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Computer aided form-finding for cable net structures

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Computer aided form-finding for cable net structures

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Content CO M PU TER AIDED FO RM -FIN D IN G FO R CABLE NET STR U C TU R ES by Tian-An Feng A Thesis Presented to the FA C U L TY OF TH E G R A D U A T E SCHOOL U N IV ERSITY OF S O U T H E R N C A LIFO R N IA In Partial Fulfillm ent of the Requirem ents for the Degree M ASTER OF B U ILD IN G SCIENCE August 1988 UMI Number: EP41416 All rights reserved INFORMATION TO ALL USERS The quality of this reproduction is dependent upon the quality of the copy submitted. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if material had to be removed, a note will indicate the deletion. UMI EP41416 Published by ProQuest LLC (2014). Copyright in the Dissertation held by the Author. Dissertation Publishing Microform Edition © ProQuest LLC. All rights reserved. This work is protected against unauthorized copying under Title 17, United States Code ProQuest LLC. 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, Ml 4 8 1 0 6 -1 3 4 6 U N IV E R SITY O F S O U T H E R N C A L IF O R N IA THE GRADUATE SCHOO L UN IV ER SITY PARK LO S A N G ELES. CA LIFO R N IA 9 0 0 0 7 This thesis, written by T ian -A n Feng under the direction of h..i.S...Thesis Committee, and approved by all its members, has been pre sented to and accepted by the Dean of The Graduate School, in partial fulfillment of the requirements for the degree of of^BuHr^di na Sci ence Dean Date. THESIS C O M M ITTE E Chairman To my aunt, Shu-zhen, and the memory of my uncle, Ku-nien Chang ii ACKNOWLEDGMENTS I would like to thank my thesis committee members, Prof. Marc Schiler, Prof. James Ambrose, and especially Prof. Goetz Schierle, who led me to explore the fascinating research field and provided me with careful guidance I have benefited from the course work I have done under the direction of Prof. Earl Sheffield at the D epartm ent of Program ming and Data Processing. I want to thank Sigma Xi, the Scientific Research Society, for their funding of the research which provided me with great encouragement. I would also like to thank Dean Harris and the School of Architecture, as well as The G raduate School of USC, for their substantial support for my study through a teaching assistantship, the James Knowles Memorial Scholarship, and the G raduate Professional Scholarship. I am very grateful to my cousin and his wife, Dr. and Mrs. Cheng-Yang Chang. Their sponsorship provided me with the opportunity to further my education in the United States. CONTENTS Page Chapter 1 IN T R O D U C T IO N 1 1-1 B A C K G R O U N D IN FO R M A T IO N 1 1-1.1 Cable Net Structures 1 1-1.2 Development of The Structures 2 1-2 TH E AIM OF THIS STUDY 7 1-3 TH E RESEA R CH M ETH ODO LO GY 8 Chapter 2 C O M PU TER AIDED FO RM -FIN D IN G PROCESS W ITH ID G P 9 2-1 G E N E R A L 9 2-2 PRINCIPLES OF S T R U C T U R A L ANALYSIS & FO R M -FIN D IN G PROCESS 11 2-3 DESCRIPTIO N AND USE OF IDGP 19 2-3.1 Loading and Running IDGP 20 2-3.2 The Input of IDGP 21 2-3.3 The O utput of IDGP 21 iv 2-4 TR ITR S AN D ITS APPLICATIONS 2-5 ISAP A N D ITS APPLICATIONS 2-6 D E M O N STR A TIO N OF TH E FO R M -FIN D IN G PROCESS 22 23 25 Chapter 3 T H E DEVELOPM ENT OF ID G P 31 3-1 G E N E R A L 31 3-2 M ATH SIM ULATION OF TH E S T R U C T U R E GEOM ETRIES 32 3-2.1 Coordinate Systems 32 3-2.2 Joint 32 3-2.3 Member 34 3-2.4 Boundaries 34 3-3 PR O G R A M M IN G AND TESTING 37 3-3.1 Program m ing 37 3-3.2 Testing 38 Chapter 4 CO NCLUSIO NS 39 A PPEND IXES 40 1. TWO DIM ENSIONAL S T R U C T U R A L IMAGE DISPLAY 41 2. T H R E E DIM ENSIONAL S T R U C T U R A L IMAGE DISPLAY 52 REFERENCES 59 v TABLES Page 1. Data Set 16 2. Coordinates & Displacement A fter Form -finding R u n 17 3. The Change In Edge-Cable Forces IS 4. Data Set 27 vi FIGURES Page 1. The Large Space With Minimal Obstruction 2 2. A Bridge Built By N atural Ropes 3 3. The Golden Gate Bridge 3 4. The Tokyo Olympic Swimming Stadium 4 5. The G erm an Pavilion In Montreal (1967) 5 6. The G erm an Pavilion (under construction) 6 7. The Form -finding Process With IDGP 10 8. Joint Displacement 12 9. N ew ton-Raphson Method 13 10. Plan of The Structure 16 11. The Process of A Form -finding R un With IDGP 20 12. Plan of The Structure 25 vii 13. Axon, X-View & Y-View of The Structure 26 14. Joints, Members, Boundaries & Net 33 15. 