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Optimization in structural design using complex method Pathan, Abdul Nabi 1970

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OPTIMIZATION.IN STRUCTURAL DESIGN USING COMPLEX METHOD by ABDUL NABI PATHAN B.E. (CIVIL), N.E.D. Engineering College Karachi (Pakistan), 1966 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n the Department of C i v i l Engineering We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA September, 1970 In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the require-ments for an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y a v ailable for reference and study. I further agree that permission for extensive copying of t h i s thesis f o r scholarly purposes may be granted by the Head of my Department or by h i s representatives. I t i s understood that copying or p u b l i c a t i o n of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of C i v i l Engineering The U n i v e r s i t y of B r i t i s h Columbia Vancouver 8, Canada Date September 29, 1970. ABSTRACT The optimum design, with respect to cost, of reinforced concrete structures, s a t i s f y i n g Building Code Requirements (ACI 318-63), -is investigated, using Box's Complex Method. Variables considered are: geometry, topology, member sizes and material properties. The optimum design, with respect to volume of sing l e span, p i n -connected, plane trusses, i s investigated, using Box's Complex Method. Variables considered are: member sizes and nodal co-ordinates. The f e a s i b i l i t y of the Complex Method i s probed by checking the r e s u l t s , e i t h e r by conducting exhaustive search or comparing them with solutions obtained with l i n e a r programming methods. TABLE OF CONTENTS CHAPTER Page I. INTRODUCTION ...... ...... . '. .. .. 1 1.1 Subject 1 1.2 Object and Scope of Study 1 1.3 B r i e f Resume of Past Work 2 1.4 Organization of Presentation 3 I I . • OPTIMIZATION PROBLEM AND ITS SOLUTION 4 2.1 Optimization Problem 4 2.2 Optimization Algorithm (Box's Complex Method) 6 2.3 Complex Method: Observations and Modifications 11 2.4 . Complex Algorithm Modules . 13 III.. COST OPTIMIZATION OF REINFORCED CONCRETE CONCRETE STRUCTURES 15 3.1 Constraints . 15 3.2- Objective Function . 16 3.3 Analysis and Design Procedure ........... 18 3.4 Examples . .................... . . 18 IV. VOLUME OPTIMIZATION OF PLANE TRUSSES • 42 4.1 Constraints . ... ................... 42 4.2 Objective Function 43 4.3 Solution Procedure 43 i i CHAPTER Page 4.4 Examples ............. 44 V. CONCLUSIONS • 74 BIBLIOGRAPHY . .. • 75 i i i LIST OF TABLES TABLE Page 1 Structure A 24 2 Structure B 27 3 Structure C 32 3A Structure C 33 4 Structure D 36 5 Structure D 39 5A Structure D 39 6 Structure D 40 6A Structure D 41 7 Example 1 49 7A Example 1 50 7B Example 1 50 8 Example 2 49 8A Example 2 50 8B Example 2 50 9 Example 3 52 9A Example 3 53 10 Example 4 56 10A Example 4 57 10B Example 4 57 11 Example 5 60 11A Example 5 61 12 Example 6 64 12A Example 6 65 12B Example 6 65 13 Example 7 68 13A Example 7 69 13B Example 7 69 14 Example 8 72 14A Example 8 73 14B Example 8 73 LIST OF FIGURES FIGURE Page 3.1 Cost Parameters C^ to C^ Structure A 17 3.2 23 3.3 Structure B 23 3.4 Structure C 29 3.5 Structure D 37 4.1 Example 1 and 2 49 412 Example 3 52 4.3 Example 4 56 4.4 Example 5 60 4.5 Example 6 64 4.6 Example 7 68 4.7 Example 8 72 NOTATIONS {x} (y) {gX> {h X} {gy> {h y} {r} {g G} {x}3 {x} (gf {x} {x}> m W f L M y i x. 8 i f X . 1 c. 1 W X . 1 N x. l E x p l i c i t v a r i a b l e s I m p l i c i t v a r i a b l e s Lower bound of e x p l i c i t v a r i a b l e Upper bound of e x p l i c i t v a r i a b l e Lower bound of i m p l i c i t v a r i a b l e Upper bound of i m p l i c i t v a r i a b l e I n i t i a l point Pseudo-random deviate Centroid of f e a s i b l e points Feasible point Infeasible point Centroid of a l l points except the worst Worst point New point O v e r - r e f l e c t i o n factor Number of e x p l i c i t v a r i a b l e s Number of i m p l i c i t v a r i a b l e s Number of points i n the complex Objective function Length of a truss member Number of bars Cross-sectional area of a bar Slab span Beam span Girder span V I N s N b N g TLX TLY WD WL W„ f 1 c y S.r. MITER MITER1 P u,v a a Depth of slab Depth of beam Depth of g i r d e r Width of beam Width of g i r d e r Lesser dimension of column Greater dimension of column Number of spans of slabs Number of spans of beams Number of spans of g i r d e r s T o t a l length i n the d i r e c t i o n i n which the slabs (or g i r d e r s ) span T o t a l length i n the d i r e c t i o n i n which the beams span Imposed dead load Imposed l i v e load T o t a l load Maximum allowable compressive s t r e s s Maximum allowable t e n s i l e s t r e s s Crushing st r e n g t h of concrete Y i e l d s t r e s s of r e i n f o r c i n g s t e e l Y i e l d s t r e n g t h of s t e e l Slenderness r a t i o Number of cycles f o r f i r s t stage o p t i m i z a t i o n Number of cycle s f o r second stage o p t i m i z a t i o n Load a p p l i e d at a node Ab s c i s s a and ordinate of a node Stress i n the bar. Allowable s t r e s s i n the bar. v i i ACKNOWLEDGMENT The author wishes to express h i s sincere thanks to Professor Samuel L. Lipson, for the guidance and assistance received i n the preparation of this work. Special thanks are due to Dr. D. L. Anderson for many valuable suggestions. The f i n a n c i a l support of the Commonwealth Scholarship and Fellowship Committee i s g r a t e f u l l y acknowledged. Gratitude i s also expressed to the U.B.C. Computing Center f o r the use of i t s f a c i l i t i e s . CHAPTER I INTRODUCTION 1.1 Subject More r a t i o n a l , more economical design has long been the goal of the s t r u c t u r a l engineer. T r a d i t i o n a l l y , t h i s had been achieved by t r i a l designs. During the l a s t decade, however, the art of s t r u c t u r a l design has undergone d r a s t i c changes. The impetus for such changes has come from aerospace industry, where the most common governing c r i t e r i o n f o r optimum design i s weight; subject to the s a t i s f a c t i o n of c e r t a i n func-t i o n a l requirements and relevant constraints. The emphasis i n the design of structures for buildings i s not the same as i n the design of aerospace structures. Building structures are generally "one-shot" ventures. As such, the structure i s not subject to f u l l - s c a l e t e s t i n g and the t o t a l cost of design must be borne by the sin g l e structure alone. Hence, the cost of design must be kept within a very l i m i t e d budget. 1.2 Object and Scope.of Study The object of t h i s i n v e s t i g a t i o n i s to demonstrate the f e a s i b i l i t y * of Box's Complex Method [1] for obtaining the optimum cost design of reinf o r c e d concrete structures, and minimum volume design of p i n -connected plane trusses. Numerals i n square brackets r e f e r to entries i n Bibliography. 2. The reinforced concrete structures are l i m i t e d to one-way slabs supported on beams, girders and columns. Compression s t e e l has not been used i n any of the f l e x u r a l members. A l l columns are subject to uni-a x i a l bending only, and are l i m i t e d to rectangular t i e d sections with symmetrical bar arrangement. The s t r u c t u r a l frame-works studied are subject to uniformly d i s t r i b u t e d gravity loads. The Building Code Requirements of American Concrete I n s t i t u t e (ACI 318-63) for Reinforced Concrete [2] d i c t a t e the various aspects of design. The plane trusses are s i n g l e span and pin-connected, with loads applied at the j o i n t s . Annular sections, with diameter to thickness r a t i o of 20, have been assumed to provide continuous s i z i n g of the cross-sectional areas of the bars. The stresses are governed by A.I.S.C. Sp e c i f i c a t i o n s [11]. 1.3 B r i e f Resume of Past Work In the recent past, several authors have investigated the pos-s i b i l i t y of using the mathematical programming techniques i n optimum design of structures. A concise but b r i l l i a n t d e s c r i p t i o n of such e f f o r t s i s provided by Lucien Schmit i n his survey paper "St r u c t u r a l Synthesis 1959-1969, a Decade of Progress [4], " together with attendant l i m i t a t i o n s . Almost a l l the investigations c i t e d , have been directed towards achieving minimum-weight structures. The emphasis on the cost optimization of structures i s only recent. Goble and DeSantis [5] have used the mathematical techniques to achieve optimum design with respect to cost of Mixed Steel Composite Girders. L i t t l e attention has been paid, however, to the optimum design of reinforced concrete structures. 3. Louis H i l l [7] and Graham [8] have applied mathematical techniques to optimize the cost of d i s c r e t e elements of reinforced concrete b u i l d i n g s . A l l these sophisticated approaches are characterized by i n t r i -cacies of formulation, complexities of inputs and large machine times; thus rendering t h e i r a p p l i c a t i o n to p r a c t i c a l b u i l d i n g structures well nigh an i m p o s s i b i l i t y . It was not u n t i l 1969 that Russel [9] used Box's Complex Method to optimize the cost of a s t r u c t u r a l roof system, and demonstrated i t s f e a s i b i l i t y to p r a c t i c a l problems. It i s hoped that the present i n v e s t i g a t i o n w i l l further demon-st r a t e the advantages and the promises of the Complex Method, as a p r a c t i c a l approach to the s o l u t i o n of a broad range of p r a c t i c a l design problems. 1.4 Organization of Presentation Chapter II includes a comprehensive des c r i p t i o n of Box's Complex Method and the modifications that were found necessary to achieve better optima. Chapters III and IV show, re s p e c t i v e l y , how design problems of cost optimization of reinforced concrete structures and volume optimi-zation of plane trusses can be formulated as programming problems. Several i l l u s t r a t i v e examples are included i n Chapters III and IV; the r e s u l t s of the study are summarized and compared; and several con-clusions drawn. CHAPTER II OPTIMIZATION PROBLEM AND ITS SOLUTION 2.1 Optimization Problem A general constrained non-linear programming problem may be defined as follows: Given the objective function f = f (x , x , ... , x ) (2.1) 1 z m Subject to the constraints g. < x. < h. ; I °i - l - l < v < h y : k g k - y k k where y k = y k ( x 1 5 x 2, ... 