . 5 1 6 . - 5 0 . 2 3 0 4 9 . 6 5 5 2 0 1 - 3 3 7 . 6 9 2 1 5 6 . 6 1 2 - 7 2 . 0 5 2 2 - 1 9 6 . 9 2 1 7 7 . 2 6 5 1 3 2 . 0 9 1 2 2 1 2 5 4 . 4 6 7 - 1 2 2 . 0 9 1 3 4 . 9 2 0 2 1 1 2 . 7 3 8 - 4 9 . 6 5 3 0 . 0 2 4 1 - 3 1 5 . 5 9 6 7 2 . 0 6 4 - 2 2 7 . 4 7 0 _ 2 j : . 1 5 6 . . 5 . 1 6 - 1 3 2 . 0 8 6 - 3 7 . 7 0 9 2 6 1 1 5 0 . 1 6 2 - 3 4 . 9 2 2 3 4 . 8 3 6 2 5 9 . 8 0 4 0 . 0 8 . 0 7 0 2 8 1 - 2 5 4 . 4 6 7 2 2 7 . 4 8 1 - 3 4 8 . 4 0 4 2 - 1 1 2 . 7 3 8 3 7 . 7 1 7 - 1 7 . 1 2 8 3 0 1 - 0 . 0 0 0 - 3 4 . 8 3 7 0 . 0 2 0 . 0 0 0 - 8 . 0 7 0 0 . 0 3 2 1 - 1 5 0 . 1 6 2 3 4 8 . 4 1 1 0 . 0 2 - 5 9 . 8 0 4 1 7 . 1 2 7 0 . 0 T O T A L H E I G H T O F T R U S S E Q U A L S 2 0 9 7 8 . 1 7 6  H E I G H T P E R F O O T O F S P A N F O R A L L M E M B E R S E Q U A L S 1 7 4 . 8 2 T O T A L H E I G H T O F T O P C H O R O E Q U A L S 1 0 2 0 0 . 0 0 _ T D I A L _ H E J G H T _ O F _ B O T . T . O M . _ C H O R D E Q U A L S 6 4 8 . 8 . 4 . 1 T O T A L L E N G T H O F T R U S S M E M B E R S I N F E E T E Q U A L S 4 7 0 . 7 7 T O T A L L E N G T H O F T R U S S H E B M E M B E R S E Q U A L S 2 3 0 . 7 7 H E I G H T O F H E B M E M B E R S E Q U A L S 4 2 8 9 . 7 6 2  R A T I O O F W E B W E I G H T T O T O T A L T R U S S W E I G H T E Q U A L S 0 . 2 0 4 5 W E I G H T P E R F O O T O F S P A N O F C H O R D M E M B E R S E Q U A L S 1 3 9 . 0 7 _MEjmT_ P t ^ _ F O .a . T _ . O F _ S P . A N _ O F _ W E B _ M E , M B E R . S _ E Q U A L S _ 3 . 5 . . 7 . 5 . C O S T C O E F F I C I E N T S U S E D A R E A S F O L L O W S -W E B M A T E R I A L C O S T P E R P O U N D E Q U A L S * 0 . 0 8  T O P C H O R D M A T E R I A L C O S T P E R P O U N D E Q U A L S S 0 . 0 9 B O T T O M C H O R O M A T E R I A L C O S T P E R P O U N D E Q U A L S » 0 . 0 9 _ C O S T _ Q F _ P R E R A R A . T . I O N _ P E R W E B . . M E M B E R E Q U A L S $ . 6 . 0 0 C O S T O F W E B J O I N T S P E R . M E M B E R E Q U A L S S 2 5 . 0 0 C O S T O F P R E P A R A T I O N P E R C H O R D M E M B E R E Q U A L S « 1 5 . 0 0 C O S T P E R C H O R D S P L I C E E Q U A L S t 4 0 . 0 0  C O S T P E R S Q U A R E F O O T O F W A L L C L A D D I N G E Q U A L S * 1 . 5 0 R O O F C O S T P E R S Q U A R E F O O T B A S E D O N J R U S S . S P A C I N G U P P E R B O U N D S O F S P A C I N G I N C R E M E N T S A R E A S F O L L O W S ^ 1 6 . 9 9 2 4 . 0 0 3 1 . 0 0 3 8 . 0 0 4 5 . 0 0 C O S T P E R S Q U A R E F O O T O F R O O F F O R E A C H I N C R E M E N T I S A S F O L L O W S - -117-0.17 0.25 0.34 0.36 0.42 ROOF COST PER SQUARE FOOT BASED ON PANEL LENGTHS UPPER BOUNDS ON PANEL LENGTHS ARE AS FOLLOWS -7.00 8.00 11.50 12.50 20.00 COST PER SQUARE FOOT OF ROOF FOR EACH PANEL INCREMENT IS AS FOLLOWS -0.40 0.50 0.70 0.80 1.00 NUMBER OF CHORD PIECES ASSUMED EQUALS 6 NUMBER OF CHORO SPLICES ASSUMED EQUALS 4 COST OF WEB MATERIAL EQUALS S 343.18 COST OF TOP CHORO MATERIAL EQUALS » 918.00 COST OF BOTTOM CHORD MATERIAL EQUALS t 583.96  WEB LABOUR COST EQUALS 1 102.00 COST.FOR WEB MEMBER JOINTS EQUALS t 425.00 CHORD LABOUR COST EQUALS * 90.00 CHORD SPLICE COST EQUALS * 160.00 COST OF WALL CLADDING FOR ONE BAY EQUALS ( 1110.64 COST OF ROOFING QUE TO TRUSS SPACING