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The ultimate load capacity of square shear plates with circular perforations : (parameter study) 1985

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THE ULTIMATE LOAD CAPACITY OF SQUARE SHEAR PLATES WITH CIRCULAR PERFORATIONS (PARAMETER STUDY) by ANTHONY GEORGE MARTIN B.ApSc, U n i v e r s i t y Of B r i t i s h Columbia, 1981 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n FACULTY OF GRADUATE STUDIES Department of C i v i l E n g i n e e r i n g We accept t h i s t h e s i s as conforming to the r e q u i r e d standard. THE UNIVERSITY OF BRITISH COLUMBIA SEPTEMBER 1985 © A n t h o n y George M a r t i n , 1985 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 l m e n t of the requirements f o r the advanced degree at the UNIVERSITY OF BRITISH COLUMBIA, I agree that the 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 for r e f e r e n c e and study. I f u r t h e r agree that permission f o r extensive copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s or her R e p r e s e n t a t i v e s . I t i s understood that copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed without my w r i t t e n p e r m i s s i o n . Anthony G. M a r t i n , P. Eng. Department of CIVIL ENGINEERING THE UNIVERSITY OF BRITISH COLUMBIA 2070 Wesbrook P l a c e , Vancouver, Canada. V6T-1W5 Date: September, 1985 ABSTRACT The incremental s t r u c t u r a l a n a l y s i s program NISA83 was used to i n v e s t i g a t e v a r i o u s parameters a f f e c t i n g the u l t i m a t e c a p a c i t y of square p l a t e s with c i r c u l a r p e r f o r a t i o n s s u b j e c t e d to uniform shear s t r e s s . Both non l i n e a r m a t e r i a l p r o p e r t i e s and n o n l i n e a r geometry were taken i n t o account i n determining the u l t i m a t e i n - plane c a p a c i t i e s and b u c k l i n g c a p a c i t i e s of p e r f o r a t e d shear p l a t e s . The parameters i n v e s t i g a t e d d u r i n g t h i s study were the hole s i z e f o r a c o n c e n t r i c l o c a t i o n , and the hole l o c a t i o n f o r a constant r a t i o of hole diameter t o p l a t e width of 0.2. In a d d i t i o n v a r i o u s doubler p l a t e s were s t u d i e d to determine the most e f f e c t i v e shape to r e s t o r e a shear p l a t e to i t s o r i g i n a l u l t i m a t e i n - p l a n e c a p a c i t y . For the f i r s t two parameters, the a n a l y s i s was separated i n t o three p a r t s . The u l t i m a t e i n - p l a n e c a p a c i t y , e l a s t i c b u c k l i n g c a p a c i t y and the u l t i m a t e e l a s t i c - p l a s t i c b u c k l i n g c a p a c i t y was determined f o r each combination of the two parameters. These were used t o i d e n t i f y the importance of both e l a s t i c b u c k l i n g and no n l i n e a r m a t e r i a l c o n t r i b u t e to the reduced u l t i m a t e p l a t e c a p a c i t i e s . The r e s u l t s from p l a t e s w i t h a c o n c e n t r i c a l l y l o c a t e d h o l e of v a r y i n g s i z e showed e x c e l l e n t c o r r e l a t i o n with other p u b l i s h e d experimental and a n a l y t i c a l r e s u l t s f o r both the in - p l a n e c a p a c i t y and the 3-dimensional b u c k l i n g c a p a c i t i e s . V a r i a t i o n of the center l o c a t i o n of a hole of a standard s i z e p r o v i d e d some s i g n i f i c a n t r e s u l t s . L i t t l e change was found i n the u l t i m a t e in-plane c a p a c i t y f o r a l l hole l o c a t i o n s . On the i i other hand, the e l a s t i c b u c k l i n g c a p a c i t y was r a i s e d by 50% a f t e r moving the hole from the p l a t e t e n s i o n d i a g o n a l to the compression d i a g o n a l . F i n a l l y , from the u l t i m a t e e l a s t i c - p l a s t i c ' b u c k l i n g c a p a c i t y r e s u l t s i t was concluded that the c o n c e n t r i c p r o v i d e s lower bound c a p a c i t y f o r a l l other hole l o c a t i o n s . The in-plane a n a l y s i s of the optimum doubler p l a t e s i z e showed wide and t h i n p l a t e s to be more e f f e c t i v e than narrow and t h i c k p l a t e s . A doubler p l a t e with the same t h i c k n e s s as the p l a t e and twice the diameter of the h o l e i s recommended to r e s t o r e the p e r f o r a t e d p l a t e to i t s o r i g i n a l i n - p l a n e c a p a c i t y . In order to a i d i n the tedious task of checking the input data and to provide a convenient way of d i s p l a y i n g the r e s u l t , a f u l l graphic p o s t - p r o c e s s o r was developed as p a r t of t h i s t h e s i s . The program NISPLOT used c o l o r g r a p h i c s a v a i l a b l e at the UBC C i v i l E n g i n e e r i n g l a b to process the output from NISA83. I t was w r i t t e n i n FORTRAN 77, u t i l i z i n g s u b r o u t i n e s from a commercial g r a p h i c s package, DI3000, to o b t a i n d e v i c e independent g r a p h i c s . NISPLOT generated p l o t s of the nodes and element mesh for each data check. When a complete a n a l y s i s was c a r r i e d out by NISA83, nodes, element mesh, d e f l e c t e d shape, and c o l o r s t r e s s f i l l p l o t s were generated. 9 , fe.fs- TABLE OF CONTENTS Page ABSTRACT i i TABLE OF CONTENTS i v LIST OF TABLES v i i LIST OF FIGURES v i i i ACKNOWLEDGEMENTS x i NOMENCLATURE x i i CHAPTER 1. INTRODUCTION 1 1 .1 Background 1 1.2 Purpose and Scope 3 2. THEORETICAL BACKGROUND 5 2.1 The U l t i m a t e Behavior of Shear P l a t e s 5 3. COMPUTER PROGRAMS , 9 3.1 NISA83 9 3.1.1 General Background 9 3.1.2 Time Step Increment 9 3.1.2.1 Convergence and Divergence 10 3.1.2.2 Constant Load C o n t r o l 11 3.1.2.3 Constant A r c l e n g t h C o n t r o l 12 3.1.2.4 I t e r a t i o n Technique 13 3.1.2.5 A p p l i e d Increment and I t e r a t i o n A l g o r i t h m 14 3.1.3 N o n l i n e a r M a t e r i a l 16 3.1.4 Element L i b r a r y 18 3.1.4.1 2-Dimensional Plane S t r e s s Element 19 3.1.4.2 3-Dimensional P l a t e S h e l l Element 21 3.2 NISPLOT 27 3.2.1 General D e s c r i p t i o n 27 3.2.2 S t r e s s F i l l Routine 29 i v 3.2.3 V i s i b l e Surface P l o t t i n g 31 3.2.4 M e t a f i l e s 32 3.2.5 Flow Charts 35 4. PLATE ANALYSIS 39 4.1 V a r i a t i o n of Hole S i z e 40 4.1.1 P l a t e Geometry 40 4.1.2 F i n i t e Element Model 40 4.1.3 Re s u l t s 44 4.1.3.1 In-plane Y i e l d i n g 44 4.1.3.2 3-Dimensional E l a s t i c B u c k l i n g ... 47 4.1.3.3 3-Dimensional E l a s t i c - P l a s t i c B u c k l i ng . 51 4.2 V a r i a t i o n of Hole L o c a t i o n 56 4.2.1 P l a t e Geometry 56 4.2.2 F i n i t e Element Model 57 4.2.3 Re s u l t s 60 4.2.3.1 In-plane Y i e l d i n g 60 4.2.3.2 3-Dimensional E l a s t i c B u c k l i n g ... 61 4.2.3.3 3-Dimensonal E l a s t i c - P l a s t i c B u c k l i ng 65 4.3 Optimume Doubler P l a t e 70 4.3.1 P l a t e Geometry 71 4.3.2 F i n i t e Element Model 71 4.3.3 R e s u l t s 73 4.3.3.1 In-plane Y i e l d i n g 73 4.4 Convergence with Mesh Refinement 75 5. CONCLUSIONS 79 REFERENCES 83 APPENDIX A D e r i v a t i o n of C o n s i s t e n t Shear Load Vector .... 84 for the B i c u b i c Isoparametric Element APPENDIX B ASCE Suggested Design Guides f o r Beams with Web Holes 88 v APPENDIX C M o d i f i c a t i o n of NISA80 at U.B.C 90 APPENDIX D Program L i s t i n g s 97 D.1 NISPLOT 97 D. 2 MESHGEN 112 APPENDIX E Communications Programs 118 E. 1 WORDSTAR Output on the MTS Xerox 9700 118 E.2 T r a n s f e r of a VAX/VMS F i l e to the UBC/MTS System 119 v i LIST OF TABLES Table No. T i t l e Page No. 2.1 D e f i n i t i o n of P l a t e Slenderness F a i l u r e Modes 7 3.1 I n t e r p o l a t i o n F u nctions f o r B i l i n e a r Plane S t r e s s Element 21 3.2 I n t e r p o l a t i o n F u nctions f o r B i c u b i c Element 25 4.1 F a i l u r e Mode C l a s s i f i c a t i o n 65 A.1 Four Cubic Shape Fu n c t i o n s along the B i c u b i c Element Boundary s=1, — 1 <r< 1 86 v i i LIST OF FIGURES Fi g u r e No. T i t l e Page No. 2.1 F u l l P l a t e , I d e a l E l a s t i c - P l a s t i c Behavior 7 2.2 P e r f o r a t e d PLate E l a s t i c - P l a s t i c 3ehavior 7 3.1 A p p l i e d Increment and I t e r a t i o n A l g o r i t h m 15 3.2 I n i t i a l S t r e s s - S t r a i n Diagram f o r M i l d S t e e l 17 3.3 E l a s t i c - I d e a l - P l a s t i c M a t e r i a l Model in U n i a x i a l Tension Test 18 3.4 B i l i n e a r Plane S t r e s s Isoparametric Element 20 3.5 B i c u b i c , Isoparametric, Degenerated, P l a t e S h e l l Element 24 3.6 Comparison of the C l a s s i c a l Concept and Degeneration F i n i t e Element Formulation 26 3.7 S u b d i v i s i o n of the 16 Node Isoparametric Element i n t o 16 Sub-Regions. Each Sub-Region i s F i l l e d with a Color According to the S t r e s s L e v e l at the Gauss I n t e g r a t i o n Point i n the Sub-Region 30 3.8 L i s t i n g of the M e t a f i l e Source F i l e "mtr.log" that Resulted in the Compound P i c t u r e i n F i g u r e [3.9] 33 3.9 Compound P i c t u r e using a M e t a f i l e 34 3.10 Flow Chart of the Frames P l o t t e d by NISPLOT 35 3.11 Flow Chart of the Subroutine I n t e r a c t i o n i n the Program NISPLOT 36 4.1 P e r f o r a t e d P l a t e Showing 1/4 F.E. Model 41 4.2 1/4 P l a t e Model using 3x3 Element Mesh 43 4.3 In-plane S t r e s s D i s t r i b u t i o n , 1/4 Model . .' 45 4.4 Comparison of the Ultimate In-plane Shear C a p a c i t i e s of C o n c e n t r i c a l l y P e r f o r a t e d P l a t e as C a l c u l a t e d by the F i n i t e Element Method and the ASCE Design Proposal given by Equation [4.2] 46 4.5 V a r i a t i o n of E l a s t i c B u c k l ing C o e f f i c i e n t with Co n c e n t r i c Hole S i z e 49 v i i i F i g u r e No. T i t l e Page No. 4.6 E l a s t i c B u c k l i n g Mode,Concentric Hole 50 4.7 Decreasing E l a s t i c - P l a s t i c B u c k l i n g C a p a c i t y of a P e r f o r a t e d P l a t e with I n c r e a s i n g A p p l i e d Load 52 4.8 Load D e f l e c t i o n Curve of a Simply Supported P e r f o r a t e d P l a t e with Various C o n c e n t r i c Hole S i z e s . 52 4.9 V a r i a t i o n of U l t i m a t e E l a s t i c - P l a s t i c B u c k l i n g Capacity of Simply Supported P e r f o r a t e d P l a t e with C o n c e n t r i c Hole S i z e 54 4.10 E-P B u c k l i n g with von-Mises S t r e s s 55 4.11 P l a t e Geometry and Loading used in the A n a l y s i s of the V a r i a t i o n of Hole L o c a t i o n Parameter 56 4.12 . F i n i t e Element Model of Half the P l a t e 58 4.13 F i n i t e Element Model of the T o t a l P l a t e 59 4.14 U l t i m a t e In-plane Capacity R e s u l t s f o r V a r i o u s Hole L o c a t i o n s Normalized to the C o n c e n t r i c Hole U l t i m a t e In-plane C a p a c i t y 61 4.15 Y i e l d at 90% U l t i m a t e In-plane Load 62 4.16 F u l l P l a t e Model, von-Mises S t r e s s 63 4.17 E l a s t i c B u c k l i n g C a p a c i t y F a c t o r s f o r V a r i o u s Hole L o c a t i o n s Normatized to the C o n c e n t r i c Hole Capa c i t y 64 4.18 P r o f i l e of the Tension Diagonal 66 4.19 P r o f i l e of the Compression Diagonal 67 4.20 U l t i m a t e E l a s t i c - P l a s t i c C a p a c i t y f o r V a r i o u s Hole L o c a t i o n s Normalized to the C o n c e n t r i c Hole . U l t i m a t e E l a s t i c - P l a s t i c C apacity 68 4.21 E-P B u c k l i n g at 90% Ultimate C a p a c i t y 69 4.22 Geometry and Loading of P e r f o r a t e d P l a t e with Doubler P l a t e 72 4.23 F i n i t e Element Mesh of P e r f o r a t e d P l a t e with Doubler P l a t e 72 ix F i g u r e No. T i t l e Page No. 4.24 E f f e c t i v e C a p a c i t y R e s t o r a t i o n Factor vs Doubler P l a t e Area f o r V a r i o u s Doubler P l a t e Diameters 74 4.25 Spread of Y i e l d Zones f o r Standard and R e i n f o r c e d P e r f o r a t e d P l a t e s 75 4.26 Convergence of the E l a s t i c B u c k l i n g Load with Mesh Refinement fo r a C o n c e n t r i c a l l y Holed P l a t e with D/b =0.2. t/6=0.0l 78 A.1 B i c u b i c Isoparametric P l a t e S h e l l Element with Uniform Shear Loading along One Edge 84 B.1 U l t i m a t e In-plane C a p a c i t y of Shear Web with a C i r c u l a r P e r f o r a t i o n as Proposed i n Reference [ 1 1 ] . 89 x ACKNOWLEDGEMENTS The author expresses h i s g r a t i t u d e to h i s a d v i s o r Dr. S. F. Stiemer f o r h i s v a l u a b l e advice d u r i n g the r e s e a r c h and t h e s i s p r e p a r a t i o n . He would a l s o l i k e to thank Dr. P. O s t e r r i e d e r f o r o r i g i n a l l y suggesting the t o p i c and f o r h i s v a l u a b l e h e l p d u r i n g the a n a l y s i s . Much g r a t e f u l a p p r e c i a t i o n i s due to the N a t i o n a l Research C o u n c i l of Canada f o r t h e i r f i n a n c i a l support throughout t h i s work. x i NOMENCLATURE A l i s t of symbols used r e p e t i t i v e l y i n t h i s t h e s i s i s given here. Conventional mathematical i n d i c e s are not i n c l u d e d . Symbol D e s c r i p t i o n A e q u i v a l e n t hole l e n g t h Ai cross s e c t i o n a l area of doubler p l a t e {Dd~D)td Ah cross s e c t i o n a l area of c i r c u l a r p e r f o r a t i o n Dt b • width of p l a t e B(un) operator matrix D diameter of c i r c u l a r p e r f o r a t i o n Dd diameter of doubler p l a t e E Young's modulus ( f o r s t e e l , 200,000. MPa.) F(un) i n t e r n a l f o r c e s i n c o n f i g u r a t i o n n H e q u i v a l e n t hole height J 2 second i n v a r i a n t of the d e v i a t o r i c s t r e s s e s k e l a s t i c b u c k l i n g c o e f f i c i e n t K{un) incremental s t i f f n e s s matrix u s i n g displacements un Kr(un) e l a s t i c s t i f f n e s s matrix Kg{?'n) geometric s t i f f n e s s matrix N(r,s) element shape f u n c t i o n s P load vector A P increment i n the load v e c t o r f o r one time step Q{un) out-of-balance load v e c t o r P - F(u n) r,- radius from the center of the hole to node t (polar c o o r d i n a t e s ) R d i s t a n c e from center of p l a t e to cent e r of p e r f o r a t i o n S f a c t o r e d r a t i o of hole diameter to p l a t e width 0.9D/6 x i i Symbol D e s c r i p t i o n S(u") s t r e s s matrix Sr.S..,S, d e v i a t o r i c s t r e s s e s t t h i c k n e s s of p l a t e t, t h i c k n e s s of doubler p l a t e e l a s t i c b u c k l i n g s t r e s s of f u l l p l a t e e l a s t i c b u c k l i n g s t r e s s of p e r f o r a t e d p l a t e exact e l a s t i c b u c k l i n g s t r e s s of p e r f o r a t e d p l a t e u l t i m a t e i n - p l a n e c a p a c i t y of p e r f o r a t e d p l a t e with doubler p l a t e r„ u l t i m a t e e l a s t i c - p l a s t i c b u c k l i n g s t r e s s of f u l l p l a t e r„ u l t i m a t e e l a s t i c - p l a s t i c b u c k l i n g s t r e s s of p e r f o r a t e d p l a t e Ty y i e l d shear s t r e s s of s t e e l or in-plane u l t i m a t e s t r e s s of f u l l p l a t e Ty in-plane u l t i m a t e s t r e s s of p e r f o r a t e d p l a t e T x y , T y z , T z x shear s t r e s s e s u exact e q u i l i b r i u m displacement v e c t o r u" approximate e q u i l i b r i u m displacement vec t o r A u n increment i n approximate e q u i l i b r i u m displacement vect o r a f a c t o r A load f a c t o r v Poisson's r a t i o (0.3) t r m mean s t r e s s cr„ y i e l d s t r e s s ( f o r s t e e l , 300. MPa.) °~zzi°'yy}0'zz normal s t r e s s e s 1* doubler p l a t e c a p a c i t y r e s t o r a t i o n f a c t o r x i i i 1 INTRODUCTION 1.1 Background F l a t r e c t a n g u l a r p l a t e s are very common s t r u c t u r a l component in modern b u i l d i n g s and o f f s h o r e s t r u c t u r e s and are o f t e n subjected to a pure uniform shear s t r e s s . These p l a t e s are normally r e f e r r e d to as shear p l a t e s or shear webs. Cost e f f e c t i v e n e s s r e q u i r e s optimal use of space and weight of a s t r u c t u r e . T h i s o f t e n r e s u l t s i n one or more holes being cut out of components such as shear webs of g i r d e r webs. These holes then allow u t i l i t y pipes or d u c t i n g to pass through the web and reduce the o v e r a l l s t r u c t u r a l weight. These shear p l a t e s with p l a i n or r e i n f o r c e d c i r c u l a r h o l e s are c a l l e d p e r f o r a t e d shear p l a t e s . E a r l y e l a s t i c in-plane s t r e s s a n a l y s i s of square p e r f o r a t e d shear p l a t e s was done by WANG [ 1 ] , who a p p l i e d a f i n i t e A i r y ' s s t r e s s f u n c t i o n over the domain of the p l a t e . His r e s u l t s show that the s t r e s s c o n c e n t r a t i o n s i n c r e a s e d very r a p i d l y with the hole s i z e r a t i o D/b (hole diameter to p l a t e width). He c a r r i e d out an i n v e s t i g a t i o n i n t o the s t r e s s d i s t r i b u t i o n produced with the a d d i t i o n of reinforcement around the h o l e . He found that the c i r c u m f e r e n t i a l s t r e s s i s s i g n i f i c a n t l y decreased about the h o l e , but the doubler p l a t e i n c r e a s e s the r a d i a l shear s t r e s s from that of the u n r e i n f o r c e d p e r f o r a t e d p l a t e . The s t a b i l i t y of very s l e n d e r p e r f o r a t e d shear p l a t e s was f i r s t i n v e s t i g a t e d by ROCKEY, ANDERSON and CHEUNG [ 2 ] , They used the f i n i t e element method to e s t a b l i s h the r e l a t i o n s h i p between the e l a s t i c b u c k l i n g load of the p l a t e s and the hole s i z e r a t i o , Djb, f o r both clamped and simply supported boundary c o n d i t i o n s . 1 They a l s o i n v e s t i g a t e d the r e l a t i o n s h i p between the hole reinforcement shape and the e l a s t i c shear b u c k l i n g c o e f f i c i e n t , jfc. In a d d i t i o n to t h i s a n a l y t i c a l work they c a r r i e d out an experimental study on p e r f o r a t e d shear p l a t e s . Since t h e i r a n a l y t i c a l model was only able to handle an e l a s t i c m a t e r i a l model, they chose the p l a t e slenderness r a t i o small enough so that n e i t h e r the t e s t p l a t e nor the a n a l y t i c a l p l a t e would be subject to m a t e r i a l y i e l d i n g . The combined e f f e c t s of the m a t e r i a l p l a s t i c i t y and buc k l i n g s t a b i l i t y were f i r s t d e a l t with by UENOYA and REDWOOD [3]. Using a 2-dimensional i n - p l a n e f i n i t e element program they c a l c u l a t e d the e l a s t i c - p l a s t i c s t r e s s d i s t r i b u t i o n due to shear l o a d i n g . These s t r e s s e s were then i n s e r t e d i n t o a R a y l e i g h - R i t z energy e x p r e s s i o n . By minimizing the p o t e n t i a l energy of t h i s system they were able to determine the e l a s t i c - p l a s t i c b u c k l i n g c a p a c i t y of the p e r f o r a t e d shear p l a t e . T h i s method was a p p l i e d to a one quar t e r model of a square shear p l a t e f o r both simply supported and clamped boundaries. The method was l a t e r extended to r e c t a n g u l a r beam webs with combined shear and moment l o a d i n g for both c i r c u l a r and r e c t a n g u l a r p e r f o r a t i o n s . URENOYA and REDWOOD c o r r e l a t e d t h e i r a n a l y t i c a l r e s u l t s with an experimental study c a r r i e d out c o n c u r r e n t l y , and with past experimental work. T h e i r two par t s o l u t i o n p r o v i d e d e x c e l l e n t r e s u l t s f o r p e r f o r a t e d p l a t e s with small p e r f o r a t i o n s . However, as the hole s i z e r a t i o , D/b> became gr e a t e r than 0.7, the r e s t r i c t e d number of terms used i n the F o u r i e r s e r i e s approximation of the d e f l e c t e d shape produced unexpected r e s u l t s . 2 For the clamped p l a t e they found that by i n c r e a s i n g the hole s i z e the u l t i m a t e e l a s t i c b u c k l i n g c a p a c i t y i s i n c r e a s e d rather than decreased, as had been expected. The r e s u l t from UENOYA and REDWOOD show that any c i r c u l a r p e r f o r a t i o n i n a shear p l a t e has a s i g n i f i c a n t e f f e c t on the s t a b i l i t y and u l t i m a t e l o a d c a p a c i t y of the p l a t e . They a l s o note that with i n c r e a s i n g hole s i z e there i s a c o n s i d e r a b l e increase i n the amount of p l a s t i c i t y developed i n the p l a t e before the u l t i m a t e c a p a c i t y i s reached. 