UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Optimization of rectangular plane frames using box's complex method. Burke, Benedict Paul 1972

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
831-UBC_1972_A7 B87.pdf [ 5.14MB ]
Metadata
JSON: 831-1.0050542.json
JSON-LD: 831-1.0050542-ld.json
RDF/XML (Pretty): 831-1.0050542-rdf.xml
RDF/JSON: 831-1.0050542-rdf.json
Turtle: 831-1.0050542-turtle.txt
N-Triples: 831-1.0050542-rdf-ntriples.txt
Original Record: 831-1.0050542-source.json
Full Text
831-1.0050542-fulltext.txt
Citation
831-1.0050542.ris

Full Text

OPTIMIZATION OF RECTANGULAR PLANE FRAMES USING BOX'S COMPLEX METHOD  by Benedict P. Burke .E., National University of Ireland(Cork), 1968  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in the Department of  Civil Engineering  We accept this thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA August, 1972  7  In p r e s e n t i n g  t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r  an advanced degree at the U n i v e r s i t y the  of B r i t i s h Columbia, I agree t h a t  L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e  and  study.  I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e copying o f t h i s t h e s i s f o r s c h o l a r l y purposes may by h i s r e p r e s e n t a t i v e s .  be  granted by  permission.  Department The U n i v e r s i t y of B r i t i s h Vancouver 8, Canada  Department or  I t i s understood t h a t copying or  of t h i s t h e s i s f o r f i n a n c i a l g a i n written  the Head of my  Columbia  s h a l l not be  publication  allowed without  my  i  ABSTRACT  This study is concerned with the minimum cost design of a multi-storey building.  The building consists of rectangular steel plane frames which are  evenly spaced and support identical floors and a roof. The frame spacing, the column positions within the frame, and the number of intermediate beams spanning between frames are optimized using Box's Complex method and the optimum solution verified by an exhaustive search procedure. Member sizes for the frame and floor system are determined by a fully-stressed design criterion, for the AISC code, within the limits of a discrete set of member properties. The optimum design for several frames with various widths and heights is determined and the influence of the above variables, and the effect of cost parameters on the optimum solution is illustrated and the results discussed.  if TABLE OF CONTENTS Page CHAPTER 1  INTRODUCTION . .  1  1.1  Brief Review of Structural Optimization. . . . . .  1  1.2  Summary of Related Work  3  1.3  Plan of Development  CHAPTER 2  . . . . . . . . . . . . . .  THE DESIGN PROBLEM  . . . . .  6  2.1  Problem Formulation  2.2  Indentification of Variables . . . . . . . . . . .  6  2.3  Constraints on the Variables .  7  2.4  Objective Function  . . . . . .  8  2.5  Automated Solution  . . . . . .  9  2.6  Design Criterion  12  OPTIMIZATION PROBLEM  15  3.1  Optimization Solution Techniques . . . . . . . . .  15  3.2  Box's Complex Method  . . . . . . . . . .  16  3.3  Modifications of the Complex Method. . . . . . . .  19  3.4  Convergence Criteria  21  CHAPTER 3  CHAPTER 4  . . . .  4  . . . . . . . . . .  6  RESULTS  23  4.1  Presentation of Results of Optimization Scheme . .  23  4.2  Influence of Analysis-Design Cycles  32  4.3  The Number and Type of Variables  32  4.4  Uniqueness and Rate of Convergence  36  4.5  Cost Model Variation  4.6  Algorithm Performance and Capabilities  .  . . . . . 36 37  i i1 TABLE OF CONTENTS (cont'd) Page CHAPTER 5 5.1  PARAMETER STUDY  44  Influence of Frame Spacing (x,) and Number of Intermediate Beams (x£)  .  44  5.2  Influence of Topology on Objective Function . . . . .  45  5.3  Influence of Roof System  45  SUMMARY OF CONCLUSIONS AND RECOMMENDATIONS  58  CHAPTER 6 REFERENCES  60  APPENDIX A  Flow Diagrams  APPENDIX B  List of Steel Sections  .  6 2  67  iv LIST OF TABLES Table No. I II  Title Problem Parameters  Page . . . .  25  Parameters for Optimization Problems  26  III  Results for Optimization Problems  29  IV  Average Intermediate Beam Spacing  V  Cost Function Parameters  • • . . . . .  34 35  V  LIST OF FIGURES Figure No.  Title  Page  2.1  Modular Components of Program  JO ...  2.2  Flow Diagram for Binary Search Procedure .  13  3.1  Simplex Method with Two Variables  18  4.1  Basic Structure Showing Variables  24.  4.2  Comparison of Exhaustive Search and Complex Solutions for Structure No. 1  39  4.3  Convergence Rates for Structure No.l  40  4.4  High, Medium and Low Labour Costs for Structure No. 1. . .  41  4.5  Roof System Costs for High, Medium and Low Labour Costs. .  42  4.6  Costs for Frame Only - High, Low and Medium Labour Costs .  43  5.1  Exhaustive Search for Structure No. 1 - Weight Criterion Used 47  5.2  Variation of Cost with Frame Spacing and No. of Bays . . .  5.3  Variation of Weight with Frame Spacing and No. of Bays . . tt 49  5.4  Optimum Design Costs vs. No. of Bays (Structure No. 12). .  50  5.5  Optimum Design Costs vs. No. of Bays (Structure No. 14). •  50  5.6  Variation of Cost with Frame Spacing and No. of Storeys- •  51  5.7  Variation of Weight with Frame Spacing and No. of Storeys  5.8  (a)  (b)  Frame Spacing =30': Material Cost Breakdown for Structure No. 1 Labour Cost Breakdown for Structure No. 1. . . .  48  ;  52  '53  Frame Spacing = 20': Material Cost Breakdown for Structure No. 1 Labour Cost Breakdown for Structure No. 1. . . .  (c)  5.9  54  Frame Spacing = 10':  Material Cost Breakdown for Structure No. 1 Labour Cost Breakdown for Structure No. 1. . . . ^5 (a) - (c) Frame and Roof System Costs for Structure No. 1 56 (d) - (f) Frame and Roof System Weight for Structure No. 1 57  vi ACKNOWLEDGEMENTS The author wishes to thank his supervisor Prof. S.L. Lipson for his advice and guidance. He would also like to thank Dr. N.D. Nathan for many helpful discussions, and Dr. D.L. Anderson for his guidance during the preparation of this thesis. The financial support of the Research Council of Canada in the form of a Research Assistantship is gratefully acknowledged. Finally many thanks are due the author's wife, Jackie, for her encouragement and support.  1 Chapter 1 INTRODUCTION Structural optimization has grown rapidly in the last 15 years.  Two  avenues of advancement have emerged: (1)  analytical optimization;  (2)  mathematical programming techniques  Automated analysis-design schemes utilizing stress ratio methods and energy criterion were used in analytical optimization.  In the second method  algorithms and numerical optimization methods were adapted to problems of structural synthesis. The purpose of this study is to examine rectangular plane frames. A design is defined by a set of parameters, some of which are preassigned while others are to be determined. Many designs are possible which will satisfy the constraints.  The design giving the minimum objective function, while satisfying  the constraints, is the design sought. The automation of a scheme to determine a good solution is demonstrated, though no guarantee of finding a local or global minimum is given. 1.1  Brief Review of Structural Optimization Papers by Wasiutynski and Brandt (1), and Sheu and Prager (2) made a  comprehensive survey of the state of the art up to 1968.  References 3, 4 also  provide excellent reviews. At the beginning of the century Michel! dealt with the development of an optimum configuration for "trusses" under certain loads (5). The optimum configuration consisted of an orthogonal network of tension and compression members. Later Cox (6) and Chan (7) applied the concept of "Michel 1 Structures" to many problems of design optimization.  2 Using a mode of simultaneous failure a degree of success was  achieved  in the optimization of structural components. Shanley (8) and Gerard (9) and other pioneers of structural optimization almost always formulated the problem in terms of equations.  This implies that certain constraints will be c r i t i c a l  at the optimum design.  The fully stresses design is an extension of the sim-  ultaneous failure approach (10), and assumes that in the optimum structure every component is subject to maximum allowable stress under at least one load condition. Klein (11) showed that some minimum weight structural design problems could be viewed as mathematical programming problems. inequalities could be used directly.  Instead of equations,  Schmit (12) demonstrated the feasibility  of using together the methods of mathematical programming and matrix structural analysis to provide a continuous automatic process for minimum weight design. He (12) defined structural synthesis "as the rational directed evolution of a structural system which, in terms of a defined objective function, efficiently performs a set of specified functional purposes". A general class of structural synthesis problems can be defined in proper mathematical form: Given the preassigned parameters and a set of distinct load conditions, find the vector of design variables (x), such that the objective function f(7) is minimized (or maximized) subject to a collection of inequality constraints on the design variables h .(30  1  0  ; j = 1, 2, ..., m.  J  where the functions h.("x) are such that unsatisfactory behaviour of the structure is precluded. A large class of structural design optimization problems take the form of nonlinear mathematical problems. Moses (13) introduced the idea of sue-  3 cessive linearizations.  A wide variety of techniques are now being used i n -  cluding gradient projection methods, steepest descent methods and direct search methods. These methods have and are being successfully applied to a wide variety of practical structures under different load conditions. In recent years there has been a renewed interest in the analytical approach to structural optimization.  The derivation of optimality criterion for  a variety of design conditions has been dealt with extensively (14,15,16). Sheu and Prager (2) point out that while mathematical programming methods have been successful, the optimal design of complex structures taxes the capacity of present digital computers. On the other hand the analytical treatment of simple problems provides an insight into the analytical nature of optimality criteria. 