EQUALS $ 1641.60  COST OF ROOFING DUE TO AVERAGE PANEL SPACING EQUALS * 1824.00 TOTAL COST EOUALS t 7198.38 TOTAL COST PER SQUARE FOOT EQUALS $ 1.5786 PRINT OUT OF PROP MATRIX AND COST VECTOR 20362.33 8.0000 0.0970 38.0000 5.4545 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.6129 23260.07 10.0000 0.0801 45.0000 5.0000 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.6056 21581.02 8.0000 0.0878 38.0000 6.6667 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.6097 OOOO 0.0812 38.0000 5.0000 0.0 0±0 <K0 0^ 0 0,0 0^ 0 0^ 0 0^ 0 1.5786 21692.81 8.0000 0.0810 38.0000 5.4545 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.5922 20810.65 10.0000 0.0740 38.0000 6.0000 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.5813 24069.54 8.OOOO _0J.Q828„ _.4.5.0000. _6.6667_ 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.60 42 22174.62 8.0000 0.0747 38.0000 6.0000 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.5829 22007.64 8.0000 0.0777 38.0000 5.4545 0.0 0.0 0.0 0.0 0.0 . 0.0 0.0 0.0 1.5886 21417.90 8.0000 0.0884 38.0000 6.6667 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.6083 14613.02 6.0000 0.1025 24.0000 20.0000 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 2.2271 22940.43 6.0000 0.1036 - 45.0000 20.0000 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 2.2233 22348.63 8.0000 Q..Q933 38.0000 12.0000 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 2.0411 15176.84 10.0000 0.0784 24.0000 10.0000 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.9620 21253.83 8.0000 0.0931 38.0000 10.0000 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.9189 18340.84 14.0000 0.0660 31.0000 7.5000 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.7809 14891.72 19297.49 24291.70 26310.64 17737.20 .0000 .0000 OOOO OOOO OOOO 0.0631 24. 0.0629 31. -QjlOMO 31, 0.0771 45, 0.0698 45. 0.0697 31. OOOO 6. OOOO 5. OOOO 5. OOOO 6. OOOO 5. OOOO 5, 6667 OOOO oooo_ 6667 OOOO 4545 0.0 0.0 _0.0_ o.o 0.0 0.0 0.0 0.0 0.0_ 6.6 0.0 0.0 0.0 0.0 _0.0_ o.o 0.0 0.0 0.0 0.0 __Q,0_ 0.0 0.0 . 0.0 0.0 0.0 ,0,0 0.0 0.0 0.0 0.0 0.0 o.o_ 0.0 0.0 0.0 0.0 0.0 i--0_ 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.7364 1.6954 J L . A 8 2 1 _ 1.6813 1.6490 1.6444 22851.59 8.OOOO 0.0996 45.0000 6.6667 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.6333 24978.64 12.0000 0.0711 45.0000 5.0000 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.6305 23757.64 8.0000 0.0913 5^.0000 6.6667 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.6241 23467.75 12.0000 0.0607 38.0000 5.0000 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.6217 11884.92 10.0000 0.1000 24.0000 6.0000 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.6211 27594.12 8.0000 0.0684 45.0000 6.6667 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.6202 21542.39 12.0000 0.0715 38.0000 6.0000 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.6154 23871.18 8.0000 0.0876 45.0000 6.6667 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.6149 _J>JL&837 _45,_O.OJK> _6.666.7 .0,0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.6129 STOP 0 EXECUTION TERMINATED (SIGNOFF 

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