3D Curved Boundary 36 16. O rganization of IDGP 37 viii ABSTRACT A n interactive computer aided form -finding process for cable net structures is presented. The form -finding process is form ed by the coordination of existing programs TR ITR S (fo r structural analysis) and ISAP (for data generation, editing and display), as well as the interactive data generation program, IDGP, which was developed by the author. The development of IDGP is introduced. The principles of the structural analysis for the form -finding process is explained. ix 1. INTRODUCTION 1-1 BACKGROUND INFORM ATION "With its roots in both technology and nature, lightw eight architecture is a contemporary discipline that enables designers to respond to humankind’s needs fo r aesthetic satisfaction, shelter and the e n jo y m e n t o f natural environments." from The First International Conference on Lightweight Structures in Architecture, 1986 1-1.1 Cable Net Structures Lightweight structures include cable net, membrane, shell, and folded structures, as well as space frames, domes, arched, stayed and trussed systems. Among them, cable net structures offer greater formal 1 e x p re s s iv e n e s s a n d s t r u c t u r a l f a n ta s y , a n d possess t r e m e n d o u s capacity for spanning greater distances and enclosing larger areas with minimal obstruction in comparison with any other structures. Fig. 1. In cable net structures, loads are taken by the axial tension resistance of cables. With a given dimension of member size, a cable can offer m uch greater tension resistance in comparison to other materials. This im portant characteristic of the cable net structures allows the structures to optimize the use of materials, and therefore, minimize the waste of resources. 1-1.2 Development of The Structures The earliest structure with a similar nature to the cable net structures Fig.l The Large Space with Minimal Obstruction 2 was probably tentlike, made of animal skins or thin branches. The discovery of knotting and weaving made fabric tents, fishing nets, and sails possible. N atural ropes were the first materials used forconstructingsm all suspension bridges. Fig.2. With the availability of iron and steel as structural materials, the first forms of modern suspension bridges appeared. Prestressed cables were used to form the structural geometry. M any of these bridges, such as the Golden Gate Bridge, the George Washington Bridge were built in the United States during the first half of this century. Fig.3. The achievements in analysis, design, and construction of suspension bridges introduced a new era of m odern tensile architecture. During the past few decades, a new and rapid movement in cable net structures has been developed. A good example of this Fig.2 A Bridge Built By N atural Ropes Fig.3 The Golden Gate Bridge 3 Prestressed network supporting the roof Main cables Anchor ->c Anchor Edge beam Struts OO Fig.4 The Tokyo Olympic Swimming Stadium 4 development is the Tokyo Olympic Swimming Stadium. It illustrates how the structural principle of suspension bridges was incorporated into the design of other building structures. Fig.4. • The history of this new development can be briefly seen through four phases. In the first phase, the emphasis was on the solution of technical problems. In the second phase, architects were deeply engaged in exploring the expressive implications of the forms that the structures could provide. The completion of the G erm an pavilion at Expo’67 in Montreal, was an indication of the arrival of the third phase. Fig.5. During that time, architects and engineers paid more attention to the interrelations between form and structure. In particular, the interactive effects of the final shape for the structure versus level of prestress, and the determ ination of structural geometry versus the running directions of cables and the locations of cable intersections. As a result of these findings, the nature of the structures has been conceived and thoroughly understood in terms of this Fig.5 The German Pavilion in Montreal, 1967 5 unique form-finding process. Fig.6. Prior to the fourth phase , the form -finding process was done by physical modeling, which is called experimental architecture. In fact, it is always d ifficult to efficiently and accurately find the form of cable net structures through physical modeling. However, the difficulty is Fig.6 The German Pavilion (under construction) 6 challenged by m odern advances in computer technology. Incorporated application programs have partially substituted m anual operations for the form -finding processes for cable net structures. This achievement has occurred in the fourth phase of the development of cable net structures, which began about two decades ago. 1-2 THE AIM OF TH IS STUDY The available computer aided form -finding programs for cable net structures have facilitated form -finding solutions for some particular structural forms. Ref.6, Ref.7, R ef.8. However, these programs can not be applied to general cases but only to specific forms. Architects who are interested in the design of such structures have not yet significantly benefited from these findings. Therefore, there is a need for a computer aided form -finding processor which can be used by architects with fundam ental knowledge of computers to solve general cases. The aim of this study is to develop a tool which may used by architects and engineers to determ ine the form of the cable net structures. 