4 = 8 k ( x l ' X2> ••• ' x m > ( 2- 5 )' hk=X(xl' V ' Xm } > ( 2- 6 ) (g and h are lower and upper l i m i t s respectively) determine the e x p l i c i t v a r i a b l e s (x., , x„, ... , x ) so as to optimize f. 1 z m In the context of Equations (2.1) through (2.6) a s t r u c t u r a l optimization problem i s as follows: = 1,2, ... , m (2.2) = 1,2, ... , I (2.3) , x J (2.4) The objective function f, of the structure, may be expressed as a function of m t o p o l o g i c a l and s t r u c t u r a l variables (x^, x^, ... > x m which are the design v a r i a b l e s . The i m p l i c i t variables (y^> ••• » ^£ may be the functions determined by the f l e x u r a l , shear and bond require-ments, or stress l i m i t a t i o n s of the b u i l d i n g code. In any design, i t i s assumed that the structures are constructed or f ab ri cated from a v a i l a b l e materials, and that the section properties of the members are such that the structures are acceptable or r e a l i z e a b l e from a p r a c t i c a l point of view. Hence, i n r e a l i t y , the objective func-t i o n f i s defined f o r d i s c r e t e values of design parameters; thus ren-dering i t an integer programming problem. It w i l l be appropriate to d i s t i n g u i s h between f i n i t e optimization problems and a n a l y t i c optimization problems. The former are well-defined as regards the degree and the form of n o n - l i n e a r i t y of the objective function, the i m p l i c i t v a r i a b l e s and t h e i r constraints. A l l abstract mathematical problems can be classed as f i n i t e . The a n a l y t i c optimization problems are not well-defined as regards the degree and the form of n o n - l i n e a r i t y of the objective function. Moreover, the i m p l i c i t v a r i a b l e s are extremely d i f f i c u l t to formulate i n terms of the e x p l i c i t v a r i a b l e s . Most of the e x p l i c i t v a r i a b l e s are interdependent. Almost a l l the s t r u c t u r a l optimization problems are a n a l y t i c i n nature. Thus i f the objective function of a continuous slab i s the t o t a l quantity of s t e e l Q , then •I M ds o kf d y where i s the t o t a l length; M i s the moment and i s a function of span, number of spans and the nature of applied loading; k i s a parameter and i s a function of f , f , d, modulus c y of e l a s t i c i t y of s t e e l , s t r e s s - s t r a i n c h a r a c t e r i s t i c s of concrete, the code l i m i t a t i o n s on the percentage of s t e e l and the method of design (whether e l a s t i c or u l t i -mate strength); and d i s the e f f e c t i v e depth of a section. The i n t e g r a t i o n i s required to take into account the d i s t r i b u t i o n of s t e e l across the length. I t i s obvious that a f i n i t e form of Q g i s l a r g e l y indeterminate and that i t i s highly non-linear. Besides, the i m p l i c i t constraints imposed by the code cannot be stated e x p l i c i t l y . These i n t r i c a c i e s , notwithstanding, the f i n i t e formulation techniques have been, and w i l l continue to be used to tackle the s t r u c t u r a l optimization problems. Thoughts, along the same l i n e s , have been expressed by Sheu and Prager [6], i n the concluding remarks section of t h e i r recent l i t e r a t u r e review. 2.2 Optimization Algorithm (Box's Complex Method) by M. J . Box [1]. The method, c a l l e d the Complex Method, was devised An algorithm used to determine the optimum vector {x} was devis for f i n d i n g the optimum of a general non-linear, constrained, objective function. This optimum i s characterized by some of the variables l y i n g at the l i m i t s of t h e i r permissible ranges. The objective function may. not n e c e s s a r i l y be continuous. The method consists of two main operations: generation of a complex, and a subsequent directed improvement of i t . A E u c l i d i a n m-dimensional space E i s considered, a point i n m the space being represented by the column vector {x} ; where It i s assumed that a bounded domain E within E i s defined b7 m m a set of m constraints of e x p l i c i t v a r i a b l e s . It i s further assumed that corresponding to each vector {x}, there may be an i m p l i c i t or a (x} T = (x± X 2 (2.7) coupling vector {y} where {y} = (y± y 2 ••• (2.8) which further bounds the domain by a set of I constraints. The e x p l i c i t constraints are assumed of the form {g X} < {x} < {h x} (2.9) while the i m p l i c i t constraints are of the form \ {g y} < {y} < {h y} ; (2.10) 8. where Y i = Y i ^ X l ' X 2 ' ' XnP ' 1 = 1 > 2 > - ( 2 . 1 1 ) £ = g Y(x , x ... , x ) ; i-= 1,2,...,£ (2.12) l l 1 z m h i = h I ( xl' x 2 ' •'• ' x m ) 5 i = l » 2 , . . . , £ (2.13) The bounded f e a s i b l e domain E i s defined as m E m = {{x} 1 {g X} < {x} < {h X} ; {g y} < {y} < {h y}} (2.14) Every point i n E m i s c a l l e d a f e a s i b l e point. It i s assumed that an i n i t i a l r - i * f e a s i b l e point, {x} i s a v a i l a b l e . An i n i t i a l complex of (n-1) points where n > m+1 i s set up as {x} = {g X} + {r} {h X - g X} (2.15) Where r i s 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). Any point of the Complex, so selected, must s a t i s f y the e x p l i c i t constraints, but need not s a t i s f y a l l the i m p l i c i t con-r - i * * r i f s t r a i n t s . If {xl i s an i n f e a s i b l e point, i t i s replaced by {xj where {x} f = {g C} ± 0.5{g C - x**} (2.16) and {g C} i s the centroid, of those points of the Complex, found or already rendered f e a s i b l e (the i n i t i a l point, being f e a s i b l e , i s also included). This process i s repeated u n t i l a f e a s i b l e point i s found. 9. The objective function f, i s evaluated at each point and the r i W point with the worst function value i s located. The worst point IxJ r i N i s replaced by a new point Ixj ; where {x} N = {g*} + a{g* - x W} , (2.17) {g } i s the centroid of a l l points of the Complex excluding the worst point and a i s an " o v e r - r e f l e c t i o n " factor ( a > 1.0). If any co-ordinate of {x}^ v i o l a t e s some e x p l i c i t constraint, i t i s reset to a value z(a very small value) in s i d e the appropriate r i N constraint. If i x j v i o l a t e s any of the i m p l i c i t constraints, i t i s replaced by { x } ^ where {*} = . (g } ± 0.5{g - x ] (2.18) u n t i l a f e a s i b l e point i s found. If t h i s improved f e a s i b l e point also happens to be the worst, the process of improvement i s c a r r i e d out by halving a. This directed sequence of improving the Complex i s continued u n t i l i t collapses into a point.. 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 e n t s t a r t i n g point and a d i f f e r e n t random number base. The values of a and n suggested by Box are 1.3 and 2m r e s p e c t i v e l y . The stopping c r i t e r i o n suggested by Box i s a conservative one, namely, 10. that the program s h a l l stop i t s e l f when f i v e consecutive equal function evaluations have occurred; which give values of f which are "equal" to the accuracy of the computer wordlength being used. The technique, aforementioned, i s e s s e n t i a l l y a rudimentary ascent/descent procedure derived from the Constrained Simplex Method of Spendley et^ al [3]. The d i r e c t i o n of t r a v e l becomes "into the adjacent Complex, which i s obtained by discarding the point of the current Complex, corresponding to the worst function value, and replacing i t by i t s mirror-image i n the plane (or hyperplane) of the remaining points." In this way steep ascent/descent procedure i s defined, subject to continuous review. The' mode of improvement of the Complex i s akin to the directed f e a s i b l e d i r e c t i o n methods. The f e a s i b l e d i r e c t i o n methods provide a means of generating an improving sequence of acceptable designs. Let the sequence of designs be of the form p+1 = p + p (2.19) J where the vector V^^~ represents the (j+1) ^  t r i a l design, the vector ~ i , . th , . , , th , ,. _. D represents the j design, and the j design modification i s repre--H sented by a. S . J If at the design represented by D , a l l the ine q u a l i t y constraints are s a t i s f i e d ; then S can be taken as the d i r e c t i o n of steepest descent or ascent. A close examination of Equations (2.17) and (2.19) brings into sharp focus the basic s i m i l a r i t i e s ; i f S i s replaced by i g - x ], a. 11. taken constant (and equal to a), and D replaced by {g } (the centroid of improved p o i n t s ) . Hence Box's Complex Method could be described as a S t a t i s t i c a l l y Directed Feasible D i r e c t i o n Method of Constrained Optimization. 2.3 Complex Method: Observations and Modifications While the generation of a complex i s accomplished once, i t s improvement i s a gradual process. Thus the number of cycles determines the type of optimum achieved. How close the "achieved optimum" i s to the global optimum, or even a true l o c a l optimum, constitutes the e f f i c i e n c y . During the course of i n v e s t i g a t i o n , the following parameters were found to be p i v o t a l , as regards the e f f i c i e n c y of the Complex Method: i . I n i t i a l point i i . O v e r - r e f l e c t i o n factor i i i . Random number base ( i . e . the shape of the complex). The following modifications have been made to incorporate the e f f e c t s of p i v o t a l patameters: a. The optimization programming routine i s run stagewise. The i n i t i a l stage consists of generation of a complex and i t s improvement for a s p e c i f i e d number of cycles. The best point at the l a s t termination i s used as the i n i t i a l point for the second stage. The complex i s set up again, with the help of a new random number base. Improvement i s effected for some 12. s p e c i f i e d number of cycles, with a changed by a very small amount. Successive stages may be performed i n a s i m i l a r way i f deemed necessary, b. While e f f e c t i n g improvement of the worst point, i f the new point, replacing the worst, happens to be the worst also; a i s made to assume the following succession of values: a ct —., 0 , - — , and - a . If the extreme value of - a does not lead to a better point, the search i s abandoned. The best point, generated up to t h i s stage, i s used as an i n i t i a l point and a new complex i s set up. Invariably, the number of cycles, dictates the termination. The value of a equal to 1.3 i s substantiated by the investigations of the author; while n (the number of points of the complex) need not be greater than m+1. I t i s asserted that, under no circumstances, n be taken less than 5. A s i g n i f i c a n t deviation, from the basic postulates of the Complex Method, i n respect of the generation of a f e a s i b l e complex, without an i n i t i a l f e a s i b l e point, has been attempted i n some of the problems. It may be borne i n mind that an i n i t i a l complex, generated with the help of pseudo-random numbers, does s a t i s f y a l l the e x p l i c i t constraints but need not s a t i s f y a l l or any of the i m p l i c i t constraints. The f e a s i -b i l i t y of the complex i s accomplished with the help of an i n i t i a l f e a s i b l e point, as outlined i n Section 2.2. The necessity of an i n i t i a l point could be obviated, however, i f the changes i n the e x p l i c i t variables 13. would bring about the f e a s i b i l i t y of the complex. The problem, i n v e s t i -gated i n Section 3.4, i s an example of such a modified approach. On the other hand, i f the optimization problem i s formulated i n such a manner that i m p l i c i t variables are eliminated, the technique reduces to a much simpler form. This l a t t e r approach has been u t i l i z e d i n some of the problems. 2.4 Complex Algorithm Modules A concise d e s c r i p t i o n of the basic constituents of the Optimi-zation Algorithm i s given herein. Appendices A and B, res p e c t i v e l y , describe the flowchart and Fortran IV l i s t i n g of the Algorithm. The Complex algorithm i s contained i n program BOX which i n turn controls the various subroutines as shown and outlined below: BOX RANDOM IMCAL CHBDX FUN SCAN IMPROV BOX Controls the input and output and d i r e c t s the whole optimi-zation scheme. RANDOM Performs a l l the operations necessary to generate a f e a s i b l e complex. IMCAL Calculates the i m p l i c i t variables and t h e i r constraints i f necessary. In case of s t r u c t u r a l engineering problems, this i s e s s e n t i a l l y an analysis module. 