1.2  Summary of Related Work In 1965 Brown and Ang (17) did an extensive study of the minimum weight  design of planar truss and frame structures. The problem was formulated in a standard form as a nonlinear programming problem. The design variables considered were member section properties and were assumed to be continuous variables. After the optimization, standard structural shapes were selected. structure.  The objective function was taken as total weight of the  The A.I.S.C. code set limits on stress and displacement for a  multiplicity of service load conditions. sidered.  No configuration variables were con-  They showed that the minimum weight design for a statically indeter-  minate structure under several load conditions did not necessarily yield a fully stressed design; and suggested the design space was not necessarily convex. Nakamura and Cornell (18) used a method of successive linearizations to solve the problem of design optimization of rectangular plane frames. Member section properties and configuration were considered as variables. Total struc-  4  ture weight was taken as the objective function. A.I.S.C. stress limits.  Constraints were typical  A single load condition only appears to have been  used. The solution for the geometry variable converged to almost equal spacing of columns. A conclusion of the study was that the design space was convex. Cornell (4) pointed out that the accurate elastic analysis methods used by Nakamura and Cornell was time consuming and thus not wholly satisfactory. Soosaar (19) in his investigation of the cost optimization of topology and geometry for planar frames, used approximate frame design methods such as portal and cantilever methods for wind load, and an inflection point assumption for gravity loads.  The frame was thus reduced to a set of statically deter-  minate structures and so minimum weight members were fully stressed.  A contin-  uous set of member sizes were assumed, as well as functional relationships among member properties.  Constraints on stress followed the A.I.S.C. code.  Soosaar's objective function was total cost.  The cost function included the  cost of member existence (topology cost) and cost due to member weight. The design space was found to be irregular and proved to be sensitive to design changes near the optimum. A two stage solution technique was used in accordance with an established hierarchy of variables. The topology solution was found f i r s t by classical optimization followed by a quadratic programming algorithm to determine the geometry. For any value of topology and geometry variables, the corresponding member sizes were also determined. Soosaar did not include frame spacing as a variable and ignored the problem of floor system design. 1.3  Plan of Development The structure studied is examined in detail in Chapter 2.  Variables  relating to the problem are identified and the constraints and objective func-  5 tion are specified.  Outlines of an automated solution and design criteria are  presented. Chapter 3 indicates the various methods of solution of nonlinear constrained optimization. The algorithm to be used is presented and modifications outlined.  The studied problem is set up in mathematical programming terms.  The results of the optimization procedure are presented in Chapter 4, and compared to an exhaustive search solution. Chapter 5 examines the results of design parameter data. Finally, Chapter 6 presents conclusions and recommendations relating to further research and development.  6 Chapter 2 THE DESIGN PROBLEM 2.1  Problem Description The problem studied is the design for minimum cost of a multi-storey  building consisting of a number of evenly spaced rectangular plane frames which support identical floors and a roof.  The frames are to be composed of wide  flange steel beam and column sections. The same grade of steel is to be used throughout. The floor and roof systems consist of reinforced concrete slabs supported either directly on the frames, or on intermediate beams running between frames. The structure can be subjected to any one or several of the following load cases: (i) (ii) (iii) (iv) (v)  full live load and full dead load; checkerboard live load and f u l l dead load; alternate checkerboard live load and full dead load; full deal load; ( i ) + wind load;  (vi)  ( i i ) + wind load;  (vii)  ( i i i ) + wind load;  (viii)  (iv) + wind load.  The members are designed by elastic theory in accordance with a slightly modified version of the A.I.S.C. 1969 code. The floor slabs are designed for maximum span and the slab depth i s constant throughout the floor. 2.2  Indentification of Variables The parameters necessary to define and design the structure must be either  7  (1) preassigned, or (2) determined. (1)  Preassigned Parameters (i)  Structure Configuration H, the total height of the structure; N s , the number of storeys; h.j, the individual storey heights where i = l,...N s ; N^, the number of bays in the frame; L, the distance between outer columns in the frame;  (ii)  Material Properties F , yield stress of steel; E, the elastic modulus of the steel;  (iii) (iv) (2)  Load conditions Unit cost parameter values.  Variables to be Determined (i) (ii)  x-j, frame spacing; X g , the number of intermediate beams; one intermediate beam is placed at each column, and the others are evenly spaced between these;  (iii) (iv)  Xg,...*j,  column positions within the frame;  x.+l, slab thickness;  w  (v) 2.3  Xj+2,...x k , member sizes,  Constraints on the Variables There are two types of constraints to be considered: geometric constraints  and behavioural design constraints. Variables  x-j,...Xj  are subject to geometric constraints, entered as upper  and lower bounds. The remaining variables are governed by behavioural design constraints.  These constraints - stress limits - are implicit, and cannot be  I  8  written as explicit functions of the design variables for this problem. 2.4  Objectivej Function  i  The objective function represents a basis for choice between accept oie alternative designs, and is a measure of the optimality of the structure,  ^mce  i •  weight is easily quantified i t is often taken as the objective function of  j  structural design optimization. of a good c i v i l engineering  In general, cost and not weight, is the measure  design.  i  The total weight or cost of the frame i t s e l f is often taken as the o;i  jective function (17,18,19). This type of objective function does not provide for a satisfactory comparison of structures with different configuration.  A  unit weight or junit cost as objective function does, and so the design of the floor system becomes a part of the problem. The minimization of the cost per square foot of the total structural i system is the objective of this study. The cost per square foot is given by C  =  Dcjl£.{ClMl  +  ( C 2 +C 3> N c+ ( C 4 + C 9> W 2 + C 4 W 3  + ?C5(Ng+x2) + 2C6Hx] +  C^N^  i  + CgLx^ + C 1Q (N b +l)}  (2.1)  i  j  The costjparameters included in the objective function are: C-j, cost per l b . of column material; C 2 , cost per column for preparation; Cg, cost per column for splicing; i  C^, cjost per l b . of beam material; Cg, qost per end for beam preparation; Cg, c^ost per sq. f t . of wall cladding; C 7 , cost per sq. f t . for formwork;  9 Cg, cost per sq. f t . per inch of thickness for in-situ reinforced concrete floors; Cg, cost per l b . of beam material for beam joints; C-jQ, cost per column footing. The other parameters included in the objective function are: W-|, weight of column material per frame; w^, weight of beam material per frame; W3, weight of beam material in floor system per bay. N , the number of columns per frame; c  Ng, the number of beams per frame; D, the summation of slab depth over a l l floors per bay. The cost model chosen for this study reflects the most important cost items in a design, but the model i s easily expanded to include other costs. It is not intended to be a model of present pricing methods. Cost functions including material costs and fabrication costs are d i f f i c u l t to obtain. The costs associated with design and construction are only part of the overall cost which would usually include maintenance costs, insurance costs and many others. Estimating costs for design and construction is d i f f i c u l t .  The cost of  labour, job location and market conditions are but a few of the variables i n volved.  Even with a final total cost for a structure i t is d i f f i c u l t to separate  purely structural costs from e l e c t r i c a l , mechanical and other costs. C.I.S.C. (20) in a recent publication points out that the structural frame of a building costs between 10-20% of the total construction costs; the mechanical and electrical sections vary between 20-50%; and so i f 10% more on the frame saves 10% of the mechanical the building is cheaper. 2.5  Automated Solution The solution i s divided into two parts;  c  V  <5P  11 (1) variables relating to geometry and topology are optimized using Box's Complex method; (2) variables relating to the frame and floor system are determined by a fully stressed design criterion, within the limits of a discrete set of member properties. A computer programme for the solution of the problem was developed. It was written in modular form and so any particular subroutine can be easily replaced by another to solve a slightly different problem. The modular form also facilitates the solution of analysis-design problems. Figure 2.1 illustrates the main components of the programme. The number of independent geometry variables may be reduced by specifying geometric symmetry of the structure.  Column positions to the l e f t of centre  are then included as independent variables while those to the right are mirrored in automatically. The analysis of a statically indeterminate structure requires that member properties - area, and moment of inertia - be known. In the automated scheme an i n i t i a l structure is set up with a l l section properties for a l l members being equal - areas are arbitrarily set equal to 10 sq. i n . and moments of inertia to 30 i n 4 . The geometry and topology variables, determined in Box's Complex method, define a configuration for the structure. designed for a fully stressed condition.  The structure is then analysed and Finally the procedure will yield a  combination of geometry, topology, slab depths and member sizes which give an optimum cost. To evaluate the objective function a complete analysis and design is made. This is by far the most time consuming portion of the programme and so i t is desirable to reduce the number of function evaluations.  