7 1-3 THE RESEARCH METHODOLOGY The methodologies employed in this study are computer program ming, data processing and programs coordination, as well as com m unication between a m ain-fram e computer and IBM personal computer. A interactive data generation program has been developed to constitute a user friendly computer aided form -finding processor. The name of this program is IDGP. The development of IDGP included the following phases: - Development of a theoretical model - Computer program ming - Program testing - Development of sample structures. 8 2. COMPUTER AIDED FORM-FINDING PROCESS W ITH IDGP 2-1 GENERAL There are two subjects in computer applications that have been intensively studied, and the results have provided the design of cable net structures with great promise for the improvement of form -finding process. The two subjects arecom puter aided structural analysis and simulations and com puter aided draftin g and graphics. However, the results of the studies have not provided a computer aided sequential operation for the form -finding process. In other words, the process of computer aided structural analysis and the process of com puter aided graphics are segregated. The process of data generation, inputting and outputting was still perform ed manually. The subject of this study is to incorporate the existing application program for structural analysis and simulations, TRITRS, and application program for structural image displaying and data editing, ISAP, in 9 com bination with the data generation program, IDGP, to simulate the conventional form -finding process for the structures. As illustrated in Fig.7, the form -finding process with IDGP is a man- m achine interactive process of w hich segregate programs are coordinated to perform d ifferen t tasks sequentially. a in I a x / a o u t p u t Fig.7 The Form-finding Process With IDGP 10 2-2 PRINCIPLES OF STRUCTURAL ANALYSIS & FORM -FINDING PROCESS* A general method of analysis for cable structures is stiffness analysis which leads to a stiffness-m atrix solution. In carrying out the analysis and the computation the following assumptions are made to simplify the problem: 1) The cable is treated as being completely flexible; i.e., it cannot sustain any bending moment. 2) The cable is considered incapable of taking any compressive forces. 3) The cladding does not contribute to the stiffness of the structures. 4) The intersection of two or more cables is treated as a joint. 5) The cable elements ( members ) lie along straight lines between the joints. The method can be best explained by considering a joint q in any space cable net connected to an adjoining joint p through element i. The pretension stress in a typical element is Fi and its length is . The initial coordinates of the joint p and q are Xp .Yp . Z and Zq, respectively. As shown in Fig.7, a system of forces P Xg , Pyg and P Zg is applied at joint q. The joint p and q deflect through up , vp , wp and uq, vq, wq, respectively, along the x, y, and z axes. The force F ■ and the length S ■ change by amounts Z \F { . and AS,- and become F-’ and S-', respectively. * The inform ation contained in the section partially based on Ref.5 11 The equations of equilibrium at joint q, before the application of loads, can be written as SUM F./S,. (Xp - X q) = 0 SUM F ./S,. (Yp - Y q) = 0 SUM F f ./S( . (Z p ~ Z q) = 0 The length of the element, S■ can be expressed as S i = [ ( • V V 2 + {Yp~Yq) 2 + { Z p-Z q) 2 ] V 2 ( 2 _ 2 ) Under application of loads, the equations will change as follows: SUM [ F / / S r {Xp + Up - X q - Uq)\ + Pxq = 0 (2-3a) SUM [F .’/S / (Yp + Vp - Yq - Vq )] + P .^ = 0 (2-3b) SUM [ F / / S / (Zp + Wp ~Zq - Wq)) + = 0 (2-3c) s / = [ ♦ Up - x q - Uq)2 + (yp + -y 9 - f / + ( Z p + WP ~z q ' V )2 ] 7 / 2 (2 ' 4) (2-la) (2-lb) (2-lc) S traight w eightless cable e lem e n ts.^ V p Initial position Deflected position Fig.8 Joint Displacement 12 Also, F = Fj- + £\Fi and = t (S/ - 1 = ^ A - (Sf-’/Sf - - 1) (2-5) where EA^ is the extensional stiffness of the <th element. Equations 2-2 and 2-4 can be combined to give Sj ’ = S( (1 + 2a- + bt-)7 / 2 (2-6) where a f = (1/ S 2 )[ (X p - X q)(Up - Uq) + (Yp - Yq)(Vp - Vq) + <Zp ~ Z q * Wp ~ bi = (1/ s 2 )[( UP - Uq)2 + {Vp - Vq)2 + ( ^ - Wq)2 ] Expanding the right-hand side of Eq. 3-6 and making the appropriate Typical nonlinear resp o n se curve - .02) (0) P AU' Fig.9 Newton-Raphson Method 13 substitution in Eq. 3-5, we get A F ,- = EA. (a,. + b-/2 - a ? /2 -a .b;/2 + a / / 2 + ... ) (2-7) Om itted the rest of derivations, we have R xq - - Sum (EA,. - F,0 [ (Up - Uq) ^ / S, + (Xp - Jr?)d,./<2Si) ] R y? = - Sum (EA,. - F,) [ (V p - K ?)c,/S , + {Yp - Y J i J V S , ) ] R zq - - Sum (E A , - F,) [ twp - H '?)cJ ./Si + (Zp - Zq)d ,/(2S,) J ( 2-8 ) where c t- = a f- + b;/2 - /2 d. = b f- - 3a-2 - 3a (b ■ + 5 a / For a cable net structure