14. CHBDX Checks the e x p l i c i t bounds of va r i a b l e s . FUN - Calculates the objective function. In case of s t r u c t u r a l engineering problems, t h i s i s e s s e n t i a l l y a design module. SCAN Scans the Complex to f i n d the worst point. IMPROV Performs a l l the operations necessary to e f f e c t improvement of the worst point of the Complex. CHAPTER III COST OPTIMIZATION OF REINFORCED CONCRETE STRUCTURES 3.1 Constraints Constraints,.that govern the topology and the design of r e i n f o r -ced concrete structures, a r i s e from general considerations and the s t i p u l a t i o n s of the b u i l d i n g code. The general constraints include a r c h i t e c t u r a l , f u n c t i o n a l and other constraints that a r i s e from p r a c t i c a l considerations. The range of a v a i l a b i l i t y of construction materials, the p r a c t i c a l r e a l i z e a b i l i t y of the geometric proportions of the s t r u c t u r a l components, and contem-porary p r a c t i c e s , are examples of these constraints. The constraints, that are imposed by the b u i l d i n g code, are, by and large, i m p l i c i t i n nature. The method of design, stress l i m i t a t i o n s , load f a c t o r s , minimum amount of reinforcement, and spacing of bars, are some of the examples of these types of constraints. The following parameters have been taken e x p l i c i t l y bounded: span and depth of slab: span, width and depth of beams and girders; dimension of columns; concrete strength and the y i e l d point of s t e e l . A l l clauses, except 909 (control of deflection) and 1505 (design strengths for reinforcement), of the Building Code Requirements f o r Reinforced Concrete (ACI 318-63) [2], pertaining to Ultimate Strength Design, have been s t r i c t l y adhered to, and incorporated as an i n t e g r a l part of a normal design procedure. 16. A l l structures are designed for a uniformly d i s t r i b u t e d l i v e load WL, and an added dead load WD, as well as the dead load of the structure i t s e l f . 3.2 Objective Function A general objective cost function f, of a reinforced concrete structure, consisting of slabs, beams, girders and columns, can be written as: f = ( C l + C 2 f ; ) Q 1 + (C 3 + C 4 f y ) Q 2 + Q 3C 5 + Q 4C 6 + Q 5C ? + where constants C. to C 0 are c a l l e d the cost parameters. Constants C^ to C^ per t a i n to the unit costs of concrete and s t e e l i n d o l l a r s as shown i n Figure 3.1; while constants C^ to Cg re f e r to the cost per contact square foot for various s t r u c t u r a l components as follows: C^ refe r s to the s o f f i t of slab, Cg r e f e r s to the s o f f i t of beams and gir d e r s , Cj r e f e r s to the sides of beams and gir d e r s , C D refers to the sides of columns: o and 0, to Q, are the calculated quantities per sq. f t . of the bu i l d i n g area as follows: i s the quantity of concrete i n cu. yd., C>2 i s the quantity of s t e e l i n l b s . , i s the contact area of the formwork i n sq. f t . of the slab s o f f i t , 18. i s the quantity of formwork i n sq. f t . of s o f f i t of beams and g i r d e r s , i s the quantity of formwork i n sq. f t . of sides of beams and g i r d e r s , and i s the quantity of formwork i n sq. f t . of sides of columns. 3.3 Analysis and Design Procedure The analysis i s based upon the c o e f f i c i e n t s given by Winter e_t al [10] f o r continuous prismatic sections. Wherever.necessary, there co-e f f i c i e n t s have been supplemented using s t i f f n e s s analysis. The columns are designed f o r uninaxial bending only. I n t e r i o r columns are designed ^b ^b for a moment of W^  Q^" while the e x t e r i o r columns for a moment of W^  ; where W„, i s the t o t a l load on the secondary beam, and JL i s the span of 1 b the beam. The main reinforcement of slabs and beams consists of str a i g h t bars only. The cut-off lengths of the bars are based on.coefficients given by C.R.S.I. [12]. In most of the cases,.these c o e f f i c i e n t s have been modified and rendered a l i t t l e conservative. The structures studied consist of: A. Continuous slabs on knife-edge supports. B. Continuous beams on knife-edge supports. C. Continuous one-way slabs r e s t i n g on l i n e s of beams which rest d i r e c t l y on knife-edge supports. 3.4 Examples 19 D. Continuous one-way slabs r e s t i n g on l i n e of beams supported by girders which i n turn rest on columns. In other words, th i s i s a complete one-storey structure. A l l these problems have been arranged i n the order of increasin; number of v a r i a b l e s . Each problem describes concisely the number and the form of v a r i a b l e s , and the constraints. A two stage optimization has been used i n every case, with 30 cycles i n the f i r s t stage and 50 cycles i n the second stage; except fo r structure D, f o r which the number of cycles are 60 and 100 respec-t i v e l y . The number of points i n the Complex n has been taken as m+1 i n a l l the cases; where m i s the number of v a r i a b l e s . The value"of o v e r - r e f l e c t i o n factor 0i has been taken as 1.3 f o r the f i r s t stage and 1.4 f o r the second. The basic fundamentals of Box's Complex Method have been adhere to i n most cases. Departures or modifications, i f any, have been d i s -cussed i n p a r t i c u l a r cases. The cost parameters f o r structures A to C are the same, and are c l • $17. 90 per cubic yard. C2 = $ 1. 30 per cubic yard. C3 = $ 0. 05 per pound. C4 " • $ 0. 03 per pound. C5 = $ 0. 40 per square foot C6 = $ 0. 70 per square foot 20. = $ 0.40 per square foot. The cost parameters for structure D have been varied. The r e s u l t s of optimization have been tabulated f o r each st r u c -ture. Exhaustive search has been c a r r i e d out for most of the problems and the r e s u l t s have been tabulated, and comparison made, to show the f e a s i b i l i t y of using the Complex Method. 21. A. Continuous Slabs on Knife-edge Supports The variables considered are: the number of spans of s l a b s — a n integer, the depth of s l a b — a multiple of 1/4". the crushing strength of c o n c r e t e — a multiple of 500 p . s . i . the y i e l d strength of s t e e l — a multiple of 5000 p . s . i . Figure 3.2 shows a t y p i c a l section of the structure. Table 1 describes the general data, the constraints, and the r e s u l t s of optimi-zation of various problems studied, using three d i f f e r e n t i n i t i a l points i n each case. It may be noted that the constraints imposed on the depth of the slab are not e x p l i c i t but a function of the span of the slab. The r e s u l t s , of an exhaustive search for every case, are also tabulated. Although the i n i t i a l points i n a l l seven problems are wide apart, the f i n a l r e s u l t s show that: 1. Since there i s no cost associated with added supports, and the cost of forms i s independent of the s i z e of span, the optimum design for any length of slab should have the maximum permis-s i b l e number of spans. This i s borne out i n each c a l c u l a t i o n i n Table 1. 2. the maximum difference between the lowest and the highest cost fo r these designs i s 7 percent (Problem 1); 1. N s 2. T s 3. f' s 4. f y, 22. 3. the slab thickness varies among the various designs, for most cases, within narrow l i m i t s , but i s generally near the lower constraint; 4. f and f are generally at t h e i r lower constraints; but c y i n several cases are s i g n i f i c a n t l y higher. It appears, therefore, that the hierarchy of the design para-meters i s span length, slab thickness and material properties. T s f t TLX F i g . 3.2 TLY F i g . 3.3 S e c t i o n A-A OO T A B L E 1 PROBLEM NUMBER 1 2 3 4 5 6 7 General Data TLX f t . WD #/P' WL #/o' 900 40 50 1,320 50 60 1,800 60 60 2,210 45 80 2,800 55 85 2,925 65 45 2,960 60 75 N min. s 50 60 75 85 100 90 90 N max. s 100 120 150 170 200 195 185 CO f' min. c 2,000 2,000 2,000 2,000 2,000 ; 2,000 2,000 +J •H CJ 4 J C" •H. f max. c 6,000 6,000 6,000 6,000 6,000 j 6,000 6,000 •l-t iH H CX 4-1 f min. V 30,000 30,000 30,000 30,000 30,000 30,000 30,000 . m C o J f max. V 80,000 80,000 80,000 80,000 80,000 80,000 80,000 -CJ J T min. s. . - " T max. s £ /35 s I /20 s £ /35 s £ /20 s £ /35 s £ /20 s £ /35 s £ /20 s " £ /35 s £ /20 s £ 735 ' s.. £ /20 s £ /35 s £ /20 s N • ~ s 60 80 100 130 140 150 160 rH to f . c 4,500 5,000 4,000 4,500 4,000 4,500 4,000 rH .. O 'rl 4 J •H C rH f . y T s 75,000 6.00 60,000 . 8.25 45,000 8.50 65,000 9.00 65,000 9.75 65,000 8.25 65,000 9.25 S5 N s 100 120 150 170 200 195 185 4 J PH . rH rt C •H-f c 2,000 . 2,000 • 2,500 2,000 2,000 2,000 2,000 f v 35,000 30,000 80,000 30,000 30,000 30,000 30,000 J • T s 3-25 4.50 4.50 5.50 5.50 5.25 6.50 N s — 60 80 100 130 140 150 160 SCHEME rH rt f c 5,000 4)500 . 4,500 3,500 4,500 6,000 4,000 . SCHEME CM O z •H 4 J •H c M f y T s 60,000 6.50 55,000. 8.00 - 65,000 8.00 65,000 8.75 55,000 9.50 45,000 8.25 45,000 8.75 1 1 N s 100 120 150 170 200 195 185 PH i f c 2,000 2,000 '2,000 2,000 2,000 6,000 2,000 171 rt c Tl j f V 35,000 30,000 30,000 30,000 30,000 40,000 30,000 J T s 3.50 4.00 4.75 5.50 5.50 5.25 6.50 o 1—1 - .. N s 60 80 100 130. 140 150; 160. OPTIMIZAT: CO o rH « •rl U •rl a rH f' c f •y T s 3,500 65,000 6.50 3,500 45,000 7.50 4,500 50,000 7.50 5,500 70,000 8.75 3,500 45,000 8.75 3,500 55,000 7.75 3,000 70,000 8.50 z N s 100 120 150 170 200 195 185 ; cn +J P* rH rt a •rl • rU ' f V c 6,000 2,000 2,000 2,000 2,000 2,000 2,000 BOX f y •-T s 35,000 3.25 30,000 \ 4.00 30,000 4.75 30,000 5.50 30,000 5.50 30,000 5.25 35,000 6.00 Exhaustive N s f * c 100 2,000 120 2,000. 150 2,000 170 2,000 . 200 2,000 195 2,000 . 185 2,000 Search f y 35,000 30,000 30,000 30,000 30,000 30,000 30,000 J • • T . s 3.50 4.50 . 4.75 5.50 5.50 5.25 6.50 OST/ SFT. Optimi-zation SrhpiTip Pt.No.l Pt.No.2 Pt.No.3 0.96131 0.96131 1.02650 1.09972 1.12109 1.12109 1.24297 1.17211 1.17211 1.26555 1.26555 1.26555 1.34991 1.34991 1.34991 1.30396 1.37564 1.30396 1.45247 1.45247 1.46705 -u Search Procedure 0.96131 1.09972 1.17211 1.26555 1.34991 1.30396 1.45247 25. B. Continuous Beams on Knife-edge Supports The variables considered are: the number of spans of beams—an integer, the depth of beam—an integer, the width of beam—an integer. the crushing strength of c o n c r e t e — a multiple of 500 p . s . i . the y i e l d strength of s t e e l — a multiple of 10,000 p . s . i . Figure 3.3 shows a t y p i c a l section of the structure. Table 2 summarizes the general data, the constraints and the r e s u l t s of optimi-zation of various problems studied, using three d i f f e r e n t i n i t i a l points i n each case. The r e s u l t s , of an exhaustive search, have also been tabulated. Although the i n i t i a l points i n a l l eight problems are widely d i f f e r e n t , the f i n a l r e s u l t s show that: 1. as i n the case of continuous slabs, the optimum design, i n a l l cases, converged to the upper constraint on N^; 2. the lower constraint on B, has been reached i n a l l but one b case, and t h i s i s near the lower constraint. This i s to be expected, since i t produces a minimum volume of concrete; 3. the maximum difference between the lowest and the highest cost i s 11.3 percent (Problem 4); 4. the beam depth varies among the various designs, f o r most cases, within narrow l i m i t s ; 26. 