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Matrix Computer Analysis of Curvilinear Grid Systems Scholars' Mine Doctoral Dissertations Student Theses and Dissertations 1967 Matrix computer analysis of curvilinear grid systems David Leon Fenton Follow this and additional works at: https://scholarsmine.mst.edu/doctoral_dissertations Part of the Civil Engineering Commons Department: Civil, Architectural and Environmental Engineering Recommended Citation Fenton, David Leon, "Matrix computer analysis of curvilinear grid systems" (1967). Doctoral Dissertations. 1872. https://scholarsmine.mst.edu/doctoral_dissertations/1872 This thesis is brought to you by Scholars' Mine, a service of the Missouri S&T Library and Learning Resources. This work is protected by U. S. Copyright Law. Unauthorized use including reproduction for redistribution requires the permission of the copyright holder. For more information, please contact [email protected]. MATRIX COMPUTER ANALYSIS OF CURVILINEAR GRID SYSTEMS BY DAVID LEON FENTON . · A DISSERTATION submitted to the faculty of THE UNIVERSITY OF MISSOURI AT ROLLA in partial fulfillment of the requirements for the Degree of DOCTOR OF PHILOSOPHY IN CIVIL ENGINEERING Rolla, Missouri 1967 Approved by PLEASE NOTE: Not original copy. Indistinct type on several pages. Filmed as received. Universitv Hicrofilms ii ABSTRACT A general equilibrium-stiffness method of matrix structural analysis was adapted and applied to the solution of the member end forces and moments of each of the members in a curvilinear structural grid system. A structural system of this nature is commonly used as the supporting framework for a "steel-framed dome", in addition to being a basic structural component in many aerospace applications. An integral part of the development of the analysis was the development of a computer program to perform the many complex operations required to obtain the solution. The engineer, by supplying the appropriate structural data and load data. to the computer program, is able to obtain the forces and moments at the ends of each of the members in the structural system corresponding to the six possible degrees of freedom of each one of the joints, or nodal points as they are referred to. David Leon Fenton iii ACKNOWLEDGEMENT The writer wishes to express his appreciation to Dr. J. H. Senne, his advisor, and to Dr. J. L. Best, for their careful and thorough review of this dissertation. The writer also wishes to express his gratitude to Barry Gregory for his valuable assistance in the develop­ ment of the computer program. Finally, the writer is deeply grateful to his wife, Janice, who, in addition to typing this dissertation while being full-time mother to his two wonderful children, Scott and Susan, has shown so much consideration and understanding during these years in school. iv TABLE OF CONTENTS Page ABSTRACT. • • • • • • • • • • • • • • • • • • • • • ii ACKNOWLEDGEMENT • • • • • • • • • • • • • • • • • • iii LIST OF FIGURES • • • • • • • • • • • • • • • • • • vi LIST OF TABLES. • • • • • • • • • • • • • • • • • • vii LIST OF SYMBOLS • • • • • • • • • • • • • • • • • • viii Chapter 1. INTRODUCTION. • • • • • • • • • • • • • 1 Chapter 2. GENERAL FORMULATION • • • • • • • • • • 6 2.1. Method of Formulation • • • • • • • • • 6 2.2. Flexibility and Stiffness Matrices ••• 7 2.3. Assembly of System Stiffness Matrix • • 16 2.4. Pinned-End Rigid Foundation Attachment. 20 2.5. Intermediate Span Loads • • • • • • • • 22 Chapter 3. SEGMENTAL CIRCULAR MEMBER • • • • • • • 24 3.1. Geometric Properties of a Segmental 24 Circular Member • • • • • • • • • • • • 3.2. Load-Displacement Equation Derivation • 25 3.3. General Energy Expression • • • • • • • 27 3.4. Deflection Calculations • • • • • • • • 28 3. 5· Flexibility Matrix ••••••••••• 33 3.6. Three Dimensional Axis Rotation Transformation Matrix • • • • • • • • • 36 Chapter 4. APPLICATION OF ANALYSIS • • • • • • • • 44 4.1. Procedure • • • • • • • • • • • • • • • 44 4.2. Mathematical Models • • • • • • • • • • 4.3. Example Computer Analysis • • • • • • • ~~ 4.4. Experimental Model ••••••••••• 57 4.5. Experimental Results •••••••••• 60 Chapter 5. COMPUTER PROGRAM FOR CURVILINEAR 67 GRID SYSTEM • • • • • • • • • • • • • • Introduction to the Program • • • • • • 67 Input Data. • • • • • • • • • • • • • • 68 Output Data • • • • • • • • • • • • • • 69 MACS Subroutine Package • • • • • • • • 71 MACS User Instructions ••••••••• 73 Flow Diagrams and Computer Program. • • 76 v Chapter 6. CONCLUSIONS AND FUTURE INVESTIGATIONS • • 115 6.1. Conclusions • • • • • • • • • • • • • • • 115 6.2. Topics for Further Investigation. • • • • 116 BIBLIOGRAPHY. • • • • • • • • • • • • • • • • • • • • 120 VITA. • • • • • • • • • . • • • • • • • • • • • • • 121 APPENDIX A. COMPUTER SOLUTIONS FOR THE MATHEMATICAL MODEL. • • • • • • • • • • • • • • • • • 122 APPENDIX B. COMPUTER SOLUTIONS FOR THE EXPERIMENTAL MODEL. • • • • • • • • • • • • • • • • • 157 vi LIST OF FIGURES Figure Page 2.1 Fundamental member load displacement. 