three equations such as Eqs. 2-8 will be required per joint, and this will lead to a set of 3N simultaneous equations, where N is the total num ber of cable joints, with the supporting structure being treated as rigid. The solution of these equations by a suitable iterative method will give the values of U,V and W. T h e su b stitu tio n o f these values into Eqs. 2-7 will give the change in forces. It can be seen from the preceding analysis that the problem of analyzing cable structures reduces to the solution of nonlinear Eqs. 2-9. These equations when assembled for the whole or part of a structure can be expressed in the matrix form as (boldface letters indicate a matrix.) K * U = - P + R (2-9) where K = square stiffness matrix for structure; consists of coefficients of unknowns U, V, W, etc., in either Eqs. 2-8. U = Vector of unknowns U, V, W P = load vector 14 R = column vector containing residual terms Rxq , Ryq , Rzq , etc. One of the methods to find the solution for the above problem is the New ton-Raphson method. The com putations in this method are based on the instantaneous stiffness of the structure, derived anew in each iterative cycle. The steps involved are given below and represented graphically in Fig-8. 1. Assume = 0 2. Solve y S °) * U(J) = - P to evaluate V (1K 3. With the aid of the obtained value of compute 4. Solve = R ^ ^ to obtain J \ U ^ ^ , which is the correction to U (1) . Thus evaluate U (2) = U(J) + , and R^2^ . 5. Solve k/ 2^ * U^2^ = R^2^ , and so on. This procedure is carried on until the value of / \ U or R becomes smaller than a specified limit. The above introduced structural analysis is perform ed by the program TRITRS. In order to process the analysis, an assumed data set is required to define the initial geometry of the structure. The assumed data is usually generated by m anual computeations as shown in Table 2. A main feature of IDGP is to generate the assumed data autom atically based on the user’s input of the mesh dimensions and boundary conditions as shown in Table 4. For each iteration of computation, TRITRS produces a set of adjusted data against the set of data of the previous iteration. This set of data provides the inform ation of joint coodinates, the change of forces, ect. 15 Fig.10 Plan of The Structure Edge cables .371 .33 29, .20 ,38, .34 .30] 2 4 2 3 22 35. .39 28] 27. .26 25. 4 0 32 36, 2 3 22 2 4 25 Edge cables Nodes 1,5 ,2 1 ,2 5 are supports Member num bers are shown circled Pretension specified for members 1 to 16 50 kips (222.4 kN) Pretension specified6 for members 17 to 40 5 kips (22.24 kN) Extensional stiffness EA for members 1 to 16 30,000 kips (133.44 MN) Extensional stiffness EA for members 17 to 40 5000 kips (22.24 MN) Table 1 Data Set S u c h a loop will not stop until the required accuracy of the com putation has been reached. A lode No. Assumed coordinates, ft x y z Ax, ft Az,ft 1 - 20.0 - 20.0 - 4 .0 0.0 0.0 0.0 2 - 10.0 - 18.5 - 2.0 -0.0101 0.0017 0.0507 3 0.0 - 18.0 0.0 0.0 0.0052 0.0 4 10.0 - 18.5 2.0 0.0101 0.0017 - 0.0507 5 20.0 - 20.0 4.0 0.0 0.0 0.0 6 - 18.5 - 10.0 - 2 .0 0.0017 - 0.0101 0.0507 7 - 10.0 - 10.0 - 1.0 - 0.0101 - 0.0101 - 0.0548 8 0.0 - 10.0 0.0 0.0 -0.0101 0.0 9 10.0 - 10.0 IX ) 0.0101 -0.0101 0.0548 10 18.5 - 10.0 2.0 - 0.0017 -0.0101 - 0.0507 1 1 - 18.0 0.0 0.0 0.0052 0.0 0.0 12 - 10.0 0.0 0.0 - 0.0101 0.0 0.0 13 0.0 0.0 0.0 0.0 0.0 0.0 14 10.0 0.0 0.0 0.0101 0.0 0.0 15 18.0 0.0 0.0 - 0.0052 0.0 0.0 16 - 18.5 10.0 2.0 0.0017 0.0101 - 0.0507 17 - 10.0 10.0 1.0 -0.0101 0.0101 0.0548 18 0.0 10.0 0.0 0.0 0.0101 0.0 19 10.0 10.0 - 1.0 0.0101 0.0101 - 0.0548 20 18.5 10.0 - 2 .0 -0.0017 0.0101 0.0507 21 -2 0 .0 20.0 4.0 0.0 0.0 0.0 22 - 10.0 18.5 2.0 - 0.0101 - 0.0017 - 0.0507 23 0.0 18.0 0.0 0.0 - 0.0052 0.0 24 10.0 18.5 - 2 .0 0.0101 - 0.0017 0.0507 25 20.0 20.0 - 4 .0 0.0 0.0 0.0 Table 2 Coordinates & Displacement A fter Form -finding R un 17 The simple cable net structure shown in Fig.9 is used to illustrate the analytical procedure introduced above. The geometry of the supporting system, as well as the pretension stress in the cables are specified, and required to compute the net geometry. The joint coordinates in the nets have to be assumed and suitably m odified by a process of successive iteration, to be finally in equilibrium with the forces. The coordinates of the edge cables are assumed by taking each of them to be parabolic in shape and considering each to lie in a separate single plane. The assumed coordinates and the changes A X , A Y > obtained from the third iteration are given in Table 2. The change in edge-cable forces is shown in Table 3. The data used for this case may be found in Table 1. Table 3 The Change In Edge-Cable Forces Member Change in Member Change in number force, kips (kN) number force, kips (kN) 1 1.2920 (5.75) 5 1.2920 (5.75) 2 0.7037(3.11) 6 0.7037 (3.11) 3 0.7037(3.11) 7 0.7037 (3.11) 4 1.2920 (5.75) 8 1.2920 (5.75) 18 2-3 DESCRIPTIO N & U SE OF IDG P