5 . f' and f are at t h e i r lower constraints, but, i n several c y cases, are s i g n i f i c a n t l y higher. Hence, the hierarchy of the design parameters i s span length, beam width, beam depth and.material properties. COST/FT. co n> cu H O 3* TJ H O n rt> n. c H CD CO N O 3* rr rt (0 H» H« s p a I CO (D 3* o c cn rr < CD BOX'S OPTIMIZATION SCHEME Pt. No. 3 F i n a l I n i t i a l Pt. No. 2 F i n a l I n i t i a l Pt. No. 1 F i n a l I n i t i a l E x p l i c i t Constraints o fD O 3 cu ro rt i-( 0) pi o os r1 *rj hrj a a a o o o • • • U S J P cu _ H cr cr Hj Hi 2! r i - c r bd H Hi H i 2S c r c r o - c r bd H Hi H I 3 c r c r ^ o - c r w H H I H I a a* c r vs o - c r as H H I M I a c r c r v ; n - c r bd H H I Hi a c r c r ^ o - c r bd H h c r c r bd bd H H i-c f c r c r c r ^ H I H I a a o - o - c r c r 9 X 9 9 3 X 9 3 9 9 (D H-X 3 9 9 9 3 X 5 4> O v l O -P-Ln -P-O ON •P-•P- O -P-00 OJ o •P-O •P-O •P-00 OJ o ro to -P-o oo o O0 Ul ON o to o V I o 00 o ON tO ON •p-ro ui ui NO co oo o o o o o o o o o o o o o o o o o o o o o o o o O Ui o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o O Ui o o o o o o o o o o o o o o o o o o o o o o o o ON o N ) H H O Ui VO O O to o o o to to to v l o to Ul OJ M O to Ul •p-o OJ o OJ o OJ o -P-00 Ul o •p-00 OJ o OJ o ON o OJ Ul ON o to o o OJ o V ] Ul vo v l v j N M O o o o o o o o o o o o o o o o o o o o o o o o o o o o Ul o o o o o o o o o o o o o o o o O Ui o o ON o o o o o o o o o o U l o o o o o o o o o Ul o ui o o o o o o o o o o o o o o o o o o o o o o o o to M OO O Ul ON o o o o o o oo -t- to to -p-Ul v l o Ul VO v l •P- ro vo to 00 00 o o o o o o o o o o o o Ul ON o o ON o o o o o o o oo o o o Ul •p-o o o o o o Ul M O tO o o o o Ul ON o o OJ o o o o o o o OJ o o o Ul OJ o o V I o o o o o o o o o -p-o o v l o o o o •P-o o o OJ ro o o Ul o o o OJ o o o o OJ Ul to ON o ro o v l o OJ o o o o o o o o o o o o o o o o o o o o o O o ui o o O Ul o o o o o o o Ul ro V l o •p-CO O M ro OJ ON o o o o o o o o o o o o ON o o o Ul o o o o U l o o OJ Ul o o U l U l o o U l o o o o o O CO O -P-o o ON o o o OJ o o o o o o o o o •P-VO o o -p-O Ul O Ul O o o o o o ON o o o OJ o o o o o o o OO O 0 o o Ul o o o 4> o o o o oo Ul ON o ro o o OJ o Ul o o o o o o o o o o o o o o o o o o o o o o ON 00 ro H v i O Ul Ul O O ON o o o ON ro Ul Ul o O -P- -P-ON 00 OO o o o o o o o o o o 00 o o Ul Ul o o ON o o o o U l o o OJ Ul o o Ul Ul o o Ul o o o o o o o to o o ON o o o OJ o o o o Ul -P-O to O O OJ U l o o Ul Ul o o -p-o o o o ON OJ o o o OJ o o o o o o o o o o o •P-Ul -P-O tO o o o o U l OJ o o ON o o o o OJ Ul ON o ro o o OJ o o OJ O Ui o o o o o o o o o o o o o o o . o o o o O -P-o , ro O , o O Ul o o o o o o o ON V I v j M to to Ul ON o OJ to ON OJ o to Ul 00 00 •p-o ro 00 -p-o ro o OJ ON OJ o ro ro 00 OJ o to Ul o Ul o OJ o Ul o ro o VI o OJ o VO to to to ON ro v i o o o o O Ul M O O -P-O O O O O O O O O o o o o o o o o o o o Ui h-i o o o o o o o o o o o o o o o o o o o ui o o o o o o o o o o o o o o o o o o o o o o o o o o . o o o o o o o o o o o o o o o o o o -p-o O ro oo o o o o o o ON VO -P-OJ -P- -P-H' tO IO o o ro Ul o o v l o o o o OJ w o o o o o to v l o o OJ o o o o o o o ro Ul o o OJ Ul o o v l o o o o Ul O ON o o o o OJ o o o OJ o o o o o o o ro o o o OJ ON o o ON O P-O Ui o o o o 00 o OJ o ro ro 00 to Ul o 00 o Ul o to o V I o OJ o o o o o o o o o o o o o o o o o o Ul o o o o o o o o o o o o o o o o o o o o o o to o Ln Ul H O O ON O O O to Ul V I o OJ to 00 OJ o o to Ul -P- Ui o o OJ 00 o o ro Ul OJ Ul ON o IO VD -p-o ro 00 OJ 00 j v o OJ o Ul o ro o v l o OJ o ON OJ o v l v l Ul VO VO v l o o o o O Ul M O O Ul O O O O O o o o o o o o o o o o o o o o o O v l O Ui o o o o o o o o o o o o o o O Ul h-1 O O O o o o o o o o o o o o o o o o o o o o o Ul M o o o o o o o o o o o o o o o o o o o o o o o o Ul o H H U 00 OJ OJ o o o o o o •LZ 28. C. Continuous Slabs Supported on Line of Beams (Beams rest on knife-edge supports) This structure i s the combination of structures A and B. . The variables considered are: the number of spans of s l a b — a n integer, the depth of s l a b — a multiple of 1/2". the number of spans of beams—an integer, the depth of beam—an integer, the width of beam—an integer, a multiple of 500 p . s . i . a multiple of 10,000 p . s . i . Figure 3.4 shows the plan and a section. The optimization procedure, of various problems studied, can be divided into two cases: I. where a l l the basic s t i p u l a t i o n s of Box's Complex Method are unaltered; and I I . where some deviations have been made. .CASE I Table 3 summarizes the general data, constraints and the r e s u l t s of optimization of various problems studied, using three d i f f e r e n t i n i t i a l points. The r e s u l t s of an exhaustive search, for some problems, have also been tabulated. Because of the p r o h i b i t i v e computing cost of carrying out an exhaustive search, such an undertaking had to be l i m i t e d to few problems only. Section A-A to VO 30. CASE II Table 3A summarizes the r e s u l t s of optimization, of some of the problems studied under Case I, using alternate or modified formulation of the Complex Method, when the necessity of the i n i t i a l points i s done away with. As discussed i n Section 2.3, the necessity, of an i n i t i a l f e a s i b l e point, can be obviated provided that the s a t i s f a c t i o n of i m p l i c i t con-s t r a i n t s i s guaranteed i n easy yet l o g i c a l fashion,' by e f f e c t i n g changes i n one or several of the e x p l i c i t v a r i a b l e s . The i m p l i c i t variables a r i s e from f l e x u r a l and shear considerations and are: 2 y^ = Bd and = Bd, where B and d are, r e s p e c t i v e l y , the width and the e f f e c t i v e depth of the concrete section of the s t r u c t u r a l component. For a p a r t i c u l a r layout and the material properties, an easy yet l o g i c a l way to s a t i s f y y^ and y^, simultaneously, would be to e f f e c t changes i n the si n g l e parameter d. Should d (or T, the t o t a l depth) reach i t s upper constraint, y^ and y^ could s t i l l be s a t i s f i e d by increasing B. Thus the Complex could be rendered f e a s i b l e . An a l t e r -nate approach, though not explored, could be to e f f e c t f e a s i b l e changes i n N g or i n d i v i d u a l l y or i n combination. A c r i t i c a l examination of r e s u l t s of optimization tabulated i n Tables 3 and 3A shows that: 1. Upper constraint of N i s the most active one. 31. 2. Slab thickness reaches i t s lower constraint. 3 . Beam width B, reaches i t s lower constraint. b 4. Number of spans of slab (N g) do not h i t the upper l i m i t , suggesting that any increase i n N , beyond c e r t a i n l i m i t , would c a l l f o r added l i n e or l i n e s of beam, which would penalize the objective function. 5. and f are generally at t h e i r lower constraints, but are s i g n i f i c a n t l y higher i n several cases. 6 . The maximum difference between the lowest and the highest cost for the designs i s one percent. 7 . The depth of beam does not reach either the upper l i m i t or the lower l i m i t . I t i s concluded, therefore, that the hierarchy of design para-meters i s span of beams, slab thickness, beam width, slab span, material properties and the beam depth. The objective function i s the cost per sq. f t . of the covered area. COST/ SFT. N O rt rt 2 3 I z a o o i-h H> bd H i-3 ^ O - c r cn U tO H M r - 1 ! — * ON ON ON Ul Ul 4> Search Technique z a c r to X-X-X-X-X-X-X-X-X-X-X-X-X-X-X-X-X-X-BOX'S OPTIMIZATION SCHEME Pt. No. 3 F i n a l i-h cxl H H 2 Z o - c r a* cn c r co ^ o OJ o SO NJ U l U l O O O o o o o o o o o o CO . I n i t i a l W H H Z Z c r c r to c r cn Mi Mi M H H Z Z ^ o - c r c r to •* » ON o ro o 00 o o o o o o o o o o o o VO ON Pt. No. 2 F i n a l M I Hi W H H Z Z •< O - c T c r c o c T c o 00 o ro ro O O O O O O O O o o o o M ro ro m o m I n i t i a l Mi Mi ^ o - c r Ui o Ui ro o o 00 o o o o o o o o o o o o o o Pt. No. 1 F i n a l z z T CD Oo O ro I-1 ro ro ON Ul J> o O O o O O O O O o o o o ro o ro ON o o o o I n i t i a l M , rh W 1-3 H Z Z *-< o - c r c r c D c r to E x p l i c i t Constraints k M> H , Ml Mi W W H H 1-3 f-3 g" 5 * *?> M <<: o - o - cr cr cr cr co co cr cr t o 3 a> 3 H-3 3 03 . 3 3 3 X 3 - 3 3 - 3 03 03 3 X 3 X 03 X 00 -P-ro U l o o o o o o ui ro o vo o o o o oo o ON ro ro 4> Ui O ro O O O O O O O O O O O O O O O O O O o o o o o o o ro O ro M ON 00 -P* o o o o (0 O 3 03 CD rt r( 03 03 § s: H H P t) f f ;»<_ X I] O f t rt g t i l W 2 td W ro to «. »# •P- oo O ON o o o o o o M M M ON ON ON 4> Ul Ul o to to to Ul Ul o o o o o o o o o o o o ON o oo M ro O to K> ON Ul o o o o o OO o o o o o ro o o ON o •p-o -P- H1 00 -p-ro o o o U l o o O O O O On O -p-o to ro O ON o Ul to o -p-00 •p-Ul o o o Ul o o o o o o o o ON o O Ui o o o o o o o o o o -p-00 oo o to to to Ul Ul poo o o o o O o O O ON o Ul o Ul ro o J> o CO O o O O o • o o o o o o o Ul o h-1 O Ul ON O Ul to o -p-00 VO o O o Ul o o o o o o o o o OO o ON to to J> to Ul o to o o Ul o o o o o o o o o o o o o o o o o o o o o o o o ON o oo o o h-* NO J> O ON o o o o o o ON ON ON -p- Ul Ul X-X- 3f * 00 o to to to Ul .p-o to O 0 o h-> to ro ui u i O ro o -p-o o o o o o o o o o o o o r-1 00 00 VO O --4 O Ln O O O O O o o o Ul o 00 Ul o o o o o o o o o o o o CO o o o o o o o o o o o o o M 00 Ui M O U i 00 o ro H1 NJ to r-1 Ul o o o Ul o o o o o o o o 00 o h-1 ro -P- o o 00 o ON ro ro J> to Ul o to o h-1 o Ul o o o o o o o o o o o o o o o o o o o o O O O O h-1 00 O vo ro O O O oo J> ** ^ M ON NJ o ON o o O O O o Ul Ui Ul VO 00 00 -P-O ro M to to . M Ul o o o o o o O O O O O O ON o 00 o to to ro -p- Ul o o o o o o o o o o o o ON 00 O vo ON o 00 I-1 -p-ON Ul CO o Ul . • . o o o o Ul o o o o o Ul to OJ o to M to ro 00 Ul o o o o o o o o o o o o ON o Ul o Ul M O0 CO O0 o to r-> to to Ul On o o o Ul o o o o o o o o Ul o 00 o o o o o o o o o o o o o ON o CO Ul o o co o o o o o o o o o o o o co o O0 o ON to to o to Ln O to O M O Ln o o o o o o o o o o o o o o o o O O O O O O O O ON O oo o o o ro o co u i o o o o o o oo o Ul Ul Ul CO ~-J Jr * Jr to 00 Ul o 00 CO o o o o o o o o o o o o M NJ CO Ul O OO o ui o o o o o o o o o o Ul o to .p-o ON ON ON h-1 I—1 O X-JS-X-X-X-X-X-X-X-X-X-X-Ul o ro .p-o o o o o o o o o o o o ro -p-o o o o o o o CO o o o o o o o h-1 Ul .p- -p- .p-Ul ON Ul o ro o ro Ul o o o o o o o o o o o o ON o 00 o ro M O to to Ln o o o o o o o o O O O O ON o ON oo o 00 ON CO o Ul o o Ul o o o o o o o Ul NJ UJ O O o o o o o o o NJ 00 o o O J> o o M NO CO Ul O 00 o o o o o o co o o o o o CO O M NJ O 4> NO -P- ON oo O o o o o o o o o NJ Ul o o o o r-1 NJ 00 Ul o •> ON O 00 o u i o o o o ON o o U l o o o 00 o ON NJ NJ h-* O NJ Ul NO I—1 o o o CO Ul o ON o c OJ O OJ NJ o o o o o o o o o o o o o H1 CO ON 4> o 00 ON Ul oo o Ul o o Ul o o o o o o o ON OJ o NJ (-1 NJ NJ Ul Ul o o o o o o o " o o o o o o "1= C T " 00 00 O Ui o o o o o o o o o o ON 0J o NJ M O NJ NJ Ul o o o o o o o o O o o o Ul o Ul Ul OJ Ul ON o O Ui o o o o o o o o U l o u i o 4^ O NJ h-1 o NJ I—1 Ul O o o Ul o o -o o o o o o o OJ to ON o o o o o o o o o o o o o 00 L o o o o o o o o o o o o o o o o o o o o o o o o 00 o OJ H" VO O Ul O O O 00 O0 ON O oo m o o o o o o o o o o O0 o o o o to o I—1 Ln NJ NJ O O o o o o o o o o o o o o o o o o o o 00 VO 00 o oo oo 00 U l ON o o o o o ON o OO o ON NJ o o o o o o o o o o o o NJ o M O Ui O M NJ I-1 o Ul o o O O O O O O o o o o ON o OJ o o o NJ O Ui OJ o o o o o o TABLE 3A PROBLEM NUMBER GENERAL DATA VARIABLES N s Nb T s Bb f * c f y COST/ SFT. 2 o o ^» =s= O O =tt= <r CM o rH rH O O I  I  VO i-H XI p-i I  I  I-l h J p H . H S B Constraints min. max. 70 140 30 60 5.00 10.00 20.0 50.0 12.0 24.0 2,000 6,000 30,000 70,000 I n i t i a l i -zers 0.22331 0.77885 0.55664 131 128 129 60 60 60 5.00 5.00 5.00 25.0 22.0 23.0 12.0 12.0 12.0 2,000 2,000 2,000 30,000 30,000 30,000 1.65 1.64 1.64 4 a a o o ~-o o =s= =«= O CN rH rH O O II I  LO oo X! >-i II II i J HJ Q i-l H H [3 JS Constraints min. max. 50 100 30 60 5.00 10.00 20.0 50.0 12.0 20.0 2,000 6,000 30,000 70,000 I n i t i a l i -zers 0.22331 0.77885 0.55664 \ 90 85 85 N X 60 60 59 5.00 5.00 5.00 21.0 23.0 21.0 12.0 12.0 12.0 2,000 2,000 2,000 30,000 30,000 30,000 1.59 1.58 1.60 7 • "a o o ^» o o =s= =s= O CN rH rH O O II || CO LO X >-< II II I J h i fi J H H 12 ts Constraints - 1 . • v - — r ^ " min. max. * 50* 100 30 60 5.00 10.00 12.0 50.0 10.0 20.0 2,000 6,000 30,000 70,000 I n i t i a l i -zers 0.22331 0.77885 0.5.56,64 x 75 75 75 60 60 59 5.00 5.00 5.00 21.0 21.0 22.0 10.0 10.0 10.0 2,000 2,000 2,000 30,000 30,000 30,000 1.45 1.45 1.46 34. D. One-storey Structure The structure studied, herein, consists of one-way slabs, beams, girders and columns. It i s subjected to uniformly d i s t r i b u t e d gravity load only. The variables considered are: 1. N g the number of spans of g i r d e r s — a n integer. 