1.2 Purpose and Scope In the research of t h i s t h e s i s the Nonlinear Incremental S t r u c t u r a l A n a l y s i s program, NISA83 [ 7 ] , i s used to c a r r y out a parameter study on square shear p l a t e s with c i r c u l a r p e r f o r a t i o n s . The a n a l y s i s c o n s i d e r s three separate l o a d i n g c a p a c i t i e s ; in-plane y i e l d i n g , e l a s t i c b u c k l i n g and e l a s t i c - p l a s t i c b u c k l i n g , for the two major parameters, hole s i z e and hole l o c a t i o n . The a n a l y s i s of the t h i r d parameter, optimum doubler p l a t e shape, only c o n s i d e r s the u l t i m a t e in-plane c a p a c i t i e s . F i n a l l y , an e s t i m a t i o n of the accuracy of the e l a s t i c b u c k l i n g a n a l y s i s i s o b t a i n e d by r e f i n i n g the element model i n order to e s t a b l i s h the convergence r a t e and exact e l a s t i c b u c k l i n g c a p a c i t y . Included as part of t h i s t h e s i s i s the development and implementation of a c o l o r g r a p h i c s post processor c a l l e d NISPLOT. The program was o r i g i n a l l y developed to check the input data. However, i t was l a t e r extended to provide post p r o c e s s i n g f o r NISA83 output. NISPLOT i s used throughout the research to provide simple p l o t s of the model nodes and element meshes for data 3 checks. In a d d i t i o n , i t i s used to i l l u s t r a t e and summarize some of inf o r m a t i o n p r o v i d e d by the f i n i t e element a n a l y s i s . The program c u r r e n t l y runs on a VAX 11/730 under the EUNICE op e r a t i n g system. I t i s w r i t t e n i n FORTRAN 77, making use of the gra p h i c s package DI3000, by P r e c i s i o n V i s u a l s [ 8 ] , 2 THEORETICAL BACKGROUND 2.1 THE ULTIMATE BEHAVIOR OF SHEAR PLATES The t h e o r e t i c a l behavior of square p l a t e s subjected to uniform shear l o a d i n g along the boundaries can be separated i n t o two types of behavior. I f the square shear p l a t e has no i n i t i a l i m p e r f e c t i o n s i t s u l t i m a t e c a p a c i t y i s governed by the l e s s e r of the u l t i m a t e m a t e r i a l shear s t r e s s or the e l a s t i c shear b u c k l i n g s t r e s s of the p l a t e . According to the v o n - M i s p s v i e l d c r i t e r i a the u l t i m a t e m a t e r i a l shear s t r e s s i s Ty = cry/\/3. ROCKEY [2] d e f i n e s the e l a s t i c shear b u c k l i n g s t r e s s by equation [2.1]. k = 9.34 f o r simply supported square p l a t e By equating the u l t i m a t e m a t e r i a l shear s t r e s s to the e l a s t i c b u c k l i n g shear s t r e s s , a simple e x p r e s s i o n , equation [2.2], f o r d e f i n i n g the balance or t r a n s i t i o n p o i n t , i n terms of the p l a t e slenderness ( t / b ) , i s ob t a i n e d . T h e o r e t i c a l l y , at t h i s t r a n s i t i o n p o i n t the m a t e r i a l w i l l y i e l d at e x a c t l y the same load l e v e l that the p l a t e b u c k l e s . Equation [2.2] i s a l s o used to d e f i n e the slenderness r a t i o above which a p l a t e i s con s i d e r e d stocky, or below which i t i s cons i d e r e d s l e n d e r . The u l t i m a t e c a p a c i t y of a stocky p l a t e i s governed by the u l t i m a t e m a t e r i a l shear s t r e s s . A slender p l a t e has a lower e l a s t i c b u c k l i n g s t r e s s than m a t e r i a l y i e l d s t r e s s , so i t s u l t i m a t e c a p a c i t y i s governed by equation [2.1]. I l l u s t r a t i o n of the e f f e c t of sle n d e r n e s s on f a i l u r e mode of i d e a l stocky and slender p l a t e s are shown i n f i g u r e [2.1]. 5 CT y/3 1 2 ( 1 - 0 . 3 2 ) E b cry _ 9.35TT2JE 0.2614 (2.2) In r e a l i t y however, p l a t e s are never p e r f e c t l y f l a t and the l o a d i n g i s never p e r f e c t l y uniform. With the i n t r o d u c t i o n of i n i t i a l i m p e r f e c t i o n s in the p l a t e surface or e c c e n t r i c i t i e s i n the l o a d i n g . The t r a n s i t i o n between y i e l d i n g and b u c k l i n g i s no longer a s i n g l e p o i n t . Now the t r a n s i t i o n from one f a i l u r e mode to the other takes p l a c e over a range of slenderness v a l u e s , depending on the degree of d e v i a t i o n from the p e r f e c t c o n d i t i o n . However, the purpose here i s not to examine the e f f e c t s of i m p e r f e c t i o n s on the unperforated p l a t e . The a n a l y s i s w i l l assume an i n i t i a l l y p e r f e c t l y f l a t p l a t e with uniform shear l o a d i n g . The e f f e c t s of imperfect c o n d i t i o n s w i l l be the s u b j e c t of f u t u r e s t u d i e s . The e f f e c t of i n t r o d u c i n g of a c i r c u l a r hole in a p l a t e i s s i m i l a r to that of i m p e r f e c t i o n s . There i s no longer a d i s t i n c t p o i n t to d i s t i n g u i s h between e l a s t i c b u c k l i n g and m a t e r i a l y i e l d i n g . For intermediate slender p l a t e s , s i g n i f i c a n t m a t e r i a l y i e l d i n g occurs around the h o l e . T h i s causes a r e d i s t r i b u t i o n of the s t r e s s e s and change i n the l o a d path. The net r e s u l t of t h i s i s the r e d u c t i o n of the p l a t e b u c k l i n g c a p a c i t y . In t h i s t r a n s i t i o n r e g i o n , r e f e r r e d to as zone I I , the u l t i m a t e c a p a c i t y of the p l a t e i s c o n t r o l l e d by the u l t i m a t e e l a s t i c - p l a s t i c b u c k l i n g s t r e s s . U n l i k e the e l a s t i c b u c k l i n g s t r e s s i n equation [2.1], no a n a l y t i c a l s o l u t i o n e x i s t e d for the zone II u l t i m a t e c a p a c i t i e s . T h e r e f o r e , s o l u t i o n s f o r the u l t i m a t e e l a s t i c - p l a s t i c 6 b u c k l i n g c a p a c i t i e s i n zone II must be determined n u m e r i c a l l y . F i g . 2.1: F u l l P l a t e , I d e a l F i g . 2.2: P e r f o r a t e d P l a t e , E l a s t i c - P l a s t i c Behavior E l a s t i c - P l a s t i c Behavior Table 2.1: D e f i n i t i o n of P l a t e Slenderness F a i l u r e Modes ZONE FAILURE MODE I II III m a t e r i a l y i e l d i n g e l a s t i c - p l a s t i c b u c k l i n g e l a s t i c b u c k l i n g The o b j e c t i v e of t h i s r e s e a r c h i s to i d e n t i f y the u l t i m a t e p l a t e c a p a c i t i e s in each zone. By r e s t r i c t i n g the displacements or v a r y i n g the m a t e r i a l p r o p e r t i e s only one f i n i t e element model i s r e q u i r e d f o r each parameter change. The m a t e r i a l c a p a c i t y of zone I i s determined by r e s t r i c t i n g the model to in-plane displacements only. The e l a s t i c b u c k l i n g c a p a c i t y of zone III i s 7 e v a l u a t e d by a p p l y i n g a small l o a d increment to the model and then performing a b i f u r c a t i o n a n a l y s i s on the r e s u l t i n g s t i f f n e s s m atrix. F i n a l l y the u l t i m a t e c a p a c i t y i n zone II i s e s t a b l i s h e d by a f u l l three dimensional e l a s t i c - p l a s t i c u l t i m a t e a n a l y s i s . However, i n order to observe the o u t - o f - p l a n e behavior a small l a t e r a l l o a d i s a p p l i e d at or near the cent e r of the p l a t e . 8 3 COMPUTER PROGRAMS 3.1 NISA83 Nonli n e a r f i n i t e element programs have been a v a i l a b l e f o r some time and are e a s i l y a c c e s s i b l e . The program, NISA83 used i n t h i s study, c o n t a i n s s e v e r a l s p e c i a l f e a t u r e s which make the program w e l l s u i t e d f o r the e l a s t i c - p l a s t i c i n s t a b l i t y problem a s s o c i a t e d with p e r f o r a t e d shear p l a t e s . 3.1.1 General Background NISA83 (Nonlinear Incremental S t r u c t u r a l A n a l y s i s ) i s i n s t a l l e d on the, U n i v e r s i t y of B r i t i s h Columbia C i v i l E n g i n e e r i n g Department's, VAX 11/730. The program i s a development of the I n s t i t u t s f u r B a u s t a t i k , U n i v e r s i t a t S t u t t g a r t , under the d i r e c t i o n of Dr. Hafner, Dr. Ramm and Dr. S a t t e l e over the p r e v i o u s ten y e a r s . NISA83 i s w r i t t e n in standard FORTRAN 77 and designed to be supported by a v a r i e t y of main frame computers. A number m o d i f i c a t i o n s were necessary to get the program o p e r a t i n g on the U.B.C. VAX. A comprehensive l i s t of a l l changes made to NISA83 su b r o u t i n e s are i n c l u d e d i n Appendix C. 3.1.2 Time Step Increment In a n o n - l i n e a r a n a l y s i s i t i s not always p o s s i b l e to determine the f i n a l e q u i l i b r i u m p o s i t i o n of the s t r u c t u r e by a p p l y i n g the u l t i m a t e load i n one step. There maybe more than one f i n a l e q u i l i b r i u m c o n f i g u r a t i o n , or s e v e r a l load paths l e a d i n g to the same e q u i l i b r i u m c o n f i g u r a t i o n . L i k e other n o n - l i n e a r programs, NISA83, f o l l o w s the l o a d d e f l e c t i o n path by breaking the problem in to a s e r i e s smaller increments. These increments are r e f e r r e d to as time steps 9 Each time step i s considered as a separate problem. At the beginning of the time step the e q u i l i b r i u m displacements, f o r c e s and s t r e s s e s are a l l known. Depending on what increment c o n t r o l i s used the end of the time step i s d e f i n e d by e i t h e r i n c r e a s i n g the l o a d or by incrementing the load-displacement v e c t o r by some cons t a n t . Within each time step s e v e r a l i t e r a t i o n s of the s o l u t i o n may be r e q u i r e d before convergence to the d e f i n e d new e q u i l i b r i u m c o n f i g u r a t i o n i s a s t a b l i s h e d . 3 .1.2.1 C o n v e r g e n c e A n d D i v e r g e n c e As NISA83 completes each i t e r a t i o n w i t h i n a time step, checks are made f o r convergence or divergence of the s o l u t i o n . The NISA83 c r i t e r i a used to d e f i n e i f the new e q u i l i b r i u m c o n f i g u r a t i o n has converged or, the s o l u t i o n has d i v e r g e d are as f o l l o w s : Convergence - i f the change in the displacement v e c t o r between i t e r a t i o n s i s l e s s than the user s p e c i f i e d value ( d e f a u l t RTOL = 0.001) e = i L _ ! i < RTOL IMI Divergence - i f a f t e r 8 e q u i l i b r i u m i t e r a t i o n s the out of balance lo a d v e c t o r i s s t i l l l a r g e r than incremental a p p l i e d l o a d v e c t o r IIAPII < - i f a f t e r a f t e r a s p e c i f i e d number of i t e r a t i o n s the convergence requirement have not been s a t i s f i e d ( d e f a u l t 15 i t e r a t i o n s ) By using the r e s t a r t o p t i o n the a n a l y s i s can be c o n t i n u e d 10 from any l a s t known e q u i l i b r i u m c o n f i g u r a t i o n . I f at the end of an i t e r a t i o n the s o l u t i o n s a t i s f i e s the convergence c r i t e r i a NISA83 updates the r e s t a r t f i l e with the new e q u i l i b r i u m d i splacements and s t r e s s e s . However, i f a s o l u t i o n f a i l s to converge by one of the c r i t e r i a above, the program stops. Thus, whenever the program stops or complets i t s task the r e s t a r t f i l e c o n t a i n s the l a s t known e q u i l i b r i u m c o n f i g u r a t i o n . By r e d e f i n i n g any of the time step parameters and r e s t a r t i n g NISA83 the a n a l y s i s i s c ontinued from t h i s l a s t c o n f i g u r a t i o n . 3 .1 .2 .2 C o n s t a n t Load C o n t r o l The c o n s t a n t load increment i s the most common time step c o n t r o l used i n f i n i t e element programs. The method allows the user to s p e c i f y the load l e v e l used to d e f i n e the end c o n d i t i o n of each time ste p . The load l e v e l i s h e l d constant throughout the time step, while the s o l u t i o n i t e r a t e s to the new c o n f i g u r a t i o n u n t i l convergence or divergence i s e s t a b l i s h e d . The c o n s t a n t load increment method works w e l l f o r problems where the n o n l i n e a r i t i e s remain small w i t h i n each time step. Normally the l o a d i s s p e c i f i e d i n constant increments along the l o a d path. As the a p p l i e d l o a d approaches the u l t i m a t e load, the increments are reduced i n magnitude. I f a load l e v e l i s s p e c i f i e d t h a t i s higher than the u l t i m a t e c a p a c i t y of the problem the program i s unable to converge to an e q u i l i b r i u m s o l u t i o n . For t h i s case the u l t i m a t e load i s d e f i n e d as the average of the l a s t known e q u i l i b r i u m load step and the l o a d of the d i v e r g e n t s t e p . Problems are encountered with t h i s method when the l o a d l e v e l approaches a b i f u r c a t i o n p o i n t , or the behavior becomes 11 h i g h l y n o n l i n e a r . In the these cases small changes in the load increment produce l a r g e deformations. These deformations i n turn caused a f u r t h e r r e d u c t i o n in the system s t i f f n e s s r e s u l t i n g i n s t i l l higher deformations. Thus, convergence to the e q u i l i b r i u m s o l u t i o n c l o s e to the u l t i m a t e c a p a c i t y becomes very d i f f i c u l t u s i n g t h i s method. 3.1.2.3 Constant A r c l e n g t h C o n t r o l One of the s p e c i a l f e a t u r e s of NISA83 i s the Riks-Wempner Constant A r c l e n g t h time step c o n t r o l . T h i s method l e t s the user s p e c i f y the i n i t i a l a r c l e n g t h to be used as a r e f e r e n c e . In each time step the a p p l i e d load v e c t o r i s assumed a v a r i a b l e along with the displacement f i e l d . The l o a d l e v e l i s s c a l e d w i t h i n each time step so the normal of the incremental displacement v e c t o r p l u s incremental load vector i s equal to the d e f i n e d r e f e r e n c e a r c l e n g t h . T h i s method i s extremely powerful i n i t s a b i l i t y to f o l l o w the l o a d d e f l e c t i o n path. As e x p l a i n e d b e f o r e , when us i n g the constant l o a d time step, once the s p e c i f i e d l o a d l e v e l i s higher than the u l t i m a t e load i t i s impossible f o r the program to converge to an e q u i l i b r i u m s o l u t i o n . However, with the constant a r c l e n g t h c o n t r o l l e d method, and the l o a d l e v e l as an unknown, the a p p l i e d load increment i s reduced u n t i l e q u i l i b r i u m i s o b t a i n e d . T h i s allows the program to o b t a i n an e q u i l i b r i u m s o l u t i o n even i f the s t i f f n e s s matrix has gone n e g a t i v e . T h i s mean that i n the b u c k l i n g shear p l a t e problem, the program i s a b l e to f o l l o w the load path to the p l a t e u l t i m a t e b u c k l i n g 12 c a p a c i t y and then on i n t o the p o s t - b u c k l i n g r e g i o n . 3.1.2 . 4 I t e r a t i o n Technique In the two time step c o n t r o l methods above, the r e q u i r e d end c o n d i t i o n s are s p e c i f i e d . Both methods assume that with each i t e r a t i o n the s o l u t i o n w i l l improve and e v e n t u a l l y the convergence requirement w i l l be s a t i s f i e d . In NISA83 the user can s e l e c t one or a combination of the M o d i f i e d Newton-Raphson and the Standard Newton-Raphson i t e r a t i o n methods. The f i r s t method uses l e s s CPU time but the second method converges more r a p i d l y i f the tangent s t i f f n e s s matrix i s changing r a p i d l y with each i t e r a t i o n . The d i f f e r e n c e between the two i t e r a t i o n techniques i s i l l u s t r a t e d by st e p i n g thougth a time step i t e r a t i o n . A s s o c i a t e d with each time step are three c o n f i g u r a t i o n s . (1) i s the l a s t known e q u i l i b r i u m c o n f i g u r a t i o n and a l l the i n f o r m a t i o n on s t r e s s e s , s t r a i n s , and displacements i s known. (2) i s the next unknown e q u i l i b r i u m c o m f i g u r a t i o n on the load path. Only the end c o n d i t i o n (load l e v e l or arc l e n g t h ) i s known i n t h i s p o s i t i o n . (n) i s some p o s i t i o n i n between (1) and (2). The procedure to get from c o n f i g u r a t i o n (1) to (2) i s as f o l l o w s : Step 1 c a l c u l a t e the unbalanced f o r c e s f o r the time step. Q(ul) = P2 - (3.1) Step 2 formulate the tangent s t i f f n e s s matrix i n c o n f i g u r a t i o n (1) K(u l) = Ke(u l) + Kg(u l) (3.2) 13 Step 3 solve f o r the incremental displacements Au=[^(u 1)]" 1P 2 (3.3) Step 4 f i n d the t o t a l displacement v e c t o r f o r the new c o n f i g u r a t i o n (n) u n = u1 + Au (3-4) Step 5 c a l c u l a t e the i n t e r n a l f o r c e s i n c o n f i g u r a t i o n (n) F(un) = J [B{un)]T S{un) dv (3.5) Vol Step 6 check f o r convergence or divergence, s e c t i o n [3.1.2.1] Step 7 c a l c u l a t e the remaining unbalanced f o r c e s f o r t h i s time step Q(un) = P 2 - F{un) (3.6) Step 8 form the new tangent s t i f f n e s matrix K(un) (Standard Newton-Raphson) or use the o l d s t i f f n e s s matrix K^u1) (Modified Newton-Raphson) Step 9 solve f o r the next displacement increment and continue with step 4 t o 9 u n t i l l convergence. 3.1.2.5 A p p l i e d Increment and I t e r a t i o n A l g o r i t h m F i g u r e [3.1] show how the v a r i o u s increment c o n t r o l s , i t e r a t i o n techniques and the r e s t a r t o p t i o n are combined to pro v i d e an e f f i c i e n t a l g o r i t h m . T h i s a l g o r i t h m i s a p p l i e d to a l l the i n - p l a n e and e l a s t i c - p l a s t i c b u c k l i n g a n a l y s e s . The approach p r o v i d e s an e f f i c i e n t method of r a i s i n g the load l e v e l to the beginning of the n o n l i n e a r p o r t i o n of the l o a d path. I t a l s o p r o v i d e s an accurate procedure t o f o l l o w the load path i n t h i s r e g i on beyond i n t o the p o s t - b u c k l i n g range. The constant load step c o n t r o l i s combined with the m o d i f i e d 14 MODIFIED NEWTON-RAPHSON WITH CONSTANT LOAD 1 / Deflection RESTART Deflection F i g . 3.1: A p p l i e d Increment and I t e r a t i o n Algorithm 15 Newton-Raphson i t e r a t i o n technique fo r the f i r s t p art of the a n a l y s i s . The l o a d l e v e l i s s p e c i f i e d to approximately 80% of the estimated u l t i m a t e c a p a c i t y . Since the m a t e r i a l y i e l d i n g i s c o n f i n e d to s m a l l areas around the hole most problems w i l l behave almost e l a s t i c a l l y up to t h i s l o a d . Then the program parameters are changed so that the constant load step c o n t r o l i s combined with the Standard Newton-Raphson i t e r a t i o n procedure. Then NISA83 i s r e s t a r t e d and the a n a l y s i s continues from the l a s t known e q u i l i b r i u m c o n f i g u r a t i o n . The method was found to be adequate f o r up to 95% of the u l t i m a t e load, using increments i n the order of 2% of the c u r r e n t s p e c i f i e d l o a d l e v e l . A f t e r the l o a d step c o n t r o l f a i l s to converge, the constant a r c l e n g t h time step c o n t r o l i s invoked. T h i s time step c o n t r o l combined with the Standard Newton-Raphson i t e r a t i o n i s then used to f o l l o w the l o a d path fo r the r e s t of the a n a l y s i s . T h i s method i s able to f o l l o w the true l o a d - d e f l e c t i o n path u n t i l the the p l a t e buckles and beyond i n t o the p o s t - b u c k l i n g r e g i o n . 3.1.3 N o n l i n e a r M a t e r i a l In order to i d e n t i f y the p o i n t at which e l a s t i c deformations stop and p l a s t i c deformations s t a r t , f o r v a r i o u s s t r e s s s t a t e s , a y i e l d c r i t e r i a i s used. The von-Mises y i e l d c r i t e r i a , used by NISA83 to i d e n t i f y p l a s t i c s t r a i n s , i s w e l l accepted as a reasonable model of the e l a s t i c p l a s t i c behavior of s t e e l s . M a t h e m a t i c a l l y the c r i t e r i a can be expressed by equation [3.7] 16 *9/y/3 = J 2 1 / 2 = y/l/2 (5| + SJ + S?) + r%, + rj, + r* (3.7) = C _ _ — (7, x x x : m m 5 * = - °"i m NISA83 a l s o a l l o w s the user to s e l e c t the usual m a t e r i a l parameters E, <ry , and v along with s t r a i n hardening modulus Eh • For m i l d s t e e l i f the s t r a i n s are between 0.002 and 0.02, then the s t r e s s - s t r a i n curve i s almost h o r i z o n t a l . Most s t r a i n hardening e f f e c t s occurs at s t r a i n l e v e l s above 0.02, as shown i n f i g u r e [3.2]. Since a l l of the p l a t e s c o n s i d e r e d were expected to have s t r a i n s below t h i s l e v e l no s t r a i n hardening i s c o n s i d e r e d . Thus, the m a t e r i a l model commonly known as " e l a s t i c i d e a l p l a s t i c " i s used throughout the a n a l y s i s . Under u n i a x i a l t e n s i o n of t h i s m a t e r i a l model give the s t r e s s - s t r a i n diagram shown i n f i g u r e [3.3]. 400 - o 0.000 0.005 0.010 0.015 0.020 0.025 Strain m m / m m I n i t i a l S t r e s s - S t r a i n Diagram f o r M i l d S t e e l F i g . 