This will be discussed  in Chapter 4 in relation to the optimum number of analysis-design cycles.  2.6  Design Criteria The design philosophy adopted is a deterministic one.  Although used in  common practice, Schmit (4) points out that in view of the uncertainties with respect to load levels and yield strengths, i t would be more rational to treat these quantities as random variables. This implies the use of a probability based design philosophy. The reinforced concrete slabs are designed using ultimate strength design methods for full dead and live loads. Each floor is assumed to consist of a continuous one way slab of constant depth.  A yield stress of 60,000 psi is used  for the reinforcement and the cylinder strength of concrete at 28 days is taken as 3,000 psi.  The percentage of main steel reinforcement is arbitrarily chosen  as 0.67%. The frame members are designed by elastic analysis methods in accordance with the A.I.S.C. 1969 code, with the exception of the interaction formula for members subject to bending and compression.  To simplify the automatic design  process the interaction formula was taken as  / t p i l . O a b  (2.2)  where, fa = axial compressive stress in member; ra = allowable compressive stress; f^ = bending stress in member; F = allowable bending stress. b  Equation (2.2) is used in the code only when the ratio of axial to a l lowable compressive stress is less than, or equal to 0.15.  When this ratio is  exceeded the section chosen for the member must satisfy similar formulae, one of which includes an amplification factor. When the slab is supported directly by the frame intermediate beams are  13 C(l)  £eoh<s>n p ^ > p e r f y  entered  I - Member n^>.  fi = N<s>. o| y>&am <QBC)C\C>XXS in takHe. W^OL= Ks. cf column SecV"t<5r>s  Cft.)" Property enreve^ V ( n j = Fr<^>p«rflj chosen  O  CCI3)  M<7  Search  TV<?c:e^ure  14  acting as tie beams between the frames and are provided only at the columns. Their moments of inertia about the major axis are made equal to the moments of inertia about the minor axis of the columns immediately below them. When the slab is carried on the intermediate beams they are designed as simply supported bending members. The design is automated to handle members subject to any of the following: (i) tension; (ii) (iii) (iv)  compression; bending; tension and bending;  (v) compression and bending. The flow diagram for each of these conditions is given in Appendix A. For each of these conditions a member property is calculated and a section is chosen from a table. The table consists of wide flanged beam and column sections. A l i s t of sections included is provided in Appendix B. The section property chosen will be equal to or greater than the property required.  Because of this the tables are arranged in ascending order of values  for area, section modulus and as far as possible moment of inertia.  It follows  from this that the tables are arranged in ascending order of weight also.  There-  fore the section whose property is equal to or greater than the calculated property is the section of least weight in the table capable of sustaining the forces applied. A binary search procedure is facilitated by this arrangement of the table. Figure 2.2 shows the flow diagram for the binary search method. Three iterations are permitted thus reducing by a factor of eight the number of sections to be examined. It should be noted that member weight is not considered in the analysis.  15 Chapter 3 OPTIMIZATION PROBLEM 3.1  Optimization Solution Technique The type of optimization problem that presents i t s e l f in structural  engineering is a nonlinear one. By nonlinear is meantthat the objective function or the constraints are nonlinear. Generally both are so in structures. Objective functions and constraints that are both linear give rise to what are called linear programming problems. A f i r s t approximation to a nonlinear problem might be to apply successive linearizations. Moses (13) used this technique.  Kelley (21) and  Linear programming methods, which are well  developed in operations research, can then be used. However in problems with pronounced nonlinearities these methods may or may not work well. Most numerical optimization techniques may be considered broadly as i t e r ative methods. Such methods require an i n i t i a l point to be specified. There are two kinds of iterative methods - iterative direct search methods and gradient methods. Iterative direct search methods do not require partial derivatives of the objective function with respect to the variables, but depend on previous values of the objective function and information from earlier iterations. two classes:  (i) sequential; ( i i ) linear.  There are  Sequential methods use a set number  of specified or random points in the space of the independent variables to locate an improved point. Box's Complex method falls into this category (22,23,24). Direction vectors are used in the linear method throughout the search.  Having  set out in one direction the results obtained will govern which way to go next (25,26). The other type of iterative method was the gradient method. This method  16 requires partial derivatives of the objective function with respect to the variables to be calculated.  The partial derivatives are usually f i r s t order  but may be of higher order i f desired. Box, Davies and Swann (22) point out that the use of numerical differentiation in the gradient method will result in the selection of a poorer direction than would be obtained using analytical differentiation; that gradient methods will require extra function evaluations near the current point; and they further suggest that such methods are inferior to the best direct search methods, with the exception of the method of Stewart, 1967 (27).  Stewart used a modified  steepest descent method with step sizes for differencing chosen to approximately balance off the effects of truncation error in the differential  approximation  and error due to cancellation of significant figures. There are also tabular direct search methods which divide the region between constraints into a grid and evaluate the objective function at each node. The coordinates of the node giving the least value of the objective function are then considered to be the variables necessary to produce a minimum. Instead of dividing the region into a grid, a random search could be used to examine a large number of points and the point which gives the least value is said to be the minimum (22). 3.2  Box's Complex Method Spendley, Hext and Himsworth devised a "Simplex" method for unconstrained  minimization in 1962(23).  In 1965 M.J. Box modified the method to find con-  strained minima. The modified method was called the "Complex" method. An illustration of how the simplex method works follows. Figures 3.1 show how three equidistant i n i t i a l points in a two variable space can, by replacing the point with the worst value of the objective function by a point with a better value, eventually approach the minimum point "A".  It should  be noted that the three points in the complex are equidistant throughout the  17 procedure; thus the last three points 7, 9, 10 will rotate about "A", never actually hitting i t .  This method indicates how an approach is made in a  direct search procedure. The procedure for the complex method may be viewed as: • (i) choosing an i n i t i a l point; (ii) (iii)  generating a complex of points; successively replacing the worst point in the complex with a better point;  (iv)  applying a convergence criterion or c r i t e r i a .  A feasible i n i t i a l point must be provided with n coordinates, where n is the number of explicit variables. Constraints are also provided and are inequalities of the form l  -\ — x i 1 u i »  i = l ,2.. .m  where m is the total number of variables, explicit and implicit; and a. and  ,  the lower and upper bounds respectively for the ith variable, are either constants or functions of the explicit functions; and where the implicit variables x  n+l* xn+2' '* *,xm  are functions of the explicit variables X-| *  *n.*  The complex consist of k points where k >^ n+l.  The i n i t i a l point is pro-  vided and the remaining (k-1) points in the complex are generated so that each point has coordinates x  i  = A  i+ r i ( u i ~ V »  1 = 1  2  » >---  n  where r . are psuedo-random deviates rectangularly distributed in the interval (0,1). This point must necessarily satisfy the explicit constraints but may not satisfy the implicit constraints.  If an implicit constraint is violated, the  centroid of a l l points generated so far is located and the point violating the  18  *£' ^places V;"a" k nooo (f) ^> n e * t - ^ r e ^ e s l " ,,'<=>" , is ^r&afesr" } but" cannot' i?e refH^crasl r<zjp\acecA i n s t e p .  Figure  3.1  'Simple* KeH\£>d  unH\ ~Tvo£> Variables  19 implicit constraint is moved halfway towards the centroid. By repeating this process a point is found inside a l l bounds. Finally a l l k points in the complex will be located. The objective function is evaluated for each point and the point with the greatest value identified. The centroid of a l l other points is located and the worst point (one with greatest value) i s replaced by a point which i s on the produced line joining the centroid and the worst point, and which is a times as far from the centroid as the worst point. If the new point satisfies a l l the constraints then the objective function is evaluated there.  However i f the new point does not satisfy an explicit con-  straint, i t is placed inside the violated bound and the objective function evaluated.  If an implicit constraint is violated or i f the value of the objective  function at the new point is not an improvement on the value at the replaced point, then the point is moved halfway to the centroid and the process repeated. This process of improving the complex continues, with the worst point being replaced by a better point each time.  Eventually the method converges to  an area of the complex in which further improvement is not possible. The method is then considered to have found a minimum - at least a local one. Box has suggested (28) that appropriate values for a and k might be 1.3 and 2n respectively.  