2. Nb the number of beam-spans—an integer. 3. T g the depth of g i r d e r s — a n integer. 4. B g the width of g i r d e r s — a n integer. 5. T b the depth of beams—an integer. 6. Bb the width of beams—an integer. 7. T s the depth of s l a b — a multiple of 1/2". 8. B c the l e s s e r side of column—an integer. 9. T c the greater side of column—an integer. 10. f' c the 500 crushing strength of c o n c r e t e — a multiple of p . s . i . 11. f the y i e l d point of s t e e l — a multiple of 10,000 p.s The general data of the structure i s : TLX = 400.00 f t . , TLY = 360.00 f t . WD = 50 l b s / s q . f t , , WL = 80 lbs/sq. f t . The cost parameters used are: c l $17.90 ; s - $ 0.60 C2 $ 1.30 ' C6 = $ 1.00 C3 $ 0.11 ; c ? = $ 1.00 C4 0.01 ' C8 = $ 0.75 35. The s o l u t i o n procedure i s b a s i c a l l y s i m i l a r to Case I I of the s t r u c t u r e C, that i s to say, no i n i t i a l p o i n t s have been used. The Complex i s rendered f e a s i b l e by e f f e c t i n g changes i n T , B , T, , B , T , B and T . s c c Table 4 describes the r e s u l t s of the o p t i m i z a t i o n procedure when the technique i s i n i t i a t e d using d i f f e r e n t random number bases. No exhaustive search has been c a r r i e d out to a s c e r t a i n the e f f i c i e n c y of the Complex Method. The program explores two p o s s i b i l i t i e s , namely, secondary beams at the t h i r d and at the quarter p o i n t s of the g i r d e r s . Figure 3.5 shows the plan and a s e c t i o n of the s t r u c t u r e when the beams are at the quarter po i n t of the g i r d e r s . A s i m i l a r arrange-ment can be v i s u a l i z e d when the beams are at the t h i r d p o i n t s . A clo s e look at Table 4 b r i n g s i n t o sharp focus the f o l l o w i n g p e c u l i a r i t i e s : 1. Secondary beams at the t h i r d p o i n t of g i r d e r s produce a somewhat cheaper design. 2. The number of spans of beams always reaches the upper l i m i t . 3. The number of spans of g i r d e r s does not n e c e s s a r i l y reach the upper l i m i t . 4. Depth of slab reaches the lower l i m i t . 5. Width of beams reaches i t s lower l i m i t i n almost a l l cases. 6. Width of g i r d e r s reaches the lower l i m i t . 7. The depths of beams and g i r d e r s , and the column dimensions, show a wide s c a t t e r . OQ era x J era U l u u u u u u t. S e c t i o n A-A v i 38. 8. f tends to reach i t s upper constraints. y . 9. f f o r most cases, varies between 4,500 p.s.i'. and c 5,500 p . s . i . 10. The difference between the best design, as regards the cost per sq. f t . of covered area and the worst design i s : 4 percent when the secondary beams are at the t h i r d point of gird e r s ; 2 percent when the secondary beams are at the quarter point of girder. Hence from the foregoing i t appears that the hierarchy of design parameters i s configuration, number of spans of beams, depth of slab, widths of beams and gi r d e r s , number of gir d e r s , depths of beams and gird e r s , material properties and the column dimensions. Cost Parameter Study The cost parameter study of structure D has been made. Table 5 describes the parameters. Table 6 and 6A summarize the f i n a l r e s u l t s of the optimization scheme, using four d i f f e r e n t i n i t i a l i z e r s i n every case. Table 5A shows the f i n a l values of the cost per sq. f t . corres-ponding to various cases. Changes i n concrete costs of about 20 percent (Type V) produced a change i n t o t a l cost of about 2 percent, while change i n s t e e l cost of about 35 percent (Type VI) produced a change i n t o t a l cost of about 7 percent. INITIALIZERS 0.22331 F i n a l P o i n t s CO rt CO ro •P- o rt • 3* • cr t3 ro rt Co co g • cn Ln o to , ro to o ON to o ro . J V ro Ln O o o ON o o o o 0 0 o Co rt CO • ro OJ o r i • a . • cr ro rt fo cn g • cn Ln O ro OJ O J o -P-•P-to o to -p-ro -P-Ln Ln O O v l O o o o ON Ln 0.55789 F i n a l P o i n t s r t co ro -p- o rt • t r • cr 13 (D r t Co cn g • cn Ln o ro O J ro o ON OJ to o ON o tO  Ln O O ON o o o o v l 0 0 Co rt CO ro O J o H • Cu • cr T3 ro rt Co cn g Ln o to ON to o -p-VO -p-o O J o VO o Ln o o v l o o o o ON 0.78850 F i n a l P o i n t s Co rt CO ro •P- o r t • a' • cr T3 ro rt Co cn g • cn Ln o IO o ro O •P-ON ro o h-1 — I o ro I—1 o o o o Ln O O O O 0 0 O Co r t co ro OJ o H • CU • cr T3 ro r t co cn g • cn Ln o to ON to o L n OJ to o 0 0 o to o Ln O O ON o o o o VI OJ Co rt CO ro -p- o r t • • cr u ro rt co cn g • cn 0.45321 F i n a l P o i n t s L n o to ON ro o ON ro to o O J o to VD Ln O O v l O O o o v l ON Co r t CO ro oo o i-j . cu • cr T3 ro rt co cn g • cn Ln o to oo O J o •p-Ln to O ro o ro o o o o ON o o o o ON O J n o P cn r t H Co H-3 g Co O o Ln O IO o 0 0 o to -P-to -P--P-o ON o o o v l o o o o B 3 Ln O to O to O O J o to o to o 0 0 o o o o -P-o o o o > M r 1 M CO cra cr TO oo td O O CO H CO H > t"1 M •9C TABLE 5 Para-meters Type c l C2 C3 C4 C5 C6 C7 C8 I 17.90 1.70 0.11 0.01 0.60 1000 1.00 0.75 II 17.90 1.30 0.11 0.03 0.60. 1.00 1.00 0.75 III 17.90 1.30 0.11 0.01 0.60 1.50 1.50 0.75 IV 20.00 1.30 0.11 0.01 0.60 lvOO 1.00 0.75 v 15.00 1.30 0.11 0.01 0.60 1.00 1.00 0.75 VI 17.90 1.30 0.11 0.01 0.60 1.00 1.00 0.75 TABLE 5A Case Parameter Changed F i n a l Values (Cost/SFT.) Best Point I C2 2.71-II C4 • 3.22-III C6' C7 2.97 . IV C l 2.71 V C l 2.58 VI Co .2.81 3 Ill i—i H Type I n i t i a l i z e r s I n i t i a l i z e r s I n i t i a l i z e r s Constraints VARIABLES 0.44225 0.22113 0.45321 0.78635 0.44225 0.22113 0.45321 0.78635 0.44225 0.22113 0.45321 0.78635 Constraints VARIABLES 1/3 pts. 1/4 pts. 1/3 pts. 1/4 pts. 1/3 pts. 1/4 pts. 1/3 pts. 1/4 pts. 1/3 pts. 1/4 pts. 1/3 pts. 1/4 pts. 1/3 pts. 1/4 pts. 1/3 pts. 1/4 pts. 1/3 pts. 1/4 pts. 1/3 pts. 1/4 pts. 1/3 pts. 1/4 pts. 1/3 pts. 1/4 pts. min. max. VARIABLES 1—1 VD O h-1 VD O I—1 M O O VD O h-1 M O O h-1 VD o I—1 h-1 o o O O t—1 M O O VO O O O O VO h-1 O Ui a 1—1 t—1 NJ NJ 1—1 h-1 O NJ h-> h-1 NJ NJ NJ NJ H> NJ NJ M 1—1 NJ NJ 1—1 h-1 NJ NJ M M NJ NJ M f—1 NJ t—1 h-1 t—1 O O 1—1 M NJ M h-1 h-1 NJ NJ h-1 NJ 00 a cr Ul Ul o o Ul Ul o ui Ul Ul O O Ul Ul O Ul Ul Ul o o Ul Ul O Ul Ul Ul o o On On O O Ul Ul o o Ul Ul O O Ul Ul O O Ul Ul O Ul 5.0 10.0 H CO 22.0 21.0 25.0 27.0 24.0 20.0 24.0 22.0 30.0 20.0 24.0 25.0 23.0 21.0 23.0 21.0 28.0 20.0 24.0 25.0 25.0 20.0 24.0 25.0 20.0 50.0 H 14.0 12.0 1.2.0 12.0 12.0 15.0 12.0 13.0 12.0 13.0 12.0 14.0 12.0 14.0 15.0 13.0 12.0 14.0 18.0 16.0 12.0 14.0 j 12.0 12.0 12.0 20.0 cd cr 45.0 53.0 51.0 68.0 45.0 54.0 46.0 57.0 55;0 47.0 47.0 61.0 50.0 52.0 48.0 57.0 52.0 52.0 54.0 69.0 47.0 49.0 47.0 62.0 30.0 80.0 H 00 12.0 12.0 12.0 12.0 12.0 12.0 12.0 12.0 12.0 12.0 12.0 12.0 12.0 12.0 12;0 12.0 12.0' 12.0 12.0 12.0 12.0 12.0 12.0 12.0 12.0 24.0 bd 00 16.0 13.0 13.0 21.0 12.0 24.0. 12.0 12.0 12.0 18.0 12.0 15.0 23.0 14.0 18.0 14.0 12.0 18.0 -O vl o o 12.0 14.0 13.0 14.0 12.0 24.0 bd n 23.0 18.0 18.0 21.0 18.0 24.0 19.0 24.0 24.0 18.0 18.0 25.0 23.0 21.0 18.0 23.0 18.0 18.0 18.0 20.0 19.0 23.0 18.0 20.0 18.0 40.0 H n 5,000 5,500 4,500 5,000 5,000 6,000 5,000 5,000 3,500 6,000 5,000 4,500 6,000 5,000 4,500 4,500 4,500 6,000 5,500 5,000 5,500 5,500 6,000 3,500 3,000 6,000 l-h o -60,000 50,000 60,000 60,000 60,000 70,000 50,000 50,000 60,000 50,000 50,000 40,000 60,000 60,000 60,000 50,000 60,000 60,000 60,000 70,000 60,000 60,000 70,000 60,000 40,000 70,000 Hi VI 2.97 3.15 2.99 3.29 2.93 3.17 3.04 3.21 3.22 3.39 3.30 3.53 OJ OJ NJ OJ ON O 3.27 3.37 2.74 2.84 2.88 2.94 NJ NJ CO vl M H1 2.74 2.82 i i COST/SFT. I n i t i a l i z e r s NJ to OJ OJ Ol Ol vi oo VO v l 00 00 Ol o Ol OJ NJ I n i t i a l i z e r s NJ NO OJ OJ Ol Ol v l 00 VO v l oo 0 0 Ol o 4> Oi 0 J NJ I n i t i a l i z e r s NJ NJ OJ OJ Oi Ol v l 00 VD v l 00 00 Ol o Ol OJ NJ H V! Tt ft) O O 3 cn rt OJ H-3 < > I—I > f W CO 4>.Oo Tl T) rt rt cn cn H1 M -P- OJ T) rt rt cn cn t—1 M 4> OJ Tl Tl rt rt cn cn -P- OJ Tl T) rt rt cn cn -P- OJ Tl T) rt rt cn cn M I—1 -P- OJ T) Tl rt rt cn cn -P> OJ TJ T) rt rt cn cn -P- OJ |D T3 rt rt cn cn -PvOJ T) Tl rt rt cn cn 4>- OJ td Tl rt rt co cn I—' I—1 4>- OJ •a T) rt rt cn CD 4>- OJ •d Tl rt rt cn cn B g Ol H' X 3 vo O VD O o o o o o o VD O o o o o o o VD O VD O O O O Oi 00 h-1 M Ol Ol Oi On Ol Ol Ol Ol On On Oi Oi On On Oi On Ol On O O O O o o o o o o o o o o o o o o o o o o o o O Oi o o NO NO 00 NJ NJ NJ •P- v l NJ NJ M 00 NJ NJ O Oi NO NJ M ON NJ NJ On vl NJ NJ On Oi NJ NJ O OJ NJ NJ O OJ NJ NJ OJ On NJ NJ NJ Ol NJ NJ On Ol NJ O O o o o o o o o o o o o o o o o o o o o o o o o o o o NJ -P- 4>- NJ CA 0 J OJ NJ NJ h-1 O NJ o o o o o o o o o o o o o o o o o .o o o o o Oi 4 v ON v l On Oi vo CT* Ol h-1 v l Ol j v NJ 00 Ol Ol v l NJ ON Ol 4>- M ON Ol OJ 0 J Ol 4> NO 00 Ol -P-NJ 4> Ol -P-Ol 4S ON 4 ^ O VD Ol 4V 4>- 00 00 OJ O.O O O O O O O O O O O o o o o o o o o o o o o o o o o 00 OJ NO o o OJ OJ NJ h-> 4> NJ o o o o o o o o o o o o o o o o o o o o 00 td NJ 00 4V OJ 4>- NJ I—1 OJ NJ NJ NJ 4?- 4V h-1 NO -P- M ON 00 NJ h-1 hr> ON NJ NJ NJ 4>- 00 4 v NJ h-1 4 ^ NJ o o o o o o o o o o o o o o o o o o o o o o o o NJ NO OJ 4> NJ NJ I—1 I—1 NJ NJ OJ h-1 NJ h-> OJ VO NO NJ 4> 4V M NJ VO OJ NJ NJ NJ h-> NJ NJ H1 OJ NJ NJ NJ 4> NJ NJ o o 1—' NO vo o NO 00 00 4 ^ h-1 O 00 O O O O o o o o o o o o o o o o o o o o o o o o Cd Oi Ol ON -P- 4V 00 4> Oi Ol Ol On Oi 4 v Oi •P- -P-O On O O O O O Oi O O o o O Ol O O o o O Ol O O o o o o o o o o O Ol O O o o O Ol o o o o O Ol o o o o O Ol O O o o On Ol o o o o Oi o o o o o On Oi o o o o o o o o o o Ml o -V l V I o o ON ON o o ON ON o o v l ON o o v j V I o o V l ON o o V l ON o o V l ON o o ON v l o o ON ON o o Ol Ol o o ON ON o o v l 4> o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o VO 00 O Ol VO 00 I—1 On VO 00 O v l VD 00 4> H" ON Ol 00 VD ON ON Ol NJ ON VI VD NO ON Ol ON 00 00 v l v l h-1 00 v l 01 NJ VD v l O VO v l VI v l o o o C/3 H CO CHAPTER IV VOLUME OPTIMIZATION OF PLANE TRUSSES 4.1 Constraints Constraints that have been imposed are i n the form of l i m i t s on the bar areas and the nodal co-ordinates. Constraints on the allowable stresses are ei t h e r e x p l i c i t l y defined or i m p l i c i t l y r e l a t e d to the Slenderness r a t i o (S.r.) of the bars as dictated by A.I.S.C. Manual [11]. For the sake of s i m p l i c i t y , pipes with a diameter to thickness r a t i o of 20, have been selected for the bars. If a i s the cro s s - s e c t i o n a l area of a bar and L i t s length, then Slenderness r a t i o (S.r.) becomes: S.r. 1.09 — (4.1) The allowable compressive stress 0"C i s ac - [ l - (• ) ] F (4.2) F.S. i f i n e l a s t i c buckling i s the c r i t i c a l condition, or a c 149,000 ( S . r . ) 2 (4.3) i f e l a s t i c bucling i s c r i t i c a l , where 43. C c E Modulus of e l a s t i c i t y of s t e e l , F y Y i e l d strength of s t e e l , F.S. Factor of safety and i s given by F.S (4.4) The maximum allowable t e n s i l e stress CJ i s given by o t = 0.66F (4.5) y 4.2 Objective Function The objective function f, i s the volume of the structure. It i s a function of cross-sectional areas and the nodal co-ordinates, and i s given by where M = t o t a l number of bars. 4.3 Solution Procedure The S t i f f n e s s Method has been used f or the analysis of the st r u c -tures. Since the Complex Method i s a constrained optimization procedure, the bounds on the areas of the members are a r b i t r a r i l y set to f a c i l i t a t e the generation of a Complex. f = M £ a.L. -i " i i (4.6) 44. The s o l u t i o n procedure i s as follows: 1. Generate the Complex with the bar areas and the nodal co-ordinates as variables using the a r b i t r a r i l y chosen bounds; 2. Since the points generated may not s a t i s f y the i m p l i c i t con-s t r a i n t s of s t r e s s , an adjustment i s necessary. This has been made i n a manner s i m i l a r to that by Estrada [14], as follows. An analysis i s performed f o r each design, and the r e s u l t i n g stresses i n a l l bars are calculated. These are then compared with the allowable stresses f o r each bar, and the bar with the greatest overstress i s selected i n each case. A l l the areas are then increased by the r a t i o of this maximum actual stress to the allow-able s t r e s s . This w i l l , generally, produce some areas which may exceed the upper bounds. But these upper bounds were a r b i t r a r i l y chosen at the beginning only to s t a r t the process, so that they are now discarded. In th i s fashion the ent i r e i n i t i a l complex i s made f e a s i b l e ; 3. Once a f e a s i b l e Complex i s generated, the scanning of the worst point and the improvement of the Complex can be performed by f o l -lowing the standard procedure. It i s not guaranteed that the optimum w i l l be global. 