The perform ance of Interactive Data G eneration Program , IDGP, is designed to simulate the m anual data generation process for the form -finding purpose. For any valid input, the program autom atically generates pre- form at ted output to feed TR ITR S for structural analysis, or to feed ISAP for structural image display. The form at of output is designed in according with the required input form at for TR ITR S and ISAP. In order to carry out the structural analysis of the form -finding process, a coarse net must be generated for initial form -finding run. The coarse net is form ed by m em bersand boundaries. The members and boundaries are defined by joints. The joints are defined by X,Y,Z coordinates. IDGP is designed to generate the corse net for a given structure by providing coordinates of joints, identifications for joints and members. A fter completion of first form -finding process run, a fine net is requested for next form -finding run which produces the final shape of the structure. The procedure is illustrated in Fig.11. IDGP is designed to be run on the IBM personal computers. The program consists of three d ifferen t functional portions: input, algorithms and computations, and output. The details of how to use the program will be elaborated in following sections. 2-3.1 Loading & Running IDGP To load the program on a micro computer, type "idgp" after the DOS 19 T F - ;_rF^ £f£NStf^TfeS t n e I A F M V A & K e y f 0 (MPUT O JpATA lr4fliT - p T&fTRS FOfZ. *Srqzi}CX\Jti&L A nALX^IA .J . / — \ . s, v \ h 1 — — L '-> — i L T2> PepfMg- A «0i (W T 7 a- l ^rgOMST^ ^5P. TUB 5TF01XUKer X t W ^ e w e ^ A t s ^ T H e i ^ T A 5ET* FCf^L a a z &z &Gr w e j 0 IN P U T ^ C U TpU y Fig.11 The Procedure of A Form -finding R u n With IDGP prompt. Then, the first of three levels of input will be ready for entering data. The input will be described in the following section. The program will not be executed until all three levels of input are completed. A fter all the data has been processed, the program will ask the user to type in a 20 filenam e on which the output data will bessved. The user should specify which disk drive this file will be saved on. 2-3.2 T he Input of IDG P The input portion of the program requests the user to enter the data that defines a structure. The required input data is organized in three levels. In the first level, the user is asked to specify whether the structure is two dimensional (2-D) or three dimensional (3-D). In the second level, the user is requested to select boundaries (curve or straight line) of the structure. In the third level, the user is asked to enter the X, Y, Z coordinates of the structural supports, net dimensions in X and Y directions, as well as, the boundary conditions (rigid or non-rigid). The input coordinates must be in the three dimensional Cartesian coordinate system. Negative values are not accepted by IDGP. 2-3.3 The Output o f IDG P The output includes the following information: 1) X, Y, Z coordinates of the structural supports 2) boundary condition: rigid or non-rigid 3) identification num bers of the joints 4) Degree of freedom of joint in X,Y and Z coodinates (0=fee, l=fixed) 5) X, Y, Z coordinates of the joints (intersections) 6) identification num bers of the elements 7) connectivity of elements (from joint i to j) 21 The data sets are contained in the file with the name that the user specified. These data sets are required to determine the structure. The data sets can also be used for the structure image display or for structural analysis through ISAP or TRITRS. 2-4 TRITRS AND ITS APPLICATIONS TR ITR S is a structural analysis program developed by Dr. Eberhard Haug. TR ITR S computes shapes and load responses of cable net structures. And it may also be applied to any two or three dimensional truss type structures in which the members take axial forces only. The method for structural analysis has been introduced in preceding sections. It is written in the F O R T R A N IV language and runs on m ain-fram e computers. The program performs the following three basic operations: 1) Calculation of the initial shape of the structure. 2) Calculation of the new structural shape after changes in the member properties or prestress forces. 3) Calculation of deflections and member forces due to applied static loads, differential tem perature changes, or turnbuckle tightenings (change in length of members). Required input is categorized in three groups. These groups are Structure D efinition Input, Loading & Member Change Input, and Convergence Checking Input. The format for each input are specified. TR ITR S is limited in problem size. The lim itationcan be found through the calculation of the required array dimension. The form ula for this 22 calculation is 16*NJ + 5*NE + 5*MPROP + 2*NACD + NJW + NMW + 3*NJ*MBAND where NJ = num ber of joints NE = num ber of members M PROP = num ber of types of members NACD = num ber of displacement convergence checks if NACD>0; otherwise 1 NA CF = num ber of force convergence checks if NACF>o; otherwise 1 NMW = num ber of members to be observed during interm ediate iterations NJW = num ber of joints to be observed during interm idiate iterations M BAND = one-half bandw idth of the structure stiffness m atrix = 3*(maximum difference between joint num bers of the ends of any member) + 3 The program provides a print-out of the coordinatesof the joints before and after loading, unstressed length of members, and member forces. 