When k >_ n+l, i t ensures the complex will not collapse  prematurely into a subspace giving a local minimum. The over-reflection factor a greater than one provides the method with a system of expansion, while the move halfway towards the centroid allows for the complex to contract. The method for finding the i n i t i a l complex ensures that the problem is scaled to the order of each variable, and the use of the psuedo-random numbers gives a set of points which are sufficiently different from each other. 3.3  Modifications of the Complex Method The automated solution substantially follows the method proposed by Box,  with a few minor modifications. (i) included.  The choice of an alternate method of finding an i n i t i a l complex is It is a non-random method proposed by Mitchell and Kaplan (29).  point P 2 j is put equal to the i n i t i a l point P-j, except for the j which is put equal to the lower bound; likewise the point P2J+1 except at the j  The  coordinate  1 S  equal to P-j  coordinate which equals the upper bound.  If any point thus generated fails to satisfy the implicit constraints then the j  coordinate is relocated halfway towards the i n i t i a l  coordinate.  This process is repeated until ultimately the i n i t i a l complex is formed. This method for generating an i n i t i a l complex ensures a variety of points in different parts of the complex. They suggest that "a good i n i t i a l point leads to a good i n i t i a l complex" thus improving the rate of convergence to a minimum. It should be observed that the f a c i l i t y of changing the i n i t i a l complex by changing the psuedo-random number i n i t i a l i z e r  is removed and a good new  i n i t i a l point must be found in order to check out previous minima. For the problem studied the non-random method gave slightly worse results than the random method. (ii)  The other modification involves the reflected point.  If this point  does not satisfy a l l the implicit constraints or i f the value of the objective function has not improved then the point will be moved halfway towards the centroid three times, i f necessary.  If the point is s t i l l unsatisfactory i t is  moved halfway to the centroid on the other side. This is done twice.  Should  the point s t i l l be unsatisfactory the centroid is used as the point.  If this  f a i l s , the best point is used as an i n i t i a l point and a new complex generated, repeating the complete procedure. This allows the method to continue even though the complex may be stuck in a subspace. By keeping the best point as the i n i t i a l point in the new  com-  21 plex, the p o s s i b i l i t y of improvement i s greatly enhanced and the complex i s once more well d i s t r i b u t e d throughout  the design space.  An upper and lower l i m i t i s s p e c i f i e d f o r the r a t i o of standard deviation to the best point.  If the r a t i o f a l l s between these bounds a r e s t a r t i s pos-  s i b l e ; however i f a s p e c i f i e d number of i t e r a t i o n has been completed there w i l l be no new complex generated  and the method w i l l continue u n t i l some convergence  c r i t e r i o n i s reached.  3.4  Convergence C r i t e r i a There are f i v e suggested  c r i t e r i a by which the complex method may  terminate.  Convergence i s assumed i f : (i)  the coordinates of the centroid do not change by some s p e c i f i e d  amount, f i v e i t e r a t i o n s in succession; o r , (ii)  the function value of the centroid does not vary by some s p e c i f i e d  amount, f i v e i t e r a t i o n s in succession; o r , (iii)  the function value of the best point does not change by a s p e c i f i e d  percentage of the previous best point value, f i v e i t e r a t i o n s in a row; o r , (iv)  the standard deviation of the function values of a l l the points i n  the complex divided by the best value i s less than or equal to a s p e c i f i e d amount. (v)  The method i s arranged to terminate i f the maximum permitted number  of i t e r a t i o n s i s reached. Box  (22) recommended use of c r i t e r i o n ( i v ) .  The f i r s t c r i t e r i o n generally governs when the curvature of the objective function i s s l i g h t near the minimum.  The design space studied in this problem  is generally very f l a t and so this terminating c r i t e r i o n i s l i k e l y . When the function value at the centroid does not change, the space considered has, in general, pronounced curvature near the minimum.  22 The third criterion - failure to improve the function value of the best point - will occur when the design is locked into a subspace containing either a global or local minimum. A flat space will also keep this criterion active, while a highly irregular one is not likely to, except in the case of the method becoming trapped in a subspace. When the standard deviation criterion governs, the complex has contracted into one area of the space and is approaching a minimum, either local or global. The final criterion is the safety valve of the system, ensuring termination when the method shows no sign of converging. The fourth criterion relating to the standard deviation is the most useful in that i t clearly indicates the rate of convergence and the regularity of the objective function. A different criterion will govern for different problems and there is no guarantee that any particular criterion will govern in any particular problem. All five criteria are possible terminating criteria for any problem, though the probability of any one terminating the procedure can be decreased by specifying a low value for the corresponding control parameter.  23 Chapter 4 RESULTS 4.1  Presentation of Results of Optimization Scheme Several problems were considered in this study to examine the following  factors in the optimization scheme: (i) the influence of the number of analysisdesign cycles; ( i i ) the influence of the number and type of variables; ( i i i ) the uniqueness and convergence rate of the optimization solution; (iv) the effect of variation in the cost model; (v) the capabilities of the algorithm. Sixteen basic structures were used and are described in Table I.  Figure 4.1  shows the basic structure parameters. Optimization results are shown in Tables II and III. There were 49 optimization computer runs. included. 1-15  All convergence criteria'were  The standard deviation criterion was quite severe for structures  (0.0005), but was relaxed to 0.005 for structure 16.  Convergence was  achieved 7 times - 5 for the standard deviation criterion, and twice because of failure to improve the objective function of the best point.  It should be  noted that 16 other runs would have "converged" i f the standard deviation criterion for structure 16 had been applied throughout. The solution given by the complex method for structure 1 is compared in Figure 4.2 to that found from an exhaustive search conducted in the design space with a coarse grid. Frame spacing variable, x^, was  increased by an increment of 10 f t . from  a lower bound of 10 f t . to an upper bound of 40 f t . beams,  The number of intermediate  was chosen for the grid so that beams were 5, 10 and 15 f t . apart.  Later beams at 6 and 12 f t . centres were included, and a further set of values found for x-j = 25 f t . When the search indicated an area in which the minimum was located, a design was evaluated there for x, = 25 f t . and x, = 8.  24  TABLE I Problem Parameters  btr. No.  Width  N  b  Sym  Ht.  N  s  X  NLC  l  1  60.  1  /  12.  1 -  -  2  60.  1  /  24.  2  3  60.  2  /  24.  2  4  60.  3  /  12.  T  5  60.  2  /  12.  1  -  6 7  60.  3  /  30.  2  60.  -  /  210.  17  -  8  90.  3  /  24.  9  90, 3  24.  Load Types  L.L  h loor D.L  L.L  Roof D.L  d  ACTIVE VARIABLES x x l x2 3 X4 x5  2  0.025  36  2  0.040  0.030 0.040 0.030  33  1  0.100  0.025  36  -  2  1,2 1,2  0.060  0.030  36  3  7  3  1,2,3  0.060 0.030  0.060 0.030  36  3  / / - / / / / / / / / - / - / -  -  3  1,2,8  0.060  0.030 0.030 0.020  36  1  / /  -  2  1.5  0.075 0.040 0.040 0.020  33  1  /  1  1  0.075 0.040  33  1  / /  -  4  1  5  0.150  0.060 0.100  33  1  / -  /  -  2  1,2  0.100  0.025  36  2  /  /  /  -  3  1,2,8  0.060 0.030  0.030 0.020  36  1  3  1,2,8  0.060 0.030 0.030 0.020  36  1  0.100  0.025  1  1  0.100  0.025 0.100  1  1  2  10  90.  3  -  11  90.  3  /  12.  12  80.  -  /  42.  3  13  80.  3  /  42.  3 -  14  108.  -  /  78.  6  -  -  3  1,2,8  0.060 0.030 0.030 0.020  15  60.  1  /  12.  1 -  -  1  1  16  150.  5  /  15.  1 20  21  3  1,2,3  24.  N  36  1  2 2 1 2  Yield Stress  / / -  -  /  Nb =1,2,3  -  -  / /  /  1  / /  / -  -  36  1  / /  /  0.060 0.030  36  3  / / -  0.060 0.030  36  2  -  0.040 0.020 0.050  / /  REMARKS  -  Nb = 2,3,4  Nb = 3,4,5,...9  -  -  NOTE: Nb = No. of bays in frame; Ng = No. of storeys; NLC = No. of load conditions; Nd = No. of analysis-design cycles; L.L = live load; D.L. = dead load; Sym = symmetry.  TABLE II Parameters for Optimum Problems (see bottom of table for definition of terms) Str. No. No. BOUNDS ON VARIABLES Objective No. Vari- Points in framebp acmg.x-, No. of Interm. Beams,x0 Column Posn.,x^ Column Posn.,x„ Function ables Complex L u 1 u Type L u L L u  REMARKS  -  -  -  1  R=2  -  -  -  1  R=2  4  11  -  -  -  -  30.  7  13  15.  45.  -  -  10.  40.  4  11  10.  25.  6  10.  40.  4  11  10.  25.  3  6  10.  40.  4  11  10.  25.  -  d  3  6  10.  40.  4  11  10.  25.  -  e  3  6  10.  40.  .11  10.  25.  -  -  5a  2  4  8.  50.  3  16  -  -  b  2  4  8.  50.  3  16  -  -  -  6a  2  4  8.  50.  b  2  4  8.  50.  -  -  7a  3  6  10.  40.  5  13  b  2  4  10.  40.  5  13  c  2  4  10.  40.  5  8a  3  6  10.  25.  9a  4  8  10.  25.  1a  2  4  10.  40.  4  11  b  1  2  10.  40.  -  2a  2  4  10.  40.  3a  3  6  10.  4a  3  6  b  3  c  . .  4  .  R=2 1  R=2  1  R=2,Nd=2  1  R=8,Nd=2  1  R=2,Nd=l  1  R=2,Nd=2  1  R=2,Nd=3  -  1  R=2  -  1  Non-Random  5.  27.5  5.  27.5  -  10.  25.  -  -  1  Nb=3  -  -  -  1  Nb=2  13  -  12  23  .15.  37.5  -  12  23  15.  -  Material Costs  . . . . . .  75  Cost  Nb=l 1 1  ro  cr,  TABLE II (cont'd) BOUNDS ON VARIABLES Objective No. Str. No. No. Vari- Points ir Frame Spacing,x., No. of Interm. Beams,x0 Column Posn. ,x^Column Posn. ,x„Function 1 ables Complex L u ' L u L u L u Type 3 75. . 15. 10. 25. 10 a 3 6 11 a 3  6  10.  6  16  6  16  10. 10. 10.  40. 40.  -  4  High Labour Costs  5  Low Labour Costs Weight  b c  3 3  6  10.  6  10.  40.  6  16  d  3  6  10.  40.  6  10.  40.  e  3  6  10.  40.  6  16 16  10.  f  3  10.  40.  6  g h  3  6 6  10.  40.  6  16 16  10. 10.  40. 40.  3  6  10.  40.  6  16  10.  40. 40.  12 a 3  6  10.  40.  