8 2.2 Member end-load coordinates. 9 2.3 Simple curvilinear grid. 16 2.4a Fixed-end forces and moments on the 23 loaded member. 2.4b Equivalent nodal loads. 23 3.1 Geometry of the segmental circular member. 24 3.2 Moments and forces on major axes. 26 3.3 Typical member orientation. 37 3.4 The three required rotations. 39 4.1 Mathematical Model-l h l , all 46 r=6 foundation attachments fixed. 4.2 Mathematical Model-l h _ l , with both y;-b pinned and fixed foundation attachments. 47 4.3 Basic coordinate dimensions. 48 4.4 Geometry of l, 3, 7, 9 node points. 49 4.5 Required geometry for ~ • 51 4.6 Experimental model with pinned foundation attachments. 59 4.7 Free body diagram of Member-12. 61 4.8 Photograph of Experimental Model 63a vii LIST OF TABLES Table Page 3.1 Partial derivatives for energy expressions. 28 4.1 Member coordinates for experimental model. 63 4.2 Summary of Test No. 1. 64 4.3 Summary of Test No. 2. 65 4.4 Summary of Test No. 3. 66 5.1 Table of Fortran symbols. 102 viii LIST OF SYMBOLS P' Member end-loads in system coordinates. Member end-displacements in system coordinates K' Appropriate member stiffness matrix in system coordinates F Flexibility matrix px' py' pz Member end-forces and moments in member coordinates. m mx' my ' z H Equilibrium matrix. * Rigid body end-displacements. K Appropriate member stiffness matrix in member coordinates. T Transformation matrix. K" System stiffness matrix s One half of the cord length of the member. R Radius of curvature of the member. e Perpendicular distance from radius focus to the end of the member. One half of the total angle subtended by the member. 9 Member end rotation e General reference angle N Normal force on any cross-section of a member. A Cross-sectional area of a member. ix M , M , M General moments at any point along X y Z a member with respect to member coordinate axes. MT' MYN Moments with respect to the major axis of the cross-section. IN Equivalent polar moment of inertia. IyN Moment of inertia about major "y" axis. I Moment of inertia about major "z" z axis. E Modulus of elasticity G Shear modulus. A Direction cosine. 0(. Rotation of major axis of a member about member x-axis. Angle between system x-axis and the projection of the member x-axis on the system xz-plane Vertical angle between the member x-axis and its projection on the system xa-plane. 1 Chapter 1 INTRODUCTION The appearance of the dome structure in today's modern space age society is as accepted and as appro­ priate as it was hundreds of years ago. Today's modern materials and advancements made in construction methods have made the dome structure one of the most economical space-structure systems in use today. There is an ever growing trend to enclose larger and larger areas and in so doing man is endeavoring to control his environment in order to eliminate the influence of weather on his daily activities. By enclosing large areas such as shopping centers the loss of business due to unfavorable weather can be reduced. The same is true of athletic stadiums, civic centers and auditoriums, green houses, opera houses and a multitude of other applications. The present demands, in addition to demands of the future where domes spanning a mile or more may be desired, present today's engineer with a tremendous challenge. The analysis of these structural systems will require talented and creative engineers with more than average ability, and, in addition, there will be a continuing need for the development of more efficient and sophisticated methods of analysis. 2 Modern matrix analysis of large structural systems together with the use of the high speed computer is becoming as common to the structural engineer's world today as the slope-deflection and moment-distribution methods have been in the past. The analysis of space frames, as well as flat grids, consisting of straight members has been well documented. 1 ' 2 ' 3 On the other hand it is felt that there is a definite need for a clean straight forward approach to the analysis of space­ structure systems composed of curved members. Much of the work that has been done with curved members deals with two dimensional structure such as arch and multi-arch frames.5 In the area of members curved in space the following approaches have been taken, a matrix procedure using a trial and error approach was suggested by, Baron,? in the paper by Eisemann, Woo, and Namyet1 , it is suggested that curved members or members with variable cross-sections could be approximated by the introduction of additional rigid joints along the length of the member resolving the member into a series of straight
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