4.4 Examples A t o t a l of eight structures have been studied. The structures 1 to 7 have been investigated by Estrada [14], while the l a s t one has been studied by Johnson and Brotton [15]. Both investigators have used 45. l i n e a r programming techniques to obtain the minimum volume. Incor-poration of d e f l e c t i o n constraints have not been attempted i n any of these cases. A si n g l e stage optimization has been performed throughout, the number of cycles being 50 for examples 1 to 3, and 300 f o r 4 to 8. The number of points i n the complex have been taken as m+l, where m i s the t o t a l number of v a r i a b l e s . The value of a i s 1.3 throughout the i n v e s t i -gation. Each problem gives a b r i e f d e s c r i p t i o n of the structure, loading, allowable stresses and the number of v a r i a b l e s . The r e s u l t s of the optimization procedure have been tabulated. A separate table summarizes the f i n a l designs at s u i t a b l e i n t e r v a l s of the optimization procedure. The stresses i n the bars corresponding to the best design have been tabulated separately along with the allowable stresses. In a l l examples, the f i n a l r e s u l t s have been compared and several conclusions are drawn. 46. EXAMPLE I The structure investigated i s a three-bar truss as shown i n F i g . 4.1. It has been discussed at great length by Reinschmidt et a l [13]. It i s subjected to two loading conditions as follows: #1 : P, = 15.0k ; P, = -25.98k l u l v #2 : P, = -20.0k. l u The stress l i m i t s are e x p l i c i t l y defined as follows: a = 20.0 k . s . i . a =. -15.0 k . s . i . (constant) c Variables considered are: 1. a the area of the bars ( a l l three bars are required to be equal). 2. u„ the abscissa of j o i n t 1. 1 J Table 7 describes the constraints and the f i n a l values. A com-parison of the f i n a l designs, generated by the Compl ex Method with that obtained by Estrada, using l i n e a r programming techniques, shows that the Complex Method leads to a s l i g h t l y better optimum. There i r e s i g n i f i c a n t differences i n the values of the design parameters (bar areas and nodal co-ordinate of j o i n t 1). Table 7A gives the values of best designs at i n t e r v a l s of 10 cycles. The volume corresponding to the design at the end of f i r s t 10 47. cycles i s about 1 percent greater than the best value. The convergence rate, thereafter slows down very r a p i d l y , u n t i l the design at the end of 40 cycles, which i s p r a c t i c a l l y the same as the design at the end of 50 cycles. Table 7B shows the stresses i n the bars corresponding to the f i n a l design which i s not f u l l y stressed. 48. EXAMPLE II This example i s the same as Example I i n a l l respects, except for the allowable stresses, which are as follows: a = 20.0 k . s . i . t a = -10.0 k . s . i . (constant), c The r e s u l t s of the optimization have been described i n Tables 8, 8A, while Table 8B shows the stresses i n the bars corresponding to the best design. There i s very l i t t l e difference between the f i n a l designs accom-plis h e d by the Complex Method and that obtained by Estrada using Linear Programming Method. The volume corresponding to the design at the end of the f i r s t 10 cycles i s nearly the same as the f i n a l design, though values of u^ d i f f e r by 1.30 percent. The design at the end of the f i r s t 20 cycles i s the same as the f i n a l design. The Table 8B shows that only one bar i s f u l l y stressed. F i g . 4.1 TABLE 7 Variables Constraints F i n a l Values Lower Upper Estrada Author a ( a l = a2 = a 3 } 0.10 5.00 0.782 0.860 U l 50.00 200.00 171.57 121.616 Volume (cu. in.) 332.840 332.582 TABLE 8 Variables Constraints F i n a l Values Lower Upper Estrada Author a ( a 1 = a 2 = a^) 0.10 5.00 1.187 1.182 u l 50.00 200.00 145.630 147.409 Volume (cu. in.) 475.100 474.974 50. TABLE 7A Variables Values at the End of 10 Cycles 20 Cycles 30 Cycles 40 Cycles 50 Cycles a U l 0.865 123.972 0.862 122.595 0.860 121.473 0.860 121.608 0.860 121.616 Volume 335.443 333.810 332.744 332.590 L _ , ^ 1 332.582 TABLE 7B Load Stresses Bar No. Case No. 1 2 3 1 a 16.99 20.00 -0.156 X a a 20.00 20.00 -15.00 9 a -15.00 -4.30 17.45 Z a a -15.00 -15.00 20.00 TABLE 8A Variables Values at the End of 10 Cycles 20 Cycles 30 Cycles 40 Cycles 50 Cycles a U l 1.187 145.552 1.182 147.409 1.182 147.409 1.182 147.409 1.182 147.409 Volume 475.113 474.974 474.974 ' 474.974 474.974 TABLE 8B Load Stresses Bar No. Case No. 1 2 3 i 0 9.82 14.63 3.67 ± 0 a 20.00 20.00 20.00 o a -10.00 -6.30 12.77 Z a -10.00 -10.00 20.00 a 51. EXAMPLE III The structure investigated i s a three bar truss as shown i n Fi g . 4.2. It i s subjected to one loading condition as follows: #1 P. = -100.0 k. Iv The stress l i m i t s are e x p l i c i t l y defined as follows: a = 20.0 k . s . i . a = -15.0 k . s . i . (constant), c Variables considered are: 1. a l area of bar #1. 2. a2 area of bar #2. 3. a 3 - area of bar #3. 4. V l the ordinate of j o i n t #1. Table 9 describes the constraints, the f i n a l values. Table 9A describes the designs at various stages of optimization. The f i n a l design generated i s better than that obtained by Estrada who has derived the global minimum using mathematical minimization p r i n -c i p l e s . He found the mathematical minimum volume to be 882 cu. i n . , and the value of v^ to be 66.10 i n . Values from t h i s l i n e a r i z e d i t e r a t i v e procedure are s l i g h t l y greater, as shown i n the table. The f i n a l design has a s l i g h t l y smaller volume than that obtained by Estrada, and the improvement a f t e r the f i r s t 10 cycles i s about .03%. 52. 1 100" F i g . 4.2 TABLE 9 Constraints F i n a l Values Variables Lower Upper Estrada Author a l 0.10 6.0 1.847 1.892 a2 0.10 6.0 4.147 4.180 a3 0.10 6.0 4.147 4.180 V l 30.0 90.0 67.670 66.085 Volume (cu. in.) 882.63 881.917 TABLE 9A Variables Values at the End of 10 Cycles 20 Cycles 30 Cycles 40 Cycles 50 Cycles a l 1.841 1.848 1.874 1.891 1.892 a2 4.140 4.145 4.165 4.180 4.180 a3 4.140 4.145 4.165 4.180 4.180 V l 67.889 67.653 66.735 66.089 66.085 Volume 882.216 882.142 882.051 881.918 881.917 54. EXAMPLE IV The structure;studied i s a s i n g l e span truss, symmetrical about i t s center l i n e as shown i n F i g . 4.3. It i s subjected to one loading condition as follows: #1. p, = -1000.0 k ; P 0 -1000.0 k ; l v 2v P 0 = -1000.0k. 3v The allowable compressive stress i s a function of Slenderness r a t i o of the bars. Y i e l d stress F = 36.0 k . s . i . y a = • 23. 76 k . s . i . Variables considered are: 1. a l (a 2) area of the. bars #1 and #2, 2. a3 (a 4) area of the bars #3 and #4, 3. a5 <a6> area of the bars #5 .and #6, 4. a 7 . (a 8) area of the bars #7 and #8, 5. a 9 ( a 1 Q ) area of the bars #9 and #10, 6. a l l area of the bar #11, 7. U l a bscissa of j o i n t #1, 8. V l ordinate of j o i n t #1, 9. v2 ordinate of j o i n t #2. Table 10 summarizes the constraints and the f i n a l designs. The best design generated by the Complex Method i s about 0.7 percent worse 55. than that obtained by Estrada. The design parameters compare reasonably well i n both methods. Table 10A describes f i n a l designs at the end of each 50 cycles. The fa c t that the values at the end of 50 cycles and 100 cycles are the same i s due to the operation of the programs, which i s set to output the best value only. Apparently the volume of 9201 cu. i n . was produced at, or before, the f i f t i e t h cycle and could be improved i n the succeeding 50 cycles, even though the worst point i n the complex continued to im-prove, and thus kept the process going. The same condition exists at the end of 200 and 250 cycles. Table 10B shows stresses i n the f i n a l design, i n which bars 7 and 8 are almost s t r e s s l e s s , bar 11 understressed, and the rest are a l -most f u l l y stressed. 60" 60" F i g . 4.3 TABLE 10 Variables Constraints F i n a l Values Lower Upper Estrada Author a l ( a 2 } 50.0 100.0 78.15 78.033 a 3 (a 4) 30.0 50.0 43.02 42.583 a 5 ( a 6 ) 0.1 5.0 0.26 0.232 a7 <V 0.1 5.0 0.13 0.100 a9 ( a 1 0 ) 0.1 5.0 0.-73 0.809 a l l 0.1 5.0 0.10 0.100 U l 10.0 50.0 10.00 10.006 V l 10.0 60.0 19.45 20.020 v2 15.0 60.0 51.76 53.197 Volume (cu. in.) 8,716 8,775 TABLE • 10A Variables Values at the End of 50 Cycles 100 Cycles 150 Cycles 200 Cycles 250 Cycles 300 Cycles a l ( a 2 } 78.699 78.699 77.179 77.878 ' 77.878 78.033 a 3 (a 4) 42.844 42.844 41.631 42.201 42.201 42.583 a 5 (a 6) 1.587 1.587 0.121 0.269 0.269 0.232 a7 <*8> 0.102 0.102 0.100 0.100 0.100 0.100 a9 ( a l O } 3.532 3.532 1.735 0.721 0.721 0.809 a l l 0.878 0.878 0.100 0.100 0.100 0.100 u l 10.000 10.000 10.134 10.000 10.000 10.006 V l 18.871 18.871 20.588 20.296 20.296 20.020 V2 50.299 50.299 53.512 54.052 54.052 53.197 Volume 9,201 9,201 8,826 8,779 8,779 8,775 TABLE 10B Load Bar • Stress Allow, stress Case No. No. 0 0 a 1 -21.39 -21.49 2 -21.39 -21.49 3 -20.96 -21.16 4 -20.96 -21.16 5 -10.25 -10.34 1 6 -10.25 -10.34 7 0.0053 23.76 8 0.0053 23.76 9 -13.47 -13.56 10 -13.47 -13.56 11 17 .'67 23.76 EXAMPLE V The structure, investigated, herein, i s a sing l e span truss symmetrical about i t s centre l i n e as shown i n F i g . 4.4. It i s subjected to one loading condition as follows: P 0 = -1000.0 k 3v The allowable compressive stress i n the bars i s a function of the Slenderness r a t i o . F = 36.00 k . s . i . y G 0 = 23.76 k . s . i . The variables considered are: area of the bars #1 and #2, area of the bars #3 and #4, area of the bars #5 and #6, area of the bars #7 and #8, area of the bar #9, abscissa of the j o i n t #1, ordinate of the j o i n t #1,. ordinate of the j o i n t ill. Table 11 describes the constraints and summarizes the r e s u l t s In addition to the constraints mentioned e x p l i c i t l y , an i m p l i c i t con s t r a i n t v.. < v_ has been imposed. 1. a l ( a 2 } 2. a 3 (a 4) 3. a5 ( V 4. a ? (a g) 5. a9 6. U l 7. 8. V l V2 59. The f i n a l design generated by the Complex Method i s almost the same as that obtained by Estrada as regards the volume and the nodal co-ordinates. However, there are s i g n i f i c a n t differences between the bar areas. Table 11A describes the designs generated by the Complex Method at various stages. The volume corresponding to the design at the end of,the f i r s t 10 cycles i s about 2.5 percent greater than the volume corresponding to the best design, while design parameters d i f f e r s i g -n i f i c a n t l y . The design at the end of 50 cycles i s almost the same as the f i n a l design, while no improvement occurs from the end of 100 cycles to the end of 300 cycles. F i g . 