6  21  10.  30.  -  -  1  b  3  6  10.  40.  6  21  10.  35.  2  4  10.  40.  6  21  -  -  -  1  c  -  1  4 6  -  -  -  -  10.  35.  -  -  1  10.  40.  6  21  10.  35.  10.  35.  10.  35.  10.  35.  -  1  21  -  13 a b c d  3 2 2  e  1  4 4  -  -  6  10.  40.  -  2  -  -  -  -  4 interm. beams  -  40. 40.  40.  REMARKS  6 1 1  R=2, Nd=2 R=8, Nd=2  1  R=2, Nd=2  1  R=2, Nd=l  1  1  R=2, Nd=3 Nb=4, Nd-1 Nb=3, Nd=l Nb=2, Nd=l x^lO, x2=10  1  x1=20  1  x2=10  1  x1=20, x2=10  cont'd  ro  TABLE II (cont'd) BOUNDS ON VARIABLES Objective No. Str. No. No. of Interm. Beams,x Column Posn. ,x Column Posn. ,Xy, Function Frame Spacing,x 0 0 n No. Vari- Points ir 1 L L u L u L u Type u ' ables Complex  REMARKS  6  12  10.  40.  8  28  1  Nb=9, Nd=l  b  5  10  10.  40.  8  28  1  Nb=8, Nd=l  c  5  10  10.  40.  8  28  1  Nb=7, Nd-1  d  4  8  10.  40.  8  28  1  Nb=6, Nd=l  e  4  8  10.  40.  8  28  1  Nb=5, Nd=l  f  3  6  10.  40.  8  28  1  Nb=4, Nd=l  g  3  6  10.  40.  8  28  2  4  8.  50.  3  16  2  4  8.  50.  3  16  2  4  b  2  4  c  2  4  d  2  4  e  2  4  14 a  15 a b 16 a  Note:  -  -  -• -  •  1  -  -  -  -  -  1  Nb=3, Nd=l Nd=3 Nd=3, 6 restarts  1  2 restarts  1  3 restarts  1  2 restarts  1  1 restart  1  3 restarts  R = Random no. generator (=2, unless otherwise stated); Nd = no. of analysis-design  cycles (=2, unless otherwise stated).  L = lower bound; u = upper bound.  ro CO  TABLE III Results of Optimization Problems (see bottom of table for definition of terms) Str. No.  INITIAL DESIGN x  l  x  2  x  3  x  4  x  5  Ubj.Fun.  X  x  l  FINAL DESIGN x4 3  x  5  Time (sec)  NI  Obj.Fun.  NE  S.D. Best Value  TerminatCriterion  1a  11.  7  -  -  -  5.3266  22.086  9  -  -  -  4.5247  85  26  56  b  11.  -  -  -  -  5.3266  22.162  -  -  -  -  4.7411  142  26  112  2a  11.  7  -  -  -  5.0255  33.23  10  4.2220  149.  20  50  3a  25.  9  25.  -  -  3.6933  23.110  9  24.666  -  -  3.6200  100  11  32  0.0030  GTE  4a  39.9  5  15.  -  -  4.5240  37.666  9 22.280  -  -  3.8816  300  14  -  0.00157  GTE  b  11.  7 20.  -  -  4.4969  30.770  8  16.962  -  -  3.8957  300  18  27  0.0228  GTE  c  11.  7  20.  -  -  4.4969  26.600  9  22.600  -  -  3.8090  300  46  96  0.000777  GTE  d  11.  7  20.  -  -  4.4969  24.998  9  22.638  -  -  3.8535  300  18  26  0.0015  GTE  e  11.  7  20.  -  -  . 4.4969  26.170  9  23.318  -  -  3.8689  300  14  -  0.0036  GTE  5a  10.  15  -  -  -  4.7431  21.637  9  -  -  -  3.7960  181  13  35  ^0.0005  •<-  b  10.  15  -  -  -  4.7431  26.277  14  -  -  -  3.8677  300  18  27  0.00161  GTE  6a  11.  -  10.  4.8085  49.995  -  17.759  -  -  3.8038  500  11  -  0.011  GTE  b  11.  -  10.  0.6834  13.781  -  19.137  -  -  0.4501  500  8  -  0.074  GTE  7a  20.  7  20.  4.0633  28.200  9  14.567  -  4.0021  1200  5  13  0.0071  GTE  b  20.  7  -  3.7226  21.035  8  -  •-  -  3.6712  1200  9  11  0.0032  GTE  c  20.  8  -  4.4586  19.264  9  -  -  -  4.4425  1200  11  34  0.0048  GTE  8a  15.  14  35.  3.4525  18.377  13  29.690  -  17  26  0.0055  GTE  . .-. . . . . . - . .  3.2785  750.  <0.0005  -<-  «-  +-  II  cont'd  vo  TABLE III (cont'd) Str. No.  INITIAL DESIGN l  x2  x  9a  15.  15  35.  55.  10 a  15.  -  20.  40.  11 a  11.  8  25.  b  11.  8  25.  c  11.  8  25.  d  11.  8  25.  e  11.  8  25.  f  39.9  16  30.  g h  20.  10  40.  11.  8  25.  12 a  20.  10  20.  b  20.  10  30.  c  20.  10  -  -  -  15.  b  15.  10  15.  c  -  10  15.  d  15.  -  15.  13 a  e  X  3  15.  X  FINAL DESIGN Obj.Fun.  4  X  l  x  2  X  3  x  4  X  5  Time (sec)  NI  Obj.Fun.  NE  S.D. Best Value  TerminatCri tenon  3.4547  23.152  11  21.246  51.797  -  3.2761  450  19  19  0.0067  GTE  -  1.3540  21.480  -  29.081  59.810  -  1.1675  380  23  53  -  U.K constant for 5 iter.  -  -  7.0774  39.997  14  28.347  5.2768  300  15  31  0.002  GTE  2.9206  22.476  14  28.722  2.5088  300  14  -  0.001  GTE  7.9012  11.0934  8 25.515  7.4355  300  15  31  0.0074  GTE  4.3029  24.554  13 29.896  3.4649  200  15  28  0.0107  GTE  4.3029  28.996  14  3.4863  300  22  26  0.0099  GTE  3.6617  39.904  13 29.783  3.5595  300  19  36  0.0064  GTE  3.9048  22.073  13  30.014  3.4312  300  40  52  0.0026  GTE  4.3029  21.831  13  29.420  3.4590  300  15  30  0.0111  GTE  -  -  3.4333  31.219  19  14.594  3.3847  1200  23  47  0.00067  GTE  3.5331  30.205  13 27.381  3.3752  1200  32  60  0.0008  GTE  3.4780  39.999  15  3.4199  750  17  76  <0.0005  •4-  -  -  3.8824  300  12  0.0063  GTE  3.4335  300  11  0.0052  GTE  3.4447  300  11  0.0036  GTE  3.6567  300  14  0.0096  4.3091  290  7  GTE O.F. constant for 5 iter.  4.8329 3.8509 3.6727 4.4695 4.3091  27.122  13.173  26.980  -  -  -  -  24.354  -  18 24.034  -  -  -  -  15 23.087  -  •a  23.824 15.  —  23  -  cont'd  TABLE III (cont'd)  Str. No.  INITIAL DESIGN X  x  l  2  x  3  x  4  x  5  FINAL DESIG N Obj.Fun.  x  l  x2  x  3  x  4  X  5  Time (sec)  NI  Obj.Fun.  S.D. Best Value  NE  TerminatCriterion  14 a  20.  19  13.  23.  37. 47.  3.1615  22.505  20  12.927  26.022  34.886 45.149  3.1402  1200  5  -  0.0211  GTE  b  20.  17  15.  28.  40.  3.0844  20.  17  15  28.  40,  3.0844  1200  5  16  0.019  GTE  c  20.  19  13.  30.  42.  3.0973  27.057  17  13.959  27.309  46.076  3.0810  1200  7  16  0.013  GTE  d  20.  19  23.  35.  -  3.0783  31.793  17  12.377  31.974  -  3.0522  1200  6  18  0.010  GTE  e  20.  17  28.  48.  -  3.0207  20.  17  28.  48.  -  3.0207  1200  5  17  0.035  GTE  f  20.  17  34.  -  -  3.0570  28.200  15 24.134  -  -  3.0191  1200  10  15  0.0196  GTE  g  20.  17  40.  -  -  3.1067  24.296  19  -  -  3.0787  1200  13  27  0.00054  GTE  15 a  20.  7  -  -  -  4.5259  32.310  9  -  -  -  4.2779  114  18  45  <0.0005  -<-  b  20.  7  -  -  -  1.1300  21.131  9  -  -  -•  1.0728  300  41  116  0.00109  GTE  16 a b  -  -  30.  60.  -  3.1356  -  -  30.  60,  -  3.1356  300  5  -  -  GTE  -  -  22.5  60.  -  3.1871  -  -  27.239  60.301  -  3.1817  300  11  -  0.00917  GTE  c  -  -  15.  60.  -  3.2994  -  -  27.239  60.301  -  3.1777  300  9  -  0.01794  GTE  d  -  -  37.5  52.5  -  3.3061  -  -  23.344  52.06  -  3.2183  300  6  -  0.05126  GTE  e  -  -  45.  67.5  -  3.3018  . . .-  -  22.171  56.528  3.1934  300  .9  -  0.00801 .  35.868  NOTE; O.F. = Objective function;  GTE = Global time exceeded on computer;  NI  = No. of iterations by Box routine;  S.D.= Standard deviation;  NE  = No. of evaluations of objective function;  BEST= Best O.F. value in complex.  The standard deviation convergence criterion parameter equals 0.0005 for problems 1 to 15; and equals 0.005 for problem 16.  . GTE  32 The algorithm solution is verified by the exhaustive search for structure 1(a) and 1(b). 4.2  Influence of Analysis-Design  Cycles  Results from the analysis and design of several multi-storey frames under a multiplicity of loading conditions indicate that the design was almost stable after 3 or 4 analysis-design cycles.  In some cases one or two member sizes a l -  ternated between subsequent cycles which meant that stress constraints were not satisfied.  This is not serious and in engineering design practice is acceptable.  When one analysis-design cycle is used the design may stabilize after several iterations in Box since one design only is stored.  The analysis-design portion  of the programme-is the most time consuming and so reduction of the number of function evaluations, or the number of analysis-design cycles would be desirable. By increasing the number of analysis-design cycles, stability of the design is enhanced; but for a given computer time the number of function evaluations (and the number of iterations in Box) is reduced. This decreases the possibility of finding a lower value of the objective function. Structures 4(c)-(e) and ll(f)-(h) show the effect of the number of analysisdesign cycles.  For each structure one analysis-design cycle gave the lowest  optimum value of the objective function with approximately three times as many iterations in each case, and four times as many function evaluations for structure 4 and approximately twice as many for structure 11. It is interesting to note that for structure 4(c)-(e) the method converged in each case to similar configurations; while for 11(g) and (h) the configurations are almost the same, but different for 11(f). 4.3  The Number and Type of Variables For structure 1 the method converged faster with two variables than with  one variable. This is shown in Figure 4.3.  The large number of function evalu-  33 ations required for the one-variable problem indicates that the rate of improvement of the best value was slow. An i n i t i a l complex was regenerated 6 times without improvement of the best point which implies the design space is quite flat near the optimum. The influence of the number and type of variables was examined in structure 13.  Comparison of 13(a)-(e) shows that the inclusion of a l l variables  did not significantly improve the best value of the objective function over 13(c) and 13(d).  In the case of 13(e) the two point complex collapsed onto  the i n i t i a l point and completed 7 iterations before ."converging". The complex "converged" because the number of analysis-design cycles was one; the design did not stabilize and though both points in the complex had the same coordinates two distinct structures were designed with correspondingly different objective function values.  For structure 13(a) the number of points in the complex was  4 and in 12 iterations the function value was reduced by approximately 25%. The conclusions to be drawn from this is that a problem with one variable should have a minimum complex size of 3 or 4, otherwise the method may break down. In 13(a)-(d) the optimum column positions were almost the same, which indicates that the objective function value variation is caused principally by x^ and Xg. Column Positions:  The results of problem 11 and 16 show that an arrangement of  columns with equal or near equal spacing gives a good design cost. agreement with Nakamura (18) and Soosaar (19).  This is in  Soosaar showed that equal spacing  of columns gave a cost within 5% of the optimum, regardless of the number of storeys. For the two tall structures considered, 7 and 14, with Nb = 3, the column positions at the termination of the optimization scheme were quite different. In structure 14 the columns were almost equally spaced, while in the taller structure, 7, the columns moved towards both ends. The influence of the wind loading causes this column concentration at the edges for structure 7, which has  34 a height to width ratio of 3.5.  