2-5 ISA P AND ITS APPLICATIONS ISAP is a structural image display program that can be run on IBM personal computers. It is designed to perform three basic operations: 1) to display any geometries (structural images) in responding to inputted data sets of the Structure D efinition Input with the same form at required by TRITRS 23 2) allows user editing of the data set on the screen 3) allows user to produce output of structural images and data set through connected plotter or printer 4) generates data set for H.P. shaped structures ISAP has been developed by Dr. Goetz Schierle. 24 2-6 A DEM ONSTRATION OF THE FO RM -FINDING PROCESS A dem onstration of the com puter aided form -finding process for cable net structures is shown in this section to illustrate the coordination of ID G P with ISAP and TRITRS. Fig.11 is the plan of a hypothetical cable net structure designed to test IDGP. The data generated by the program is identical to the data obtained by m anual calculation. And the data has been used to generate the structural geometry successfully. The structure’s axonmetric, south elevation (Y-View), as well as, east elevation (X-View) are displayed in Fig.11. The data from the output of IDGP is listed in Table 4. PLAN SCALE 1 : 7 37 22 18 1 50 55 23. 49 IS 3Z 3a 40 51 54 57 61 \ l 2 45 32 22 33 53 50 44 56 47 17 6 0 19 19 23 23. ,36 43 15 i38 Fig.12 Plan of The Structure 25 Fig. 13 Axon, X-View & Y-View of The Structure AXON SCALE 1 : 7 SCALE 1 : 7 X -V IE W SCALE 1 : 7 26 )le 4 62 62 111 111 100 111 0 100 101 100 111 100 100 100 101 111 100 100 111 100 101 100 101 100 DA TA SET REM A RK S 1 0 -2 10.00 10.00 10.00 20.00 40.00 20.00 40.00 21.54 23.85 40.00 40.00 27.65 33.33 80.00 33.33 80.00 36.92 42.31 80.00 40.00 42.94 40.00 80.00 33.53 88.00 40.00 46.00 40.00 100.00 40.00 80.00 80.00 34.71 46.67 120.00 46.67 120.00 52.31 60.77 50.00 130.00 50.00 120.00 80.00 35.88 80.00 120.00 43.33 140.00 60.00 70.00 80.00 131.58 45.26 160.00 54.29 68.57 120.00 120.00 39.33 160.00 80.00 37.06 120.00 133.68 38.95 identification of analysis l = form -finding run 2=load run No. of joints and members No. of inputted joints & members joint information: Col.l joint No. Col.2 degree of freedom in X,Y,Z directions (0=free, l=fixed) Col.3 X coordinates Col.4 Y coordinates Col.5 Z coordinates 27 Table 4 (Continued) 23 100 " 160.00 120.00 35.33 24 101 200.00 42.86 65.71 25 101 160.00 135.79 32.63 26 111 210.00 40.00 65.00 27 101 200.00 80.00 38.24 28 101 200.00 120.00 31.33 29 101 240.00 31.43 62.86 30 101 200.00 137.89 26.32 31 101 240.00 40.00 59.47 32 100 240.00 80.00 39.41 33 101 240.00 120.00 27.33 34 0 260.00 80.00 40.00 35 100 246.67 120.00 26.67 36 111 273.33 40.00 53.33 37 111 240.00 140.00 20.00 38 111 280.00 20.00 60.00 1 1 2 1 2 2 5 1 3 5 10 1 4 10 12 1 5 12 14 1 6 14 18 1 7 18 22 1 8 22 25 1 9 25 30 1 10 30 37 1 element information: Col.l element No. Col.2 joint i Col.3 joint j Col.4 element kind (i.e. cable,prestress, or compression element) 28 Table 4 (Continued) 11 37 12 35 13 34 14 36 15 38 16 29 17 26 18 24 19 19 20 17 21 13 22 9 23 6 24 3 25 2 26 4 27 7 28 26 29 31 30 5 31 8 32 11 33 15 34 21 35 27 36 32 37 12 35 34 36 38 29 26 24 19 17 13 9 6 3 1 4 7 9 31 36 8 11 15 21 27 32 34 16 Table 4 (Continued) 38 16 20 39 20 23 40 23 28 41 28 33 42 33 35 43 3 4 44 4 8 45 8 10 46 6 7 47 7 11 48 11 16 49 16 18 50 13 15 51 15 20 52 20 22 53 19 21 54 21 23 55 23 25 56 24 27 57 27 28 58 28 30 59 29 31 60 31 32 61 32 33 62 33 37 identifies the end of data set 30 3. THE DEVELOPMENT OF IDGP 3-1 GENERAL The development of IDGP was conducted in four phases. These phases were the m athematical modeling phase, the program structuring phase, the coding phase and the testing phase. The program is designed to deal with data generation of both two dimensional(2-D) and three dimensional(3-D) cable net surface structures. The m athematical modeling of the program began with a 2-D math model. T hen 2-D program m ing and testing phases were conducted. A fter the completion of the program m ing cycle for 2-D ( m ath m o d elin g -stru c tu rin g - coding - testing ), then the 3-D program has been developed by extending the concepts and principles of 2-D geometry. The details of this program m ing process are introduced in following sections. 