4.4 TABLE 11 Variables Constraints F i n a l Values Lower Upper Estrada Author a l ( a 2 } 10.0 40.0 31.71 31.665 a 3 (a 4) 5.0 20.0 15.05 17.225 a 5 (a6) 10.0 40.0 31.92 31.667 a y (a g) 0.1 2.0 0.13 0.100 a9 10.0 30.0 21.36 17.224 U l 20.0 50.0 39.47 40.860 V l -50.0 -15.0 -35.03 - -36.340 V2 -50.0 -25.0 -41.44 -41.071 Volume (cu. in.) 6,843 6,846 TABLE 11A Variables Values at the End of 10 Cycles 50 Cycles 100 to 300 Cycles a l ( a 2 ) 34.953 31.915 31.665 a 3 (a 4) 11.365 17.790 17.222 a 5 (a 6) 37.229 31.884 31.667 a ? (a g) 0.100 0.100 0.100 a9 30.564 15.969 17.224 U l 35.619 41.526 40.860 V l -26.858 -36.419 -36.340 V2 -36.441 -40.605 -41.071 Volume 7,019 6,847 6,846 62. EXAMPLE VI A two-hinged truss arch, symmetrical about i t s centre l i n e as shown i n F i g . 4.5, has been investigated herein. It i s subjected to three loading conditions as follows: #1. P. = -20000 k; P • = -200.0 k; P, = -200.0 k. Iv 3v 4v #2. P, = 200.0 k. l u #3. P. = -200.0 k. 4u The allowable compressive stress i s a function of the Slenderness r a t i o of the bars. F = 36.0 k . s . i . • y a =23.76 k . s . i . The variables considered are: 1. a-L (a 2) area of the bars #1 and #2, 2. a3 ( a 4 } area of the bars #3 and #4, 3. a 5 (a 6) area of the bars #5 and #6, 4. a ? (a 8) area of the bars #7 and #8, 5. a9 ( a 1 0 ) area of the bars #9 and #10, 6. a H area of the bar #11, 7. U l a bscissa of the j o i n t #1, 8. u2 ' abscissa of the j o i n t #2, 9. V l ordinate of the j o i n t #1, 10. V2 ordinate of the j o i n t #2, 11. V 3 ordinate of the j o i n t #3. 63. Table 12 describes the constraints and summarizes the r e s u l t s . In addition to the constraints mentioned, the following i m p l i c i t con-s t r a i n t s have been imposed. U2 > U l V2 < V l - V 3 The volume corresponding to the f i n a l design generated by the Complex Method i s 0.30 percent greater than that obtained by Estrada. The values of nodal co-ordinate variables are about 3 percent d i f f e r e n t . Table 12A describes the f i n a l designs at i n t e r v a l s of 50 cycles generated by the Complex Method. Table 12B describes the stresses corresponding to the f i n a l design. V 64. F i g . 4.5 TABLE 12 Constraints F i n a l Values Variables Lower Upper Estrada Author a l ( a 2 } 10.0 20.0 14.63 14.809 a 3 (a 4) 5.0 15.0 9.43 9.751 : a 5 (a6) 5.0 10.0 7.23 7.271 a 7 (a g) 1.0' 4.0 1.61 1.458 a 9 (a 1 0) 2.0 7.0 5.91 6.125 a l l 1.0 4.0 1.55 1.586 U l 60.0 ' 80.0 60.00 60.155 U2 70.0 110.0 99.18 97.863 v l 60.0 90.0 78.35 75.393 v2 40.0 : 70.0 56.21 54.868 V 3 70.0 100.0 93.10 90.313 Volume (cu. in.) 7,301 7,327-65. TABLE 12A Variables Values at. the End of 50 Cycles 100 Cycles 150 Cycles 200 Cycles 250 Cycles 300 Cycles a l (a2> 15.189 14.870 14.834 14.834 14.766 14.809 a3 ( a 4) 10.154 9.811 9.777 9.777 9.715 9.751 a5 ( a 6> 7.129 7.231 7.305 7.305 7.302 7.271 a ? (a g) 1.286 1.671 1.539 1.539 1.471 1.458 a9 ( a 1 0 ) 6.409 5.916 6.016 6.016 6.129 6.125 a l l 1.639 1.556 1.551 1.551 1.568 1.586 U l 62.494 60.269 60.207 60.207 60.218 60.159 u2 97.242 100.004 98.643 98.643 98.371 97.863 V l 74.869 76.271 75.823 75.823 75.794 75.393 V2 55.213 55.553 54.859 54.859 54.221 54.868 V 3 89.810 91.299 90.639 90.639 90.553 90.313 Volume 7,444 7,363 7,348 7,348 7,330 7,327 TABLE 12B Load Case No. 1 Load Case No. 2 Load Case No. 3 Bar No. a a 0 a 0 0 a a a 1 -20.14 -20.16 -1.57 • -20.16 0.24 23.76 2 -20.14 -20.16 0.24 23.76 -1.57 -20.16 3 -19.68 -19.86 -19.75 -19.86 0.59 23.76 4 -19.68 -19.86 0.59 23.76 -19.57 -19.86 5 -18.79 -18.82 19.25 23.76 -14.92 -18.82 6 -18.79 -18.82 -14.92 -18.82 19.25 23.76 7 2.52' 23.76 -19.16 -19.35 -2.68 -19.35 8 2.52 23.76 -.2.68 -19.35 -19.16 -19.35 9 -19.91 • -20.13 23.76 23.76 -14.87 -20.13 10 -19.91 -20.13 -14.87 -20.13 23.76 23.76 11 -9.51 -14.37 -14.36 -14.37 -14.36 -14.37 66. EXAMPLE VII The structure investigated i s required to be geometrically symmetrical. It has a r o l l e r support on one side as shown i n F i g . 4.6. It i s subjected to two loading conditions as follows: #1. P, = -100.00 k; P 0 = -100.0 k;- P„ = -100.00 k. Iv 2v 3v #2. P a 200.0 k. l u The stresses are e x p l i c i t l y defined and are: a = 20.0 k . s . i . a = -15.0 i . s . i . c The variables considered are: 1. a l ( a 2 } area of bars #1 and #2, 2. a 3 (a 4) area of bars #3 and #4, 3. a 5 (a 6 ) area of bars #5 and #6, 4. a 7 (a g) area of bars #7 and #8, 5. a9 ( a 1 0 } area of bars #9 and #10, 6. a l l area of bar #11, 7. U l a bscissa of j o i n t #1, 8. V l ordinate of j o i n t #1, 9. ' V2 ordinate of j o i n t #2. Table 13 describes the constraints and summarizes the r e s u l t s . In addition to the e x p l i c i t constraints mentioned, the following a d d i t i o n a l i m p l i c i t constraint has been imposed: v 2 > v 1 + 1.0 The volume corresponding to the f i n a l design generated by the Complex Method i s about 0.7 percent less than that obtained by Estrada. There i s s i g n i f i c a n t difference i n the values of the nodal co-ordinates while the bar areas are reasonably close. Table 13A summarizes the f i n a l designs at i n t e r v a l s of 50 cycles The f i n a l design volume at the end of the f i r s t 50 cycles i s about 2 percent greater than the best design volume. A rather slow rate of convergence i s quite evident. Table 13B describes the actual stresses (a) corresponding to the best design and the allowable stresses. It shows that out of 11 bars about 8 of them are f u l l y stressed under at l e a s t one loading condition 68. F i g . 4.6 TABLE 13 Variables Constraints F i n a l Values Lower Upper Estrada Author a-j^  (a 2) 7.0 •12.0 10.26 10.221 a3 ( a 4 } 6.0 10.0 8.66 9.011 a5 ( a 6 } 2.0 5.0 4.10 3.8,02 a7 ( a 8 } 5.0 10.0 7.53 7.392 a9 ^ lO- 1 2.0' 5.0 2.71 3.102 a l l 3.0 6.0 4.19 4.040 U l 20.0 40.0 25.73 23.641 V l 40.0 60.0 44.37 40.321 V2 40.0 60.0 45.37 41.321 Volume (cu. in.) 5,341 5,306 69. TABLE 13A Variables Values at the End of 50 Cycles 100 Cycles 150 Cycles 200 Cycles 250 Cycles 300 Cycles a l ( a2> 10.280 10.009 10.003 10.007 10.089 10.221 a 3 (a 4) 8.466 8.255 8.263 8.265 8.701 9.011 a 5 (a 6) 4.820 4.747 4.722 4.718 4.205 3.802 a ? (a g) 7.800 7.744 7.727 7.726 7.545 7.392 a 9 ( a 1 Q ) 2.246 2.614 2.622 2.622 2.896 3.102 a l l 4.063 3.769 3.765 3.768 3.917 4.040 U l 23.318 20.000 20.003 20.040 - 21.656 23.641 V 43.395 40.000 40.004 40.004 40.093 40.321 V2 46.621 43.875 43.865 43.849 42.341 41.407 Volume 5,408 5,352 5,346 5,345 5,327 5,306 TABLE 13B Bar No. Load Case No.. 1 Load ' Case No. 2 a aa 1 -15.00 -15.00 2.81 20.00 2 -15.00 -15.00 -3.12 -15.00 3 -15.00 -15.00 -15.00 -15.00 4 -15.00 -15.00 -7.40 -15.00 5 17.13 20.00 -14.95 -15.00 6 17.13 20.00 15.02 20.00 7 16.32 20.00 20.00 20.00 8 16.32 20.00 6.37 20.00 9 -15000 -15.00 13.13 20.00 10 -15.00 -15.00 -10.79 -15.00 11 -15.00 -15.00 -0.03 -15.00 70. EXAMPLE VIII A highly redundant truss with 9 j o i n t s and 21 members, as shown i n F i g . 4.8, i s being investigated. I t has been studied and discussed by Johnson and Brotton [15]. It i s subject to two loading conditions as follows: #1. P„ = -50.0 tons. 3v #2. p r = -30.0 tons. 5v The stress l i m i t s are e x p l i c i t l y defined as: 2 a = 9.50 tons/in . 2 a = -9.50 tons/in (constant). rr c There are 16 v a r i a b l e s . Bars Nos. 1 to 4 form one group as do bars 19 to 21, while the remaining. 14 bars can be designed i n d i v i d u a l l y . The structure has a f i x e d geometry. The constraints f o r the areas 2 are set at 0.5 and 10.0 i n , as shown i n Table 14. Table 14 compares the r e s u l t s . The volumes corresponding to best designs obtained by Johnson and Brotton and the author are prac-t i c a l l y the same, while there are wide v a r i a t i o n s i n the bar areas, sug-gesting that i n the v i c i n i t y of the optimum several designs are possible. Table 14A summarizes the best designs at an i n t e r v a l of 50 cycles. The volume corresponding to the f i n a l design at the end of the f i r s t 50 cycles i s only 1.9 percent o f f the best design. The volume corresponding to the design at the end of 100 cycles i s only 0.6 percent o f f the volume 71. of the best design. The convergence i s very slow and there i s no change i n the values a f t e r 200 cycles. Table 14B summarizes the r e s u l t s . I t shows that out of 21 members, only 12 are f u l l y stressed. F i g . 4.8 TABLE 14 Variables Constraints F i n a l Values Lower Upper Johnson & Brotton Author a, to a, 1 4 0 50 10 0 3. 62 3. 776 a5 0 50 . 10 0 4. 48 4. 480 a6 0 50 10 0 0. 50 0. 500 a7 0 50 10 0 1. 21 1. 372 a8 0 50 10 0 0. 91 0. 500 a9 0 50 10 0 1. 30 1. 122 a10 0 50 10 0 1. 08 1. 247 a l l 0 50 10 0 0. 98 0. 706 a l 2 0 50 10 0 1. 56 1. 412 a l 3 0 50 10 0 4. 55 4. 539 a14 0 .50 10 0 0. 61 0. 500 a l 5 0 50 10 0 1. 48 1. 619 a l 6 0 50 10 0 0. 50 0. 500 a l 7 0 50 10 0 0. 84 0. 596 a l 8 0 50 10 0 0. 50 0. 759 a19 t 0 a21 0 50 10 0 4. 54 4. 549 Volume (cu . in.) 9, 180 9,184 73. TABLE 14A Variables Values at the End of ' 50 Cycles 100 Cycles 150- Cycles 200 Cycles 250 Cycles 300 Cycles a, to a, 1 4 3.789 3.797 3.779 3.776 3.776 3.776 a5 4.529 4.493 4.483 4.480 4.480 4.480 a6 0.515 0.503 0.500 0.500 0.500 0.500 a7 1.504 1.411 1.375 1.373 1.373 1.372 a8 0.515 0.503 0.500 0.500 0.500 0.500 a9 1.142 1.084 1.117 ' 1.122 1.122 1.122 a i o 1.228 1.311 1.256 1.246 1.246 1.247 a l l 0.927 0.636 0.703 0.707 0.707 0.706 a12 1.622 1.379 1.414 1.413 1.413 1.412 313 4.538 4.544 4.542 4.540 4.540 4.539 a14 0.515 0.503 0.500 0.500 0.500 0.500 a!5 1.740 1.650 1.622 1.619 1.619 1.619 a l 6 0.515 0.503 0.500 0.500 0.500 0.500 a17 0.812 0.522 0.592 0.597 0.597 0.596 a l 8 0.515 0.837 0.757 0.757 0.757 0.759 a19 to a21 4.651 4.570 4.552 4.549 4.549 4.549 Volume 9,355 9,237 9,191 9,184 9,184 9,184 TABLE 14B Bar No. i n Load Case Bar No. i n Load Case #1 #2 #1 #2 1 6.278 2.81 12 -7.92 -9.50 2 9.500 6.78 13 9.50 3.02 3 5.070 5.61 14 -1.99 7.91 4 2.911 3.50 15 -6.20 9.50 5 -9.500 -3.55 16 -6.43 -3.68 6 -4.14 -5.55 171 97.22 -9.50 . 7 ' 9.50 -9.31 18 -7.90 -9150 8 7.56 -3.22 19 -9.50 -4.10 9 4.95. 9.50 20 -6.67 -7.47 10 7.16 9.50 21 -3.67 -4.55 11 8.68 9.50 CHAPTER V CONCLUSIONS (1) The f e a s i b i l i t y of Box's Complex Method has been demonstrated. It has been shown that the problems of constrained optimization can be solved d i r e c t l y without resorting to penalty-function formulations. (2) The p i v o t a l parameters, a f f e c t i n g the e f f i c i e n c y of the Complex Method, have been discussed. (3) It has been demonstrated, that the necessity of an i n i t i a l point can be obviated, i f modifications to the Complex Method are made. (4) The e f f i c i e n c y of the performance of the Complex Method depends upon the number of variables and on the number of active constraints. Hence for the s t r u c t u r a l problems where the constraints are well-defined and i t i s guaranteed that the optimum would l i e at some vertex, the Complex Method could prove to be the most e f f i c i e n t algorithm. (5) No claim i s made that the Complex Method guarantees a global optimum. BIBLIOGRAPHY 1. Box, M. J . "A New Method of Constrained Optimization and a Com-parison with Other Methods," Computer Journal, V ol. 8, 1965, pp. 42-52. 2. "ACI Building Code Requirements for Reinforced Concrete (ACI 318-63)," American Concrete I n s t i t u t e , D e t r o i t , Michigan, June 1963. 3. Spendley, W.,Hext,G. R. and Himsworth, F. R. "Sequential A p p l i -cations of Simplex Designs i n Optimization and Evolutionary Operation," Technometrics, Vol. 4, 1962, pp. 441-461. 4. Schmit, L. A. "Stru c t u r a l Synthesis 1959-1969, A Decade of Progress," A Survey Paper presented at Japan-U.S. Seminar on Matrix Methods of S t r u c t u r a l Analysis and Design, August 25-30, 1969, Tokyo, Japan. 