In structure 14 the almost equal spacing of  columns reduces the bending moments due to vertical load, while the low height to width ratio of 0.72 provides adequate resistance to wind loads. Number of Intermediate Beams: If the structures are grouped according to width, the optimum number of intermediate beams per floor within each group is almost equal.  The average spacing of intermediate beams within each group is shown  below in Table IV.  Width  Average x 2  Average Spacing  60'  9.35  7.20'  80'  12.85  6.75'  90'  12.60  7.75'  108'  17.35  6.60'  TABLE IV : Average Intermediate Beam Spacing These results show good agreement with the exhaustive search done for structure 1, where the optimum intermediate beam spacing is approximately 7 f t . (regardless of frame spacing), which corresponds to the minimum slab thickness permitted. Frame Spacing:  The final design in several cases (4a, 6a, 11a, l l f , 12c) where  cost was the criterion, yielded a frame spacing close to the upper bound. Where weight was the criterion of a good design (lie) the frame spacing for the final design was close to the lower bound. This is verified by results of the exhaustive search (see Figure 5.3). Only in one case however (12c) of the 7 runs that converged, did the algorithm converge to the upper bound or lower bound for frame spacing.  Frame spacing influences the objective function more than the other variables.  TABLE V Cost Function Parameters  Objective Function Type  l ($ per lb.)  1  0.100  2  0.100  3  0.100  4  *  C  C  2 ($)  C  C  C  7 ($ per sq.ft.)  0.090  10.000  3.000  0.750  -  0.090  -  50.000  20.000  0.090  10.000  3.000  0.100  100.000  40.000  0.090  20.000  3.000  1.500  5  0.100  25.000  10.000  0.090  5.000  3.000  0.380  6  1.000  -  1.000  -  20.000  C  6 ($ per sq.ft.)  -  4 ($ per lb.)  C  5 ($)  50.000  3 ($)  -  -  -  C 9 8 ($ per ($ per sq. f t . per in.) lb.) C  0.155  -  -  -  -  -  0.030  -  ' C10 ($) 100.00  -  25.000 *  100.00  0.311  0.060  200.00  0.078  0.015  -  -  50.000  -  Remarks  Normal costs Material costs  High labour Low labour Weight  Cq is the cost per lb. .per end for beam joints except in type 3 where i t is the cost per beam joint.  cn  36 the degree of influence however seems to depend on the objective function chosen. 4.4  Uniqueness and Rate of Convergence When a minimum has been found the only way of checking whether the  minimum is local or global is to run the problem again with a new, and i f poss i b l e , completely different i n i t i a l point.  If the method converges each of  several times to the same point then i t is inferred that this is the global minimum. Such uniqueness of the optimum solution was checked for structures 4 and ll(d-h).  A unique solution did not emerge though maximum deviations from  the best solutions obtained were 2.28% and 3.44% for 4 and 11, respectively. For 4c the complex was rapidly contracting - shown by the small value for standard deviation divided by the least value of the objective function in the complex - and was near convergence. Comparison of results for 12b and 13b (with  = 3, for each) shows that  with four times more running time the optimum solution is improved by only 1.73%. The complex, at termination, has contracted about the minimum almost within the standard deviation convergence criterion l i m i t . It can be concluded that any minimum cannot be verified as a global minimum; and the general slow rate of convergence shown throughout the problems indicates a flat design space which does not agree with Soosaar's conclusion of pronounced curvature of the objective function near the minimum. 4.5  Cost Model Variation Six different objective functions v/ere formed by varying the cost para-  meters as in Table V. The influence of the type of objective function used can be seen in the results for l l ( a - f ) . The frame spacing shows the most sensitivity to the objective function  37 type.  The results for optimum frame spacing in structure 11 (a-f), concur  with the trend established in the exhaustive search for structure 1 shown in Figures 4.4, 4.5 and 4.6.  High labour costs move frame spacing to the'upper  bound; medium labour costs show l i t t l e sensitivity to frame spacing away from the lower bound; and low labour costs tend to move frame spacing slightly towards the lower bound. position of x-| =20'  In Figure 4.4 this is shown by the reversal of the  relative to x-j =30'  as the costs change.  When weight or material cost is the objective function as in 6b, l i e and T5b, frame spacing moves to the lower bound. As the frame spacing decreases the weight per sq. f t . of frame increases slightly while the floor system weight reduces so much that the total weight per sq. f t . for the whole structure is decreased. This will be shown to be true in Chapter 5. Objective functions involving labour and topology indicate that frame spacing is a more c r i t i c a l variable than either the number of intermediate beams or the column positions. When weight is the. objective function the number of intermediate beams becomes more important than the frame spacing, which tends to the lower bound for an optimum design. 4.6  Algorithm Performance and Capabilities Box's complex method in general performed quite adequately in spite of  the severe standard deviation convergence criterion applied to most problems. However the overall optimization procedure became quite slow when large structures (7,12,14) were optimized.  The major share of the time was spent in the  analysis-design portion of the scheme. The number of iterations in the complex method was reduced significantly rending the method ineffective.  For structure  12 the complex was close to convergence on termination in a l l cases; structure 14 was far from convergence except in the case of 14g. This clearly indicates that the use of the optimization scheme is limited to small structures. The amount of time used in the design-analysis portion of  38 the program for structures 7, 12 and 14 would justify consideration of approximate method of analysis for larger structures.  39  .- to \  \  \  \  \ \  \ es!  \ \ \ •  i  \  S£ «  \  VJ  a-  Si- O V*  Cxi  \  \  X X Ul  \  5*  *  - \  ti-  <3 /  3  c  V  31  /I/ in  /  / / / /  ty  <3~?  *\\\  / /  5 "0  <3  \\ *\ I  •s « ^  \\  C  V  5.5-  A  A  ® ® ® ® A  ®  ®  ®  A  A  A  ® A  ®  ®  ®  ®  ®  A  A  +  ®  ®  ®  ®  ®  ®  ®  ®  ®  @  )  Cow*r4er\ce  A  —1  1  /5"  /O  4.3  ®  A A  Figure  ®  rates  W  25  SVracvrufe U<P. I . 4^  o  41  -®  ®-  11- igh  \c\\?DUk*~  cast's  -A----A-  N\adtum l a b o u r coste  - S  Low \<a\poav czosYs,  H-  I /  / / /  /  ®, •  ®"  / /  Go  • X,,Si Or-  &•  5.0 --  W V  Si.  %  v3  .a  4 . 0 --  Z.O -•  -a— -a—  -a-  2.0-  figure. 4.4 MigK , N\e<^ium .and Lai^ labour  1o  K  42  1 S t r u c tWe- Mc?. i = 2o'  / #  2.5  X , = (o  //'  / // ///  / //  / / 7  2.0 --  /  //  A /  1  , '  3  0  "  /  1.5  / A X ' -A"'  '  s  x,-2o' x , » io'  f  .0 --  •••  a-  0.5" -IO' Picture -4-.S  15'  1  43  -® 3.? --  "0—  •A -H  High  Ic^u/  <%£fS  JAe^iiuM |<aJaj>uir tvpshs H—  0  cosH  L?oo  /  / /  3..0--  ^®  -  2.5--  X  y®  5" x.-i©'  ^a VI  -A ^ - 3 ^ ' _^a  \.5--  '^a  15  P i c t u r e 4-.L,  H i g h , Loco  <a<si NWliurn n  *,»2o' x,= s o '  44 Chapter 5 PARAMETER STUDY  The influence of the following parameters on the cost and weight per sq. f t . of the structure is examined:  (i) frame spacing; ( i i ) the number of  intermediate beams; ( i i i ) the number of bays in the frame; (iv) the number of storeys; and (v) the roof system. An exhaustive search procedure was conducted for structure 1 for this purpose, and results from the optimization of structures 12 and 14 are also i n cluded. 5.1  Influence of Frame Spacing (x^) and Number of Intermediate Beams (x 2 ) Figure 4.2 shows that for a unit cost as objective function the optimum  spacing of intermediate beams is approximately 7 f t . This span allows the slabs to be the minimum thickness permitted.  Intermediate beam spacing below  this results in the same slab thickness while the increased number of intermediate beams causes the objective function to increase. With weight per sq. f t . as the objective function the optimum spacing of intermediate beams decreases to approximately 5 f t . when x-j = 10'.  The  weight per sq. f t . of intermediate beams increases slightly in this case, but the frame weight is significantly reduced because the load applied by the intermediate beams to the frame is approaching a uniformly distributed load pattern which gives reduced bending moments in comparison to point loads. For a constant number of intermediate beams the unit cost in Figure 4.2 shows very l i t t l e sensitivity to variation in the frame spacing away from extreme values.  However the unit weight as shown in Figure 5.1 decreases signi-  ficantly with decreased spacing.  It should be noted that the difference in  unit weight between x, = 30' and x, = 10' is most pronounced for small inter-  45 mediate beam spacing. 5.2  Influence of Topology on Objective Functions  Number of bays:  Figure 5.2 shows the effect of the number of bays and frame  spacing on a unit cost objective function, when the intermediate beam spacing is constant at 10 f t .  (  The objective function is insensitive to frame spacing  away from the lower bound.  A unit weight criterion showed in Figure 5.3 that  the minimum frame spacing was the optimum, and increased rapidly with increasing frame spacing. When the number of bays was increased from 1 to 3, the axial force and bending moment per column and the moments per frame beam were reduced. This gave an 18% and 14% reduction in unit cost for x-j = 10' and x-j =40', respectively. With unit weight criterion the corresponding reductions were 65.6% and 41.6%. The final design costs from the optimization scheme for structures 12 and 14 are plotted against the number of bays in Figures 5.4 and 5.5. The i n fluence of the number of bays i s not significant.  Maximum difference in cost  is approximately 0.60% and 0.76% for structures 12 and 14 respectively.  No  figures are available for the influence of the number of bays on unit weight in this case. Number of Storeys:  Structure 1, with Nb = 3, was used as the basic structure  to study the influence of the number of storeys. Figures 5.6 and 5.7 show the influence of frame spacing and the number of bays on the objective functions, types 1 and 6. The optimum design tends to move to the upper and lower bounds for cost and weight criterion respectively. 