3.2: 17 o co CO 4 0 0 - 3 0 0 - 2 0 0 - 100 0 .000 0.005 0.010 0.015 Strain m m / m m 0.020 0.025 Fig.3.3: S t r e s s - S t r a i n Diagram f o r E l a s t i c I d e a l P l a s t i c M a t e r i a l Model i n U n i a x i a l Tension The values of the parameters used to d e f i n e the m a t e r i a l p r o p e r t i e s are; c r y = 300. MPa. E= 200,000. MPa. Eh= 0.0 MPa. v= 0.30 3.1.4 Element L i b r a r y NISA83 c o n t a i n s an e x t e n s i v e element l i b r a r y . Elements range from t r u s s , beam, and curved beam elements f o r l o n g i t u d i n a l elements and, 4 and 8 node i s o p a r a m e t r i c s with plane s t r e s s or plane s t r a i n f o r 2-dimensional a n a l y s i s to, a f a m i l y of degenerate p l a t e s h e l l elements w i t h 4,8,9 or 1 6 nodes f o r problems i n v o l v i n g membrane f o r c e s and p l a t e bending. A short d e s c r i p t i o n of the two elements used i n t h i s r e s e a r c h i s presented here. 18 3.1.4.1 2-Dimensional Plane S t r e s s Element For the two dimensional a n a l y s e s of doubler p l a t e c a p a c i t i e s a simple plane s t r e s s element i s used. T h i s 4 node iso p a r a m e t r i c element i s thoroughly documented as being w e l l - c o n d i t i o n e d and g i v i n g r e l i a b l e r e s u l t s f o r plane s t r e s s problem [10], T h i s element i s used to d i s c r e t i z e the 2-dimensional domain of the p e r f o r a t e d p l a t e with a doubler p l a t e , and determine the u l t i m a t e in- p l a n e p l a t e y i e l d c a p a c i t y . The element i s d e r i v e d from the i n t e r p o l a t i o n f u n c t i o n s given i n t a b l e [3.1]. As f o r a l l i s o p a r a m e t r i c elements the f u n c t i o n s are used f o r mapping the element from the g l o b a l x-y c o o r d i n a t e s to the l o c a l r - s c o o r d i n a t e s , and a l s o f o r i n t e r p o l a t i o n of the element displacement f i e l d , equation [3.8,3.9], In order to maintain the energy bound of the f i n i t e element f o r m u l a t i o n , 2x2 Gauss numerical i n t e g r a t i o n i s used with t h i s element. T h i s order of i n t e g r a t i o n e x a c t l y e v a luates the element s t i f f n e s s i n t e g r a l p r o v i d e d the c o o r d i n a t e t r a n s f o r m a t i o n matrix i s a c o n s t a n t . Some higher order e r r o r i s introduced i f the c o o r d i n a t e t r a n s f o r m a t i o n matrix i s not c o n s t a n t . However, i f the element mesh i s r e f i n e d the t r a s f o r m a t i o n matrix approches a constant and the e r r o r becomes n e g l i g i b l e . Thus, t h i s order of numerical i n t e g r a t i o n i s s u f f i c i e n t to ensure convergence as the (3.8) (3.9) 19 mesh i s r e f i n e d and preseve the energy bound. A f u l l d e r i v a t i o n of the complete element s t i f f n e s s matrix i s given i n refere n c e [ 1 0 ] . Table 3.1: I n t e r p o l a t i o n Functions f o r B i l i n e a r Plane S t r e s s Element Element Node No. Coordi r nate s Interpc 1 +r ) l a t i o n E 1 - r 'unction 1+s * 4.0 1-s 1 1 1 1 - 1 - 2 -1 1 - 1 1 - 3 -1 -1 - 1 - 1 4 1 -1 1 - - 1 3.1.4.2 3-Dimensional P l a t e S h e l l Element For a l l 3-dimensional and some 2-dimensional a n a l y s i s , the higher order degenerated p l a t e s h e l l element i s used. BATHE [12] found that the 16 node, b i c u b i c i s o p a r a m e t r i c p l a t e s h e l l element pr o v i d e s r e l i a b l e r e s u l t s f o r 3-dimensional a n a l y s e s . However, inorder to e l i m i n a t e v a r i a t i o n s i n r e s u l t s due to modelling changes, t h i s same element and mesh i s a l s o used f o r some i n - plane u l t i m a t e c a p a c i t y c a l c u l a t i o n s . By a p p l y i n g the a p p r o p r i a t e boundary c o n d i t i o n s to the nodes, the 3-dimensional model i s r e s t r i c t e d to in-pl a n e displacements and can be s u c c e s s f u l l y used fo r the c a l c u l a t i o n of the in-pl a n e y i e l d c a p a c i t i e s . The i n t e r p o l a t i o n f u n c t i o n s , c a l l e d "Lagrangian cubic polynomials", used f o r the 16 node element f o r m u l a t i o n are given i n t a b l e [3.2], The term "degenerated element" r e f e r s to the way element i s formulated. The c l a s s i c a l f o r m u l a t i o n of a f i n i t e element i s based on p l a t e or s h e l l theory. T h i s theory i s developed from the gen e r a l 3-dimensional f i e l d equation by making 21 some r e s t r i c t i n g assumptions ( i e . small displacements and i g n o r i n g higher order terms) to reduce i t to a 2-dimensional e q u a t i o n on the s h e l l s u r f a c e . A s t r u c t u r a l system i s then d i s c r e t i z e d u s ing the elements i n t h i s 2-dimensional domain. The degenerate f o r m u l a t i o n i s more general than the c l a s s i c a l approach. No behavior or displacement assumptions are used to reduce the 3-dimensional f i e l d equation to a 2-dimensional domain. Instead the f o r m u l a t i o n i s based on the f u l l 3- dimensional f i e l d e quations. T h e r e f o r e , any 3-dimensional s t r u c t u r a l system i s d i s c r e t i z e d d i r e c t l y by a f i n i t e element model. A schematic comparison of the two fo r m u l a t i o n s i s shown i n f i g u r e [3.6] G e o m e t r i c a l l y n o n l i n e a r problems are sol v e d using l a r g e d e f l e c t i o n theory which c o n s i d e r the second order s t r a i n displacement terms that are n e g l e c t e d i n l i n e a r a n a l y s e s . NISA83 all o w e d the user to apply e i t h e r the Updated Lagrangian or T o t a l Lagrangian l a r g e d e f l e c t i o n f o r m u l a t i o n . Here the Updated Lagrangian f o r m u l a t i o n i s used f o r a l l b u c k l i n g a n a l y s i s and a b r i e f d e s c r i p t i o n f o l l o w s . The Updated Lagrangian Formulation i s based on the element l o c a l c o o r d i n a t e system t r a c k i n g with the element incremental d e f o r m a t i o n s . At the beginning of each time step the element c o o r d i n a t e system, c a l l e d the " c o r o t a t i o n a l system", i s updated to the l a s t known e q u i l i b r i u m c o n f i g u r a t i o n . A l l i n f o r m a t i o n about s t r e s s e s , s t r a i n s and the e q u i l i b r i u m displacements are known i n t h i s c o n f i g u r a t i o n and i t i s used as a r e f e r e n c e throughout the increment. At the end of the time step, when a new 22 e q u i l i b r i u m c o n f i g u r a t i o n i s e s t a b l i s h e d , the c o o r d i n a t e system, s t r a i n s and s t r e s s e s are updated or transformed t o t h i s new c o n f i g u r a t i o n , ready to s t a r t the next time increment. With t h i s l a r g e d e f l e c t i o n theory the element c o r o t a t i o n a l system shares the same r i g i d body modes as the element, thus any s t r e s s e s or s t r a i n s c a l c u l a t e d are with r e f e r e n c e t o the l a s t known deformed e q u i l i b r i u m c o n f i g u r a t i o n of the element. A 4 x 4 Gauss numerical i n t e g r a t i o n i n the r - s plane and a 5 p o i n t Simpson i n t e g r a t i o n through the t h i c k n e s s i s used with t h i s element. Again t h i s order of i n t e g r a t i o n e x a c t l y e v a l u a t e s the element s t i f f n e s s i n t e g r a l i f the c o o r d i n a t e t r a n s f o r m a t i o n matrix i s a con s t a n t . The Simpson numerical i n t e g r a t i o n i n the th i c k n e s s d i r e c t i o n p r o v i d e s r e l i a b l e r e s u l t s i f the width to th i c k n e s s r a t i o of the p l a t e i s greater than 1 0 . 23 10 In local space (projection on to the plane t = 0) F i g . 3.5: B i c u b i c , I s o p a r a m e t r i c , Degenerated, P l a t e S h e l l Element 24 T a b l e 3.2: I n t e r p o t a t i o n F u n c t i o n s f o r B i c u b i c Element. Elem Node No. coc r r d . s F a c t I nte 1+r r p o l a 3r+1 t i o n 3r-1 Funct 1 - r i o n * 1 +s 256 3s+1 3s-1 1 -s 1 1 1 1 1 1 1 - 1 1 1 - 2 -1 1 - 1 1 1 1 1 1 - 3 -1 1 - 1 1 1 - 1 1 1 4 1 1 1 1 1 - . - 1 1 1 5 1/3 9 1 1 - 1 1 1 1 - 6 -1/3 - 9 1 - 1 1 1 1 1 - 7 -1 1/3 9 - . 1 . 1 1 1 1 - < 8 -1 "1/3 - 9 - 1 1 1 1 - 1 9 "1/3 -1 - 9 - 1 1 - 1 1 < 1 0 1/3 -1 9 - 1 - 1 - 1 1 1 1 1 -1/3 - 9 1 1 - - 1 12 1 1/3 9 < 1 1 - 1 - - 1 3 1/3 1/3 81 1 - 1 - 1 - 1 4 -1/3 1/3 -81 < - 1 1 1 - 1 5 -1/3 "1/3 81 - 1 1 - 1 - 16 1/3 -1/3 -81 • 1 - 1 < - 1 25 CLASSICAL CONCEPT DEGENERATION Assumptions Analytical reduction _3_dim—— 2 dim over thickness Discretiza tion 2 dim — p o i n t numerical j* over surface Dis- place mentL*. solution Assumptions Discretiza tion 3 dim point numerical ^ over volume 3.6: Comparison of the C l a s s i c a l Concept and Degeneration F i n i t e Element Formulation 26 3 . 2 N I S P L O T 3 . 2 . 1 General D e s c r i p t i o n A s c o m m o n l y k n o w n , o u t p u t g e n e r a t e d b y f i n i t e e l e m e n t p r o g r a m c a n b e v e r y l e n g t h y . T h e p r o g r a m N I S P L O T w a s d e v e l o p e d t o u s e t h e c o l o r g r a p h i c s c a p a b i l i t i e s o f t h e D e p a r t m e n t o f C i v i l E n g i n e e r i n g , U . B . C , t o p r o c e s s t h e o u t p u t f r o m N I S A 8 3 i n t o a m o r e c o n c i s e f o r m . N I S P L O T w a s u s e d f o r c h e c k i n g t h e i n p u t d a t a b e f o r e r u n n i n g a n a n a l y s i s . A f t e r a n a n a l y s i s i t w a s u s e d a s a c o l o r g r a p h i c p o s t - p r o c e s s o r , p r o d u c i n g e l e m e n t m e s h , 3 - d i m e n s i o n a l d e f l e c t e d s h a p e s , o r e l e m e n t s t r e s s e s ( d i s p l a y e d i n u p t o s e v e n c o l o r s ) . N I S P L O T i s w r i t t e n i n a v e r y g e n e r a l a n d m o d u l a r f o r m s o t h a t a n y c h a n g e s o r s u b s e q u e n t m o d i f i c a t i o n s t h a t a r e r e q u i r e d c a n b e c a r r i e d o u t q u i c k l y a n d e f f i c i e n t l y . T h e p r o g r a m i s w r i t t e n i n s t a n d a r d F O R T R A N 7 7 p r o g r a m i n g l a n g u a g e a n d m a k e s c a l l s t o s t a n d a r d g r a p h i c s s u b r o u t i n e s i n t h e c o m m e r c i a l g r a p h i c s s o f t w a r e p a c k a g e , D I 3 0 0 0 b y P r e c i s i o n V i s u a l s [ 8 ] , N I S P L O T r u n s o n t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a V A X 1 1 / 7 3 0 u n d e r t h e E U N I C E 4 . 1 e n v i r o n m e n t , w h i c h i s a U N I X e m u l a t o r . I t i s n e s s e s a r y t o t h e E U N I C E o p e r a t i n g s y s t e m a s o p p o s e d t o t h e n a t i v e V A X - V M S s y s t e m b e c a u s e o f t h e s t o r a g e r e s t r i c t i o n s o n t h e V A X . T h e D I 3 0 0 0 s u b r o u t i n e l i b r a r y r e q u i r e s a l a r g e a m o u n t o f d i s k s t o r a g e a n d o n l y o n e o b j e c t l i b r a r y c a n b e s t o r e d o n t h e V A X 1 1 / 7 3 0 s y s t e m d i s k . S i n c e t h e C i v i l E n g i n e e r i n g D e p a r t m e n t i s c u r r e n t l y i n t h e p r o c e s s o f c o n v e r t i n g t o t h e E U N I C E o p e r a t i n g s y s t e m , o n l y t h e E U N I C E l i b r a r y i s s t o r e d o n t h e d i s k . O n t h e o t h e r h a n d , N I S A 8 3 o n l y r u n s u n d e r t h e V A X - V M S s y s t e m . H o w e v e r a l l i n p u t a n d o u t p u t f i l e s a r e V M S t o E U N I C E 2 7 compat i b l e . L i k e any other program that makes c a l l s to the g r a p h i c s subroutines of DI3000, the o b j e c t module i s l i n k e d to the c o r r e c t device d r i v e r to c r e a t e the device dependent executable module. At the C i v i l E n g i n e e r i n g Graphics l a b the program can be l i n k e d to a Seiko D-SCAN GR-1104, S i l i c o n Graphics Inc. IRIS-1200, and a Watanabye Instrument Corp. WX4636R p l o t t e r . The image c o u l d a l s o l i n k e d to any other the device d r i v e r supported by DI3000. NISPLOT i s s e q u e n t i a l l y s t r u c t u r e d and prompts the user f o r the input data f i l e s . The d e f a u l t NISA83 output geometry, displacement, and s t r e s s f i l e s are " p l o t . g e o " , " p l o t . d i s " , and " p l o t . s t r " r e s p e c t i v e l y . The user can supply one, two or a l l three of the input f i l e s when requested, but the geometry f i l e must always be one of the f i l e s . The program checks to see i f the f i l e s s p e c i f i e d e x i s t and prompt the user i f i t does not. U n f o r t u n a t e l y , the user must-be sure that the c o r r e c t f i l e type i s given when prompted, as the program does not check the v a l i d i t y of a f i l e type. When given a l l three f i l e s as input, NISPLOT w i l l p l o t the f o l l o w i n g frames: 1) the mesh nodes and node numbers 2) the element mesh and element numbers with each element group in a new c o l o r . 3) a s o l i d c o l o r s t r e s s f i l l of the 16 node isop a r a m e t r i c p l a t e s h e l l element 4) 3-dimensional p l o t of the d e f l e c t e d shape with 2 c o l o r s , one for the near and one f o r the f a r s i d e , of the 16 node i s o p a r a m e t r i c p l a t e s h e l l element. 28 5) 3-dimensional d e f l e c t e d shape with a seven c o l o r s o l i d f i l l of the 16 node i s o p a r a m e t r i c p l a t e s h e l l element The program r o u t i n e s f o r p l o t t i n g the element nodes, element mesh and the t i t l e are u n i v e r s a l r o u t i n e s and work for any of the elements c o n t a i n e d in the NISA83 l i b r a r y . The other r o u t i n e s i n v o l v i n g p l o t t i n g the element s t r e s s e s , d e f l e c t e d shape with s o l i d f i l l , or d e f l e c t e d shape with c o l o r s t r e s s f i l l , are element dependent, and are only supported f o r the 16 node iso p a r a m e t r i c p l a t e s h e l l element with 4x4 Gauss i n t e g r a t i o n . 3.2.2 S t r e s s F i l l Routine T h i s r o u t i n e f i l l s each sub-region of a l l the 16 node elements with a c o l o r t h a t i s c o n s i s t e n t with the von-Mises s t r e s s l e v e l c a l c u l a t e d at the 4x4 Gauss i n t e g r a t i o n p o i n t s . F i r s t a l l the element s t r e s s are searched to determine the maximum and minimum valu e s of the von-Mises s t r e s s e s . Then t h i s range i s d i v i d e d i n t o s i x l e v e l s and a c o l o r assigned to each l e v e l . Next the r o u t i n e proceeds through a l l the elements as f o l l o w s . Each 16 node i s o p a r a m e t r i c element i s s u b d i v i d e d i n t o 16 sub-regions. Each region c o n t a i n s one Gauss i n t e g r a t i o n p o i n t . The boundaries of these 16 sub-regions are d e f i n e d by 25 nodes as shown i n f i g u r e [3.9]. C a l c u l a t i o n of these p o i n t s i s done using the c u b i c i n t e r p o l a t i o n f u n c t i o n s given i n t a b l e [3.2], By examining the von-Mises s t r e s s at the gauss i n t e g r a t i o n p o i n t c o n tained w i t h i n a sub-region the r o u t i n e s e l e c t s the c o r r e c t c o l o r and f i l l s the sub-region a r e a . 29 <5> 6 1—Q 6 1 9 7 O 7 20 8 O + o — t - 3 I I + 1 4 1 3 + 17 16 -I- 21 1 5 -4.-4-- 25 10 6" 1 6 I + + 11 o 9 O i 2 + 22 15 - - T - 4 + <V1 ^ 23 14 + 1 12 1 3 4 > 10 O G l o b a l Nodes # Subdivided Element Nodes (25 nodes, 16 regions) -f- Gauss I n t e g r a t i o n P o i n t s F i g 3.7: S u b d i v i s i o n of the 16 Node Isoparametric Element i n t o 16 Sub-Regions. Each Sub-Region i s F i l l e d with a C o l o r According to the S t r e s s L e v e l at the Gauss I n t e g r a t i o n Point i n the Sub-Region. 30 3.2.3 V i s i b l e Surface P l o t t i n g The subroutine VISBLE i s a simple r o u t i n e that attempts to improve the q u a l i t y of any three-dimensional p l o t by d i s t i n g u i s h i n g between s u r f a c e s that are v i s i b l e i n the viewing plane and those that are not. By examining the normal of any given surface i n the viewing plane the subroutine i s able to d e t e c t i f the top or bottom s u r f a c e i s v i s i b l e to the viewer. The hidden s u r f a c e subroutine VISBLE i s c a l l e d from both the s t r e s s f i l l r o u t i n e and the s o l i d f i l l r o u t i n e j u s t before the element sub-region i s f i l l e d . The g l o b a l x,y,z c o o r d i n a t e s of three or more p o i n t s that l i e on a sub-region s u r f a c e are passed to t h i s s u b r o u t i n e . These p o i n t s are then converted i n t o u,v c o o r d i n a t e s of the l o c a l viewing system. The subroutine then c a l c u l a t e s the c r o s s product of two v e c t o r s formed by p o i n t ( 1 ) - p o i n t ( 2 ) and p o i n t ( 3 ) - p o i n t ( 2 ) . I f the r e s u l t i s p o s i t i v e the top s u r f a c e i s v i s i b l e on the viewing plane, and i f i t i s negative the bottom s u r f a c e i s v i s i b l e on the viewing plane. By c a l l i n g the s o l i d f i l l r o u t i n e twice, once f i l l i n g a l l the v i s i b l e bottom s u r f a c e s with dark blue and then f i l l i n g a l l the v i s i b l e top s u r f a c e s with l i g h t blue, i t i s p o s s i b l e to c r e a t e p l o t s i n which the two s u r f a c e s are e a s i l y d i s t i n g u i s h e d . S i m i l a r l y , by c a l l i n g the s t r e s s f i l l r o u t i n e twice and using a d i f f e r e n t f i l l p a t t e r n fo r top and bottom s u r f a c e s a c l e a r e r i l l u s t r a t i o n r e s u l t s . The r o u t i n e works w e l l f o r the simple convex d e f l e c t e d shapes of the p l a t e problem. For complex s u r f a c e s , however, a more s o p h i s t i c a t e d a l g o r i t h m i s r e q u i r e d , but t h i s i s beyond the scope of t h i s t h e s i s . JENSSEN [6] p r o v i d e s a good r e f e r e n c e f o r f u t u r e work in t h i s area. 31 3.2.4 M e t a f i l e s A m e t a f i l e i s a device independent p l o t f i l e which i s s t o r e d on the host and may be kept as a permanent f i l e for f u t u r e r e f e r e n c e . The idea of the M e t a f i l e s i s that they can be r e c a l l e d and manipulated by anyone using the DI3000 m e t a f i l e t r a n s l a t o r without having to run NISPLOT or having access to the NISA83 output f i l e s . When the user requests that a M e t a f i l e be cr e a t e d , NISPLOT c r e a t e s and opens a new f i l e c a l l e d "NISAPLOT.MFL". A l l the subsequent frames drawn by NISPLOT on the c u r r e n t device are then s t o r e d in t h i s f i l e , as we l l as appearing on the viewing d e v i c e . These p l o t frames, or p i c t u r e s , are s t o r e d i n the standard format of the-DI3000 graphics package. When NISPLOT f i n i s h e s running i t c l o s e s the f i l e . I f a m e t a f i l e i s requested, the p l o t t i n g sequence i s mo d i f i e d . NISPLOT p l o t s the t i t l e o nly i n the f i r s t frame. The reason f o r t h i s w i l l become apparent a f t e r f u r t h e r d i s c u s s i o n of the manipulations that can be done with the t r a n s l a t o r . A d e t a i l e d d e s c r i p t i o n of the t r a n s l a t o r can be found in the DI30.00 Users' Manual [8] under the chapter " M e t a f i l e s " . Only a b r i e f e x p l a n a t i o n of p o s s i b l e manipulations i s o u t l i n e d . . To r e c a l l p i c t u r e s from the m e t a f i l e , NISAPLOT.MFL, to a new d e v i c e , the DI3000 m e t a f i l e t r a n s l a t o r l i n k e d to the device i s invoked ( i e . mtrans.seiko). The m e t a f i l e t r a n s l a t o r allows the user to d e f i n e a viewing port and window and then request any of the p i c t u r e s from the M e t a f i l e be drawn i n these viewing a t t r i b u t e s . By d i v i d i n g the screen of the viewing device i n t o two or four viewing p o r t s and r e q u e s t i n g that d i f f e r e n t p i c t u r e be 32 drawn i n each p o r t , a compound p i c t u r e i s generated. However, since the t i t l e was only drawn on the f i r s t p i c t u r e generated, i t w i l l not appear in any of the view p o r t s . To p l a c e the t i t l e on the subsequent compound p i c t u r e the user can d e f i n e a new view port f o r the f u l l d evice screen and then s e l e c t p i c t u r e one to be drawn. T h i s causes the t i t l e to appear under the t o t a l compound p i c t u r e i n s t e a d of in each s u b - p i c t u r e . One l a s t f eature about the m e t a f i l e t r a n s l a t o r i s that the commands to d e f i n e the m e t a f i l e , view p o r t s , windows and even the order of p i c t u r e s drawn can be set up i n a log f i l e . When the M e t a f i l e t r a n s l a t o r i s a c t i v a t e d , the user a s s i g n s t h i s f i l e as a command source. The t r a n s l a t o r w i l l then execute these commands in sequence. T h i s source f i l e can be set up to draw p i c t u r e s from s e v e r a l M e t a f i l e s i n sequence to d i s p l a y a s e r i e s of r e s u l t s . S E T M F 1 N I S A P L O T . M F L S E T W 1 ( - 1 . 0 1 . 0 - 1 . 0 1 . 0 ) S E T W 2 ( - 1 . 0 1 . 0 - 0 . 9 5 0 . 9 5 ) S E T V 1 ( - 1 . 0 0 . 0 0 . 0 5 1 . 0 ) S E T V 2 ( 0 . 0 1 . 0 0 . 0 5 1 . 0 ) S E T V 3 ( - 1 . 0 0 . 0 - 0 . 9 0 . 0 5 ) S E T V 4 ( 0 . 0 1 . 0 - 0 . 9 0 . 0 5 ) S E T V 5 ( - 1 . 0 1 . 0 - 1 . 0 1 . 0 ) S E T D E F A U L T W 2 S E T D E F A U L T B O X O N D R A W P 2 V 1 N O E J E C T D R A W P 3 V 2 N O E J E C T D R A W P 4 V 3 N O E J E C T D R A W P 5 V 4 N O E J E C T S E T B O X O F F D R A W P 1 V 5 W 1 F i g . 3.8: L i s t i n g of the M e t a f i l e Source F i l e "mtr.log" that R e s u l t e d i n the Compound P i c t u r e i n F i g u r e [3.9] 33 F ig. 3.9: C o m p o n d P ic ture us i ng a METAFILE 3.2.5 Flow Charts F i g u r e [3.10] shows a flow c h a r t of NISPLOT with respect to the d i f f e r e n t frames p l o t t e d . F i g u r e [3.11] shows a more d e t a i l e d flow c h a r t of NISPLOT showing how s u b r o u t i n e s i n t e r a c t . ( S T A R T ) DEFINE INPUT FILES PLOT AND NUMBER NODES PLOT AND NUMBER ELEMENTS IF (3dstr) NO PLOT ELEMENTS WITH SOLID COLOR STRESS FILL PLOT UNDEFLECTED SHAPE IN 3 - D THEN DEFLECTED S H A P E WITH ONE COLOR SOLID FILL PLOT UNDEFLECTED SHAPE IN 3 - D THEN DEFLECTED S H A P E WITH SOLID COLOR STRESS FILL F i g . 3.10 Flow Chart of the Frames P l o t t e d By NISPLOT 35 ( START ) / S E T U P / T /READNO/ MAXMIN VIEW PLTNOD - NODNUM PLTHED / R E A D E L / • PLTELE r PLTHED DRAY ELENUM DATIN ELESTR SHAPE 1 * VISBLE PLTELE DRAY LEGEND J^d5plt), / R E A D N O / VIEW r ADDDIS PLTELE ( PLTHED ADDDIS FILLEL > PLTELE DRAY ELENUM VISBLE ELESTR ( PLTELE - LEGEND (eof.ne. NO < E N D ) F i g . 3.11: Flow Chart of the Subroutine I n t e r a c t i o n i n the Program NISPLOT 36 D e s c r i p t i o n s of subroutines i n f i g u r e [3.10] Subroutine D e s c r i p t i o n SETUP asks the operator f o r the input f i l e names c o n t a i n i n g the geometry, displacements and s t r e s s e s , and opens these f i l e s READNO reads the x,y,z c o o r d i n a t e s of the nodes MAXMIN determines the maximum and minimum g l o b a l dimensions of the node c o o r d i n a t e s VIEW s e t s the frame viewing a t t r i b u t e s i e . 