31 3-2 MATH SIM ULATION OF THE STRUCTURE GEOMETRIES 3-2.1 Coordinate System s Two types of coordinate system had been used during the development of IGDP: Cartesian and Polar coordinate system. A Cartesian coordinate system is a system whereby points on a plane are identified by an ordered pair of numbers, representing the distances to two perpendicular axes. A Polar coordinate system is a system w hereby any point in a plane can be identified by its distance from the origin (r) and its angle of inclination (0). Polar coordinates can be changed into Cartesian coordinates by using the equations: x = r * cos & y = r * sin & The three dimensional Cartesian coordinate system has been used in both TR ITR S and ISAP. In order to generate the required data which are identical to TR ITR S and ISAP’s, a Cartesian coordinate system was selected for IDGP to identify the locations of structural elements. Polar coordinates were only used during the analytical process of math modeling. All Polar coordinates have been converted into Cartesian coordinates in the program. 3-2.2 Joint A joint is defined by its coordinates X, Y, Z. Each joint is also given an identification num ber and a num ber defining the degree of freedom in X,Y,Z 32 directions. The values of the coordinates must be greater or equal to zero. Fig.13. The origin of the coordinate system is located by the user. Fig.14 Joints, Members, Boundary & Net 33 3-2.3 Member A member is a line segment which connects to joints. A member is defined by a num ber and the joint num bers at each of its ends. In one of the cable running directions, the coordinates of the ends can be obtained by finding the intercepts of a line equation and a plane equation. The equations of a line and a plane are: 2-D: ( Y - Y y )/( X - X 7 ) = ( Y2 - Y 7 )/( X 2 - X 7 ) X = c 3-D: (X -X 7)/(X 2- X y) = (Y-Y7)/(Y 2-Y 7) = (Z -Z 7)/(Z 2- Z 7) X = c where C is a given constant (mesh X dimension). Fig.14 The coordinates of joints in perpendicular cable running direction are obtained by the same method, except only intercepts of nets and boundaries are needed. Members are also defined by properties describing them as cable, prestress, or compression member. 3-2.3 Boundaries The boundaries a of structure may be either a line or a curve. The m athem atical model for the a line boundary is similar to the m athem atical model for a member. The 3D curved boundary is illustrated in Fig.14. The curve selected to represent the boundary is the three dimensional 34 curve which may be stated by the following equations: X ’ = r * Cos t Y’ = r * Sin t Z ’ = k * t Where r is radios, K is slope of the line. These equations can be applied to general cases with origin of Xo, Yo,Zo by translating the above equations into the following equations: X = Xo + r * Cos t Y = Yo + r * Sin t Z = Zo + K * t The problem can be solved by substituting X, Y, Z with X^, Y j, Z j, and X 2 > Y 2 , Z 2 , which are the coordinates of two supports of a boundary. Xy = Xo + r * Cos t j Yj = Yo + r * Sin tj Zj = Zo + k * tj and X 2 = Xo + r * Cos t 2 Y 2 = Yo + r * Sin t 2 Z 2 = Zo + k * t 2 Xj, Y j, Zj, X 2, Y 2 , Z 2 are given by the user’s input. Radius r can be found through the form ula r = [( X y - X 2 )2 + ( Yj - Y2 )2 ]/ ( 4 * H ) + H 35 Where H is the perpendicular distance from the m idpoint of the chord to the curve, which represent the curvature of the boundary. It is m easured on the X-Y plane projection of the curve in a 3-D coordinate system. H is inputted by the user. Fig.15 3D Curved Boundary pLAH y - C ( y NET) \HT&gS&C1Y0Nl Of X , Y n ^ T PLAN Y= D $ X NET MTeP'Sec.To^ of- frajH O A fiq t Y t J f T c 0 36 \ i 3-3 PROGRAMMING & TESTING 3-3.1 Programming IDGP was written in the BASIC program m ing language, because the language is easy to interface with micro computers, as well as, to document the program. Therefore, future users will be able to use this program as a foundation from which a more advanced and sophisticated program may be developed. The program has four subroutines which generate the data for 2-D and 3-D structural forms with either line boundaries or curved boundaries. However, the subroutines dealing with curved boundaries is still in the development stage. The organization of IDGP is shown in Fig.15. Fig.16 O rganization of IDGP OUTPUT HA TH ■\ H a t h M a re u - _ x * Y -Y . Y *-Y | Y - t . Yi - y , _ y 7 - * ' X"- Xo +R<25T T=r Y» 4 RSk OT y __ _ Y (ufUT K Y ,Z _ ippoT P = -fcn> X,Y,H — l^fOT 2 P - L M E 3 P - l i ME '------------- -- - 'Z ’ D-CU fZVlx Z'' (N p U T \)coazp\tiAr&s op B o u u p a R 'x c b H s n - n o n i ^.iseiecricM op gauMt*«i6s M a ih m o p e £_ V=-V.+R< 2 ST Y~Y»4tZSt»T K = -f C H > H- 3V>~co«vtz ~ Z 3 r 37 3-3.2 Testing About 20 data sets have been inputted to test the perform ance of the program. The results of the non compiled IDGP for 38 joints, 62 elements with 3-D line boundary structure were identical to the results calculated manually. Fig.l 1. This problem took about 1-2 second to process using IDGP. Whereas, the m anual solution took about 6 hours to solve. T h e e x e c u t io n tim e fo r larg e c a b le net s t r u c t u r e s in c re a s e s exponentially, as would be the case in m anual solutions. For example, a 400 joints problem was solved in about 20 minutes. However, this problem could take as much as several days to solve manually. 