5. Goble, G. G. and DeSantis, P. V. "Optimum Design of Mixed Steel Composite Girders," Journal of Str u c t u r a l D i v i s i o n , ASCE, Vol. 92, No. ST6, December, 1966, pp. 25-43. ' 6. Sheu, C. Y. and Prager, W. "Recent Developments i n Optimal Struc-t u r a l Design," Applied Mechanics Reviews, Vol. 21, No. 10, October, 1968, pp. 985-992. 7. H i l l , L. A. "Automated Optimum Cost Building Design," ASCE, Journal of S t r u c t u r a l D i v i s i o n , Vol. 92, No. ST6, December, 1966, pp. 247-263. 8. Graham, J . D. "Optimum Design of Reinforced Concrete Buildings," ASCE, 2nd Conference on E l e c t r o n i c Computation, S t r u c t u r a l D i v i s i o n , Pittsburgh, 1960, pp. 89-104. 9. Russel, A. D. "Cost Optimization of a St r u c t u r a l Roof System," M.A.Sc, Thesis of U.B.C. , August, 1969. 10. Winter, G., Urquhart, L. C , O'Rourke, C. E., Nilson, A. H. "Design of Concrete Structures," New York, N.Y., McGraw-Hill Book Company, 1964, pp. 344-345. 11. "Manual of Steel Construction," American I n s t i t u t e of Steel Construction, Inc., New York, N.Y;., 1965. 12. "C.R.S.I. Handbook," Concrete Reinforcing Steel I n s t i t u t e , D e t r o i t , Michigan. 13. Reinschmidt, K. F. , C o r n e l l , C. A., and Brotchie, J . F. " I t e r a t i v e Design and Structural Optimization," Journal of Structural D i v i s i o n , ASCE, Vol. 92, No. ST6, December, 1966, pp. 281-318. 14. Estrada-Villegas, J. E. "Optimum Design of Planar Trusses Using Linear Programming," S.M. Thesis of M.I.T., 1965. 15. Johnson, D. and Brotton, D. M. "Optimum E l a s t i c Design of Redun-dant Trusses," Journal of Structural D i v i s i o n , ASCE, Vol. 95, No. ST12, December, 1969, pp. 2589-2610. A P P E N D I C E S 7 7 . APPENDIX A START D READ m,n,£, MITER, MITERI, CONSTRAINTS READ INITIAL POINT, a AND INITIALIZER PRINT BEST POINT AND THE OBJECTIVE FUNCTION SO FAR'GENERATED VES CALL BOX FOR FIRST STAGE OPTIMIZATION5 • STOP 3 TAKE BEST POINT AS INITIAL POINT a a + 0.1 CALL BOX FOR SECOND STAGE OPTIMIZATION PRINT BEST POINT AND THE OBJECTIVE FUNCTION 78. SUBROUTINE BOX GENERATE COMPLEX ± CALCULATE IMPLICIT VARIABLES {y} {x} f={g C}-0.5{g c-x""> {j>)<{Ry> ,1 {x} f={g C} +0.5{g C-x^} TAKE BEST POINT SO FAR GENERATED AS INITIAL POINT CALCULATE f SCAN THE WORST POINT {x} Z = a YES FIND C.G. OF ALL POINTS EXCEPT THE WORST POINT REPLACE THE WORST POINT BY A A NEW POINT {xJ N using a CHECK FOR EXPLICIT AND IM-PLICIT CONSTRAINTS a =a-Z/2 NO YES ITER=ITER + 1 ( R E T U R N " ) 79 tLIST OPTIM 1 A P P E N D I X B 2 C 3 C 4 C 5 C PROGRAM FOR S O L U T I O N OF G E N E R A L N O N L I N E A R O P T I M I Z A T I O N PROBLEM 6 C U S I N G B O X ' S C O M P L E X METHOD 7 C X & Y ARE E X P L I C I T & I M P L I C I T V A R I A B L E S R E S P E C T I V E L Y 8 C BOTH X & Y ARE T W O - D I M E N S I O N A L ARRAYS WHERE COLUMNS R E F E R TO P O I N T S 9 C AND ROWS R E F E R TO THE C O - O R D I N A T E S 10 C M=NU.OF E X P . V A R . L = N O . 0 F I M P . V A R . N=MO. OF P O I N T S IN THE COMPLEX 11 C GX £ HX ARE LOWER f, UPPER BOUNDS OF E X P . V A R . 12 C GY S HY ARE LOWER f. UPPER BOUNDS OF I M P . V A R . 13 C MITER IS THE N O . OF C Y C L E S FOR F I R S T S T A G E O P T I M I Z A T I O N 14 C M I T E R l IS THE N O . O F C Y C L E S FOR SECOND S T A G E O P T I M I Z A T I O N 15 C PROPER E X P R E S S I O N S FOR I M P L I C I T V A R I A B L E S AND O B J E C T I V E F U N C T I O N 16 C MUST BE W R I T T E N IN S U B R O U T I N E S IMC AL & FUN R E S P E C T I V E L Y 17 C E X P R E S S I O N S FOR B O X ' S P O S T - O F F I C E PROBLEM ARE SHOWN IN T H I S 18 C P R E S E N T A T I O N ONLY 19 D I M E N S I O N X ( 2 0 , 4 0 ) , Y ( 2 0 , 4 0 ) , G X ( 2 0 ) , H X ( 2 0 ) , G Y ( 2 0 ) , H Y ( 2 0 ) , 2 0 IX I ( 5 0 ) , S U M ( 2 0 , 4 0 ) , C G ( 2 0 , 4 0 ) , F ( 4 0 ) 21 C O M M O N / B 1 / X / B 2 / G X , H X / 0 3 / Y / B 4 / G Y , H Y / B 5 / S ' J M , C G / B 6 / F / B 7 / X I 22 R E A D ( 5 , 1 ) M , L , N , M I T E R , M [ T E R 1 23 1 F 0 R M A T I 8 I 1 0 ) 24 W R I T E ( 6 , 3 ) M , L , N , M I T E R , M I T E R l 25 3 F O R M A T ! 2 0 X , ' N 0 N L 1 N E A R O P T I M I Z A T I O N U S I N G B O X ' ' S COMPLEX M E T H 0 D ' / 2 2 26 I X , ' ' / ' N O OF E X P L . V A R . = * , I 27 2 4 , 5 X , ' N 0 OF I M P L . V A R = • , I 4 / • N O OF P O I N T S IN THE C OMPLE X= • I 4 / • NO OF 28 3 C Y C L E S FOR 0 P T I M I Z A T I 0 N : ' / 5 X , ' F I R S T S T A G E ' , I 4 / 5 X , ' S E C O N D S T AGE = ' > 29 4 1 4 ) 30 . C READ IN E X P . V A R . C O N S T R A I N T S 31 R E A D ! 5 , 2 ) ( G X ( I ) , H X ( I ) , 1 = 1 , M ) 32 2 F 0 R M A T I 2 F 1 0 . 0 ) 33 C READ IN I M P . V A R . C O N S T R A I N T S 34 R E A D ! 5 , 2 ) ( G Y ( I ) , H Y ( I ) , 1 = 1 , L ) 35 W R I T F ( 6 , 6 2 ) ( G X ( I ) , H X ( I ) , 1 = 1 , M ) 36 62 • FORMAT( ' L I M I T S OF E X P L . VAR= • / ( 1 0 X , 2 F 1 0 . 3 ) ) 3 7 W R I T E ! 6 , 6 3 ) ( G Y ( I ) , H Y ( I ) , 1 = 1 , L ) 38 63 F O R M A T ! « L I M I T S OF I M P L . V A R = ' / ( 1 O X , 2 F 1 0 . 3 ) / / ) 39 C AX 1 = I N I T 1 A L I Z E R Z = 0 V E R - R E F L E C T I ON F A C T O R 40 C READ IN AXI , Z , I N I T I A L POINT 41 37 R E A D ! 5 , 7 ) A X I , Z , ( X ( I , 1 ) , I = 1 , M ) 42 7 F 0 R M A T ( 2 F 1 0 . 0 / ( 8 F 1 0 . 0 ) ) 43 C A L L F U . N ( l ) 44 Z l = Z + 0 . 1 45 W R I T E ( 6 , 8 ) A X I , Z , (XI I , 1 ) , 1 = 1 , M ) , F ( 1 ) 46 8 F O R M A T ! ' I N I T I A L I Z E R 1 1 ' , F 1 0 . 5 , 2 X , ' A L P H A = ' , F 1 0 . ? / ' I N I T I A L P O I N T £ COR 47 IRE SPONDING O B J E C T I V E F U N C T I O N V A L J E : • / ( 12F 10 . 3) ) 48 A R = R A N D ( A X I ) 49 00 9 J = l , 5 0 50 X I ! J ) = R A N D ( 0 . ) 51 9 C O N T I N U E 52 11=1 53 C A L L B O X ( L , M , N , Z , I I , M I T E R ) 54 W R I T E ( 6 , 1 0 ) ( X ( I , 1 ) , I = 1 , M ) , F ( 1 ) 55 10 F O R M A T ! ' F I N A L V A L U E S OF E X P L . V A R . A N D O B J E C T I V E F U N C T I O N : • / 5 X , • A T o-o=<i 4i)wns 611 H'I=1 921 oa M l ( 1 4 I )X=( I ) IXV 621 e n w ' i = i 521 oa Z U (N'wiNvas n v o 0*> U l d/9a/oo*rjns/6r)/AH 4xo/<7a/A/£a/xH*xci/2e/x/ifl/Ncwwo: O i l (O2 )0V*(OZ) IX\ ! 4 (O*>)d ' (O*> , OZ>9a*( o v o z > wnsi 601 ' ( 0 Z ) A H 4 l O Z ) A 9 4 ( O Z ) X H 4 ( O Z ) X 9 4 ( 0 * > 4 0 Z ) A 4 ( 0 < 7 ' O Z ) X NOlSi\3WIQ 801 INIOd 0 i O l 0 3 A C ' « c W l M3I\ V A9 X33dW03 3H1 3 0 I N I O d ISdOM 3H1 S3jV3d3.d AOfJdWl 3NIinO>JOnS J 9 0 1 u'z ' i V w ' D A o a d w ] aNiinoaans 601 0N3 •/OI N » n i 3 > j £01 (N1N03 ~I~IV3 201 anNii INOO 12 101 aniNUNOO 801 001 o/ (r• i iwns = (r*1193 i l l 66 < r M i x + i i - r 4 i ) w n s = < r ' i >wns 86 11' 1)x = ( 1 4 1 ) w n s Lb w 4 i = i i n oa 9 0 1 96 101 01 00 56 s # o*t (1 - r * i IOD-IT'I ) x ) - ( i - r 4 i ) ' j o = ( r 4 i ) x 611 *>6 t l 4 I )X = t l 4 I ISO £ 6 w 4 i = i s n oa *70I 26 101 0 1 09 16 5 - o * ( t i - r 4 n o o - t r 4 i ) x ) + ( i - r 4 i ) s c = ( r 4 i ) x £ 1 1 06 ( I 4 I )X = (I 4 I )99 68 w41 = 1 t n 00 201 88 901 01 09 18 -?01 01 0 5 ( I X U S ' l T i r ' X U I J I 98 201 0 1 O'J ( (» ) AH* 1 9 ' ( r 4 » I A ) 31 68 t m v j M nivo *;8 i r 4 w ) x u 9 H 0 n v o £8 r=a 101 Z8 ~i*i = » HOI oa 02 18 N4z=r iz oa 08 6 bL ( t I 1 X 9 - t I )XH)*(I ) -H+(I IX€=(r*I)X £ 1 2 BL W 41=I £ 1 2 oa LL ( •0K1NVM={>I)>J 1 9L 41 = >l 1 00 6 i ( ( 1 )£ )OrJVVJ=>JV *iL N 4 2 = r & on IL 1 •0)UKV>J=(I )>J ZL ( ( I 1 ) IX )ONVb = « V \L I X / / e / 3 / 9 G / 9 3 ' H n S / ^ a / A H 4 A 9 / ^ f i / A / t y / X H 4 X 9 / 2 « / X / I 9 / I M 0 W r J 0 9 OL ( 0 S ) I X 4 ( 0 , 7 ) d ' ( 0 4 ? 4 0 2 ) 9 9 4 ( 0 , 7 4 0 2 ) HOST 69 4 ( 0 2 ) > J ' ( C Z ) A H ' ( C Z ) A 9 4 t 0 Z ) X H 4 ( 0 2 ) X 9 4 ( 0 < 7 4 0 Z ) A 4 ( 0 4 > 4 0 2 ) X N01St\'3Wia 89 X3TdW09 3 T 9 1 S V D 3 V S31V>J3N3S WOONVa 3N I i n o a f H l S 9 L9 ( I I 4 N i 4 W 4 1 ) WOONV b 3NI in0>J8nS 99 ON 3 59 dOlS ££ *>9 li. 0 1 09 £9 ( • * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * 1 29 **********************************************************i ) lVW>IOd Zl 19 ( Z l 4 9 ) 3 J . I d M 0 9 ( ( £ - 0 1 3 2 1 ) / 4 i= 3 9 V i S G I \ G J 3 S 30 (JiV3 3H1 IV i 4 X6 ) IVWbOd 11 65 ( l ) 3 4 ( W 4 l = l 4 ( l 4 l ) X ) ( n 4 9 ) 3 1 1 > J M 85 ( I b 3 i l W ' 1 1 4 1 Z 4 N 4 W 4 T 1X03 I I V O L 5 ( ( £ ' 0 1 321 ) / . =39V1S 1S>JI3 30 0N3 3H1I 95 °3 116 DO 126 J = 2 , N 117 126 SUM! I i J ) = SUM( I , J - 1 ) + X( I t J I 118 O N = N - l 119 OO 141 1=1 ,M 120 141 A B ( I ) = S U M ( I . N l / D N 121 ALPHA=Z 122 F P = F ( 1 ) 123 28 DO 129 [ = 1 , M 124 129 X ( I , 1 ) = AB( [) + ( A B ( I ) - X I I , 1 ) >*ALPHA 125 99 DO 100 K = 1 , L 126 12=0 127 111 C A L L C H B D X I M , 1 I 128 12=12+1 129 I F ( I 2 . G T . 1 0 ) 0 0 TO 30 130 C A L L I M C A L ( l ) 131 I F ( Y ( K , 1 ) . G T . H Y I K ) ) 3 0 TO 202 132 ' I F ( Y ( K , 1 ) . L T . G Y { K ) ) G 0 TO 103 133 GO TO 100 134 202 DO 112 1=1 ,M 135 112 X I I , 1) = AB( I > + <X( I , 1 ) - A B ( I ) >*0 .5 136 GO TO 111 137 103 DO 11<» 1=1, M 138 114 XI I , 1 ) = A B ( I ) - ( X ( I , 1 ) - A B ( I ) ) * 0 . 5 139 GO TO 111 140 100 C O N T I N U E 141 C A L L FUN( I) 142 F N = F ( 1 ) 143 F 1 = F N - F P 144 I F ( F 1 ) 3 0 , 3 0 , 4 1 145 30 A L P H A = A L P H A - Z / 2 . 0 146 I F ( A L P H A . L T . - Z ) G 0 TO 42 147 DO 132 1=1 ,M 148 132 XI I , 1)=AXI (I ) 149 GO TO 28 150 . 41 RETURN 151 42 RETURN 1 152 END 153 S U B R O U T I N E C H B D X ( M , J ) 154 C S U B R O U T I N E CHBDX CHECKS THE E X P L I C I T BOUNDS OF V A R I A B L E S 155 D I M E N S I O N X ( 2 0 , 4 0 ) , G X ( 2 0 ) , H X ( 2 0 ) 156 - C 0 M M 0 N / B 1 / X / B 2 / G X . H X 157 DO 1 1=1,M 158 I F ( X ( I , J ) . G T . H X I I ) ) X ( I , J ) = H X ( I ) 159 I F ( X ( I , J ) . L T . G X I I ) ( X ( I , J ) = G X ( I ) 160 1 C O N T I N U E 161 R E T U R N 162 END 163 S U B R O U T I N E I M C A L ( J ) 164 C S U B R O U T I N E IMC4L C A L C U L A T E S THE I M P L I C I T V A R I A B L E S 165 . D I M E N S I O N X ( 2 0 , 4 0 ) , Y ( 2 0 , 4 0 ) 166 C 0 M M 0 N / B 1 / X / B 3 / Y 167 C ENTER E Q U A T I O N S FOR I M P . V A R I A B L E S * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * 168 Y ( 1 , J ) = X ( 1 , J ) + 2 . 0 * X ( 2 , J ) + 2 . 0 * X ( 3 , J ) 16.9 RETURN 170 END 171 S U B R O U T I N E F U N ( N 2 ) 172 C S U B R O U T I N E FUN C A L C U L A T E S THE O B J E C T I V E F U N C T I O N 173 D I M E N S I O N X ( 2 0 , 4 0 ) , F ( 4 0 ) 174 C O M M O 1 M / B I / X / B 6 / F 175 DO 1 J = 1 , N 2 176 C ENTER EQUATION FOR OBJECTIVE FUNCTI ON**************************************** 177 F(J)=X(1,J)*X(2,J)*X(3,J) 178 1 CONTINUE 179 RETURN 180 END 181 SUBROUTINE SCAN(M,N) 182 C SUBROUTINE SCAN SCANS THE WORST POINT 183 DIMENSION XI20,40),F(40) 184 C0MM0N/BI/X/B6/F 185 17 N1=N-1 186 20 DO 22 J=l,Nl 187 IP1=J+1 188 DO 22 K=IP1,N 189 IFIFlJ).LE.F(K))G0 TO 22 190 TEMP=F(J) 191 F(J)=F(K) 192 F(K)=TEMP 193 DO 121 1=1,M 194 PTEM=X(I,J) 195 X(I,J)=X(I,K) 196 X(I,K)=PTEM 197 121 CONTINUE 198 22 CONTINUE 199 RETURN 200 END 201 SUBROUTINE BOX(L,M,N,Z,11,MITER) 202 C SUBROUTINE BOX DIRECTS THE OPTIMIZATION SCHEME 203 DIMENSION X(20,40),F(40) 204 C0MM0N/B1/X/B6/F 205 CALL RANDOM(L,M,N,I 1) 206 CALL SCAN(M.N) 207 DO 14 ITER=1,MITER 208 CALL IMPROV(L,M,N,Z,E40) 209 GO TO 14 210 40 11=11+1 211 CALL SCAN(M.N) 212 DO 17 I=1,M 213 17 XI I,1)=X(I,N) 214 CALL RANDOM(L,M,N,I1) 215 CALL SCAN(M.N) 216 14 CONTINUE 217 44 • CALL SCAN(M.N) 218 DO 38 1=1,M 219 38 X( I , 1)=X1 I,N ) 220 11=11+1 221 RETURN 222 ENO END OF FILE SSIG *3 79. APPENDIX C The examples c i t e d i n Chapter IV were coded i n FORTRAN IV and run on IBM 360/67. The t o t a l execution time for various examples was as follows: Example No. No. of Cycles Execution Time i n Sees 1 50 7.0 2 50 7.0 3 50 7.5 . 4 300 185. 5 300 135 6 300 143 7 300 45 8 300 90 

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