2-D or 3-D view, window, view p o r t e t c . PLTNOD marks the node p o i n t with "+" NODNUM numbers the node p o i n t marks PLTHED p l o t s the heading at the bottom of the page READEL reads i n the element node numbers PLTELE s e t s the l i n e s t y l e and c o l o r a t t r i b u t e s f o r each element and c o l l e c t s the node numbers f o r each l i n e i n one a r r a y , ready f o r DRAY DRAY draws the l i n e given by PLTELE ELENUM numbers the elements at t h e i r mid p o i n t READST reads i n the element s t r e s s e s ELESTR separates the 16 node i s o p a r a m e t r i c element i n t o 16 sub-regions with 25 nodes, then f i l l s each of the sub- r e g i o n s with the a p p r o p r i a t e c o l o r a c c o r d i n g to the s t r e s s l e v e l at the Gauss p o i n t i n the sub-region DATAIN i n i t i a l i z e s v a r i a b l e s used i n ELESTR SHAPE co n v e r t s the 16 node i s o p a r a m e t r i c element to a 25 node element using the element shape f u n c t i o n s f o r i n t e r p o l a t i o n VISBLE checks to see i f the sub-region to be p l o t t e d i s v i s i b l e in the viewing plane 37 Subroutine D e s c r i p t i o n LEGEND w r i t e s the s t r e s s legend i n the upper right-hand corner of the frame ADDDIS s c a l e s and adds the nodeal displacements to the o r i g i n a l nodal c o o r d i n a t e s FILLEL separates the 16 node i s o p a r a m e t r i c element i n t o 9 sub- region and f i l l s each sub-region with one of two c o l o r s depending on the response from VISBLE 38 4 PLATE ANALYSIS Numerical a n a l y s i s was performed on a s e r i e s of standard square p l a t e s with c i r c u l a r p e r f o r a t i o n s to e s t a b l i s h the e f f e c t s of three parameters on the u l t i m a t e c a p a c i t i e s of shear p l a t e s . For the f i r s t two parameters (hole s i z e and hole l o c a t i o n ) the u l t i m a t e i n - p l a n e , e l a s t i c b u c k l i n g , and u l t i m a t e e l a s t i c - p l a s t i c b u c k l i n g c a p a c i t i e s were determined. For the t h i r d parameter (the shape of a doubler p l a t e ) only the in-plane u l t i m a t e c a p a c i t i e s was determined. The p l a t e t h i c k n e s s was s e l e c t e d so that the t h i c k n e s s to width r a t i o (t/b ) was c l o s e to the i d e a l balanced slenderness r a t i o , as d e s c r i b e d i n s e c t i o n [ 2 . 1 ] . The o u t s i d e dimensions f o r the standard p l a t e , used throughout the a n a l y s i s , were 1000 mm x 1000 mm x 10 mm. T h i s p r o v i d e d a p l a t e slenderness of 1/100, which was very c l o s e to the balance r a t i o f o r the f u l l p l a t e of 1/98.7, given by equation [2.2]. T h i s slenderness ensured t h a t the r e s u l t i n g analyses d e a l t with the combined m a t e r i a l and b u c k l i n g f a i l u r e modes. The i n v e s t i g a t i o n of the f i r s t parameter, hole s i z e , was done on the standard square p l a t e d e s c r i b e d above with a c o n c e n t r i c h o l e . The diameter of the hole was v a r i e d from 0.156 to 0.90b. For the hole l o c a t i o n parameter, the center of a 0.26 diameter hole was moved about the s u r f a c e . Many d i f f e r e n t geometries and models were used d u r i n g the study of the hole s i z e and hole l o c a t i o n . In order to determine the s i g n i f i c a n c e of n o n l i n e a r m a t e r i a l or geometry on the c a p a c i t y of the p e r f o r a t e d p l a t e , three u l t i m a t e c a p a c i t i e s were c a l c u l a t e d f o r each v a r i a t i o n of the two 39 major parameters. For each model c o n f i g u r a t i o n , the u l t i m a t e i n - plane y i e l d c a p a c i t y , e l a s t i c b u c k l i n g c a p a c i t y and the u l t i m a t e e l a s t i c - p l a s t i c b u c k l i n g c a p a c i t y were c a l c u l a t e d . The in - p l a n e y i e l d c a p a c i t y i n d i c a t e the i n f l u e n c e of m a t e r i a l y i e l d i n g around the hole on the p l a t e c a p a c i t y . The u l t i m a t e e l a s t i c b u c k l i n g c a p a c i t y was used to e s t a b l i s h the change i n the e l a s t i c o u t - o f - plane s t i f f n e s s due to the presence of the p e r f o r a t i o n . F i n a l l y , the two f a c t o r s were co n s i d e r e d together with a c a l c u l a t i o n of the u l t i m a t e e l a s t i c - p l a s t i c b u c k l i n g c a p a c i t y . 4.1 V a r i a t i o n of Hole S i z e The f i r s t parameter i n v e s t i g a t e d f o r i t s e f f e c t on the ulti m a t e p l a t e c a p a c i t y was the s i z e of a c e n t r a l l y l o c a t e d p e r f o r a t i o n . R e s u l t s f o r the u l t i m a t e i n - p l a n e , e l a s t i c b u c k l i n g , and e l a s t i c - p l a s t i c b u c k l i n g c a p a c i t i e s were obtained f o r hole diameters from 0.156 to 0.96. 4.1.1 P l a t e Geometry The standard square p l a t e , 6=1000 mm and r=l0 mm, was analyzed with c o n c e n t r i c hole diameters. The r a t i o of hole diameter to p l a t e t h i c k n e s s , D/b, v a r i e d from 0.15, 0.2, 0.3, 0.4,...to 0.9. A l l m a t e r i a l parameters were h e l d constant to the s p e c i f i c a t i o n s i n s e c t i o n [3.1.3], 4.1.2 F i n i t e Element Model By o b s e r v i n g symmetry of the p l a t e geometry and l o a d i n g , and by a p p l y i n g a p p r o p r i a t e boundary c o n d i t i o n s , only one qua r t e r of the p l a t e needed to be analysed. The two d i a g o n a l s of the square p l a t e (see f i g u r e [4.1]) are two axes of symmetry f o r the p l a t e l o a d i n g and geometry. For the symmetric b u c k l i n g modes there w i l l 40 F ig . 4.1: Per fo ra ted P late Show ing 1/4 F.E. M o d e l be no r o t a t i o n about these axes, and these two axes remain p e r p e n d i c u l a r to each other f o r a l l deformations due to t h i s l o a d i n g . As shown in f i g u r e [4.2] the 16 node p l a t e s h e l l element in a 3x3 g r i d was used to model a quarter of the p l a t e . T h i s same 9 element mesh was used f o r a l l the c o n c e n t r i c hole a n a l y s e s . The boundary c o n d i t i o n s f o r one model were v a r i e d to s u i t the u l t i m a t e load c a l c u l a t i o n d e s i r e d . For 3-dimensional buckl_ing, displacement boundary c o n d i t i o n s were imposed along the p l a t e outer edge and the two axes of symmetry. Then, by r e s t r i c t i n g the displacement f i e l d to i n - p l a n e movement only the same element mesh was used f o r the a n a l y s i s of the p e r f o r a t e d p l a t e u l t i m a t e i n - p l a n e c a p a c i t y . The q u a r t e r p l a t e model made of a 90 deg. s e c t i o n of p l a t e was separated i n t o nine s l i c e s by 10 e q u a l l y spaced r a d i a l l i n e s . Nodes were then p l a c e d along these l i n e s u s i n g a p r o p o r t i o n a l spacing given by equation [4.1]. T h i s provided a dense c o n c e n t r a t i o n of nodes around the p e r f o r a t i o n where the s t r e s s g r a d i e n t s were the h i g h e s t and m a t e r i a l y i e l d i n g was most severe. (4.1) 42 4.2: 1/4 P late M o d e l us ing 3x3 E lement M e s h The c o n s i s t e n t load v e c t o r f o r a uniform shear s t r e s s was a p p l i e d along the p l a t e boundaries. Using such a d i s t o r t e d element mesh r e q u i r e d the use of the exact c o n s i s t e n t load vector f o r each element to c o r r e c t l y model the uniform s t r e s s t r a c t i o n along the p l a t e boundary. A gen e r a l f o r m u l a t i o n f o r the element c o n s i s t e n t l o a d v e c t o r , d e f i n e d i n terms of element g l o b a l c o o r d i n a t e s , was developed and i s documented i n Appendix A. The r e s u l t i n g equations [A.8] were used with the element g l o b a l c o o r d i n a t e s to c a l c u l a t e each element c o n s i s t e n t load and then t h i s was added to the t o t a l model c o n s i s t e n t l o a d v e c t o r . 4.1.3. R e s u l t s The u l t i m a t e in-plane c a p a c i t y , u l t i m a t e e l a s t i c b u c k l i n g c a p a c i t y and the u l t i m a t e e l a s t i c - p l a s t i c b u c k l i n g c a p a c i t y was c a l c u l a t e d f o r each v a r i a t i o n of the hole s i z e . 4.1.3.1 In-plane Y i e l d i n g The hole s i z e had a d i r e c t e f f e c t on the u l t i m a t e in-plane c a p a c i t y of the p e r f o r a t e d p l a t e . With i n c r e a s i n g hole s i z e there was found to be a corresponding decrease i n u l t i m a t e c a p a c i t y . The r e s u l t s of the f i n i t e element work and the ASCE [11] design proposal show an e x c e l l e n t c o r r e l a t i o n f o r the e n t i r e range of hole s i z e s . The design equation [4.2] i s d e r i v e d i n appendix B and was based on the ASCE p r o p o s a l assuming a square p l a t e and a c i r c u l a r p e r f o r a t i o n . T h i s equation along with the r e s u l t s of the f i n i t e element work have been i l l u s t r a t e d i n f i g u r e [4.4] and show almost a s t r a i g h t l i n e correspondence between dec r e a s i n g c a p a c i t y and i n c r e a s i n g hole s i z e . (4.2) >/4/352 -25 + 1 44  1.2 1.1- Hole S i ze Ratio D/b F i g . 4.4: Comparison of the U l t i m a t e In-plane Shear C a p a c i t i e s of C o n c e n t r i c a l l y P e r f o r a t e d P l a t e as C a l c u l a t e d by the F i n i t e Element Method and the ASCE Design Proposal given by Equation [4.2] 46 4.1.3.2 3-Dimensional E l a s t i c B u c k l i n g The v a r i a t i o n of the ul t i m a t e e l a s t i c b u c k l i n g c a p a c i t y with hole s i z e was s i g n i f i c a n t l y d i f f e r e n t from that of the in-plan e y i e l d c a p a c i t y . For the smaller hole s i z e s the decrease i n c a p a c i t y was almost p r o p o r t i o n a l to the h o l e s i z e . For the medium s i z e holes the e l a s t i c b u c k l i n g c a p a c i t y was much lower than the s t r a i g h t l i n e c o r r e l a t i o n found i n the i n - p l a n e y i e l d c a p a c i t y . When the hole diameter became gr e a t e r than 0.2 6 the b u c k l i n g c a p a c i t y became s i g n i f i c a b t l y lower than a s t r a i g h t l i n e c o r r e l a t i o n . The e l a s t i c b u c k l i n g c o e f f i c i e n t s f o r both the clamped and simply supported p l a t e boundaries a r e i l l u s t r a t e d i n f i g u r e [ 4 . 5 ] , and compared with other t h e o r e t i c a l r e s u l t s by UENOYA, REDWOOD [ 3 ] and MARCO [ 9 ] . The u l t i m a t e e l a s t i c b u c k l i n g c a p a c i t y i s r e l a t e d to the e l a s t i c b u c k l i n g c o e f f i c i e n t by equation [ 4 . 3 ] . UENOYA and REDWOOD used a combination of in-plane f i n i t e element s t r e s s a n a l y s i s and a R a y l e i g h - R i t z energy method t o determine the e l a s t i c b u c k l i n g c o e f f i c i e n t . The p e r f o r a t e d p l a t e s u r f a c e was d i s c r e t i z e d by the f i n i t e element method u s i n g constant s t r e s s t r i a n g l e (CST) elements. An e l a s t i c a n a l y s i s p r o v i d e d the s t r e s s d i s t r i b u t i o n throughout the domain. The s t r e s s e s were then s u b s t i t u t e d i n t o the minimum p o t e n t i a l energy e x p r e s s i o n , and an b i f u r c a t i o n a n a l y s i s p r o v i d e d the b u c k l i n g l o a d s . The d e f l e c t e d shape i n the energy expression was represented by the f i r s t e i g h t terms of a F o u r i e r s e r i e s . The (4.3) 47 r e s u l t i n g values showed a good c o r r e l a t i o n with the c l a s s i c a l s o l u t i o n f o r a f u l l p l a t e and agree with the c u r r e n t work f o r the smaller h o l e s . However, as the h o l e s i z e , D/b, becomes grea t e r 0.4 t h e i r e l a s t i c a n a l y s i s g i v e s a much higher b u c k l i n g load than the c u r r e n t f i n i t e element work or t h a t of MARCO. The e l a s t i c b u c k l i n g c o e f f i c i e n t s determined by MARCO were obtained using the same program as used i n the c u r r e n t work. A f u l l model of the p e r f o r a t e d p l a t e was used throughout the a n a l y s i s . The model was made up of 16 b i c u b i c , i s o p a r a m e t r i c , p l a t e s h e l l elements. Despite u s i n g fewer elements i n the model, the r e s u l t s he obtained were s l i g h t l y lower than the c u r r e n t work. One p o s s i b l e e x p l a n a t i o n f o r t h i s i s the use of a d i f e r e n t i n i t i a l l o a d increament before the b u k l i n g a n a l y s i s i s done. If MARCO a p p l i e d a higher i n i t i a l l o a d to the p l a t e before the b i f u r c a t i o n a n a l y s i s was done he would get lower b u c k l i n g l o a d . A l s o , i f a lower order i n t e g r a t i o n was i n h i s a n a l y s i s i t would have s o f t e n i n g the s t i f f n e s s m a t r i x , thus a s l i g h t l y lower b u c k l i n g l o a d would r e s u l t . A t y p i c a l b u c k l i n g mode f o r the q u a r t e r p l a t e model i s i l l u s t r a t e d i n f i g u r e [4.6], A along the compression d i a g o n a l , on the r i g h t - h a n d s i d e , the displacements tend to be concentrated i n the center of the p l a t e . Along the t e n s i o n d i a g o n a l , on the l e f t - hand s i d e , the displacements are more evenly d i s t r i b u t e d . T h i s i s because along the compression d i a g o n a l the p l a t e i s subjected to te n s i o n a c r o s s the boundary. T h i s t e n s i o n tends to r e s t r a i n the out- o f - p l a n e movement of the p l a t e i n t h i s a r ea. Thus, lower displacements occur i n these a r e a s . 48 1 6 Hole Size Ratio D/b F i g . 4.5: V a r i a t i o n of E l a s t i c B u c k l i n g C o e f f i c i e n t w i t h C o n c e n t r i c Hole S i z e 49 F ig . 4.6: E last i c Buck l i ng M o d e , Con cen t r i c Ho l e 4.1.3.3 3-Dimensional E l a s t i c - P l a s t i c B u c k l i n g The combined e f f e c t s of n o n l i n e a r m a t e r i a l and geometry were s t u d i e d by determining the u l t i m a t e e l a s t i c - p l a s t i c b u c k l i n g c a p a c i t y of the p e r f o r a t e d p l a t e . The s i g n i f i c a n c e of the n o n l i n e a r m a t e r i a l behavior i s d i s p l a y e d i n the t y p i c a l l o a d - d e f l e c t i o n path shown i n f i g u r e [ 4 . 7 ] , The p l a t e b u c k l i n g load was determined at v a r i o u s stages along the l o a d - d e f l e c t i o n path. As each new e q u i l i b r i u m c o n f i g u r a t i o n was e s t a b l i s h e d a b i f u r c a t i o n a n a l y s i s was performed to determine the b u c k l i n g l o a d . When the p l a t e m a t e r i a l f i r s t s t a r t s to y i e l d , the s t i f f n e s s matrix i s e f f e c t i v l y softened and the b u c k l i n g c a p a c i t y i s reduced. As the p l a t e move ou t - o f - p l a n e the post b u c k l i n g s t r e n g t h of the p l a t e i n c r e a s e the b u c k l i n g c a p a c i t y . F i n a l l y , the u l t i m a t e e l a s t i c - p l a s t i c b u c k l i n g c a p a c i t y i s reached when the m a t e r i a l y e i l d i n g has lowered the b u c k l i n g load to the same value as the c u r r e n t l o a d l e v e l . The r e s u l t i n g l o a d d e f l e c t i o n paths of four p e r f o r a t e d p l a t e s with d i f f e r e n t hole s i z e s are shown i n f i g u r e [4.8]. While the m a t e r i a l behaves e l a s t i c a l l y the l a t e r a l displacements are s m a l l . At approximately 90% of the u l t i m a t e l o a d there i s a very r a p i d change i n the p l a t e s t i f f n e s s . The d e f l e c t i o n i n c r e a s e s very q u i c k l y with only a small i n c r e a s e i n a p p l i e d l o a d . There i s a long p l a t e a u as displacements continue to increase with only a small i n c r e a s e i n a p p l i e d l o a d i n g . F i n a l l y , when the n o n l i n e a r i t i e s have lowered the b u c k l i n g c a p a c i t y s u f f i c i e n t l y the determinant of the s t i f f n e s s matrix becomes nega t i v e . T h i s p o i n t d e f i n e s the u l t i m a t e e l a s t i c - p l a s t i c b u k l i n g c a p a c i t y of the p l a t e 51 150 1 4 0 - 1 3 0 - 120 110- 1 0 0 - 9 0 - 8 0 - 70 Stable 4 Unstable • Legend Load Path Buckling Load I 12 14 i 16 6 8 10 Lateral Deflection m m i g . 4.7: Decreasing E l a s t i c - P l a s t i c B u c k l i n g C a p a c i t y of a P e r f o r a t e d P l a t e with I n c r e a s i n g A p p l i e d Load. 18 200 Full Pate D/b=0.15 D/b=0.2 D/b=0.3 0 2 4 6 Lateral Deflection m m F i g . 4.8: Load D e f l e c t i o n Curve of a Simply Supported P e r f o r a t e d P l a t e with V a r i o u s C o n c e n t r i c Hole S i z e s 8 D/b=0.5 i 1 i 1 i 1 i 10 12 14 16 i 18 - I — ' — l — ' — I — ' — l — 20 22 24 26 28 52 Although the curves have s i m i l a r c h a r a c t e r i s t i c s , f i g u r e [4.8] c l e a r l y demonstrates that the length of the p l a t e a u changes with hole s i z e . As the hole s i z e i n c rease the pl a t e a u becomes longer and at the same time the slope of the loa d - d e f l e c t i o n path becomes s t e e p e r . T h i s means i s that f o r l a r g e r holes there w i l l be more ou t - o f - p l a n e d i s t o r t i o n i n the p l a t e before the u l t i m a t e load i s reached. A l s o , a steeper slope i n d i c a t e s that the p l a t e i s s t i f f e r and w i l l s t i l l c a r r y more lo a d . Thus, p l a t e s with l a r g e r h o l e s w i l l be more d u c t i l e . F i g u r e [4.8] a l s o c l e a r l y shows that the u l t i m a t e e l a s t i c - p l a s t i c b u c k l i n g c a p a c i t y i s s u b s t a n t i a l l y reduced with i n c r e a s i n g hole s i z e . The r e s u l t s of the present study along with numerical r e s u l t s by UENOYA f o r the u l t i m a t e e l a s t i c - p l a s t i c b u c k l i n g c a p a c i t i e s of a c o n c e n t r i c a l l y p e r f o r a t e d p l a t e are i l l u s t r a t e d i n f i g u r e [ 4 . 9 ] . The r e s u l t s of the present study were obtained from l o a d - d e f l e c t i o n curves s i m i l a r to those i n f i g u r e [ 4 . 8 ] . The u l t i m a t e l o a d was d e f i n e d as the h i g h e s t load l e v e l obtained on the l o a d - d e f l e c t i o n curve. The two se t s of r e s u l t s show good agreement f o r the smaller hole s i z e , but the c u r r e n t work of the author g i v e s s u b s t a n t i a l l y lower values f o r the l a r g e r h o l e s . There may be two p o s s i b l e e x p l a n a t i o n s f o r these d i s c r e p a n c i e s . F i r s t l y , the use of the r e l a t i v e l y s t i f f CST element, used by UENOYA and REDWOOD, to c a l c u l a t e the s t r e s s d i s t r i b u t i o n may have r e s u l t e d i n an underestimation of the true s t r e s s f i e l d . If the s t r e s s e s were under estimated the R a l y l e i g h - R i t z m i n i m i z a t i o n would produce a higher p l a t e c a p a c i t y . Secondly, by r e s t r i c t i n g the displacement 53 1.2 1.1 T 1 1 1 1 1 1 1 1 1— 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Hole Size Ratio D/b F i g . 4.9: V a r i a t i o n of Ult i m a t e E l a s t i c - P l a s t i c B u c k l i n g Capacity of Simply Supported P e r f o r a t e d P l a t e with C o n c e n t r i c Hole S i z e . 5 4 F ig . 4.10: E-P Buck l i ng w i th von-M i ses Stress f u n c t i o n to the f i r s t e i g h t terms of the F o u r i e r s e r i e s , UENOYA assumes that the r e s u l t i n g displacement f i e l d may be a c c u r a t e l y represented by a combination of these modes. T h i s assumption was s u f f i c i e n t f o r the f u l l p l a t e b u c k l i n g mode to which i t was compared. However, i f higher terms become more dominate as the hole s i z e i n c r e a s e s , the displacement f u n c t i o n may not be able to a c c u r a t e l y represent the lowest b u c k l i n g mode. The r e s u l t would be to o v e r - p r e d i c t the b u c k l i n g c a p a c i t y . 4.2 ' V a r i a t i o n of Hole L o c a t i o n The second parameter s t u d i e d was the l o c a t i o n of a hole on the p l a t e c a p a c i t y . The center of a standard hole was p l a c e d i n va r i o u s l o c a t i o n s on the p l a t e s u r f a c e . For each l o c a t i o n the ul t i m a t e i n - p l a n e , e l a s t i c b u c k l i n g , and e l a s t i c - p l a s t i c b u c k l i n g c a p a c i t i e s were determined. 4.2.1 P l a t e Geometry •b/2 b/2 b/2 b/2 F i g . 4.11: P l a t e Geometry and Loading used i n the A n a l y s i s of the V a r i a t i o n of Hole L o c a t i o n Parameter 56 A standard hole with a diameter of 0.2b was centered at v a r i o u s l o c a t i o n s i n the p l a t e . The geometry and l o a d i n g are shown in f i g u r e [4.11]. The parameters used to d e f i n e the hole l o c a t i o n , ANG. and R/b, were v a r i e d as f o l l o w s ; ANG=45 to 135 deg., 5/6=0.15 and 0.3. The hole only needed to be c o n s i d e r e d i n one quarter of the p l a t e area due to the symmetry in l o a d i n g and geometry. 4.2.2 F i n i t e Element Model Two b a s i c element meshes were r e q u i r e d to do the a n a l y s i s fo r the e c c e n t r i c hole l o c a t i o n s . When the hole center was l o c a t e d along a p l a t e diagonal there i s an a x i s of symmetry about t h i s d i a g o n a l . By a p p l y i n g the a p p r o p r i a t e displacement boundary c o n d i t i o n s along t h i s a x i s , only a one h a l f - p l a t e model was r e q u i r e d . However, i f the hole was not l o c a t e d on a d i a g o n a l there was no a x i s of symmetry and a f u l l p l a t e model r e q u i r e d . The h a l f model was used whenever p o s s i b l e as i t used s i g n i f i c a n t l y l e s s CPU time f o r the a n a l y s i s . The h a l f and f u l l p l a t e models were generated by the same method. The hole c e n t e r was s e l e c t e d as the o r i g i n of a p o l a r c o o r d i n a t e system. R a d i a l l i n e s were then e s t a b l i s h e d to each corner of the p l a t e , d i v i d i n g the model i n t o q u a r t e r s e c t i o n s . Each s e c t i o n was then modeled by using nine i s o p a r a m e t r i c elements i n a 3x3 g r i d . T h i s r e s u l t e d i n a h a l f and f u l l model made up of e i g h t e e n and t h i r t y - s i x elements r e s p e c t i v e l y . The nodal s p a c i n g i n each q u a r t e r of the model was determined i n a s i m i l a r manner as was used f o r the center hole models. With the p o l a r c o o r d i n a t e system each quarter was d i v i d e d i n t o nine s e c t i o n s by ten e q u a l l y spaced r a d i a l l i n e s . Ten nodes were then 57 CC F ig . 