38 4. CONCLUSIONS The findings of this research have shown that the conventional physical modeling process for form -finding for cable net structures can be greatly improved by the interactive computer sim ulation process. ID G P is a pow erful tool which can facilitate the form -finding process in terms of its efficiency, flexibility, and accuracy. This tool may be used by architects and engineers to generate the required data for solving a form - finding problem in seconds, which may otherwise take hours to generate manually. The coordinated operation of IDGP, ISAP and TR ITR S enables architects and engineers to quickly simulate and m odify the fo rm o f a cable net structure, and obtain the final shape with the specifications of required structural elements. Moreover, this computer aided form -finding process provides greater accuracy for a solution, since there is a greater probability of operation and calculation error by the conventional process. Fu tu re development of IDGP could include the addition of a new module to generate a fine mesh over a coarse net as shown in Fig.12. Thus, the final shape could be obtained after another form -finding run. This further developm ent could lead the form -finding process to be a fully computer aided process. 39 APPENDIXES ( SAMPLE ST R U C T U R E S ) 1. TWO DIM ENSIONAL STR U C TU R ES I ] I CHANGE OF NETS \ t | I 41 i CHANGE OF NETS .9 42 CHANGE o f n e t s 43 CHANGE OF BOUNDARY CHANGE O F BOUNDARY CHANGE O F BOUNDARY CHANGE OF BOUNDARY I 47 CHANGE OF BOUNDARY & NETS S B ! CHANGE OP BOUNDARY & NETS CHANGE OF BOUNDARY & NETS 50 CHANGE OF BOUNDARY & NETS i i c o n i » • a a •a a a a 'a a _ T r . •»;»»*# ■.»'»»» a a a a a a a a a.*, t « £ » ■ * > * • > * ' * iiiiii *• a a • a •*- laaaa'M,*-*'** a a a* u a i a a j i i r i u a i i i a a ; * ■ a a*a • a a a a •« »a. i a a n ii« > « j « i« t« * ■ » « i a a j* ; m a a a a ia a .a a a aa aaaaV a a a a a a- a a a a a aa.aaaaa.a a aa a a* a aa aaax * * > ,# .. _ aaaaaaaa.aaaaaxafaXa 51 2. T H R E E DIM ENSIONAL STR U C TU R ES < J to 52 SCALE 1 : 6 .B 9 1 54 TEST4-AX0N SCALE 1 : 5 .7 5 U 1 U 1 TEST5-AX0N SCALE 1 : 6 .5 7 9 x. reS T6-A X0N SCALE 1 : 5 .5 3 5 U 1 'J TEST6 SCALE 1 : 7 .6 3 5 U 1 00 REFERENCES 1. Drew, Philip. Tensile A rchitecture. Westview Press, 1979 2. Drew, Philip. Frei Otto: Form & Structure. Westview Press, 1976 3. Otto, Frei. Tensile Structures. The MIT Press, 1973 4. Schierle, G. Goetz. Lightw eight Tension Structures. U niversity of California, Berkeley, 1968 5. Krishna, Prem. Cable - Suspended Roofs. M cGraw-Hill Book Com pany, 1978 6. Grilndig, Lothar. Bahndorf, Joachim. F orm finding of A Roof Structure For A H ealth SPA. LSA86, 1986 7. Barnes, Michael R. Com puter-Aided Design of Cable & M em brane Structures. With A pplications To Expo 88 & DODC. R iy a d h . LSA86, 1986 59 8. W akefield, David. TENSYL: The Development of An Intergrated CAD System For Stressed M embrane Structures. LSA86, 1986 9. Plastock, Roy A. Kalley, Gordon. Theory & Problems of Com puter G raphics. Mcgraw-Hill Book Com pany, 1985 10. Chasen, Sylvan H. Geometric Principles & Procedures For Computer Graphics A pplications. Prentice-Hall, Inc., 1978 11. Haug, Eberhard. The TR ITR S M A N N A L . 1974 12. Schierle, G. Goetz. The ISAP M A N N A L . 1986 13. International Symposium On A rchitectural Fabric Structures, Proceedings. Orlando, Frorida, 1984 14. Roland, Conrad. Frei Otto - Spannw eiten. Berlin, 1965 15. International Symposium On Widespan Surface Structures, Work D ocum ents. Institute for Lightw eight Structures, Stuttgart, 1976 16. Institute for Lightw eight Structures, Net In N ature & Technics. Stuttgart, 1975 60

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Asset Metadata

Creator Feng, Tian-An (author)

Core Title Computer aided form-finding for cable net structures

Degree Master of Building Science

Degree Program Building Science

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag engineering, architectural,OAI-PMH Harvest

Language English

Contributor Digitized by ProQuest (provenance)

Advisor Schierle, Goetz (committee chair), Ambrose, James (committee member), Schiler, Marc (committee member)

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c17-782790

Unique identifier UC11348276

Identifier EP41416.pdf (filename),usctheses-c17-782790 (legacy record id)

Legacy Identifier EP41416.pdf

Dmrecord 782790

Document Type Thesis

Rights Feng, Tian-An

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the au...

Repository Name University of Southern California Digital Library

Repository Location USC Digital Library, University of Southern California, University Park Campus, Los Angeles, California 90089, USA