5.3  Influence of Roof System A breakdown of the costs and weight of the exhaustive search for struc-  ture 1 is shown in Figures 5.8 and 5.9. The influence of the roof system i s  46  particularly noticable when weight is the objective function.  It causes the  weight per sq. f t . of the total structure (see Fig. 5.1) to increase with i n creased spacing of frames.  47  N  0 \ \ \  \ \ \  \  \  \  \  \  x  \  ^ \  \ \  \  \ \  \  \  \  /  \  \  \ i  V  N  \  \  C o  in  \  a)  C C CP  /  LU  fe  /  /  /  ;  6. /  O  /  o iri  c6  (sen) ^j'bs  i  \ \  ^M^^fA  5  48  i  1  1  o  I N  «  I  o  T" i  /  o  /  a  /  tf  ///  U-  :£ '3  /  /  '7'  /  /  /  U-  0 c c  / //  /  . 0 <5  ///  /  5  /  /  /  /  <5  /  /  *  / /  /  in  o  in I "  \  •\ \  \ \ \  \  \ \  \  \\ \ \ ^ \ \ \\ \  \  V  \  w \ \  2-  V^ri^icn o\ Weight VMH\ F/aw< g-p  50  3>A1 ®  341  f  \ \  \  \  \  3  Fl^gfe 6>.4 Optimum Des ugh VS.  Goz\s  o$ Q>#<jS  S»5 +  sr  ^3  3.05- +  \  \  \ \  7  vs. Mg- &\ feays  9  \ \  \  4.50  \ \ \  \  \ \  \\  \  -V-  S.SO  1 IO  Pv2^rne S>p<ag.in<3 <  figure 5.C  VanruahgN.  Cost"  52  \  \  \ \  \  \  \  \  \ \  \  \  \  \  \  \  \  \  \  \ ^\  \  £  8  \  \  \ \  eft tl zz  VP  Q  •3  3  5  \  c c xt  <5  \  V q  ^  0 zz  V  \ \  a)  e  0  \  \  0  4-  \ \  o  \ \ \  \ \  \  0  l  CP <  <5  • V) C  ^  * £ £  £  I I I  J  m H* 4  -Lin  3"  -  I i i \  ni i  11  /  11  /  t 1  1  In  w  A)  2 ^ vi  v.  V  11 ti 1  Q  in  o  d  c J5 "<3  i  3 .  •s-  vP  cD  c  \  o I CD  1  "2  J_  v>  A—  V3  4in  q <s  In  in Q  6'  £  2  _£ J I  Hi i. \ i i  c  3  3 1  \  \  9  /  / -to  /  I / /  1 I I  /  r  d  . oa  4  4-IP  0 In O  LO  n  J in s.  VI  VI  U-  <3  \  4-W  \  \  I  V  Co  VA i  in d  4  U-  vl  S  cQ ID  3  \ i  \  \  cr  /  z  4-  t o  0  t  10 in  S  \  5 VO  \  c  3  I  \ 5*1  o  0  6  1  57 S  \\ I\  2a  f  4-i/>  In  N  N  CP  \  2  \  \  \  \  --  vn  ml*  *2  (A  \  /  cQ  LL  X  in  1  \  \ /  \ \ \ In  ifi Si"?  In  c3  tz  3  \  \  ^  t-cf  \  in  I  58  Chapter 6 SUMMARY OF CONCLUSIONS AND  (i)  RECOMMENDATIONS  The f e a s i b i l i t y of an automated scheme f o r the optimum design of  structures with rectangular plane frames, intermediate f l o o r beams and concrete floors has been demonstrated.  The method uses exact methods of analysis coupled  with a modified AISC code for design and considers d i s c r e t e member s i z e s . (ii)  The c a p a b i l i t i e s of the optimization scheme i s limited by the amount  of computer time available and computer storage capacity. (iii)  It cannot be proven that the optimum design found by the complex  method i s a local or a global minimum, however exhaustive search techniques did not disclose any points with a better design. (iv)  The f l o o r system i s an integral part of the structure and should be  included in the optimization scheme.  It has been shown by an exhaustive search  procedure that the optimum frame spacing based on a unit weight c r i t e r i o n i s s i g n i f i c a n t l y d i f f e r e n t when the f l o o r system i s included in the problem. (v)  Regardless of the objective function used the design space near the  minimum appears to be f l a t .  Soosaar (30) found that the design space had a more  pronounced curvature near the minimum f o r a cost objective function than f o r a weight c r i t e r i o n . system and this may (vi)  However Soosaar considered only the frame and not the f l o o r account f o r the difference in r e s u l t s .  For structures with low height to width ratios equal column spacing  gives objective function values very close to those f o r the optimum column position.  For t a l l e r structures with a height to width r a t i o greater than 2 the i n -  t e r i o r columns tended to move towards the edges at optimum spacing.  This shows  agreement with the results of Nakamura (18). (vii)  The number of intermediate beams did not greatly e f f e c t the objective  59 function. mum  In general  i t was  found that intermediate  beams spaced to give mini-  slab thickness gave optimum r e s u l t s . (viii)  The  type of objective function used i s important.  It was  demon-  strated that a weight and a cost c r i t e r i o n lead to widely d i f f e r e n t optimum designs with the major difference occurring in frame spacing. value system the optimum frame spacing occurred  For a weight  at the lower bound; while a  high labour cost objective function gave the upper bound as the optimum frame spacing.  For future work in this area the following points are recommended: (i)  To reduce computer run time, approximate methods of analysis and  design checked by exact analysis-design procedures every 20th i t e r a t i o n or at termination, should be (ii)  say,  included.  The geometry variable r e l a t i n g to column positions should be re-  placed by a topology variable related to the number of columns in the structure. (iii)  The number of intermediate  optimization scheme and when optimized procedure.  beams could be treated in a separate entered  into the main optimization  sub-  60 REFERENCES 1.  Z. Wasiutynski and A. Brandt, "The present state of knowledge in the field of optimum design of structures", Applied Mechanics Review, 16, pp. 341-350 (1963).  2.  C.Y. Sheu and W. Prager, "Recent developments in optimal structural design", Applied Mechanics Review, 21, pp. 955-992 (1968).  3.  L.A. Schmit, "Structural synthesis. 1959-1969. A decade of progress", A paper presented at Japan-U.S. Seminar on Matrix Methods of Structural Analysis and Design, August 1969, Tokyo, Japan.  4.  W.Z. Cohn, Editor, "An introduction to structural optimization", University of Waterloo, Study No. 1, 1969.  5.  A.G.M. Michel 1, "The limits of economy of material in frame structures", Philosophical Magazine, Series IV, Vol. 8, No. 47, London, November 1904.  6.  H.L. Cox, "The design of structures of least weight", Pergamon Press, Oxford, 1965.  7.  A.S.L. Chan, "The design of Michell optimum structures", College of Aeronautics, Report No. 142, 1960.  8.  F.R. Shanley, "Weight-Strength analysis of aircraft structures", McGraw-Hill Book Co., Inc., New York, 1952.  9.  G. Gerard, "Minimum weight analysis of compressive structures", New York University Press, New York, 1956.  10.  R.A. Gellatly, "The role of optimization in the design of aircraft structures" A paper presented at the Third Air Force Conference on Matrix Methods in Structural Mechanics, October 1971, Wright Patterson A.F.B., U.S.  11.  B. Klein, "Direct use of extremal principles in solving certain optimization problems involving inequalities", Operations Research, Vol. 3, 1955, pp. 168-175.  12.  L.A. Schmit, "Structural design by structural synthesis", Proceedings of the Second National Conference on Electronic Computation, Structures Division, ASCE, Pittsburgh, PA., September 1960, pp. 105-132.  13.  F. Moses, "Optimum structural design using linear programming", Journal of Structural Division, ASCE, No. ST4, December 1964.  14.  W. Prager and J.E. Taylor, "Problems of optimal structural design", Journal of Applied Mechanics, 35, pp. 102-106, 1968; AMR 21 (1968), Rev. 6597.  15.  C.Y. Sheu and W. Prager, "Minimum-weight design with piecewise constant specific stiffness", Journal of Optimization Theory and Applications 2, pp. 179-186, 1968.  16.  V.B. Venkayya, "Design of optimum structures", Computers and Vol. 1, pp. 265-309, August 1971.  Structures,  61 17.  D.M. Brown and A. Aug, "Structural optimization by nonlinear programming", Journal of Structural Division, ASCE, Vol. 92, No. ST6, December 1966, p. 319.  18.  Y. Nakamura, "Optimum design of framed structures using linear programming", S.M. Thesis, Dept. of Civil Engineering, M.I.T., 1966.  19.  K. Soosaar and C A . Cornell, "Optimization of topology and geometry of. structural frames", A paper presented at the ASCE Joint Speciality Conference on Optimization and Nonlinear Problems, Chicago, April 1968.  20.  Canadian Institute of Steel Construction Publication, "Steel Building Design", October 1971.  21.  J.E. Kelley, "The cutting-plane method for solving convex problems", SIAM Journal, Vol. 8, No. 4, December 1960, pp. 703-712.  22.  M.J, Box, D. Davies and W.H. Swann, "Optimization Techniques", I.C.I. Monograph No. 5, Oliver and Boyd Ltd., Edinburgh and London, 1968.  23.  W. Spendley, G.R. Hext and F.R. Himsworth, "Sequential application of simplex designs in optimization and evolutionary operation", Technometrics, 4 (1962), p. 441.  24. J.A. Nelder and R.Mead, "A simplex method for function minimization", Computer Journal, 7 (1965), pp. 308-313. 25.  H.H. Rosenbrock, "An automatic method for finding the greatest or the least value of a function", Computer Journal, 3 (1960), pp. 175-184.  26.  R. Hooke and T.A. Jeeves, "Direct search solution of numerical and statistical problems", Journal of Association of Computer Machinery, 8 (1961), pp. 212-221.  27.  G.W. Stewart I I I , "A modification of Davidson's minimization method to accept difference approximations of derivatives", Journal of Association of Computer Machinery 14 (1967), pp. 72-83.  28.  M.J. Box, "A new method of constrained optimization and a comparison with other methods", Computer Journal, * (1965), pp. 42-52.  29.  R.A. Mitchell and J.L. Kaplan, "Nonlinear constrained optimization by a nonrandom complex method", Journal of Research, National Bureau of Standards, Vol. 72C, No. 4, October-December, 1968.  30.  K. Soosaar, "Optimization of topology and geometry of structural frames", Sc.D. Thesis, M.I.T., May 1967.  62  APPENDIX A  Enter" voiH-> Unburn  E>uiUd Shffncss IX.  I I  TjeSi^n  member  E  Calculate  w&d&er  t YeS tAes^o&d Sachem  Asso^le. structure. \a\rc$est  members  Is  Avne.  "he Y^p'd ri£ fe f rv3»se  ?  "Yes  cycles  =>lere. ?  Yes  £]_ouJ_bia^ram -ftpr Xn<al<js^.-Resign Procedure  <2?tn-  Calculate, dilatable,  r<yv3iU s t r e s s ,  I  c?{  ~Ten9i^i~i  lc? ^ l l o c o .  (C^lcul^te  Member  ferule,  <al|c5oocalpl<.  S^r-eSS  Ccmpv^siue  •  Choose  Y-veui  Sectvc>v\  —».  /  Calculate 6»ll<?u)<5ible. bending stress Ffc> ;  I  Secirck -ha(ale. £<s>v •appv-opyi^he section  Check U€Jb a?mpctcf~r»ess  Check, sh&ar Stress  Nlewber S u b j e c t '  ^ P u r e ^bendm  66  G^lculcite &\\c*Ach\e> ber>d>iN3 stress.  stress J[^>v Section  j  CWoose hejui section  Caiculoile. cz>rnp/essn« Stress,-£s*  Yes  Rou  /^ch^n  Wo  Di<flgr<arr> .