4.12: F in i te E lement M o d e l of Ha l f the P late F ig . 4.13: F in i te E lement M o d e l of the Tota l Plate p l a c e d along each r a d i a l l i n e with p r o p o r t i o n a l spacing as given by equation [4.1]. T y p i c a l examples of the r e s u l t i n g f i n i t e element mesh f o r the h a l f and f u l l models are i l l u s t r a t e d i n f i g u r e [4.12] and [4.13]. These element meshes were used through-out the analyses, with some d i s t o r t i o n to accommodate the p l a t e geometry. 4.2.3 R e s u l t s 4.2.3.1 In-plane Y i e l d i n g R e s u l t s from the in-plane y i e l d i n g a n a l y s i s are i l l u s t r a t e d i n f i g u r e [4.14]. As the 0.2b diameter hole was moved over the p l a t e surface there was l i t t l e change i n the u l t i m a t e in-plane y i e l d c a p a c i t y . In f a c t only one l o c a t i o n had more than a 2% change. T h i s l o c a t i o n had the hole near the p l a t e boundary and had a 4% change i n c a p a c i t y . The model d e f i n e d by the parameters, 72/6 =0.3, ANG.=90, D/b =0.2, produced a s i g n i f i c a n t l y lower c a p a c i t y than the other models. Y i e l d i n g s t a r t e d around the hole i n t h i s model as with other models. Once s u f f i c i e n t y i e l d i n g had occu r r e d between the inner hole boundary and the outer p l a t e boundary, the p l a t e reached i t s u l t i m a t e c a p a c i t y . The f a i l u r e mechanism was s i m i l a r to the other c o n f i g u r a t i o n s but y i e l d i n g was r e s t r i c t e d to the small area between the hole boundary and the p l a t e outer boundary. Therefore t h i s c o n f i g u r a t i o n was co n s i d e r e d a l o c a l m a t e r i a l f a i l u r e r a t h e r than an u l t i m a t e p l a t e c a p a c i t y l i m i t . In theory t h i s l o c a l m a t e r i a l f a i l u r e mode may be of i n t e r e s t , but i n r e a l i t y shear webs would not l i k e l y experience t h i s type of f a i l u r e . Most shear webs have a flan g e or s t i f f e n e r l o c a t e d on a l l s i d e s . T h i s would p r o v i d e a mechanism f o r 6 0 '6 ID •> v • 1 * ' 0.962 * .•••«•.. N .•0 .980 / •-.0.980 S- .^"6.989 1.03 '•0 .989 / 0.995' - ' " . ' 0 5 9 5 R/b=0.3 10 ^R/b.= O.I5 \, b / 2 \ b / 2 . F i g . 4.14: Ult i m a t e In-plane Y i e l d C a p a c i t y R e s u l t s f o r Va r i o u s Hole L o c a t i o n s Normalize to the C o n c e n t r i c Hole Ul t i m a t e In-plane C a p a c i t y r e d i s t r i b u t i o n of f o r c e s . The areas of hig h s t r a i n would t r a n s f e r f o r c e s i n t o the boundary s t i f f e n e r s . These f o r c e s would t r a v e l through the s t i f f e n e r s and then be t r a n s f e r r e d back i n t o the p l a t e at re g i o n s of lower s t r a i n . 4.2.3.2 3-Dimensional E l a s t i c B u c k l i n g U n l i k e the in-plane c a p a c i t y , s i g n i f i c a n t changes i n the u l t i m a t e e l a s t i c b u c k l i n g c a p a c i t y occur when the hole l o c a t i o n moves away from the center of the p l a t e . I f the hole was moved from the or t e n s i o n d i a g o n a l to the compression diagonal the e l a s t i c b u c k l i n g c a p a c i t y could be i n c r e a s e d by as much as 50%. A number of r e s u l t s f o r the e l a s t i c b u c k l i n g c a p a c i t i e s are i l l u s t r a t e d i n f i g u r e [4.17]. A l l the c a p a c i t i e s are given as a f a c t o r of the e l a s t i c b u c k l i n g c a p a c i t y of a p l a t e with a c o n c e n t r i c h o l e so that d i r e c t comparisons can be made between l o c a t i o n s . I f the hole was l o c a t e d i n the t e n s i o n d i a g o n a l the 61 62 F ig . 4.16: Fu l l P late, von -M i se s Stress f a c t o r ranged from 0.992 to 1.017. T h i s i s e s s e n t i a l l y constant. However, along the compression d i a g o n a l the f a c t o r i n c r e a s e d to 1.52. T h i s means that there was a 52% i n c r e a s e i n the e l a s t i c b u c k l i n g c a p a c i t y by moving the hole from the t e n s i o n d i a g o n a l to thecompresion d i a g o n a l . R/b=0.3 R/b = 0 . ! 5 b / 2 •b / 2 F i g . 4.17: E l a s t i c B u c k l i n g C a p a c i t y F a c t o r s for V a r i o u s Hole Hole Locations Normalize to the C o n c e n t r i c Hole C a p a c i t y . The e l a s t i c b u c k l i n g mode of the p l a t e shown i n f i g u r e [4.18] and [4.19] i l l u s t r a t e s , the d i f f e r e n t p r o f i l e s of the t e n s i o n diagonal and compression d i a g o n a l r e s p e c t i v e . The displacement p r o f i l e along the t e n s i o n d i a g o n a l was very smooth, s i m i l a r i n shape to a simple s i n e wave. However, acr o s s the compression d i a g o n a l , the p r o f i l e appeared to be made up of more complicated shapes. The displacements were more pronounced at the hole boundary then at the edges of the p l a t e . At the c o r n e r s of the p l a t e the displacements were very small and examination of the eigenvector showed that some were a c t u a l l y negative i n these areas. 64 4.2.3.3 3-Dimensional E l a s t i c - P l a s t i c B u c k l i n g The r e s u l t s of the c a l c u l a t i o n s f o r the u l t i m a t e e l a s t i c - p l a s t i c b u c k l i n g c a p a c i t y are d e t a i l e d i n f i g u r e [4.20]. The c a p a c i t i e s are expressed as a f a c t o r of the c o n c e n t r i c a l l y p e r f o r a t e d p l a t e u l t i m a t e e l a s t i c p l a s t i c c a p a c i t y . The u l t i m a t e e l a s t i c - p l a s t i c b u c k l i n g p l a t e c a p a c i t y f o r each hole l o c a t i o n i s r e l a t e d to the magnitude of change i n the e l a s t i c b u c k l i n g c a p a c i t y . The r e s u l t s showed that i f there was l i t t l e or no change i n the e l a s t i c b u c k l i n g c a p a c i t y , the p l a t e would undergo a t y p i c a l e l a s t i c - p l a s t i c b u c k l i n g (zone II) f a i l u r e . The combined e f f e c t s of non l i n e a r m a t e r i a l and geometry would produced an u l t i m a t e e l a s t i c - p l a s t i c c a p a c i t y lower than i f e i t h e r of the two f a c t o r s were c o n s i d e r a l o n e . However, i f the e l a s t i c b u c k l i n g c a p a c i t y was i n c r e a s e d s i g n i f i c a n t l y , as when the hole i s moved to compression d i a g o n a l , the b u c k l i n g c a p a c i t y was so much higher than the in-pl a n e y i e l d c a p a c i t y t h a t the p l a t e would f a i l without b u c k l i n g . Thus the p l a t e would experience an i n - p l a n e y i e l d f a i l u r e (zone I ) . Examples of models that underwhen zone I and a zone II f a i l u r e modes, are given i n t a b l e [4.1]. Table 4.1: F a i l u r e Mode C l a s s i f i c a t i o n Model Character i st i c s F a i l u r e D/b R/b ANG. MPa. MPa. MPa. C l a s s A 0.2 0.3 45 136.9 133.5 128.3 zone II 2C 0.2 0.0 - 138.4 131.3 126.2 zone II E 0.2 0.3 135 136.9 207.5 138.8 zone I 65 F ig . 4.18: P ro f i l e of the Tens ion D iagona l : P ro f i l e of the Comp r e s s i o n D i agona l *6 • b/2 - j . b / 2 F i g 4.20: Ultimate E l a s t i c - P l a s t i c C a p a c i t y f o r Vario u s Hole L o c a t i o n s Normalized to the C o n c e n t r i c Hole Ultimate E l a s t i c - P l a s t i c B u c k l i n g C a p a c i t y Since the e l a s t i c b u c k l i n g c a p a c i t y of the p l a t e with e c c e n t r i c hole always higher than that of a p l a t e with a c o n c e n t r i c hole and, no examples of e l a s t i c b u c k l i n g (zone I I I ) f a i l u r e were found i n t h i s a n a l y s i s . The e l a s t i c b u c k l i n g c a p a c i t y was never s i g n i f i c a n t l y lower than the in-pla n e c a p a c i t y . However, t h i s does not mean that a zone I I I f a i l u r e c o u l d not occur. If the slenderness r a t i o of the p l a t e were decreased, the e l a s t i c b u c k l i n g c a p a c i t y would be reduced, while the i n - p l a n e c a p a c i t y would remain unchanged, f o r c i n g the p l a t e i n t o a zone I I I f a i l u r e mode. The c o n c e n t r i c a l l y p e r f o r a t e d p l a t e p r o v i d e s a lower bound value f o r a l l other hole l o c a t i o n s . With reference to f i g u r e [4.20], a l l but one hole l o c a t i o n produced an u l t i m a t e 68  c a p a c i t y f a c t o r greater than that of the c o n c e n t r i c h o l e . The hole l o c a t i o n that r e s u l t e d i n a lower u l t i m a t e c a p a c i t y was the same one that had a low in - p l a n e c a p a c i t y . As d i s c u s s e d in s e c t i o n [4.2.3.1] t h i s low value was due to a l o c a l m a t e r i a l f a i l u r e . I t i s u n l i k e l y that t h i s mode of f a i l u r e w i l l occur i n r e a l i t y , s i n c e the fl a n g e s and s t i f f e n e r s around the web w i l l provide a mechanism f o r the r e d i s t r i b u t i o n of f o r c e s . I f l o c a l m a t e r i a l f a i l u r e s are prevented, then the u l t i m a t e in-plane c a p a c i t y , f o r any hole l o c a t i o n , w i l l be approximately equal to the in- p l a n e y i e l d c a p a c i t y of a p l a t e with c o n c e n t r i c h o l e . The e l a s t i c b u c k l i n g a n a l y s i s i n d i c a t e d that the c o n c e n t r i c hole had the lowest c a p a c i t y . The u l t i m a t e e l a s t i c - p l a s t i c b u c k l i n g c a p a c i t y was governed by a combination of the in-pla n e y i e l d c a p a c i t y and e l a s t i c b u c k l i n g c a p a c i t i e s . Moving the hole away from the center of the p l a t e i n c r e a s e d the e l a s t i c b u c k l i n g c a p a c i t y and had l i t t l e e f f e c t on the i n - p l a n e c a p a c i t y . Thereforethe minimum u l t i m a t e e l a s t i c - p l a s t i c b u c k l i n g c a p a c i t y for any l o c a t i o n of a given hole s i z e i s given by the c o n c e n t r i c hole c o n f i g u r a t i o n . 4.3 Optimum Doubler P l a t e The f i n a l parameter i n v e s t i g a t e d i n the study was the e f f e c t i v e n e s s of the a d d i t i o n of a doubler p l a t e around the p e r f o r a t i o n . The o b j e c t i v e of the doubler p l a t e was to lower the s t r e s s e s around the hole, l i m i t i n g the m a t e r i a l y i e l d i n g and r e s t o r i n g some or a l l of the p l a t e s in-plane y i e l d c a p a c i t y . The goal of t h i s study was to ev a l u a t e the e f f e c t i v e n e s s of v a r i o u s doubler p l a t e shapes. 70 4.3.1 P l a t e Geometry The standard square p l a t e with a c o n c e n t r i c 0 . 2 6 diameter c i r c u l a r p e r f o r a t i o n was used throughout t h i s part of the study. A doubler p l a t e i n the c i r c u l a r shape was attached to the p e r f o r a t e d p l a t e around the h o l e . The diameter and t h i c k n e s s of the doubler p l a t e was v a r i e d . The diameter v a r i e d from 1 . 3 D to 2 .425Z>and the t h i c k n e s s from 0 . 2 5 * to 1 . 5 0 t . 4.3.2 F i n i t e Element Model L i k e the other models f o r the c o n c e n t r i c a l l y p e r f o r a t e d p l a t e only the one-quarter model was r e q u i r e d . The axes of symmetry were again the p l a t e d i a g o n a l s . The same displacement boundary c o n d i t i o n s were a p p l i e d along these axes of symmetry and a l l displacements were r e s t r i c t e d to i n - p l a n e movements only. The one-quarter p l a t e model shown in f i g u r e [ 4 . 2 3 ] c o n s i s t e d of an 8 x 5 mesh of plane s t r e s s elements d i s c u s s e d i n s e c t i o n [ 3 . 1 . 4 . 2 ] , The model was generated by d i v i d i n g the quarter p l a t e i n t o e i g h t s e c t i o n s by nine e q u a l l y spaced r a d i a l l i n e s . Six nodes were then p l a c e d on each r a d i a l l i n e . The f i r s t node placed on each l i n e was set at the hole boundary. The next four nodes on each r a d i a l l i n e were set a constant r a d i i of 1 . 3 £ J 1 . 7 5 D , 2.0D and 2 . 4 2 5 D . The l a s t node was then l o c a t e d a l o n g the outer p l a t e boundary. The doubler p l a t e was .modeled by s p e c i f y i n g a t h i c k e r p l a t e fo r the i n t e r n a l element r i n g s . For the s m a l l e s t diameter doubler p l a t e s only the f i r s t element r i n g was t h i c k e n e d . For l a r g e r doubler p l a t e s i z e s two or three element r i n g s were thickened. T h i s allowed the same model to be used throughout the a n a l y s i s with minimal changes between models. 71 T JL X\ \Dd \ \ 2 0 0 \ E = 200 000 MPa (Ty - 300 MPa v = 0.3 t = 10 500 500 F i g . 4.22: Geometry and Loading of P e r f o r a t e d P l a t e with Doubler P l a t e F i g . 4.23: F i n i t e Element Mesh of P e r f o r a t e d P l a t e with Doubler P l a t e 72 4.3.3 Results 4.3.3.1 In-plane Y i e l d i n g The doubler p l a t e a n a l y s i s showed th a t , given the same c r o s s s e c t i o n a l areas, a wide, t h i n doubler p l a t e was more e f f e c t i v e than a narrow, t h i c k p l a t e . In f i g u r e [4.24] a p l o t of the e f f e c t i v e c a p a c i t y r e s t o r a t i o n f a c t o r vs. a nondimensional doubler p l a t e area i s shown f o r v a r i o u s doubler p l a t e diameters. As the doubler p l a t e diameter r a t i o , $ was in c r e a s e d there was a s i g n i f i c a n t i n crease i n the c a p a c i t y r e s t o r a t i o n f a c t o r . The c a p a c i t y of a p l a t e w i t h doubler p l a t e i s given by equation [4.4]. I f the c a p a c i t y r e s t o r a t i o n f a c t o r i s 1.0, the p l a t e was r e s t o r e d to i t s o r i g i n a l u n p e r f o r a t e d c a p a c i t y . In f i g u r e [4.24] i t i s shown that a doubler p l a t e of Dd = 2.0D and Ad/A=1 »0 has a c a p a c i t y r e s t o r a t i o n f a c t o r of almost 1.0. S o l v i n g f o r the doubler p l a t e t h i c k n e s s * td> g i v e s the parameters f o r an optimum doubler p l a t e s i z e as, Dd=2.0D, td=t. Uy = fy + 1> (rv ~ fy) (4.4) Without the doubler p l a t e , y i e l d i n g s t a r t e d at the inner boundary of the p e r f o r a t i o n arid propagated around the hole and up i n t o the body of the p l a t e . Once the y i e l d i n g had extended from the inner p e r f o r a t i o n boundary to the outer p l a t e edges, the p l a t e became unstable and the u l t i m a t e l o a d had been attaned. With the a d d i t i o n of a doubler p l a t e there was a s i g n i f i c a n t m o d i f i c a t i o n to the y i e l d p a t t e r n . Again, y i e l d i n g s t a r t e d at the p e r f o r a t i o n boundary, but i t d i d not extend i n t o the p l a t e . Instead, a second y i e l d zone developed at the outer doubler p l a t e boundary. T h i s y i e l d i n g spread r a p i d l y from the edge of the 73 doubler p l a t e to the p l a t e boundary as the l o a d l e v e l i n c r e a s e d . Once the y i e l d i n g had extended inward to the inner p e r f o r a t i o n boundary there was no f u r t h e r i n c r e a s e i n p l a t e c a p a c i t y . A comparison of the two u l t i m a t e y i e l d p a t t e r n s , f o r the standard and the r e i n f o r c e d p e r f o r a t e d p l a t e s as d e s c r i b e d above -is given in f i g u r e [4.25], 0.5 1.0 Doubler Plate Area AA/A F i g . 4.24: E f f e c t i v e C a p a c i t y R e s t o r a t i o n F a c t o r vs Doubler P l a t e Area f o r V a r i o u s Doubler P l a t e Diameters 74 doubler plate reinforcement Dd = 400 j t d = 20 without reinforcement F i g . 4.25: Spread of Y i e l d Zones f o r Standard and R e i n f o r c e d P e r f o r a t e d P l a t e s 4.1 Convergence with Mesh Refinement An approximate s o l u t i o n method i s enhanced i f i t can be r i g o r o u s l y proven t h a t , as the step s i z e or element s i z e i s reduced, the method w i l l render the exact s o l u t i o n . The approximate method should p r o v i d e some s o r t of bound on the exact s o l u t i o n and show an asymptotic convergence to the s o l u t i o n . The f i n i t e element formulation s a t i s f i e s a l l of these requirements. From f i n i t e element theory i t can be shown that the s t r a i n energy of the f i n i t e element s o l u t i o n p r o v i d e s a lower bound value f o r the system s t r a i n energy. Furthermore i t has been shown that an eigenvalue a n a l y s i s of the f i n i t e element s t i f f n e s s matrix w i l l p r o v i d e an upper bound value f o r the system's e l a s t i c b u c k l i n g l o a d . The theory a l s o s t a t e s that the f i n i t e element s t r a i n energy w i l l a s y m p t o t i c a l l y converge to the exact s o l u t i o n l i k e (l/n) p . where n i s the number of elements i n any one d i r e c t i o n 75 and p depends on the governing d i f f e r e n t i a l e q u a t i o n of the continuum problem and the i n t e r p o l a t i o n f u n c t i o n s used to formulate the element. Proofs of these p r o p e r t i e s can be made under c e r t a i n c o n d i t i o n s . The c o n s i s t e n t load must be a p p l i e d over the system domain as w e l l as between elements and exact i n t e g r a t i o n of the element area i s assumed. The c o n s i s t e n t load requirement between elements and on the element boundaries has been s a t i s f i e d throughout the a n a l y s i s . The l o a d i n g between elements was a u t o m a t i c a l l y s a t i s f i e d by the f i n i t e element f o r m u l a t i o n . Along the element boundaries the c o n s i s t e n t shear load v e c t o r f o r each element s u b j e c t e d to uniform shear was c a l c u l a t e d and a p p l i e d . Exact i n t e g r a t i o n r e q u i r e s a l o t of CPU time i f numerical i n t e g r a t i o n i s used. The accuracy of the i n t e g r a t i o n must be s u f f i c i e n t to e x a c t l y evaluate the element s t i f f n e s s i n t e g r a l , i n c l u d i n g a l l term i n the determinant of the J a c o b i a n c o o r d i n a t e t r a n s f o r m a t i o n matrix. Terms in the Jacobian matrix may be to the second and t h i r d power and the determinant w i l l have terms to the f o u r t h , f i f t h and s i x t h power. Combining t h i s with other terms i n the element s t i f f n e s s i n t e g r a l would r e q u i r e i n t e g r a t i o n be s u f i c i e n t to f u l l y evaluate a polynomial of order nine or ten. Instead of e x a c t l y i n t e g r a t i n g the s t i f f n e s s i n t e g r a l f o r a l l cases, BATHE [ 1 0 ] has shown i t i s s u f f i c i e n t t o use a reduced i n t e g r a t i o n , however, the i n t e g r a t i o n must be s u f f i c i e n t to e x a c t l y evaluate the s t i f f n e s s i n t e g r a l i f the determinant of the Jacobian matrix i s a c o n s t a n t . Since exact i n t e g r a t i o n of the element s t i f f n e s s i n t e g r a l i s 76 not r e a l i s t i c a lower order i n t e g r a t i o n i s used. Using a lower order i n t e g r a t i o n the f i n i t e element s o l u t i o n may s t i l l a s y m p t o t i c a l l y converge to the exact v a l u e . However, the r a t e of convergence w i l l be governed by the i n t e g r a t i o n e r r o r and not the order of the element. By assuming that there was an asymptotic convergence rate with mesh refinement, the accuracy of the e l a s t i c b u c k l i n g c a p a c i t i e s c a l c u l a t e d by NISA83 were e s t i m a t e d . The work was performed using the standard 1/4 p l a t e model of a simply supported p e r f o r a t e d p l a t e with a hole diameter of 0.26. The element mesh f o r the p l a t e was v a r i e d from 1x1 to 6x6. A b i f u r c a t i o n a n a l y s i s was done on each of these permutations to approximate the e l a s t i c b u c k l i n g c a p a c i t y of the p e r f o r a t e d p l a t e . A value was then assume f o r the exact e l a s t i c b u c k l i n g c a p a c i t y of the p l a t e and the r e l a t i v e e r r o r c a l c u l a t e d f o r the b u c k l i n g c a p a c i t y give by each mesh. A Log-Log p l o t of r e l a t i v e e r r o r vs number of elements i n one d i r e c t i o n was set up and s t r a i g h t l i n e passed through the p o i n t s u s i n g a l e a s t squares f i t a l g o r i t h m . A new exact s o l u t i o n was then assumed and t h i s process was repeated. By keeping r e c o r d i n g of the cumulative e r r o r of each l e a s t squares f i t a s s o c i a t e d w i t h an assumed exact s o l u t i o n , the best value f o r the exact e l a s t i c b u c k l i n g c a p a c i t y was e s t a b l i s h e d . The r e s u l t s of t h i s work are shown in f i g u r e [4.26]. The slope of the l i n e i n f i g u r e [4.26], r e p r e s e n t i n g the convergents r a t e , i s approximately -3. T h e r e f o r e , the convergence r a t e f o r t h i s element and model i s n to the power -3 or (1/n) 3 . 77 A l s o given by the f i g u r e i s the r e l a t i v e e r r o r a s s o c i a t e d with each mesh. I t shows that the 3x3 element mesh has an e r r o r of 5%. A more r e f i n e d g r i d would have p r o v i d e d a more accurate s o l u t i o n f o r the u l t i m a t e e l a s t i c and e l a s t i c - p l a s t i c b u c k l i n g c a p a c i t i e s , but would r e q u i r e more storage and CPU time which was not a v a i l a b l e on the VAX 11/730. The f u l l p l a t e model with the present mesh already used the c a p a c i t y of the VAX. By using a c o n s i s t e n t element mesh, with v a r y i n g degrees of d i s t o r t i o n depending on hole s i z e and l o c a t i o n , the c a l c u l a t e d u l t i m a t e c a p a c i t i e s that express the same r e l a t i v e e r r o r . Therefore, r e s u l t s from a l l the analyses were d i r e c t l y compared. Any changes i n u l t i m a t e c a p a c i t i e s was a t t r i b u t e d to the parameters s t u d i e d and not due to changes i n the modeling technique. 0.001 Number Of Elements In One Direction n F i g . 4.26: Convergencs of the E l a s t i c B u c k l i n g Load with Mesh Refinement f o r a C o n c e n t r i c a l l y Holed P l a t e with Z>/6 = 0.2, 1/6 = 0.01 78 5 CONCLUSIONS The behavior of a square shear p l a t e with a c i r c u l a r p e r f o r a t i o n at i t s u l t i m a t e l o a d can be d e s c r i b e d by one of three d i f f e r e n t f a i l u r e modes. The parameter which determines the f a i l u r e mode i s the p l a t e slenderness r a t i o , t/b ( t h i c k n e s s / width). For stocky p l a t e s the u l t i m a t e c a p a c i t y i s l i m i t e d by the in-plane m a t e r i a l y i e l d c a p a c i t y . For slender p l a t e s the e l a s t i c b u c k l i n g c a p a c i t y of the p l a t e i s much lower than the m a t e r i a l in-plane y i e l d c a p a c i t y . The u l t i m a t e c a p a c i t y i s t h e r e f o r e c o n t r o l l e d by the e l a s t i c b u c k l i n g c a p a c i t y . F i n a l l y , f o r intermediate slender p l a t e s , both the i n - p l a n e y i e l d c a p a c i t y and the e l a s t i c b u c k l i n g c a p a c i t y are of the same magnitude. The f a i l u r e mode i s a f u n c t i o n of both m a t e r i a l y i e l d i n g and b u c k l i n g so the u l t i m a t e c a p a c i t y i s governed by the e l a s t i c - p l a s t i c b u c k l i n g c a p a c i t y . No a n a l y t i c a l s o l u t i o n e x i s t s f o r determining the u l t i m a t e c a p a c i t y of p e r f o r a t e d p l a t e s with t h i s type of f a i l u r e . T herefore, numerical methods are r e q u i r e d to estimate these c a p a c i t i e s . The program NISA83 was used to c a r r y out a parameter study on p e r f o r a t e d p l a t e s . For each parameter c o n f i g u r a t i o n the program c a l c u l a t e d the u l t i m a t e i n - p l a n e y i e l d c a p a c i t y , e l a s t i c b u c k l i n g c a p a c i t y and the u l t i m a t e e l a s t i c - p l a s t i c b u c k l i n g c a p a c i t y . The r e s t a r t option of the program and a b i l i t y to change the time step c o n t r o l and i t e r a t i o n methods c o n t r i b u t e d to the e f f i c i e n c y of these c a l c u l a t i o n s . The "constant a r c l e n g t h " time step c o n t r o l proved to be w e l l s u i t e d to f o l l o w the l o a d - d e f l e c t i o n path of the p l a t e i n t o the post b u c k l i n g r e g i o n . 79 Some minor changes were r e q u i r e d to get NISA83 running on the C i v i l E n g i n e e r i n g VAX 11/730. The computer speed and storage c a p a c i t y was s u f f i c i e n t to handle the n o n l i n e a r problem modeling a 1/4 p l a t e . However, when the f u l l p l a t e model was used the CPU time requirement became l a r g e and the memory storage became c r i t i c a l . I f a much l a r g e r n o n l i n e a r problem were attempted a l a r g e r computer would be r e q u i r e d The p l o t t i n g program, NISPLOT, was developed f o r the output from NISA83. Although, the program ran under the EUNICE o p e r a t i n g system i t was found to be completely compatible with the VMS output f i l e s from NISA83. The i n f o r m a t i o n that i s presented i n a g r a p h i c a l form f o r both data checks and p o s t - p r o c e s s i n g displacements or s t r e s s e s make a g r a p h i c s program a n e c e s s i t y f o r any f i n i t e element program. I t i s reccommended that NISPLOT (or a s i m i l a r program) be extended to i n c l u d e a l l the elements i n the NISA83 l i b r a r y . The f i r s t parameter i n v e s t i g a t e d i n the study was the v a r i a t i o n of a c o n c e n t r i c hole s i z e . The r e s u l t s f o r the u l t i m a t e i n - p l a n e y i e l d c a p a c i t y showed a s t r a i g h t l i n e c o r r e l a t i o n between i n c r e a s i n g hole s i z e and de c r e a s i n g p l a t e c a p a c i t y . The r e s u l t s were a l s o c o r r e l a t e d to the ASCE Suggested Design G u i d e l i n e s . I t was found that these design r u l e s tended to overestimate the i n - p l a n e c a p a c i t y of the p l a t e . The e l a s t i c b u c k l i n g c a p a c i t i e s c a l c u l a t e d f o r each parameter v a r i a t i o n were i n agreement with other p u b l i s h e d r e s u l t s . For h o l e s l a r g e r than 0.4 of the p l a t e width, the e l a s t i c b u c k l i n g c a p a c i t y was reduced s i g n i f i c a n t l y below a 80 s t r a i g h t l i n e c o r r e l a t i o n . T h i s r e d u c t i o n should be taken i n t o account i n the design of any web where b u c k l i n g c o u l d occur. F i n a l l y , the u l t i m a t e e l a s t i c - p l a s t i c b u c k l i n g c a p a c i t i e s were c a l c u l a t e d for each hole s i z e . The r e s u l t s were compared to other p u b l i s h e d work. The c u r r e n t r e s u l t s showed that the c a p a c i t y of the p l a t e have been s l i g h t l y overestimated by o t h e r s . The u l t i m a t e e l a s t i c - p l a s t i c b u c k l i n g c a p a c i t y was lower than e i t h e r the in-plane y i e l d or the e l a s t i c b u c k l i n g v a l u e s . Y i e l d i n g of the m a t e r i a l around the h o l e reduces the p l a t e b u c k l i n g l o a d thus lowering the u l t i m a t e p l a t e c a p a c i t y . The second parameter i n the a n a l y s i s was the hole l o c a t i o n . The u l t i m a t e in-plane y i e l d c a p a c i t y was c a l c u l a t e d f o r each l o c a t i o n . The r e s u l t s i n d i c a t e that there i s l i t t l e v a r i a t i o n i n the p l a t e c a p a c i t y with hole l o c a t i o n . The c a p a c i t y of a p l a t e with a c o n c e n t r i c hole seemed to p r o v i d e a good approximation of a p e r f o r a t e d p l a t e even with a hole i n any l o c a t i o n except c l o s e to the p l a t e edge. If the hole was l o c a t e d too c l o s e to the boundary of the p l a t e there was the p o s s i b i l i t y that l o c a l m a t e r i a l y i e l d i n g between the hole and the near boundary c o u l d reduce the p l a t e in-plane c a p a c i t y . The c a l c u l a t i o n of the e l a s t i c b u c k l i n g l o a d f o r each v a r i a t i o n i n hole l o c a t i o n y i e l d e d some unexpected r e s u l t s . The e l a s t i c b u c k l i n g load was found to i n c r e a s e by up to 50% i f the hole was moved from the p l a t e t e n s i o n d i a g o n a l to the compression d i a g o n a l . The c o n c e n t r i c hole produced the lowest e l a s t i c b u c k l i n g c a p a c i t y of any of the hole l o c a t i o n s . 81 The u l t i m a t e e l a s t i c - p l a s t i c b u c k l i n g c a p a c i t i e s appeared to be a combination of the f i r s t two c a p a c i t i e s . By moving the hole away from the center of the p l a t e the u l t i m a t e p l a t e c a p a c i t y would be governed by the in-pla n e y i e l d c a p a c i t y , i f the e l a s t i c b u c k l i n g load became s i g n i f i c a n t l y higher than the in-pla n e y i e l d c a p a c i t y . Since the c o n c e n t r i c hole provided the lowest e l a s t i c b u c k l i n g c a p a c i t y i t was not a s u r p r i s e to f i n d t h a t t h i s l o c a t i o n a l s o had the lowest u l t i m a t e e l a s t i c - p l a s t i c b u c k l i n g c a p a c i t y . Thus the p l a t e with a c o n c e n t r i c hole pr o v i d e s a lower bound value f o r the u l t i m a t e e l a s t i c - p l a s t i c c a p a c i t y of a p l a t e s with the same hole s i z e . The e f f e c t i v e n e s s of a doubler p l a t e as reinforcement was a l s o i n v e s t i g a t e d . The diameter and t h i c k n e s s of c i r c u l a r doubler p l a t e s were v a r i e d . The doubler p l a t e dimensions, *«//t = 1.0, and •0/6=2.0, gave a p e r f o r a t e d p l a t e the same in-pla n e y i e l d c a p a c i t y as i t would have without the h o l e . An attempt was made to e s t a b l i s h the accuracy and convergence r a t e of the element mesh used i n a l l the a n a l y s e s . The e l a s t i c b u c k l i n g l o a d was c a l c u l a t e d f o r the one q u a r t e r p l a t e model using a mesh ranging from 1x1 to 6x6 elements. By pas s i n g the best s t r a i g h t l i n e through the Log-Log p l o t of r e l a t i v e e r r o r vs number of elements i n one d i r e c t i o n , ( n ) , a convergence r a t e of (1/n)3 was determined. The e r r o r i n the 3x3 element g r i d , used throughout t h i s work, was estimated from t h i s p l o t to be 5%. 82 REFERENCE 1 Wang, Chu-Kia. " T h e o r e t i c a l A n a l y s i s of P e r f o r a t e d Shear Webs", Presented at a meeting of the ASME C i n c i n n a t i S e c t i o n , C i n c i n n a t i , Ohio Oct. 2-3, 1945. 2 Rockey, K. C , Anderson, R. G., and Cheung, Y. K. "The Behavior of Square shear Webs Having A C i r c u l a r Hole", Symp. on Thin Walled S t e e l S t r u c t u r e s , U n i v e r s i t y Colledge of Swansea, Crosby Lockwood and Sons L t d . 1969, pp.148-169. 3 Uenoya, M. and Redwood, R. G. " E l a s o - P L a s t i c Shear Buckling of Square P l a t e s with C i r c u l a r Holes", Computers and S t r u c t u r e s , Vol.8, pp. 291-300, Pergamon Press L t d . , 1978. 4 Uenoya, M. and Redwood, R. G. " B u c k l i n g of Webs With Openings", Computers and S t r u c t u r e s , Vol.9, pp. 191-199, Pergamon Press L t d . , 1978. 5 Redwood, R. G., and Uenoya, M. " C r i t i c a l Loads f o r Webs with Holes", J o u r n a l of the S t r u c t u r a l D i v i s i o n , ASCE, Vol.105, No. 105, pp. 2053-2067, Oct. 1979. 6 Janssen, T. L. "A Simple E f f i c i e n t Hidden L i n e Algorithm", Computers and S t r u c t u r e s , Vol.17, pp. 563-571, Pergamon Press L t d . , 1983. 7 H'afner, L., Ramm, E., S a t t e l e , J . M., and Stegmuller, H. "NISA80 Proqrammdokumentation Programmsystem", B e r i c h t des I n s t i t u t s f ur Baustik, U n i v e r s i t a t S t u t t g a r t , 1981. 8 P r e c i s i o n V i s u a l s Inc. "DI3000 Users Guide", P r e c i s i o n V i s u a l s , 6260 Lookout Road, Boulder, Colorado, 80301 USA, March 1984. 9 Marco, Renzo "Buckling of P l a t e s with C i r c u l a r Holes", B.Ap.Sc. T h e s i s , U n i v e r s i t y of Maitoba, 1984. 10 Bath^ Klaus-Jurgen " F i n i t e Element Procedures i n Engineering A n a l y s i s " , P r e n t i c e H a l l Inc., Englewood C l i f f s , N. J . 1982 11 Subcommitty on Beams with Web Openings "Suggested Design Guides f o r Beams with Web Holes", ASCE, J o u r n a l of the S t r u c t u r e s Div., Vol.97, pp.2707-2728, Nov. 1971. 12 Bathe^ Klaus-Jurgen and B o l o u r c h i , S a i d "A Geometric and M a t e r i a l Nonlinear P l a t e S h e l l Element", Computers and S t r u c t u r e s , Vol.11, pp. 23-48, Pergamon Press L t d . , 1980. 83 APPENDIX A D e r i v a t i o n of the C o n s i s t e n t Shear Load Vector f o r the B i c u b i c Isoparametric Element. U s i n g a h i g h e r o r d e r e l e m e n t s u c h as t h e b i c u b i c i s o p a r a m e t r i c e l e m e n t g i v e s e x c e l l e n t r e s u l t s t o a p r o b l e m m o d e l e d w i t h a s m a l l number o f e l e m e n t s . However, u s i n g fewer e l e m e n t s means t h a t e a c h e l e m e n t i s more d i s t o r t e d and h e n c e , more dependent on t h e a p p l i e d l o a d s . Work done by BATHE [ 1 2 ] h a s shown t h a t t h e b i c u b i c i s o p a r a m e t r i c e l e m e n t i s n o t a f f e c t e d as much by d i s t o r t i o n as many of t h e l o w e r o r d e r e l e m e n t s . N e v e r t h e l e s s , he does reccommend u s i n g t h e e l e m e n t c o n s i s t e n t l o a d v e c t o r i n o r d e r t o m i n i m i z e any e r r o r s c a u s e d by t h e e l e m e n t d i s t o r t i o n . The f o l l o w i n g i s t h e d e r i v a t i o n of t h e c o n s i s t e n t l o a d v e c t o r f o r t h e b i c u b i c i s o p a r a m e t r i c e l e m e n t , w i t h a u n i f o r m s h e a r a p p l i e d a l o n g one b o u n d a r y . The l o a d v e c t o r i s d e f i n e d i n t e r m s of t h e e l e m e n t c o o r d i n a t e s . The r e s u l t i n g e q u a t i o n s c a n be u s e d t o d e t e r m i n e t h e c o n s i s t e n t s h e a r v e c t o r f o r any d i s t o r t e d e l e m e n t . F i g . A . 1 : B i c u b i c I s o p a r a m e t r i c P l a t e S h e l l E l e m e n t w i t h U n i f o r m Shear L o a d i n g a l o n g One Edge 84 The d e f i n i t i o n of the i t h term of the element c o n s i s t e n t l o a d v e c t o r P i s give by equation [A.1]. Pi = JJ q(x,y)Ni(r,s)dxdy (A.l) Area where <l{xiy) = a p p l i e d t r a c t i o n on the element s u r f a c e Pi = i' Aterm of c o n s i s t e n t load v e c t o r ^ ( ^ 3 ) = shape f u n c t i o n f o r node x For the i s o p a r a m e t r i c element t h i s can be converted i n t o the r,s l o c a l c o o r d i n a t e system and the volume i n t e g r a t i o n reduced to the l i m i t s of -1 to 1. +1+1 P{ = J J q (z, y) 7Y, ( r , s) detJ dr ds (A.2) -1-1 where detJ = determinant of the Jacobian t r a n s f o r m a t i o n matrix If <j(r,s) i s uniform along along the edge s=1 and zero everywhere e l s e , equation [A.2] can be reduced t o . +1 Pi = q j Ni detJ dr i = 2,6,5,1 (4.3) -1 and ^ dN- . dr 3 j=2,6,5,l or c o n v e r t i n g the i n d i c e s to 1 through 4, gives the l i n e i n t e g r a l , +1 Pk = q f Nkiy^x^dr k = 1,2,3,4 x2 x 3 85 Table A.I: Four Cubic Shape Functions along the B i c u b i c Element Boundary s=1, -1<r<1 TV* * 1 6 f a c t . (r+1) (3r+1) (3r-1 ) (1-r) Ni 1 - 1 1 1 N2 -9 1 — 1 1 N3 9 1 1 — 1 1 1 1 1 - M u l t i p l y i n g these out and t a k i n g the d e r i v a t i v e s gives: ^ = ( ^ ) ( - 3 r 3 + r2 + 3 r - l ) ^ 4 = ( ^ ) ( 9 r 3 + 9 r 2 - r - l ) ffi.(±)(-^ + ttr+1) dN2 / - 9 \ , 9 . - « r = ( » ) ( - 9 r 2 + 2 r + 3 ) Expanding equation [A.3] . - l - l +X3fNt9-£dr + XlJNt 9Jl±dr -1 -1 (4.4) (4.5) (4.6) 86 Each of the four terms in equation [ A . 6 ] i s of the form where k dr Nk = Ak (akr* + bkr2 + ckr+d) ^ = B,(e i r 2 + / i r + ri Thus, the i n t e g r a t i o n f o r these terms need only be done once, +1 +1 Nk-g~r dr = AkBi j (akr3 + bkr2 + ckr + dk) for2 + fir + gt) dr - l = 2AkBt \ { h € l ~ a"fl) + (<**e< + ckfi + bk9l) + d k e i ^ 5 3 {A.7) S u b s t i t u t i n g the corresponding v a l u e s f o r ak,bk,cktdk,ei,fi gt from equations [A.4 ] and [ A .5 ] i n t o equation [ A . 6 ] r e s u l t s i n four simple l i n e a r e x p r e s s i o n s f o r the c o n s i s t e n t l o a d vector f o r an element with uniform shear s t r e s s along one s i d e . Px = - 0 . 5 0 0 Z J + 0 . 7 1 2 5 x 2 - 0 . 3 0 0 i 3 + 0 . 0 8 7 5 x 4 ' P2 = -0.7125Z! + 0 . 0 x 2 + 1 . 0 1 2 5 z 3 - 0 . 3 0 0 x 4 P3 = 0.300Z! - 1 . 0 1 2 5 x 2 + O.O13 + 0 . 7 1 2 5 x 4 P 4 = -0.0875X! + 0 . 3 0 0 x 2 - 0 . 7 1 2 5 x 3 + 0 . 5 0 0 x 4 , U.8) 87 Appendix B ASCE Suggested Design Guides for Beams with Web Holes The u l t i m a t e i n - p l a n e shear s t r e s s of a p e r f o r a t e d web as g i v e n by e q u a t i o n [ 1 8 ] i n Ref e r e n c e [ 1 1 ] i s g i v e n by: f » - ( ' " f ) v r E r » — a* ( § ) ' ( * m ) For a " c i r c u l a r h o l e H = 0.9D A = 0A5D S u b s t i t u t i n g the above i n t o e q u a t i o n [ B . 1 ] g i v e s : I f the s u b s t i t u t i o n S = 0.90(D/6) i s made i n e q u a t i o n s [ B . 3 ] and [ B . 4 ] and the two e q u a t i o n s a r e combined and s i m p l i f i e d , a s i n g l e e x p r e s s i o n , e q u a t i o n [ B . 7 ] , i s o b t a i n e d f o r the u l t i m a t e i n - p l a n e shear s t r e s s f o r a square p l a t e p e r f o r a t e d by a c i r c u l a r h o l e . a = 3 . o ( i - l ) (B.6) fy _^(l-S)(l/S-l) T» \Jl + 3(l/S- l ) 2 88 Hole Size Roitio D/b F i g . B.1: Ultimate In-plane Capacity of Shear Web with a C i r c u l a r P e r f o r a t i o n as Proposed i n Reference [ 1 1 ] 8 9 APPENDIX C M o d i f i c a t i o n of NISA80 a t U.B.C. NISA80 The f o l l o w i n g i s a summary of the changes that were made to the October 83 v e r i o n of NISA80, c a l l e d NISA83 on the C i v i l Engineering VAX 11/730, at U.B.C. PROGRAM NISA83C Change f i l e names to s u i t the d i s k names at U. from SLF( 1 ) = SLF(2) = SLF(3) = SLF(4) = SLF(5) = SLF(6); SLF(7): ' DRA2 ' DRA2 ' DRA2 ' DRA2 ' DRA2 ' DRA2 ' DRA2 [SCRATCH [SCRATCH tSCRATCH [SCRATCH [SCRATCH [SCRATCH [SCRATCH ]NISA20, ]NISA21, ]NISA22, ]NISA2 3, ]NISA24 ]NISA25 ]NISA26 SCR* SCR' SCR' SCR' SCR' SCR' SCR' F1='DRA2:[SCRATCH]NS01.SCR' F2='DRA2:[SCRATCH]NS 0 2.SCR' F3='DRA2:[SCRATCH]NS03.SCR' F8='DRA2:[SCRATCH]NS08.SCR' F9='DRA2:[SCRATCH]NS09.SCR' to SLF(1)='NISA20.SCR* SLF(2)='NISA21.SCR' SLF(3)='NISA22.SCR* SLF(4)='NISA23.SCR' SLF(5)='NISA24.SCR' SLF(6)='NISA25.SCR' SLF(7)='NISA26.SCR' C F1='NS01.SCR' F2='NS02.SCR' F3='NS03.SCR* F8='NS08.SCR' F9='NS09.SCR' SUBROUTINE INPUT Changes have been made to the c y l i n d r i c a l nodal input r o u t i n e s so tha t the user can s e l e c t the normal a x i s . NAXIS=0 y-z plane i s s p e c i f i e d i n p o l a r c o o r d i n a t e s x i s the nomal a x i s NAXIS=1 same as NAXIS=0 NAXIS=2 x-y plane i s s p e c i f i e d i n p o l a r c o o r d i n a t e s z i s the nomal a x i s from 20 READ(INP,1000) - - - - ,Z(N),KN,IT WRITE(IOUT,2002) - - - - ,Z(N),KN,IT 1000 FORMAT( - - - - , 1 5 , 1 2 ) 2001 FORMAT( - - - - ,5X,2HIT/) 2002 FORMAT( - - - - ,I5,2X,I5) to 20 READ(INP,1000) - - - - ,Z(N),KN,IT,NAXIS WRITE(IOUT,2002) - - - - ,Z(N),KN,IT,NAXIS 90 1000 FORMAT( - - - - ,15,12,12) 2001 FORMAT( ,5X,2HIT,3X,5HNAXIS/) 2002 FORMAT( ,I 5,2X,I 5,2X15) from C C CYLINDRICAL COORDINATES C 50 DUM = Z(N) * RAD Z(N) = Y(N) * SIN(DUM) Y(N) = Y(N) * COS(DUM) to C C CYLINDRICAL COORDINATES C 50 CONTINUE IF (NAXIS.EQ.2) THEN DUM = Z(N) * RAD Z(N) = X(N) X(N) = Y(N) * COS(DUM) Y(N) = Y(N) * SIN(DUM) ELSE DUM = Z(N) * RAD Z(N) = Y(N) * SIN(DUM) Y(N) = Y(N) * COS(DUM) END IF from DX = (X(N)-X(NOLD)) / XNUM to IF (NAXIS.EQ.2) THEN DZ = (Z(N)-Z(NOLD)) / XNUM ELSE DX = (X(N)-X(NOLD)) / XNUM END IF from C C CYLINDRICAL COORDINATES C 60 ROLD = Y(NOLD) / COS(DUMOLD) RNEW = Y(N) / COS(DUM) to C C CYLINDRICAL COORDINATES C 60 IF (NAXIS.EQ.2) THENN ROLD = X(NOLD) / COS(DUMOLD) RNEW = X(N) / COS(DUM) ELSE ROLD = Y(NOLD) / COS(DUMOLD) RNEW = Y(N) / COS (DUM) END IF SUBROUTINE MAIN Change the open statements f o r f i l e s NPLOT and NPLOT1 from unformated to formated. The reason f o r t h i s change i s the EUNIC FORTRAN 77 compiler gave an e r r o r when i t read from an unformated f i l e . T h i s problem seems to be s p e c i f i c to the U. B. C. VAX and i s r e l a t e d to running VMS with a EUNIC emulator rather than n a t i v e UNIX. from IF(IPG.EQ.I) OPEN(UNIT=NPLOT,- - - ,FORM=UFM) IF(MODEX.NE.O) OPEN(UNIT=NPLOT1,- - - ,FORM=UFM) to IF(IPG.EQ.I) OPEN(UNIT=NPLOT,- - - ,FORM=FM) IF(MODEX.NE.O) OPEN(UNIT=NPLOT1,- - - ,FORM=FM) 91 SUBROUTINE D3INP Change the write statements from unformated to formated. from WRITE (NPLOT)NUMEP,MN to WRITE (NPLOT,2010)NUMEP,MN 2010 FORMAT(2I5) SUBROUTINE PLN Change the write statements from unformated to formated. from WRITE (NPLOT) HED WRITE (NPLOT) NUMNP,NUMEG WRITE (NPLOT) X1,Y1,Z1 to WRITE (NPLOT,2000) HED WRITE (NPLOT,2010) NUMNP,NUMEG WRITE (NPLOT,2020) X1,Y1,Z1 2000 FORMAT (A72) 2010 FORMAT (215) 2020 FORMAT (1P,3E15.6) SUBROUTINE D3PLOT Changes where made to a l l writ e statements from unformated to formated output. f rom to 2000 2010 2015 2020 2030 WRITE WRITE WRITE WRITE WRITE WRITE WRITE WRITE WRITE WRITE WRITE WRITE WRITE WRITE WRITE WRITE WRITE WRITE FORMAT FORMAT FORMAT FORMAT FORMAT NPLOT) N,13,(INODE(I),I=1,13) NPLOT) N,9,(INODE(I),1=1,9) PLOT) N,5,(INODE(I),1=1,5) NPLOT) N,4,(INODE(I NPLOT) N,4,(INODE(I NPLOT) N,4,(INODE(I NPLOT) N,4,(INODE(I NPLOT) N,3,(INODE(I NPLOT) N,3,(INODE(I = 1,4) = 1,4) = 1,4) = 1,4) = 1,3) = 1 ,3) NPLOT NPLOT NPLOT NPLOT NPLOT NPLOT NPLOT NPLOT NPLOT (215, (215, (215, (215, (215, ,2000) N,13 ,2010) N,9, ,2015) N,5, ,2020) N,4, ,2020) N,4, ,2020) N,4, ,2020) N,4, ,2030) N,3, ,2030) N,3, 2X,13(15)) 2X,9(I5)) 2X,5(I5)) 2X,4(I5)) 2X,3(I5)) ,(INODE(I),1=1,13) (INODE(I) ,1=1,9) (INODE(I) ,1=1,5) (INODE(I) ,1=1,4) (INODE(I) ,1=1,4) (INODE(I),1=1,4) (INODE(I),1=1,4) (INODE(I) ,1=1,3) (INODE(I) ,1=1,3) Note; i n l i n e 3 WIRTE(PLOT)N,- - PLOT i s not a ty p i n g e r r o r , T h i s i s how i t appeared in the o r i g i n a l v e r t i o n . SUBROUTINE PLOTGEO Change the write statement from unformated to formated. f rom WRITE (NPLOT) N,IEP,(INODE(I),I=1,IEP) 92 to WRITE (NPLOT,) N,IEP, (I NODE(I),I = 1 , 1EP) 2 0 0 0 FORMAT ( 2 1 5 , 2 X,<IEP> , 1 5 ) SUBROUTINE WRITE Change the write statements from unformated to formated. from WRITE (NPLOT)PHED WRITE (NPLOT)DSI,DS2,DS3 to WRITE (NPLOT,2060)PHED WRITE (NPLOT,2070)DS1,DS2,DS3 2060 FORMAT (A70) 2070 FORMAT (1P,3E15.6) ************************************************** NISA80.2 The f o l l o w i n g i s a summary of the changes that were made to the June 84 v e r i o n of NISA80, c a l l e d NISA84 on the C i v i l E n g i neering VAX 11/730, at U.B.C. SUBROUTINE