-fear ^ygsic^n  67  APPENDIX B  ****************************************************************#****^ T A B L E OF S E C T I O N S *************************************************************************^ HF BEAM S E C T I O N S **************** SIZE SMALL 8JK6.b 10JR9 10B 1 1.5 12JR 11 12B 14 12B16.5 14B 17.2 1 2D 19 10WF21 12WF22 14B22 14B26 16U26 14WF30 1 4WF34 16WF36 16WF4 0 16WF45 16WF45 16WF50 18WF50 18WF55 2 1WF55 21WF6 2 2 1WF68 24WF68  AREA 0. 1 1 1. 95 2. 64 3. 39 3. 4 5 4. 14 4. 86 5. 05 5. 62 6. 1 9 6. 4 7 6. 48 7. 65 7. 66 8. 81 10. 00 10. 5 9 11. 7 7 13. 24 13. 25 14. 70 14. 71 16. 1 8 16. 1 9 18. 23 2 0 . 00 2 0 . 01  DEPTH  FLANGE WIDTH THICK 0. 1 1 0. 1 1 1 0. 1 11 0. 189 8. 00 2. 281 0. 206 10. 00 2. 688 9. 87 0. 204 3. 950 0. 225 3. 063 12. 00 0. 224 3. 970 1 1.91 0. 269 12. 00 4. 0 0 0 14. 00 4. 000 0. 272 4. 0 10 0. 349 12. 15 9. 90 0. 340 5. 750 31 0. 4 24 4. 0 30 12. 13. 72 0 0 0 0. 3 35 5. 89 13. 0. 4 18 5. 025 0. 345 15. 65 5. 50 0 0. 38 3 6. 733 13. 86 14. 00 6. 750 0. 453 0. 428 15. 85 6. 992 0. 5 03 1 6 . 00 7. 000 16. 12 0. 563 7. 0 39 0. 499 17. 06 7. 4 7 7 7. 073 0. 628 16. 25 18. 00 7. 500 0. 570 7. 532 0. 6 30 18. 12 2 0 . 80 8. 2 1 5 0. 522 0. 6 15 20. 9 9 3. 240 0.680 8. 2 70 2 1 . 13 0. 5 8 2 2 3 . 71 8. 961  WEB THICK 0. 111 0. 135 0. 155 0. 1 80 0. 175 0. 200 0. 230 0. 210 0. 240 0. 240 0. 260 0. 230 0. 255 0. 250 0. 270 0. 287 0. 29 9 0. 30 7 0. 346 0. 355 0. 380 0. 358 0. 390 0. 375 0. 400 0. 430 0. 416  IX 0.1 13.7 39.0 51.9 72.2 83.2 105. 3 147.3 130. 1 105.3 155.0 197.4 242.6 298.1 289.6 339.2 445 .3 515.5 583. 3 704.5 655.4 800.5 889.9 1 140.7 1326.3 1473.3 1814.5  X-AXIS SX ' 0. 1 4. 7 7. 8 10. 5 12. 0 14. 8 17. 5 21. 0 21.4 21.5 2 5. 3 28. 8 34. 9 38. 1 41.8 48. 5 56. 3 64. 4 72. 4 7 8. 9 80. 7 89. 0 98. 2 109. 7 126. 4 139. 9 153. 1  RX 0. 1 1 3. 12 3. 85 3. 92 4. 57 '4. 6 1 4. 65 5. 40 4. 81 4. 14 4. 91 5. 52 5. 6 3 6. 24 5. 73 5. 83 6. 49 6. 62 6. 64 7. 30 6. 68 7. 38 7. '4 1 8. 40 8. 53 8. 59 9. 53  I¥ 0.1 0. 3 0.6 2.0 1.0 2. 3 2. 8 2. 7 3. 7 9.7 4. 6 6. 4 8.3 8. 7 17.5 21.3 22. 1 2 6. 5 3D. 5 31.9 34. 8 37. 2 42.0 44.0 53. 1 60. 4 63. 8  tf-AXIS SY R If 0. 1 0.11 0. 3 0.42 0. 5 0.43 1.0 0.77 0. 6 0.53 1. 1 0.74 1.4 0.76 1.3 0.72 1. 8 0.3 1 3.4 1.25 2. 3 9.85 2. 6 0.9 9 3. 3 1.04 3. 2 1.07 5. 2 1.41 6. 3 1.4 6 6.3 1.45 7. 6 1. 50 8.7 1.52 8. 5 1.55 9. 8 1 . 54 9. 9 1.59 1 1. 1 1.61 10. 7 1.65 12. 9 1.71 14. 6 1.74 14.2 1.79  WT/FT 0. 1 6. 5 9. 0 11.5 11.8 14. 0 1 6. 5 1 7. 2 1 9. 0 21.0 22.0 22.0 26.0 26.0 30.0 3 4.0 36. 0 40. 0 4 5.0 45. 0 50. 0 50. 0 55. 0 55. 0 62. 0 68. 0 6 8. 0  CONT. WF BEAM SECTIONS **************** SIZE AREA DEPTH 22. 37 2UWF76 24. 71 24WF84 27WF84 24. 72 24WF9 4 27. 63 27WF94 27. 65 30WF99 29. 1 1 31. 77 30WF108 34. 13 30WF1 16 34. 71 33WF118 33WF130 38. 26 39. 70 36WF135 33WF141 41. 51 44. 1 6 36WF150 36WF 160 47. 09 36WF170 ' 49.98 53. 54 36WF182 57. 1 1 36WF194 58. 7 9 33WF200 64. 73 33WF220 36WF230 67. 73 72. 0 3 36WF245 36WF260 7 6.56 36WF280 82. 32 36WF300 88. 1 7 99. 99 TOO BIG  23. 91 24. 09 26. 6 9 24. 29 26. 9 1 29. 6 4 29. 82 30. 00 32. 86 3 3.10 35. 55 33. 3 1 3 3.84 36. 00 36. 16 36. 32 36. 4 8 33. 00 33. 25 35. 88 36. 06 36. 24 36. 5 0 36. 72 36. 99  FLANGE WIDTH THICK 8. 9 85 0. 682 0. 772 9. 015 9. 9 63 0. 636 0. 872 9. 061 0. 747 9. 990 10. 458 0. 670 13. 4 84 0. 760 10. 500 0. 850 0. 735 11. 484 0. 855 11. 510 0. 794 11. 945 0. 960 11. 535 0. 940 11. 974 12. 000 1. 020 12. 027 1. 100 12. 072 1. 180 12. 117 1. 260 1. 150 1 5.750 1. 275 15. 810 16. 475 1. 260 16. 512 1. 350 16. 555 1. 440 16. 595 1. 570 16. 655 1. 680 20.000 1. 999  WEB THICK 0. 440 0. 470 0. 436 0. 51 6 0. 49 0 0. 522 0. 548 0. 564 0. 554 0. 580 0. 598 0. 605 0. 625 0. 653 0. 680 0. 725 0. 770 0. 71 5 0. 775 0. 765 0. 83 2 0. 845 0. 885 0. 945 0. 999  IX 2096.4 2364.3 2 8 24.3 2683.0 3 266.7 3 938.5 446 1.0 4 919.1 5885.9 6 6 99.0 7795. 1 7442.2 9012.1 9738.3 10470.3 1 1281.5 12103.4 11048.2 12312.1 14 988.4 16092.2 17233.3 13819.3 20290.2 99999.9  X-AXIS SX RX 9. 68 175. 4 196. 3 9. 78 211.7 10. 69 220. 9 9. 85 242. 8 10. 87 269. 1 11. 70 299. 2 1 1.85 327. 9 12. 0 0 358. 3 13. 02 404. 8 13. 23 438. 6 14. 01 446. 8 13. 3 9 502. 9 14. 29 541. 0 14. 3 8 579. 1 14. 47 621.2 14. 52 663. 6 14. 88 669. 6 13. 71 740. 6 13. 79 8 3 5. 5 14. 88 892. 5 14. 95 951. 1 15. 00 1031.2 15. 12 1 105. 1 15. 17 9999.9 20. 00  Y-AXIS IY 5Y RY 76. 5 17. 0 1. 35 88.3 19. 6 1. 8 9 19. 2 1 .97 95. 7 132. 2 22. 6 1. 92 115. 1 23. 0 2. 0 4 1 16. 9 22. 4 2. 00 1 3 5. 1 25. 8 2. 06 29. 2 2. 12 153.2 1 70. 3 29. 7 2. 2 2 231. 4 35. 0 2. 29 207. 1 34. 7 2. 23 229. 7 39. 8 2. 35 250.4 4 1.8 2. 3 3 2 7 5. 4 45. 9 2. 42 300. 6 50. 0 2. 4 5 54. 3 2. 4 7 327. 7 355. 4 58. 7 2. 49 59 1.7 87. 8 3. 43 732. 4 99. 0 3. 4 3 873 . 9 105. 7 3. 59 944. 7 114. 4 3. 62 13 20.6 123. 3 3. 65 1127.5 135. 9 3. 70 1225. 2 147. 1 3. 73 3300.0 300. 0 9. 99  WT/F r 76. 0 84. 0 84. 0 94.0 94.0 9 9. 0 1 38. 0 116.0 118.0 1 30. 0 1 35. 9 14 1.0 1 50. 0 1 60. 0 1 70. 0 1 82. 0 19 4.0 200. 0 2 2 0.0 230.0 2 4 5. 0 260. 0 2 80.0 300.0 800. 0  WF COLUMN SIZE SMALL 6W12 4W n 8W17 5W18.5 6W20 8W24 8W28 8N32.6 8W3 5 8W40 8W48 12W53 10W54 10W60 12W65 12W72 1U W 78 12W79 12W85 14W87 14W95 14W103 14W1 1 1 14W 1 19  SECTIONS  AREA 0. 1 1 3 . 54 3. 82 5. 01 5. 4 3 5. 8 8 7.06 8 . 23 9 . 56 10.30 1 1.80 1 4 . 10 1 5 . 60 1 5 . 90 1 7 . 70 1 9 . 10 21.20 22.90 23.20 25.00 2 5 . 60 2 7 . 90 3 0 . 30 3 2 . 70 3 5.00  DEPTH 0. 1 1 6 . 00 4 . 16 8 . 00 5 . 12 6 . 20 7 . 93 8 . 06 8 . 00 8 . 12 8 . 25 8 . 50 1 2 . 06 1 0 . 12 1 0 . 25 1 2 . 12 1 2 . 25 1 4 . 06 1 2 . 38 1 2 . 50 1 4 . 00 14. 1 2 1 4 . 25 14. 3 7 1 4 . 50  FLANGE WIDTH THICK 0 . 1 1 1 0 . 111 4.000 0 . 279 0 . 34 5 4.060 0 . 308 5. 250 0 . 4 20 . 5 .0 2 5 0 . 367 6. 318 0 .398 6. 500 0 . 46 3 6 . 54 0 0 .459 7. 940 0 . 49 3 3. 027 0.558 8.077 3 . 117 0. 683 0 .576 10. 000 0 . 6 18 13. 028 0 .683 1 0 . 0 75 0 . 606 12.000 0.6 7 1 12. 040 0 . 7 18 12. 000 0 .736 12. 080 0 . 796 12. 105 0 .688 1 4 . 500 1 4 . 54 5 0 . 748 14. 575 0 . 8 13 1 4 . 620 0 . 87 3 0 . 9 38 1 4 . 650  WEB THICK 0.111 0 . 230 0.280 0 . 233 0 . 2 65 0. 258 0.245 0. 285 0.315 0. 315 0.365 0 . 43 5 0.345 0 . 368 0.415 0 . 39 0 0 . 430 0 . 428 0 . 470 0. 495 0 . 420 0. 465 0. 495 0 . 540 0 . 570  IX 0.1 21.7 11.3 56.5 25.4 4 1.5 82.5 97.3 114.0 125.3 145.3 184.0 425 .0 305.0 34 4.3 533.3 59 7 .0 851.3 663.0 723.0 967.0 1 060.0 1 170.0 1270.3 1370.0  X-AXIS SX' 0. 1 7. 3 5. 5 14. 1 9. 9 13. 4 20. 8 24. 3 2 8. 4 31.1 35. 5 43. 2 70. 7 60. 4 6 7. 1 88. 0 97. 5 121.3 107. 3 1 16. 0 138. 0 151.0 1 64. 0 176. 0 189. 3  Y-AXIS  RX 0. 1 1 2 . 48 1. 72 3 . 36 2 . 15 2 . 66 3 . 42 3 . 45 3 . 44 3 . 50 3 . 53 3. 6 1 5 . 23 4 . 39 4.4 1 5 . 28 5 . 31 6 . 09 5 . 34 5 . 38 6 . 13 6 . 17 6 . 21 6 . 23 6 . 25  IY 3. 1 2.9 3.8 7.4 8.9 13.3 18.2 21.6 34. 1 42.5 49.0 60. 9 96. 1 134.0 116.0 1 75.0 195.0 207.0 216. 0 235.0 353 .0 384.0 420.0 455.3 492.0  5Y 0. 1 1. 5 1. 9 2. 8 3. 5 4. 4 5.6 6. 6 8. 6 10. 6 12. 1 15. 0 19. 2 20. 7 23. 1 2 9. 1 32. 4 34.5 3 5 .8 38. 9 48. 2 52. 8 57. 6 62. 2 67. 1  RY 0. 1 1 0 . 92 0 . 99 1. 22 1. 23 1. 51 1. 6 1 1. 62 1. 89 2 . 03 2 . 34 2.0 3 2 . 43 2 . 56 2 . 57 3 . 01 3 . 34 3 . 00 3 . 05 3 . 07 3 . 70 3 . 71 3 . 72 3. 7 3 3 . 75  WT/FT  0. 1 12. 0 13. 0 1 7.0 1 8 .5 20. 0 24. 0 2 8 .0 3 2 . (, 35. 0 40. 0 48. 0 53. 0 54. 0 60. 0 65. 0 72. 0 78. 0 79. 0 85. 0 87. 0 9 5 .0 103. 0 111. 0 119. 0  CONT. WF COLUMN SECTIONS SIZE  AREA  3 7 . 30 14W127 14W 1 36 a o . 00 a i . 80 1UW 142 m w 150 aa. 1 0 4 6 . 50 i a w 158 49. 1 0 law 167 5 1 . 70 law 176 law 18a 54. 1 0 56. 70 1 a w 19 3 5 9 . 40 law 202 iaw211 6 2 . 10 6 4 . 40 iaw219 6 7 . 10 iaw228 69. 7 0 14W237 iaw2a6 7 2 . 30 iaW264 7 7 . 60 8 4 . 40 iaW287 iaw3ia 9 2 . 30 9 4 . 10 14W320 14W342 1 0 1 . 00 109. 00 14W370 14W398 117. 00 12 5.00 iawa26 134. 0 0 iawa55 iaw500 147. 00 iaw550 162. 00 1 7 8 . 00 iaw605 14W655 196.00 14W730 215.00 TOO BIG 990. 90  DEPTH 14. 62 14. 7 5 1 4 . 75 14. 8 8 1 5 . 00 1 5 . 12 15. 25 1 5 . 38 15. 50 1 5 . 63 15. 7 5 1 5 . 87 16. 00 16. 12 16. 25 16. 50 16. 81 1 7 . 19 16. 81 1 7.56 1 7 . 94 1 8 . 31 1 8 . 69 19. 0 5 19. 6 3 2 0 . 26 2 0 . 94 2 1.67 2 2 . 44 3 5 . 99  FLANGE WIDTH THICK 14.690 0 .998 14. 740 1.06 3 1.063 15. 500 15. 51 5 1. 128 1. 188 15. 550 1.248 15. 600 1.3 13 1 5.64 0 15. *>60 1. 378 1.438 15. 710 15. 750 1.503 1.563 1 5 . 800 15. 825 1.623 1 5 . 865 1.688 15. 910 1.748 1.8 13 15. 945 16. 025 1.938 16. 1 30 2 .093 16. 235 2 . 283 16. 710 2 .093 16. 3 65 2 .468 16. 475 2 .6 58 16. 590 2 . 84 3 3 .0 33 16. 695 16. 82 8 3 .213 1 7 . 008 3 .501 17. 206 3 . 8 18 4 . 157 17. 418 17. 64 6 4 .522 4 .9 10 1 7.889 5 . 999 20. 000  WEB THICK 0. 610 0. 6 60 0. 680 0.-695 0. 730 0 . 780 0. 820 0 . 840 0. 890 0 . 930 0. 980 1.000 1.040 1.030 1. 1 30 1. 210 1. 310 1. 420 1.890 1. 550 1. 660 1.770 1.880 2. 0 10 2. 190 2. 390 2. 600 2. 830 3. 070 0. 500  X-AXIS IX 1480. 0 1590. 0 1670. 0 1790. 0 1900. 0 2 0 2 0 .0 2 150.0 2270.0 2400.0 2540.0 2670.0 2 8 0 0 .0 2940.0 3080.0 3230.0 3530.0 3910.0 4400.0 4 14 0 .0 4910.0 5450.0 6010.0 6 6 10.0 7220.0 8250.0 9450.0 1 0 9 0 0 .0 12500. 0 1 4 4 0 0 .0 9 9 9 9 9 .9  SK  202. 0 216. 0 227. 0 240. 0 2 53. 0 267. 0 282. 0 29 6. 0 310.0 325. 0 3 3 9.0 353. 0 368. 0 38 2. 0 392.0 427. 0 465. 0 512. 0 493. 0 559. 0 638. 0 657. 0 707. 0 758. 0 8 40.0 9 33. 0 1040.0 1150.0 1280.0 9999.9  RX 6. 29 6. 3 1 6. 32 6. 37 6. 40 6. 42 6. 45 6. 49 6. 5 1 6. 54 6. 55 6. 59 6. 62 6. 65 6. 68 6. 74 6. 8 1 6. 90 6. 53 6. 99 7. 08 7. 17 7. 25 7. 35 7. 4 9 7. 64 7. 81 7. 99 8. 18 2 0 . 00  IY 523. 0 568. 0 660. 0 70 3.0 745. 0 790 .0 7 38 .0 78 3.0 3 3 3 .0 380.0 1 3 33.0 1370. 0 1123. 0 1 170.0 0 1 2 33 . 1 3 3 0 .0 1 473. 0 16 3 0 .0 1 5 4 30. 1310. 0 1990. 0 2173. 0 2 3 6 3 .0 2560. 0 2333. 0 3260. 0 3633 .0 4170. 0 4723. 0 9000. 0  Y-AXE S S Y RY 7 1.8 3 .76 77. 0 3 .77 8 5 . 2 3 .97 90. 6 3 .9 9 9 5 . 8 4 .30 10 1.0 4 .0 1 107. 0 4 .32 113.0 4 .04 118.0 4 .05 124.0 4 .06 130.0 4 .07 13 6.0 4 .03 142.0 4 .10 14 8. 0 4 . 1 1 154. 0 4 . 1 2 1 66. 0 a .14 1 8 2 . 0 4 .17 2 0 1 . 0 4 . 20 1 96. 0 4 . 17 22 1. 0 4 .24 24 1.0 4 .27 2 6 2 . 0 4 . 31 283. 0 4. 3 4 3 04 . 0 4 . 37 3 3 9. 0 a .43 3 78. 0 4 .49 4 23.0 a . 55 4 72. 0 4 .62 5 2 7 . 0 a .69 9 0 0 . 0 9 .99  WT/F  127. 0 1 3 6 .0 142. 0 1 5 0 .0 158. 0 1 6 7 .0 1 7 6 .0 1 B4.0 193. 0 202 .0 21 1 .0 219. 0 2 2 8.0 2 3 7.0 246. 0 2 64 .0 287. 0 314. 0 320. 0 3 4 2 .0 3 7 0 .0 3 9 8.0 4 2 6.0 455. 0 500. 0 550. 0 605. 0 655. 0 730. 0 80 0 .0  

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0050542/manifest

Comment

Related Items