FNAMES from SFL(1)='DRA2:"SCRATCH:NISA.SCR' SFL(2)='DRA2:"SCRATCH:NISA.SCR' SFL(3)='DRA2:"SCRATCH:NISA.SCR' SFL( 4 ) = 'DRA2:"SCRATCH:NISA.SCR' SFL(5)='DRA2:"SCRATCH:NISA.SCR' to SFL(1)='SCRATCH:NISA.SCR' SFL(2)='SCRATCH:NISA.SCR' SFL(3)='SCRATCH:NISA.SCR' SFL(4)='SCRATCH:NISA.SCR' SFL(5)='SCRATCH:NISA.SCR' from 2000 FORMAT( German text ) 2010 FORMAT( German text ) to 2000 FORMAT( E n g l i s h t e x t ) 2010 FORMAT( E n g l i s h text ) SUBROUTINE OPENRF from to SLF( 1 SLF(2 SLF(3 SLF(4 SLF(5 SLF(6 SLF(7 SLF( 1 SLF(2 SLF(3 SLF(4 SLF(5 SLF(6 SLF(7 ='DRA2: ='DRA2: ='DRA2; ='DRA2: ='DRA2: ='DRA2: ='DRA2: 'SCRATCH' "SCRATCH" "SCRATCH" "SCRATCH" "SCRATCH" "SCRATCH" "SCRATCH" NISA.RN1' NISA.RN2' NISA.RN3' NISA.RN4' NISA.RN5' NISA.RN6' NISA.RN7' 'SCRATCH: 'SCRATCH: 'SCRATCH: :'SCRATCH: ••' SCRATCH: •'SCRATCH: •'SCRATCH: NISA.RN1' NISA.RN2' NISA.RN3' NISA.RN4' NISA.RN5' NISA.RN6' NISA.RN7' 93 C's have been placed i n the f i r s t column of the second v e r s i o n of OPENRF so that i t i s not compiled by the U. B. C. Vax 11/730 Th i s second v e r s i o n i s for the Cray computer. SUBROUTINE FOPEN from CHARACTER NAME*40 to CHARACTER NAME*(*) SUBROUTINE HEDIN Mo d i f i e d the output heading to acknowledge that the work i s being done at the U. B. C. s i t e on the C i v i l Engineering Vax 11/730. SUBROUTINE INPUT Changes have been made to the c y l i n d r i c a l nodal input r o u t i n e s so that the user can s e l e c t the normal a x i s . NAXIS=0 y-z plane i s s p e c i f i e d i n p o l a r c o o r d i n a t e s x i s the nomal a x i s NAXIS=1 same as NAXIS=0 NAXIS=2 x-y plane i s s p e c i f i e d i n p o l a r c o o r d i n a t e s z i s the nomal a x i s from 20 READ(INP,1000) - - - - ,Z(N),KN,IT WRITE(IOUT,2002) - - - - ,Z(N),KN,IT 1000 FORMAT( - - - - ,15,12) 2001 FORMAT( - - - - ,5X,2HIT/) 2002 FORMAT( - - - - ,I5,2X,I5) to 20 READ(INP,1000) - - - - ,Z(N),KN,IT,NAXIS WRITE(IOUT,2002) - - - - ,Z(N),KN,IT,NAXIS 1000 FORMAT( - - - - ,15,12,12) 2001 FORMAT( - - - - ,5X,2HIT,3X,5HNAXIS/) 2002 FORMAT( - - - - ,I 5,2X,I 5,2X15) from C C to C C c CYLINDRICAL COORDINATES 50 DUM = Z(N) * RAD Z(N) = Y(N) * SIN(DUM) Y(N) = Y(N) * COS(DUM) CYLINDRICAL COORDINATES 50 CONTINUE IF (NAXIS.EQ.2) THEN DUM = Z(N) * RAD Z(N) = X(N) X(N) = Y(N) * COS(DUM) Y(N) = Y(N) * SIN(DUM) ELSE DUM = Z(N) * RAD Z(N) = Y(N) * SIN(DUM) Y(N) = Y(N) * COS(DUM) 94 END IF from DX = (X(N)-X(NOLD)) / XNUM to IF (NAXIS.EQ.2) THEN DZ = (Z(N)-Z(NOLD)) / XNUM ELSE DX = (X(N)-X(NOLD)) / XNUM END IF from C C CYLINDRICAL COORDINATES C 60 ROLD = Y(NOLD) / COS(DUMOLD) RNEW = Y(N) / COS(DUM) to C C CYLINDRICAL COORDINATES C 60 IF (NAXIS.EQ.2) THENN ROLD = X(NOLD) / COS(DUMOLD) RNEW = X(N) / COS(DUM) ELSE ROLD = Y(NOLD) / COS(DUMOLD) RNEW = Y(N) / COS(DUM) END IF SUBROUTINE DKTM There was a compile time e r r o r bacause of the m u l t i p l e d e c l a r a t i o n of the v a r i a b l e ICON. from COMMON /PRECIS/ NDP,ICON to COMMON /PRECIS/ NDP,NPY ************************************** NISA80.2 UPDATE The f o l l o w i n g i s a summary of the changes that were made to the update of the 3D-PLATE SHELL ELEMENT i n s t a l l e d Sept. 17 85. on C i v i l Engineering VAX 11/730, at U.B.C. TRANSFER FILE D3DMAIN.FOR When the Tr a n s f e r f i l e D3DMAIN.FOR was read from the IBM d i s k e t t e using the program KERMIT a non ASCII c h a r a c t e r was found on l i n e No. 1117. The ch a r a c t e r was e d i t e d from the f i l e using the PC e d i t o r , EDLIN, and repla c e d by ?. Line 1112 to 1119 SUBROUTINE D3LSS (A,G,GI,IT) C Q ****************************************************** c * * C * TRANSFORM STRESS AND STRAIN LOCAL-GLOBAL * C * LOCAL 3-DIRECTION IS ZERO ??? * C * * C * A ... VECTOR TO BE TRANSFORMED * 95 TRANSFER FILE D3DINP.FOR The same problem of a non ASCII c h a r a c t e r occured i n l i n e No. 393 of the t r a n s f e r program D3DINP.for. Again the chacter was e d i t e d from the f i l e before t r a n s f e r of the f i l e was completed to the VAX. Line 390 to 393 2020 FORMAT (1H1,15X,'E L E M E N T I N F O R M A T I O N'// 1 5X,'IEL = NUMBER OF NODES FOR THIS ELEMENT'/ 1 5X,'IPS = STRESS OUTPUT CONTROL NUMBER'/ 2 5X,'KG = NODE INCREMENT FOR GENERATION ( SECOND CARD ? ) ' SUBROUTINE D3STIF When NPAR(5) was s e l e c t e d as 1 (commplete t h i c k n e s s i n t e g r a t i o n ) the program stopped because of an e r r o r i n the v a r r i a b l e array dimesion. Which arra y was never determined, however, the v a r r i a b l e NBO i s never appears in the parrameter s t r i n g i n the subroutine D3DISD. from C 120 CALL D3DISD (DISD,DDISD,B,ALFN,EDIS,DC,DCA(1,1,N),NC(1,N), 1 HTET(1,1,N),IEL,MN,NBO,ND,IFORM,HHI) C to C 120 CALL D3DISD (DISD,DDISD,B,ALFN,EDIS,DC,DCA(1,1,N),NC(1,N), 1 HTET(1,1,N),IEL,MN,ND,I FORM,HHI) C 96 APPENDIX D Program L i s t i n g s APPENDIX D.l NISPLOT Q ********************************************************* C C U N I X V E R S I O N C C N I S A 8 3 C C P L O T c C THIS PROGRAM USES "DI-3000" TO PLOT THE FINITE ELEMENT GRID OUTPUT C AND THE DEFLECTED SHAPE FROM "NISA83" IN A 3-D FORM. C c p C R E A T E M E T A F I L E S c INPUT AND OUTPUT FILES. c GEO. INPUT FILE (FORMATED) 'INF ILE' = 1 (FROM USER) c DISP. INPUT FILE (FORMATED) 'INF ILE ' = 4 (FROM USER) c STRESS INPUT FILE (FORMATED) 'INFILE ' = 3 (FROM USER) c OUTPUT PLOTED TO 'NOUT' = 6 c OUTPUT SCRATCH FILE 'I SCR' = 7 c FULL PLATE GEO. ' ' = 8 c DEVICE TYPE 'MDEV' = 0 (METAFILES) c 'NDEV' = 1 (GRAPHICS) Q ***************************************************************************** IMPLICIT REAL*4(A-H.O-Z) COMMON / PLT / INODE(100,13,5), IEL(100,5), N(100,5). NMAX(5) COMMON / MAX / RMIN(3), RAVE(3), RMAX(3), RATIO COMMON / STR / NF(6, 16),RS(2,25),NPOINT(4, 16),FACT(16),ICOL(7 ) COMMON / SVIEW / D(3), U(3) COMMON / HEAD / PHEAD INTEGER NUMP, NUMEG LOGICAL D3STR, D3PLT, METST, PHEAD CHARACTER*45 HED. PHED CHARACTER EOF DIMENSION STRESS(16,30,5), 1X(500), Y(500), Z(500), DX(500), DY(500). DZ(500). 2RX(500), RY(500). RZ(500). STRMAX(2) NPGEO = 1 NPDIS = 4 NPSTR = 3 NOUT = 6 ISCR = 7 NPLATE= 8 MDEV = 0 NDEV = 1 PHEAD = CALL SETUP (NPGEO,NPDIS,NPSTR,NPLATE,ISCR,D3STR.D3PLT,METST) c C SET UP THE SCREEN FOR PLOTTING c CALL JBEGIN CALL JDINIT (NDEV) CALL JDEVON (NDEV) IF (METST)THEN CALL JFSOPN (3,0,0,'NISAPLOT.MFL') CALL JDINIT (MDEV) CALL UDEVON (MDEV) END IF CALL JASPEK ( 1 .RATIO) IF (RATIO.LT. 1) THEN CALL JVSPAC (-1.0, 1.0, -RAT 10,' RAT 10 ) ELSE IF (RATIO.GT. 1 ) THEN CALL JVSPAC (-1.0/RATIO, 1.0/RATIO, -1.0, 1.0) ELSE RATI0=1.0 END IF CALL JSETDB (0) 97 NISPLOT L i s t i n g R E A D I N A N D P L O T F U L L P L A T E A N D E L E M E N T M O D E L . I F ( N P L A T E . N E . O ) T H E N NVIEW=0 N P L O T = N P L A T E C A L L R E A D N O C A L L M A X M I N C A L L V I E W C A L L R E A D E L C A L L J O P E N C A L L P L T E L E C A L L P L T H E D C A L L J C L O S E C A L L J P A U S E C A L L J F R A M E E N D I F ( X , Y , Z , H E D , N U M P , N U M E G . 1 , N P L O T . I S C R ( X . Y . Z . N U M P ) ( N V I E W ) ( N U M E G . N P L O T , I S C R ) ( X , Y , Z . N U M E G , N V I E W ) ( H E D ) ( N D E V ) I S T O P ) R E A D I N F U L L M E S H N O D E P O I N T S A N D E L E M E N T S . N V I E W = 1 N P L O T = N P G E O C A L L R E A D N O ( X , Y , Z , H E D , N U M P , N U M E G , 1 . N P L O T . I S C R . I S T O P ) C A L L M A X M I N ( X , Y , Z , N U M P ) C A L L V I E W ( N V I E W ) C A L L A P L O T S U B R O U T I N E T O P L O T T H E N O D E P O I N T S . ( S U B R O U T I N E P L O T N O ) I F ( M E T S T ) T H E N C A L L J O P E N C A L L P L T H E D ( H E D ) C A L L J C L O S E C A L L J P A U S E ( N D E V ) C A L L J F R A M E P H E A D = . F A L S E . E N D I F C A L L J O P E N C A L L P L T N O D ( X , Y . Z , N U M P , N O U T ) C A L L P L T H E D ( H E D ) C A L L J C L O S E C A L L J P A U S E ( N D E V ) R E A D E L E M E N T D A T A A N D P L O T E L E M E N T ( S U B R O U T I N E P L O T E L ) C A L L R E A D E L ( N U M E G , N P L O T , I S C R ) C A L L J F R A M E C A L L J O P E N C A L L . P L T E L E ( X , Y , Z , N U M E G . N V I E W ) C A L L P L T H E D ( H E D ) C A L L J C L O S E C A L L J P A U S E ( N D E V ) I F D3STR . T R U E . R E A D I N T H E S T R E S S F I L E A N D P L O T T H E U N D E F L E C T E D S H A P E W I T H A S T R E S S C O L O R F I L L I F (D3STR) T H E N N P L O T = N P S T R C A L L V I E W ( N V I E W ) C A L L R E A D S T ( S T R E S S , N M A X , S T R M A X , N U M E G , N P L O T , I S C R ) C A L L J F R A M E C A L L J O P E N C A L L E L E S T R ( X , Y . Z . S T R E S S , S T R M A X , N U M E G . 2 ) C A L L P L T E L E ( X , Y , Z , N U M E G . 2 ) C A L L P L T H E D ( H E D ) C A L L J C L O S E C A L L L E G E N D ( S T R M A X ) C A L L J P A U S E ( N D E V ) E N D I F I F D3PLT . T R U E . T H A N P L O T T H E D E F L E C T E D A N D T H E O R I G I N A L S H A P E I N 3 - D . 98 NISPLOT L i s t i n g I F ( D 3 P L T ) T H E N N P L O T = N P D I S C A L L R E A D N O ( D X , D Y , D Z , P H E D . N U M P , N U M E G , 2 , N P L O T . I S C R , I S T O P ) I F ( I S T O P . L T . O ) G O T O 5 0 D O 3 9 I T E R = 1 . 1 0 N V I E W = 2 C A L L V I E W ( N V I E W ) C A L L A D D D I S ( X , Y , Z . D X . D Y . D Z . R X , R Y , R Z , N U M P , - 1 ) C A L L J F R A M E C A L L J O P E N C A L L P L T E L E ( X . Y , Z . N U M E G . 3 ) C A L L P L T H E D ( H E D ) C A L L J C L O S E C A L L J P A U S E ( N D E V ) C A L L A D D D I S ( X , Y , Z . D X . D Y . D Z , R X , R Y , R Z . N U M P , 1 ) C A L L J O P E N C A L L F I L L E L ( R X . R Y , R Z , N U M E G , 1 ) C A L L F I L L E L ( R X , R Y , R Z , N U M E G , 2 ) C A L L P L T E L E ( R X , R Y . R Z , N U M E G , 2 ) C A L L J C L O S E W R I T E ( N O U T , 1 1 0 0 ) R E A D ( 5 , ' ( A 1 ) ' ) E O F I F ( E O F . E C ' S ' . O R . E O F . E C ' s ' ) G O T O 5 0 c C I F D 3 S T R A N D D 3 P L T A R E T R U E P L O T T H E D E F L E C T E D S H A P E W I T H C A S T R E S S C O L O R F I L L 3 9 5 0 I F ( D 3 S T R ) T H E N I F ( E O F . E C ' C . O R . E O F . E C ' C ) T H E N C A L L J F R A M E C A L L J O P E N C A L L P L T E L E ( X , Y , Z . N U M E G , 3 ) C A L L E L E S T R ( R X , R Y , R Z , S T R E S S , S T R M A X , N U M E G , 1 ) C A L L E L E S T R ( R X , R Y , R Z , S T R E S S , S T R M A X , N U M E G . 2 ) C A L L P L T E L E ( R X , R Y . R Z , N U M E G , 2 ) C A L L P L T H E D ( H E D ) C A L L J C L O S E C A L L L E G E N D ( S T R M A X ) W R I T E ( N O U T , 1 1 0 0 ) R E A D ( 5 , ' ( A 1 ) •  ) E O F I F ( E O F . E O . ' S ' . O R . E N D I F E N D I F C O N T I N U E C O N T I N U E E N D I F E O F . E O . ' S ' ) G O T O 5 0 C L O S E P L O T R O U T I N E I F ( M E T S T ) T H E N C A L L J D E V O F ( M D E V ) C A L L J D E N D ( M D E V ) E N D I F C A L L J D E V O F ( N D E V ) C A L L J D E N D ( N D E V ) C A L L J E N D C L O S E ( U N I T = N P G E O ) C L O S E ( U N I T = I S C R ) 1 1 0 0 F O R M A T ( / , ' < R E T U R N > T O C O N T I N U E , S < R E T U R N > T O S T O P . ' ) S T O P E N D Q ******************************* ************ ********************************* C S U B R O U T I N E V I E W c **************************************************************************+* S U B R O U T I N E V I E W ( N V I E W ) I M P L I C I T R E A L * 4 ( A - H . O - Z ) C O M M O N / M A X / R M I N ( 3 ) , R A V E ( 3 ) , R M A X ( 3 ) , R A T I O C O M M O N / S V I E W / D ( 3 ) . U ( 3 ) I N T E G E R N V I E W C A L L J R I G H T ( . T R U E . ) C A L L J V U P N T ( R A V E ( 1 ) , R A V E ( 2 ) . R A V E ( 3 ) ) 99 NISPLOT L i s t i n g I F ( N V I E W . E O . 1 ) T H E N D( 1 ) = - 3 . 0 D ( 2 ) = - 3 . 0 D ( 3 ) = -1 . 0 U( 1 ) = - 1 . 0 U ( 2 ) = - 1 . 0 U ( 3 ) = 4 . 0 UMIN = RMIN( 1 ) - R A V E ( 1 ) UMAX = RMAX( 1 ) - R A V E ( 1 ) V M I N = R M I N ( 2 ) - R A V E ( 2 ) V M A X = R M A X ( 2 ) - R A V E ( 2 ) C A L L J N O R M L ( 0 . 0 . 0 . 0 . - 1 . 0 ) C A L L J U P V E C ( 0 . 0 , 1 . 0 . 0 . 0 ) C A L L JWINDO ( U M I N , U M A X , V M I N . V M A X ) C A L L J P E R S P ( - 1 0 . 0 ) E L S E IF ( N V I E W . E O . O ) T H E N UMIN = RMIN( 1 ) - R A V E ( 1 ) * 0 . 7 UMAX=RMAX( 1 ) - R A V E ( 1 ) * 0 . 7 V M I N = R M I N ( 2 ) - R A V E ( 2 ) * 0 . 7 V M A X = R M A X ( 2 ) - R A V E ( 2 ) * 0 . 7 C A L L J N O R M L ( 0 . 0 , 0 . 0 . - 1 . 0 ) C A L L J U P V E C ( 1 . 0 , 1 . 0 , 0 . 0 ) C A L L JWINDO ( U M I N , U M A X , V M I N , V M A X ) C A L L J P E R S P ( - 1 0 . 0 ) E L S E I F ( N V I E W . E O . 2 ) T H E N D U M = R M A X ( 1 ) - R M I N ( 1 ) U M I N = - 0 . 6 5 * D U M UMAX= 0 . 6 5 * D U M V M I N = - 0 . 6 5 * D U M VMAX= O . G 5 * D U M D I S T = ( R M A X ( 1 ) - R M I N ( 1 ) ) * 0 . 9 0 c 100 C O N T I N U E WRITE ( G . 1 0 1 0 ) ( D ( I ) , I = 1 , 3 ) READ ( 5 , 1 0 0 0 , E R R = 1 0 0 ) B X . B Y . B Z I F ( B X . E O . 0 . 0 . A N D . B Y . E O . 0 . 0 . A N D . B Z . E O . 0 . 0 ) T H E N E L S E D ( 1 ) = BX D ( 2 ) = BY D ( 3 ) = BZ 110 C O N T I N U E WRITE ( 6 , 1 0 2 0 , E R R = 1 1 0 ) ( U ( I ) , I = 1 , 3 ) READ ( 5 , 1 0 0 0 , E R R = 1 1 0 ) B X . B Y . B Z IF ( B X . E O . 0 . 0 . A N D . B Y . E O . 0 . 0 . A N D . B Z . E O . 0 . 0 ) T H E N E L S E U ( 1 ) = B X U ( 2 ) = B Y U ( 3 ) = B Z END IF END IF 1 0 0 0 FORMAT ( 3 G 1 2 . 6 ) 1 0 1 0 FORMAT (/ , ' NORMAL V E C T O R X . Y . Z ? ' , 3 F 1 0 . 3 ) 1 0 2 0 FORMAT ( ' UP V E C T O R X . Y . Z ? ' , 3 F 1 0 . 3 ) c C A L L JNORML (D (1 ) , D ( 2 ) , D ( 3 ) ) C A L L J U P V E C (U (1 ) , U ( 2 ) , U ( 3 ) ) C A L L JWINDO ( U M I N , U M A X , V M I N , V M A X ) C A L L J V U P L N ( D I S T ) C A L L J P E R S P ( D I S T * - 3 . 0 ) END I F C A L L J W C L I P ( . T R U E . ) R ETURN END c **************************** C S U B R O U T I N E S E T U P Q ***************************************************************************** S U B R O U T I N E S E T U P ( N P G E O . N P D I S , N P S T R , N P L A T E , I S C R , 1 D 3 S T R . D 3 P L T . M E T S T ) I M P L I C I T R E A L * 4 ( A - H . O - Z ) C H A R A C T E R CHAR L O G I C A L D 3 S T R , D 3 P L T , M E T S T , S T A T C H A R A C T E R * 2 0 I N F I L E 100 NISPLOT L i s t i n g c c c 1 0 0 1 10 1 2 0 GEOMETRY AND S C R A T C H F I L E W R I T E ( 6 , 2 0 0 0 ) C A L L I F I L E ( I N F I L E , S T A T ) IF ( . N O T . S T A T ) GOTO 100 OPEN ( U N I T = I S C R , F I L E = ' f o r 0 0 7 . d a t ' , S T A T U S = ' s c r a t c h ' ) OPEN ( U N I T = N P G E O , F I L E = I N F I L E , S T A T U S = ' o l d ' ) I F ( I N F I L E . E O . ' 2 f . g e o ' ) THEN OPEN ( U N I T = N P L A T E , F I L E = ' 2 f . p l t ' , S T A T U S = ' o l d ' ) REWIND N P L A T E E L S E N P L A T E = 0 END IF REWIND ISCR REWIND NPGEO D I S P L A C E M E N T F I L E C O N T I N U E W R I T E ( 6 , 2 0 1 0 ) READ ( 5 , ' ( A 1 ) ' , E R R = 1 1 0 ) CHAR IF ( C H A R . E O . ' y ' D 3 P L T = . T R U E . W R I T E ( 6 , 2 0 2 0 ) C A L L I F I L E ( I N F I L E , S T A T ) IF ( . N O T . S T A T ) GOTO 120 OPEN ( U N I T = N P D I S . F I L E = I N F I L E . S T A T U S = ' o 1 d ' ) REWIND NPD I S E L S E D 3 P L T = . F A L S E . END IF . O R . C H A R . E O . ' Y ' ) T H E N C C C 1 3 0 140 S T R E S S F I L E C O N T I N U E W R I T E ( 6 , 2 0 3 0 ) READ ( 5 , ' ( A 1 ) ' , E R R = 1 3 0 ) CHAR IF ( C H A R . E Q . ' y ' . O R . C H A R . E O . ' Y ' ) T H E N D 3 S T R = . T R U E . W R I T E ( 6 , 2 0 4 0 ) C A L L I F I L E ( I N F I L E , S T A T ) IF ( . N O T . S T A T ) GOTO 140 OPEN ( U N I T = N P S T R , F I L E = I N F I L E , S T A T U S = ' o l d ' ) REWIND NPSTR E L S E D 3 S T R = . F A L S E . END IF C C M E T A F I L E C WRITE ( 6 , 2 0 5 0 ) READ ( 5 , ' (A1 ) ' ) CHAR IF ( C H A R . E O . ' Y ' . O R . C H A R . E Q . ' y ' ) T H E N M E T S T = . T R U E . E L S E M E T S T = . F A L S E . END IF 2 0 0 0 FORMAT ( ' G E O M E T R I C INPUT F I L E N A M E ? ' . $ ) 2 0 1 0 FORMAT ( / / . ' D O YOU HAVE A D I S P L A C E M E N T F I L E Y / N ? ' . $ ) 2 0 2 0 FORMAT ( ' D I S P L A C E M E N T INPUT F I L E NAME? ' . $ ) 2 0 3 0 FORMAT ( / / , ' D O YOU HAVE A S T R E S S F I L E Y / N ? ' . $ ) 2 0 4 0 FORMAT ( ' S T R E S S INPUT F I L E NAME? ' , $ ) 2 0 5 0 FORMAT ( / / , ' D O YOU WANT TO C R E A T E A M E T A F I L E Y/N? ' . $ ) R E T U R N END Q *************** ******************************************* C S U B R O U T I N E I F I L E Q * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * S S U B R O U T I N E I F I L E ( I N F I L E . S T A T ) C H A R A C T E R * 2 0 I N F I L E L O G I C A L S T A T * * * * * * * * * * * * * * * * * * 1 0 1 NISPLOT L i s t i n g R E A D ( 5 , ' ( A 2 0 ) ' ) I N F I L E I N Q U I R E ( F I L E = I N F I L E , E X I S T = S T A T ) IF ( S T A T . E O . . F A L S E . ) THEN W R I T E ( 6 , * ) ' * * * * E R R O R * * * * W R I T E ( 6 , * ) ' F I L E DOES NOT E X I S T ' W R I T E ( 6 , * ) ' TRY A G A I N ' END I F RETURN END c * * * * * * * * * * * * * * * * * * * * * * * * * * * * * , * * * * , * * « * , » * * * * * * * * * * * * * * * * * * * * * * * « * C S U B R O U T I N E R E A D N O Q ******************************************************************* S U B R O U T I N E READNO ( X , Y , Z , H E D , N U M P , N U M E G , I C O R D , N P L O T , I S C R , I STOP ) I M P L I C I T R E A L M ( A - H . O - Z ) I N T E G E R N U M P , N U M E G , I C O R D , N P L O T . I S C R C H A R A C T E R * 4 5 HED D I M E N S I O N X( 1 ) , Y( 1 ) , Z ( 1 ) R E A D ( N P L O T , 2 0 0 0 , E R R = 2 4 0 , I O S T A T = I S T O P ) HED W R I T E ( I S C R , 2 0 0 0 ) HED IF ( I C O R D . EQ. .1 ) T H E N R E A D ( N P L O T . 2 0 1 0 ) NUMP,NUMEG W R I T E ( I S C R , 2 0 1 0 ) NUMP ,NUMEG END I F DO 2 2 0 1 = 1 , NUMP R E A D ( N P L O T , 2 0 2 0 , E R R = 2 4 0 , I O S T A T = I S T O P ) X ( I ) , Y ( I ) , Z ( I ) W R I T E ( I S C R , 2 0 2 0 ) X( I ) , Y ( I ) , Z ( I ) 2 2 0 C O N T I N U E 2 4 0 C O N T I N U E 2 0 0 0 FORMAT ( A 4 5 ) 2 0 1 0 FORMAT ( 2 1 5 ) 2 0 2 0 FORMAT ( 1 P . 3 E 1 5 . 6 ) R E T U R N END c * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * C S U B R O U T I N E R E A D E L c * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * S U B R O U T I N E R E A D E L ( N U M E G . N P L O T , I SCR ) I M P L I C I T R E A L * 4 ( A - H , 0 - Z ) COMMON / P L T / I N 0 D E ( 1 0 0 , 1 3 . 5 ) , I E L ( 1 0 0 , 5 ) , N ( 1 0 0 , 5 ) , N M A X ( 5 ) I N T E G E R N U M E G , N P L O T , I S C R c C READ NODES OF E L E M E N T AND S T O R E IN I N O D E ( 3 0 0 , 1 3 , 5 ) C FOR ELEMNT GROUP NUM. c DO 4 2 0 NUM=1,NUMEG R E A D ( N P L O T , 2 0 0 0 ) N M A X ( N U M ) , N U M E L W R I T E ( I S C R , 2 0 0 0 ) N M A X ( N U M ) , N U M E L DO 4 1 0 LOOP = 1 ,NMAX (NUM) R E A D ( N P L O T , 2 0 1 0 ) I E L ( L O O P , N U M ) , N ( L O O P , N U M ) , 1 ( I N O D E ( L O O P , J , N U M ) , J = 1 , N ( L O O P , N U M ) ) W R I T E ( I S C R , 2 0 1 0 ) I E L ( L O O P , N U M ) , N ( L O O P , N U M ) , 1 ( I N O D E ( L O O P , J , N U M ) , J = 1 , N ( L O O P , N U M ) ) 4 1 0 C O N T I N U E 4 2 0 C O N T I N U E WR ITE ( I S C R , 2 0 2 0 ) 2 0 0 0 FORMAT ( 2 1 5 ) 2 0 1 0 FORMAT (21 5 , 2 X , 13( 1 1 5 , : ) ) 2 0 2 0 FORMAT ( ' * * C O M P L E T E D R E A D I N G IN NODE AND E L E M . D A T A . * * ' . / ) R E T U R N END c * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * C S U B R O U T I N E R E A D S T C * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * S U B R O U T I N E R E A D S T ( S T R E S S , N M A X , S T R M A X , N U M E G , N P L O T . I S C R ) C H A R A C T E R * 4 5 SHED D I M E N S I O N S T R E S S ( 1 6 , 3 0 , 5 ) , N M A X ( 5 ) , S T R M A X ( 2 ) S T R M A X ( 1 ) = 1 0 0 0 0 . 0 S T R M A X ( 2 ) = - 1 0 . 0 READ ( N P L O T . 1 0 2 0 ) S H E D DO 7 2 0 NUM=1,NUMEG DO 7 1 0 I E L = 1 , N M A X ( N U M ) / 5 102 NISPLOT L i s t i n g READ ( N P L O T , 1 0 0 0 ) ( S T R E S S ( I , I E L , N U M ) , I = 1 , 16) WRITE ( I S C R . 1 0 0 0 ) ( S T R E S S ( I , I E L . N U M ) , I = 1, 1G) DO 7 0 0 1 = 1 , 1G IF ( S T R E S S ( I . I E L , N U M ) . L T . S T R M A X ( 1 ) ) 1 S T R M A X f 1 ) = S T R E S S ( I , I E L . N U M ) IF ( S T R E S S ( I . I E L , N U M ) . G T . S T R M A X ( 2 ) ) 1 S T R M A X ( 2 ) = S T R E S S ( I . I E L . N U M ) 7 0 0 C O N T I N U E 7 1 0 C O N T I N U E 7 2 0 C O N T I N U E IF ( ( S T R M A X ( 2 ) - S T R M A X ( 1 ) ) . L T . 0 . 0 0 1 ) S T R M A X ( 2 ) = S T R M A X ( 2 ) + 1 . 0 1 0 0 0 FORMAT ( 4 ( 4 ( 2 X , 1PE 1 2 . 5 ) / ) ) 1 0 2 0 FORMAT ( 2 X . A 4 5 ) RETURN END C S U B R O U T I N E M A X M I N SUBROUT INE MAXMIN ( X , Y , Z , N U M P ) I M P L I C I T R E A L * 4 ( A - H . O - Z ) COMMON / MAX / R M I N ( 3 ) , R A V E ( 3 ) , R M A X ( 3 ) , R A T I O D I M E N S I O N X ( 5 0 0 ) , Y ( 5 0 0 ) , Z ( 5 0 0 ) , R ( 3 ) DATA R / 1 . 0 , 1 . 0 , 1 . 0 / R M I N ( 1 ) = X ( 1 ) R M I N ( 2 ) = Y ( 1 ) R M I N ( 3 ) = Z ( 1 ) RMAX(1 )=X( 1 ) R M A X ( 2 ) = Y ( 1 ) R M A X ( 3 ) = Z( 1 ) DO 5 1 0 I = 2 , N U M P IF ( X ( I ) . L T . R M I N ( 1 ) ) R M I N ( 1 ) = X ( I I F ( X ( I ) . G T .RMAX( 1 ) ) R M A X ( 1 ) = X ( I I F ( Y ( I ) . L T . R M I N ( 2 ) ) R M I N ( 2 ) = Y ( I I F ( Y ( I ) . G T . R M A X ( 2 ) ) R M A X ( 2 ) = Y ( I IF ( Z ( I ) . L T . R M I N ( 3 ) ) R M I N ( 3 ) = Z ( I IF ( Z ( I ) . G T . R M A X ( 3 ) ) R M A X ( 3 ) = Z ( I C O N T I N U E D E L T X = RMAX( 1 ) -RMIN(1 ) D E L T Y =RMAX(2 ) - R M I N ( 2 ) D E L T Z =RMAX(3 ) - R M I N ( 3 ) I F ( ( D E L T Y / R A T I O ) . G T . D E L T X ) T H E N D E L T = D E L T Y R ( 1 ) = 1 . 0 / R A T I O E L S E D E L T = D E L T X R ( 2 ) = R A T I 0 END IF DO 5 2 0 1 = 1 , 3 R A V E ( I ) = ( R M A X ( I ) + RM IN ( I ) )/2 . 0 R M A X ( I ) = R A V E ( I ) + R ( I ) * D E L T * 0 . 5 9 R M I N ( I ) = R A V E ( I ) - R ( I ) * D E L T * 0 . 5 9 5 2 0 C O N T I N U E RETURN END C A * * * * * * * * * * * * * * . * . * * * * * * * * * * * * * * * * * * * * . * * * * * * * * . * * C S U B R O U T I N E A D D D I S S U B R O U T I N E ADDDIS ( X , Y , Z , D X . D Y , D Z , R X . R Y , RZ , N U M P . N C A S E ) I M P L I C I T R E A L * 4 ( A - H . O - Z ) COMMON / MAX / R M I N ( 3 ) , R A V E ( 3 ) , R M A X ( 3 ) , R A T I O D I M E N S I O N X ( 5 0 0 ) , Y ( 5 0 0 ) , Z ( 5 0 0 ) , NCASE = -2 NO S C A L I N G IS D O N E . NCASE = -1 ADD ONLY C O N S T . TO Z D I S P L A C E M E N T S . N C A S E = 0 S C A L E THE Z - D I S P L A C E M E N T ( D Z ) AND ADD TO O R I G I N A L C O O D I N A T E S . NCASE = 1 DO BOTH THE A B O V E . 1 D X ( 5 0 0 ) , D Y ( 5 0 0 ) , D Z ( 5 0 0 ) , R X ( 5 0 0 ) , R Y ( 5 0 0 ) , R Z ( 5 0 0 ) CONST = N C A S E * 0 . 0 6 5 * (RMAX( 1 ) -RMIN ( 1 ) ) I F ( N C A S E . E O . - 1 ) T H E N C 0 N S T = 1 6 . 0 * CONST D M U L T = 0 . 0 E L S E I F ( N C A S E . E Q . - 2 ) T H E N 1 0 3 c c c c c c NISPLOT L i s t i n g 6 0 0 C O N S T = 0 . 0 D M U L T = 0 . 0 E L S E S E A R C H THROUGH THE Z D I S P L A C E M E N T S AND THE S C A L E THEM SO THAT THEY HAVE A P P R O X . 2 0 % S C R E E N WINDOW. R M A X ( 3 ) = D Z ( 1 ) R M I N ( 3 ) = D Z ( 1 ) DO 6 0 0 I = 2 , N U M P IF ( D Z ( I ) . G T . R M A X ( 3 ) ) RMAX (3 ) = DZ ( I ) I F ( D Z ( I ) . L T , R M I N ( 3 ) ) R M I N ( 3 ) = D Z ( I ) C O N T I N U E IF ( R M A X ( 3 ) . L E , R M I N ( 3 ) ) T H E N DMULT=1 . 0 E L S E DMULT= 0 . 2 5 * ( R M A X ( 1 ) - R M I N ( 1 ) ) / ( R M A X ( 3 ) - R M I N ( 3 ) ) END IF END IF DO 6 1 0 1 = 1 ,NUMP R X ( I ) = X ( I ) + D X ( I ) R Y ( I ) = Y ( I ) + D Y ( I ) R Z ( I ) = Z ( I ) + D Z ( I ) * D M U L T + CONST 6 1 0 C O N T I N U E R E T U R N END Q *********************** C S U B R O U T I N E P L T H E D C * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * S U B R O U T I N E P L T H E D ( H E D ) I M P L I C I T R E A L * 4 ( A - H . O - Z ) ' COMMON / MAX / R M I N ( 3 ) , R A V E ( 3 ) , R M A X ( 3 ) , R A T I O COMMON / HEAD / PHEAD C H A R A C T E R * 4 5 H E D . NEWHED C H A R A C T E R * 1 F L A G L O G I C A L PHEAD D I M E N S I O N W X ( 4 ) , W Y ( 4 ) , W Z ( 4 ) c C L O C A T E THE V I E W P L A N E AND THE S E T THE T E X T A T T R I B U T E S C SO THE H E A D I N G WILL A P P E A R AT THE TOP OF THE P A G E . I F ( P H E A D . E O . I F ( R A T I O . L E . C A L L JCONVW JCONVW JCONVW JCONVW JCONVW JCONVW JCONVW JCONVW . F A L S E . ) R ETURN 1 .0 ) T H E N (-1 . 0 , R A T 1 0 , W X ( 1 ) , W Y ( 1 ) ,WZ( 1 ) ) ( O . O . R A T I 0 , W X ( 2 ) , W Y ( 2 ) , W Z ( 2 ) ) ( 0 . 0 , 0 . 0 , W X ( 3 ) , W Y ( 3 ) , W Z ( 3 ) ) ( 1 . 0 , - R A T I 0 , W X ( 4 ) , W Y ( 4 ) , W Z ( 4 ) ) (- 1 , 0 / R A T I O , 1 . 0 , W X ( 1 ) , W Y ( 1 ) , W Z ( 1 ) ) ( 0 . 0 , 1 . 0 , W X ( 2 ) , W Y ( 2 ) , W Z ( 2 ) ) ( 0 . 0 , 0 . 0 , W X ( 3 ) , W Y ( 3 ) , W Z ( 3 ) ) ( 1 , 0 / R A T I O , - 1 . 0 , W X ( 4 ) , W Y ( 4 ) . W Z ( 4 ) ) C A L L C A L L C A L L E L S E C A L L C A L L C A L L C A L L END IF C X B A S E = W X ( 2 ) - W X ( 1 ) C Y B A S E = W Y ( 2 ) - W Y ( 1 ) C Z B A S E = W Z ( 2 ) - W Z ( 1 ) C X P L A N = W X ( 1 ) - W X ( 3 ) C Y P L A N = W Y ( 1 ) - W Y ( 3 ) C Z P L A N = WZ( 1 ) -WZ (3 ) C X S I Z E = 0 . 0 6 5 * R A T I 0 * S Q R T ( C X B A S E * C X B A S E + C Y B A S E * C Y B A S E + C Z B A S E * C Z B A S E ) C Y S I Z E = 0 . 0 5 5 * S Q R T ( C X P L A N * C X P L A N + C Y P L A N * C Y P L A N + C Z P L A N * C Z P L A N ) C A L L J U P D A T WR ITE ( 6 , 2 0 0 0 ) 0 READ ( 5 , 1 0 0 0 ) F L A G I F ( F L A G . E O . ' Y ' .OR WRITE ( 6 , 2 0 0 1 ) . READ ( 5 , 1 0 0 1 ) NEWHED END IF C A L L J B A S E ( C X B A S E . C Y B A S E , C A L L J P L A N E ( C X P L A N , C Y P L A N , C A L L J S I Z E ( C X S I Z E . C Y S I Z E ) C A L L J C O L O R ( 0 ) C A L L J J U S T ( 2 , 3 ) F L A G . E O . ' y ' ) T H E N C Z B A S E ) C Z P L A N ) 1 0 4 NISPLOT L i s t i n g C A L L J3MOVE ( W X ( 2 ) , W Y ( 2 ) . W Z ( 2 ) ) C A L L J F O N T ( 1 8 ) C A L L J F A T T R ( 1 , 1 . 0 . 1 . 3 , 1 6 3 8 3 ) I F ( F L A G . E O . ' Y ' . O R . F L A G . E O . ' y ' ) THEN C A L L J F S T R G (NEWHED) E L S E C A L L J F S T R G ( H E D ) END IF C C WRITE F O O T N O T E AT BOTTOM OF PAGE C C X S I Z E = C X S I Z E / 2 . 0 C Y S I Z E = C Y S I Z E / 2 . 0 C A L L J L W I D E ( S O O O ) C A L L d d U S T ( 3 , 1 ) C A L L JCOLOR ( 2 ) C A L L J S I Z E ( C X S I Z E . C Y S I Z E ) C A L L J 3 M 0 V E ( W X ( 4 ) , W Y ( 4 ) , W Z ( 4 ) ) C A L L J F O N T ( 1 ) C A L L J 3 S T R G ( ' U . B. C . C I V I L E N G I N E E R I N G ' ) 1 0 0 0 FORMAT (A1 ) 1001 FORMAT ( A 4 5 ) 2 0 0 0 FORMAT ( / , ' Do y o u wan t a new t i t l e ? y / n ' ) 2001 FORMAT (/ , ' E n t e r new T i t l e ' , / , 1 ' B 1 2 3 4 E ' ) R ETURN END c * * * * * » * * * * * * * * * * * * * * * * * * * * * * * * * * * * ^ C S U B R O U T I N E P L T N O D C »*****«».»*•.******•«*..««**««*«****»**.»*«.*************•**.*.,*•* S U B R O U T I N E P LTNOD ( X . Y . Z . N U M P ) I M P L I C I T R E A L * 4 ( A - H . O - Z ) C H A R A C T E R * 1 YES COMMON / MAX / R M I N ( 3 ) , R A V E ( 3 ) . R M A X ( 3 ) , R A T I O I N T E G E R NUMP D I M E N S I O N X ( 5 0 0 ) , Y ( 5 0 0 ) , Z ( 5 0 0 ) C A L L JCMARK ( 2 ) C A L L JCOLOR ( 0 ) DO 7 2 0 1 = 1 , N U M P C A L L J3MARK ( X ( I ) . Y ( I ) . Z ( I ) ) 7 2 0 C O N T I N U E c C NUMBER THE NODE P O I N T S . c C A L L J U P D A T W R I T E ( 6 , 2 0 0 0 ) READ ( 5 , 1 0 0 0 ) YES I F ( Y E S . E O . ' Y ' . O R . Y E S . E O . ' y ' ) C A L L NODNUM ( X . Y . Z . N U M P ) 1 0 0 0 FORMAT ( A 1 ) 2 0 0 0 FORMAT ( / , ' NODE NUMBER ING? y / n ' , $ ) RETURN END c **************************************************************** C S U B R O U T I N E P L T E L E c **************************************************************** S U B R O U T I N E P L T E L E ( X , Y , Z . N U M E G , N W R I T E ) I M P L I C I T R E A L * 4 ( A - H . O - Z ) COMMON / P L T / I N O D E ( 1 0 0 , 1 3 , 5 ) . I E L ( 1 0 0 . 5 ) . N ( 1 0 0 . 5 ) , N M A X ( 5 ) COMMON / MAX / R M I N ( 3 ) , R A V E ( 3 ) , R M A X ( 3 ) , R A T I O INTEGER NUMEG D I M E N S I O N X ( 5 0 0 ) . Y ( 5 0 0 ) , Z ( 5 0 0 ) c C C A L L E L E M E N T P L O T S U B R O U T I N E TO C PLOT O U T S I D E OF THE ELEMENT IN S O L I D L I N E , C AND P L O T ANY I N T E R N A L L I N E S WITH DASHED L I N E S . C LOOP OVER E L E M E N T G R O U P S . c ICOLOR = 1 I F (NWRITE . E O . 3 ) I C 0 L 0 R = 2 DO 8 1 0 NUM=1,NUMEG IF (NWRITE . E O . 1) I C0L0R=NUM IF ( I C O L O R G E . 3 ) I COLOR = ICOLOR+1 C A L L J C O L O R ( I C O L O R ) 105 NISPLOT L i s t i n g I E L O = 0 OO 8 0 0 K = 1 , N M A X ( N U M ) IF ( I E L O . E O . I E L ( K . N U M ) ) T H E N I S T Y L = 3 E L S E I S T Y L = 0 I E L O = I E L ( K , N U M ) END IF IF ( N W R I T E . E 0 . 1 . O R . N W R I T E . E 0 . 2 . O R . I S T Y L . E Q . O ) THEN C A L L J L S T Y L ( I S T Y L ) C A L L ORAY ( X . Y , Z , K , N U M ) END IF 8 0 0 C O N T I N U E 8 1 0 C O N T I N U E c C WRITE E L E M E N T N O . IN THE M IDLE OF THE E L E M E N T . c C A L L E LENUM (X , Y , Z , N U M E G , N W R I T E ) RETURN END Q ****************************************^ C S U B R O U T I N E D R A Y Q **************************** S U B R O U T I N E DRAY ( X , Y , Z , K , N U M ) I M P L I C I T R E A L * 4 ( A - H . O - Z ) COMMON / P L T / I N O D E ( 1 0 0 . 1 3 , 5 ) , I E L ( 1 0 0 , 5 ) . N ( 1 0 0 , 5 ) , N M A X ( 5 ) I N T E G E R N R A Y . K . N U M D I M E N S I O N X ( 5 0 0 ) , Y ( 5 0 0 ) , Z ( 5 0 0 ) , 1 X A R R A Y ( 1 2 ) , Y A R R A Y ( 1 2 ) , Z A R R A Y ( 1 2 ) N R A Y = N ( K , N U M ) - 1 DO 3 0 0 J = 2 , N ( K , N U M ) X A R R A Y ( J - 1 ) = X ( I N O D E ( K , J . N U M ) ) Y A R R A Y ( J - 1 ) = Y ( I N 0 D E ( K . J . N U M ) ) Z A R R A Y ( J - I ) = Z ( I N O D E ( K , J , N U M ) ) 3 0 0 C O N T I N U E C A L L J 3 M 0 V E ( X ( I N 0 D E ( K , 1 . N U M ) ) , Y ( I N O D E ( K , 1.NUM) ), 1 Z ( I N O D E ( K , 1 . N U M ) ) ) C A L L J 3 P 0 L Y ( X A R R A Y , Y A R R A Y , Z A R R A Y , N R A Y ) RETURN END Q ***************************************************************************** C S U B R O U T I N E N O D N U M Q ***************************************************************************** S U B R O U T I N E NODNUM ( X . Y . Z , N U M P ) I M P L I C I T R E A L * 4 ( A - H , 0 - Z ) COMMON / MAX / R M I N ( 3 ) , R A V E ( 3 ) , R M A X ( 3 ) , R A T I O C H A R A C T E R * 3 CHAR I N T E G E R NUMP D I M E N S I O N X ( 5 0 0 ) , Y ( 5 0 0 ) , Z ( 5 0 0 ) , W X ( 2 ) , W Y ( 2 ) , WZ( 2 ) I F ( N U M P . G E . 2 0 0 ) RETURN C C A L L JCONVW ( 0 . 0 . 0 . 0 , W X ( 1 ) , W Y ( 1 ) , W Z ( 1 ) ) C A L L JCONVW ( 0 . 5 , 0 . 0 . W X ( 2 ) , W Y ( 2 ) , W Z ( 2 ) ) C X B A S E = W X ( 2 ) - W X ( 1 ) C Y B A S E = W Y ( 2 ) - W Y ( 1 ) C Z B A S E = W Z ( 2 ) - W Z ( 1 ) C A L L J B A S E ( C X B A S E . C Y B A S E . C Z B A S E ) C A L L J P A T H ( 1 ) X S I Z E = 0 . 0 1 3 * ( R M A X ( 1 ) - R M I N ( 1 ) ) Y S I Z E = 0 . 0 1 3 * ( R M A X ( 2 ) - R M I N ( 2 ) ) C A L L J S I Z E ( X S I Z E , Y S I Z E ) C A L L J C O L O R ( 1 ) C A L L J J U S T ( 3 , 3 ) DO 7 0 0 1 = 1 , N U M P W R I T E ( C H A R , ' ( 13 ) ' ) I C A L L J 3 M 0 V E ( X ( I ) , Y ( I ) , Z ( I ) ) C A L L J 3 S T R G ( C H A R ) 7 0 0 C O N T I N U E R E T U R N END 106 NISPLOT L i s t i n g C S U B R O U T I N E E L E N U M c «««**«**.«.*****.**.*.****^^ S U B R O U T I N E ELENUM ( X , Y , Z . N U M E G , N W R I T E ) I M P L I C I T R E A L * 4 ( A - H . O - Z ) COMMON / P L T / I N O D E ( 1 0 0 , 1 3 , 5 ) , I E L ( 1 0 0 , 5 ) , N( 1 0 0 . 5 ) , N M A X ( 5 ) COMMON / MAX / R M I N ( 3 ) , R A V E ( 3 ) , R M A X ( 3 ) , R A T I O C H A R A C T E R * 3 CHAR D I M E N S I O N X ( 5 0 0 ) . Y ( 5 0 0 ) , Z ( 5 0 0 ) . W X ( 3 ) , W Y ( 3 ) , W Z ( 3 ) I F ( N W R I T E . E O . 2 . O R . N W R I T E . E O . O ) T H E N RETURN END I F C C A L L JCONVW ( 0 . 0 , 0 . 0 , W X ( 1 ) , W Y ( 1 ) ,WZ( 1 ) ) C A L L JCONVW ( 0 . 5 , 0 . 0 , W X ( 2 ) , W Y ( 2 ) , W Z ( 2 ) ) C A L L JCONVW ( 0 . 0 , 0 . 5 , W X ( 3 ) , W Y ( 3 ) , W Z ( 3 ) ) C X B A S E = W X ( 2 ) - W X ( 1 ) C Y B A S E = W Y ( 2 ) - W Y ( 1 ) C Z B A S E = W Z ( 2 ) - W Z ( 1 ) C X P L A N = W X ( 3 ) - W X ( 1 ) C Y P L A N = W Y ( 3 ) - W Y ( 1 ) I F ( C X P L A N . E O . O . A N D . C Y P L A N . E O . O ) RETURN C C Z P L A N = 0 . 0 C A L L J B A S E ( C X B A S E . C Y B A S E . C Z B A S E ) C A L L J P L A N E ( C X P L A N , C Y P L A N , C Z P L A N ) C A L L J P A T H ( 1 ) C A L L J J U S T ( 2 , 2 ) X S I Z E = 0 . 0 2 2 * ( R M A X ( 1 ) - R M I N ( 1 ) ) Y S I Z E = 0 . 0 2 2 * ( R M A X ( 2 ) - R M I N ( 2 ) ) C A L L J S I Z E ( X S I Z E , Y S I Z E ) I C 0 L 0 R = 2 DO 8 4 0 NUM=1,NUMEG IF (NWRITE , E 0 . 1) ICOLOR=NUM IF ( I C O L O R . G E . 3 ) I COLOR= ICOLOR+1 C A L L J C O L O R ( I C O L O R ) I E L O = 0 DO 8 3 0 K = 1 , N M A X ( N U M ) IF ( I E L O . N E . I E L ( K , N U M ) ) T H E N I E L O = I E L ( K , N U M ) R E L X = 0 . 0 R E L Y = 0 . 0 R E L Z = 0 . 0 N N = N ( K , N U M ) - 1 DO 8 2 0 1 = 1 ,NN R E L X = R E L X + X ( I N O D E ( K , I . N U M ) ) R E L Y = R E L Y + Y ( I N O D E ( K , I , N U M ) ) R E L Z = R E L Z + Z ( I N O D E ( K , I , N U M ) ) 8 2 0 C O N T I N U E R E L X = R E L X / N N R E L Y = R E L Y / N N R E L Z = R E L Z / N N W R I T E ( C H A R . ' ( 1 3 ) ' ) I E L ( K . N U M ) C A L L J 3 M 0 V E ( R E L X , R E L Y , R E L Z ) C A L L J 3 S T R G ( C H A R ) END IF 8 3 0 C O N T I N U E 8 4 0 C O N T I N U E RETURN END c »*«***»*******•****************.***»*^^ C S U B R O U T I N E F I L L E L E c * * * * * * * * * * * * * * * * * * * * * » . * * * ^ ^ S U B R O U T I N E F I L L E L ( X , Y , Z , N U M E G , N S U R F ) I M P L I C I T R E A L * 4 ( A - H . O - Z ) L O G I C A L V I S B L E . OK COMMON / P L T / INODE( 1 0 0 , 1 3 . 5 ) , I E L C 1 0 0 . 5 ) , N( 1 0 0 . 5 ) . N M A X ( 5 ) D I M E N S I O N N O D ( 9 , 4 ) . X ( 1) , Y ( 1) . Z ( 1 ). I NUMB( 16 ) . D X ( 4 ) . D Y ( 4 ) . D Z ( 4 ) DATA NOD / 1, 2 , 3 , 5, 1 6 . 1 3 . 1 1 , 1 4 , 1 5 , 1 2 , 3 , 4 . 6 . 1 5 . 1 4 . 1 4 . 1 5 . 6 , 2 1 3 , 1 6 , 5 . 1 5 , 1 4 . 1 1 , 9 . 8 , 7 , 3 1 2 . 1 3 , 1 6 , 1 6 , 1 3 . 1 2 . 1 0 . 9 . 8 / IF ( N S U R F . E O . 1 ) T H E N 107 NISPLOT L i s t i n g I C 0 L 0 R = 4 E L S E I C 0 L 0 R = 6 END IF I N T E N = 1 6 3 8 4 C A L L J P I N T R ( 1 ) C A L L J C O L O R ( I C O L O R ) C A L L J P I D E X ( I C O L O R , I N T E N ) DO 9 5 0 NUM=1,NUMEG C C C O L L E C T A L L THE NODE NUMBERS FOR ONE ELEMENT IN TO ONE S T R I N G . C DO 9 4 0 L E L = 1 . N M A X ( N U M ) , 5 DO 9 0 0 1= 1 , 1 2 I N U M B ( I ) = I N O D E ( L E L , I , N U M ) 9 0 0 C O N T I N U E DO 9 1 0 1= 2 , 3 INUMB(1 + 11) = I N O D E ( L E L + 1 , I , N U M ) I N U M B ( I + 1 3 ) = I N 0 D E ( L E L + 2 , I , N U M ) 9 1 0 C O N T I N U E C C S T A R T F I L L I N G E L E M E N T C DO 9 3 0 1= 1 , 9 DO 9 2 0 J=1 , 4 D X ( J ) = X ( I N U M B ( N O D ( I , J ) ) ) D Y ( J ) = Y ( I N U M B ( N O D ( I , J ) ) ) D Z ( J ) = Z ( I N U M B ( N O D ( I , J ) ) ) 9 2 0 C O N T I N U E OK = V I S B L E ( D X , D Y , D Z , N S U R F ) IF ( O K ) C A L L J 3 P L G N ( D X . D Y . D Z , 4 ) 9 3 0 C O N T I N U E 9 4 0 C O N T I N U E 9 5 0 C O N T I N U E C A L L J P I N T R ( 0 ) RETURN END Q ************************** C S U B R O U T I N E E L E S T R C * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *'* * * * * * * * * * S U B R O U T I N E E L E S T R ( X , Y , Z , S T R E S S , S T R M A X , N U M E G , N S U R F ) I M P L I C I T R E A L * 4 ( A - H , 0 - Z ) L O G I C A L V I S B L E , OK COMMON / P L T / I N O D E ( 1 0 0 . 1 3 , 5 ) , I E L ( 1 0 0 , 5 ) , N( 1 0 0 , 5 ) , N M A X ( 5 ) COMMON / STR / N F ( 6 , 1 6 ) , R S ( 2 , 2 5 ) , N P O I N T ( 4 , 16 ) , F A C T ( 1 6 ) , I C O L ( 7 ) D I M E N S I O N X ( 1 ) . Y ( 1 ) , Z ( 1 ) , E L X ( 4 ) , E L Y ( 4 ) . E L Z ( 4 ) D I M E N S I O N X N E W ( 2 5 ) , Y N E W ( 2 5 ) , Z N E W ( 2 5 ) D I M E N S I O N S T R E S S ( 1 6 , 3 0 , 5 ) , S T R M A X ( 2 ) C A L L D A T I N I C 0 L D = O I N T E N = 1 6 3 8 4 IF ( N S U R F . E O . 1 ) T H E N I V A L U E = 8 E L S E I V A L U E = 1 END IF C A L L J P I N T R ( I V A L U E ) C C LOOP OVER A L L E L E M . G R O U P S . AND E A C H ELEMENT IN THE GROUP . C DO 8 9 0 NUM=1.NUMEG DO 8 8 0 NE L E = 1 , N M A X ( N U M ) / 5 L E L = ( N E L E - 1 ) * 5 + 1 C C C A L L SHAPE TO G E N E R A T E A 25 NODED ELEMENT WITH THE C SAME OUTER B O U N D A R I E S C AS THE 16 NODE I S P A R A M E T R I C E L E M . C C A L L S H A P E ( X , Y , Z , X N E W , Y N E W , Z N E W , L E L , N U M ) C C F I L L THE 16 S E C T I O N S OF THE ELEMENT WITH THE COLOR A S S O C I A T E D WITH C THE S T R E S S L E V E L IN E A C H S E C T I O N . C 108 NISPLOT L i s t i n g DO 8GO I A R E A = 1 , 1 6 ICNEW=6* ( S T R E S S ( I A R E A , N E L E , N U M ) - S T R M A X ( 1 ) + 0 . 0 5 ) / 1 ( S T R M A X ( 2 ) - S T R M A X ( 1 ) ) + 1 IF ( I C N E W . N E . I C O L D ) T H E N IF ( I C N E W . G T . 7 ) ICNEW=7 IF ( I C N E W . L T . 1 ) ICNEW=1 ICOLD= ICNEW C A L L JCOLOR ( I C O L ( I C N E W ) ) C A L L J P I D E X ( I C O L ( I C N E W ) . I N T E N ) END IF C C S T A R T F I L L I N G E L E M E N T C DO 8 5 0 J= 1 .4 E L X ( J ) = X N E W ( N P O I N T ( J , I A R E A ) ) E L Y ( J ) = Y N E W ( N P O I N T ( J , I A R E A ) ) E L Z ( J ) = Z N E W ( N P O I N T ( J , I A R E A ) ) 8 5 0 C O N T I N U E OK = V I S B L E ( E L X , E L Y . E L Z , N S U R F ) I F ( O K ) C A L L J 3 P L G N ( E L X , E L Y , E L Z , 4 ) 8 6 0 C O N T I N U E C C END LOOP OVER E L E M E N T S IN GROUP , AND END LOOP OVER A L L G R O U P S . C 8 8 0 C O N T I N U E 8 9 0 C O N T I N U E C A L L J P I N T R ( 0 ) RETURN END Q *************************** C S U B R O U T I N E S H A P E Q ***************************************************************************** S U B R O U T I N E S H A P E ( X , Y , Z , XNEW,YNEW,ZNEW, L E L . N U M ) COMMON / STR / N F ( 6 , 16 ) , R S ( 2 , 2 5 ) . N P O I N T ( 4 , 1 6 ) , F A C T ( 1 6 ) , I C O L ( 7 ) COMMON / P L T / INODE( 1 0 0 , 1 3 , 5 ) . I E L ( 1 0 0 . 5 ) , N( 1 0 0 , 5 ) . N M A X ( 5 ) D I M E N S I O N X( 1 ) , Y ( 1 ) , Z ( 1) D I M E N S I O N X N E W ( 2 5 ) , Y N E W ( 2 5 ) . Z N E W ( 2 5 ) , I N U M B ( 1 6 ) D I M E N S I O N A ( 8 ) , S F ( 16 ) DO 901 1 = 1 , 2 5 X N E W ( I ) = 0 . 0 Y N E W ( I ) = 0 . O Z N E W ( l ) = 0 . 0 • 901 C O N T I N U E C C C O L L E C T A L L THE NODE NUMBERS FOR ONE ELEMENT IN TO ONE S T R I N G . C DO 9 0 0 1 = 1 , 1 2 I N U M B ( I ) = I N O D E ( L E L , I , N U M ) '• 9 0 0 C O N T I N U E DO 9 10 1 = 2 , 3 I N U M B ( 1 + 1 1 ) = I N O D E ( L E L + 1 , I , N U M ) I N U M B ( I + 1 3 ) = I N O D E ( L E L + 2 , I , N U M ) 9 1 0 C O N T I N U E C C C A L C U L A T E NEW C O O R D I N A T E S OF A 25 NODED E L E M E N T . C DO 9 4 0 I C O O R D = 1 , 2 5 C C F IND THE S H A P E F U N C T I O N S G I V E THE L O C A L C O O R D I N A T E S R AND S. C R = R S ( 1 . I C O O R D ) S = R S ( 2 , I C O O R D ) A( 1 ) = ( 1 + R) A ( 2 ) = ( 3 *R+1 ) A ( 3 ) = ( 3 * R - 1 ) A ( 4 ) = ( 1 - R ) A ( 5 ) = ( 1 + S ) A ( 6 ) = ( 3 * S + 1 ) A ( 7 ) = ( 3 * S - 1 ) A ( 8 ) = ( 1 - S ) DO 9 2 0 J = 1 , 16 S F ( J ) = A ( N F ( 1 , J ) ) * A ( N F ( 2 , J ) ) * A ( N F ( 3 . J ) ) * 1 A ( N F ( 4 , J ) ) * A ( N F ( 5 , J ) ) * A ( N F ( 6 , J ) ) * F A C T ( J ) / 2 5 6 . 0 109 NISPLOT L i s t i n g 9 2 0 C O N T I N U E DO 9 3 0 1 = 1 , 1 6 X N E W ( I C O O R D ) = X N E W ( I C O O R D ) + S F ( I ) * X ( I N U M B ( I ) ) Y N E W ( I C O O R D ) = Y N E W ( I C O O R D ) + S F ( I ) * Y ( I N U M B ( I ) ) Z N E W ( I C O O R D ) = Z N E W ( I C O O R D ) + S F ( I ) * Z ( I N U M B ( I ) ) 9 3 0 C O N T I N U E 9 4 0 C O N T I N U E R E T U R N END Q **************************************************** C S U B R O U T I N E L E G E N D Q ***************************************************************************** S U B R O U T I N E L E G E N D ( S T R M A X ) COMMON / STR / N F ( 6 , 1 6 ) , R S ( 2 , 2 5 ) , N P O I N T ( 4 , 16 ) , F A C T ( 16 ) , I C O L ( 7 ) D I M E N S I O N S T R M A X ( 2 ) , R E L X ( 4 ) , R E L Y ( 4 ) , R E L Z ( 4 ) C H A R A C T E R * 1 7 CHAR C DATA R E L X / - 0 . 0 1 , - 0 . 1 0 , 0 . 0 , 0 . 1 0 / DATA R E L Y / - 0 . 0 2 5 , 0 . 0 . 0 . 0 5 , 0 . 0 / DATA R E L Z / 0 . 0 , 0 . 0 , 0 . 0 . 0 . 0 / C A L L J R E S E T C A L L J R I G H T ( . T R U E . ) C A L L J V P O R T ( . 4 5 , 1 . 0 , - 1 . 0 , 1 . 0 ) C A L L J V U P N T ( 0 . 0 , 0 . 0 , 0 . 0 ) C A L L JNORML ( 0 . 0 , 0 . 0 , - 1 . 0 ) C A L L d U P V E C ( 0 . 0 , 1 . 0 , 0 . 0 ) C A L L dWINDO ( - . 2 7 5 , . 2 7 5 . - 1 . 0 , 1 . 0 ) C A L L d P E R S P ( - 1 . 0 ) C A L L J W C L I P ( . F A L S E . ) C A L L d O P E N C A L L J P I N T R ( 1 ) C A L L J S I Z E ( 0 . 0 4 , 0 . 0 4 ) C A L L J d U S T ( 1 , 2 ) C A L L J C O L O R ( 0 ) C A L L J 3 M 0 V E ( - 0 . 2 7 5 , 0 . 8 7 , 0 . 0 ) C A L L J H S T R G ( ' [ B U N D ] S T R E S S * [ B L C ] T [ E L C ] M [ B L C ] P A * M M ' ) C A L L d S I Z E ( 0 . 0 2 5 . 0 . 0 2 5 ) PNTX = - 0 . 1 6 0 DO 100 1 = 1 , 7 PNTY = 0 . 8 7 - 1 * 0 . 0 7 5 51 = S T R M A X ( 1 ) + ( 1 - 1 ) * ( S T R M A X ( 2 ) - S T R M A X ( 1 ) ) / 6 . 0 52 = S T R M A X ( 1 ) + ( I ) * ( S T R M A X ( 2 ) - S T R M A X ( 1 ) ) / 6 . 0 - 0 . 1 I F ( I . E 0 . 7 ) T H E N W R I T E ( C H A R , 1 0 1 0 ) S1 E L S E W R I T E ( C H A R , 1 0 0 0 ) S 1 , S 2 END I F C A L L d P I D E X ( I C O L ( I ) , 1 5 0 0 ) C A L L d 3 M 0 V E ( P N T X , P N T Y , O . 0 ) C A L L d R 3 P G N ( R E L X , R E L Y , R E L Z , 4 ) C A L L d 3 S T R G ( C H A R ) 1 0 0 C O N T I N U E C A L L d P I N T R ( 0 ) C A L L d C L O S E C 1 0 0 0 FORMAT ( F 7 . 1 , ' T 0 ' , F 7 . 1 ) 1 0 1 0 FORMAT ( F 7 . 1 ) R E T U R N END c ******************************** C L O G I C A L F U N C T I O N V I S B L E Q **************************************************************** L O G I C A L F U N C T I O N V I S B L E ( E L X . E L Y , E L Z , N S U R F ) D I M E N S I O N E L X ( 4 ) , E L Y ( 4 ) , E L Z ( 4 ) , V X ( 4 ) , V Y ( 4 ) , V A L U E ( 2 ) C C S E E I F THE P L A N E D E F I N E D BY THE FOUR P A S S E D P O I N T S IS C V I S I B L E UNDER THE C U R R E N T V I E W I N G T R A N S F O R M A T I O N C C NSURF =1 F I L L U N D E R S I D E ( - Z ) C =2 F I L L T O P S I D E ( + Z ) C 110 NISPLOT L i s t i n g DO 100 1 = 1 , 4 C A L L UCONWV ( E L X ( I ) , E L Y ( I ) , E L Z ( I ) , V X ( I ) . V Y ( I ) ) 1 0 0 C O N T I N U E C DO 110 1 = 1 , 2 DDX1 = V X ( I + 1 ) - V X ( I ) DDY1 = V Y ( 1 + 1 ) - V Y ( I ) DDX2 = V X ( I + 2 ) - V X ( I + 1 ) DDY2 = V Y ( I + 2 ) - V Y ( I + 1 ) V A L U E ( I ) = - D D X 1 * D D Y 2 + D D X 2 * D D Y 1 I F ( N S U R F . E O . 2 ) V A L U E ( I ) = V A L U E ( I ) * - 1 . 0 110 C O N T I N U E C V I S B L E = V A L U E ( 1 ) . G T . 0 . 0 0 . O R . V A L U E ( 2 ) . G T . 0 . 0 0 C R E T U R N END Q ***************************************************** C S U B R O U T I N E D A T I N Q **************************************************************************** S U B R O U T I N E D A T I N COMMON / STR / N F ( 6 . 1 6 ) , R S ( 2 , 2 5 ) , N P O I N T ( 4 , 1 6 ) , F A C T ( 1 6 ) , I C O L ( 7 ) DATA F / 1 , 2 , 3 , 5 , 6 , 7 , 1 , 2 , 4 . 5 . 6 , 7 , 1 . 3 , 4 . 5 . 6 . 7 , 1 2 , 3 , 4 , 5 , 6 , 7 , 2 , 3 , 4 , 5 , 6 , 8 , 2 , 3 . 4 , 5 . 7 , 8 , 2 2 , 3 , 4 , 6 , 7 , 8 , 1 . 3 , 4 , 6 , 7 , 8 , 1 , 2 . 4 . 6 . 7 . 8 , 3 1 , 2 , 3 , 6 , 7 , 8 , 1 , 2 , 3 , 5 , 7 , 8 , 1 , 2 , 3 , 5 . 6 , 8 , 4 1 . 2 , 4 , 5 . 6 , 8 . 1 . 2 . 4 . 5 . 7 , 8 . 1 , 3 . 4 . 5 . 7 . 8 . 5 1 , 3 , 4 . 5 , 6 , 8 ^ / DATA RS / 1 . 0 , 1 . 0 . 6 . 5 , 1 . 0 , 0 . 0 . 1 . 0 , - 0 . 5 , 1 . 0 , - 1 . 0 , 1 . 0 , 1 - 1 . 0 , 0 . 5 , - 1 . 0 , 0 . 0 , - 1 . 0 , - 0 . 5 , - 1 . 0 , - 1 . 0 , - 0 . 5 , - 1 . 0 , 2 0 . 0 , - 1 . 0 , 0 . 5 , - 1 . 0 , 1 . 0 , - 1 . 0 , 1 . 0 , - 0 . 5 , 1 . 0 . 0 . 0 , 3 1 . 0 , 0 . 5 , 0 . 5 , 0 . 5 , 0 . 0 , 0 . 5 , - 0 . 5 , 0 . 5 , - 0 . 5 , 0 . 0 , 4 0 . 0 , 0 . 0 , 0 . 5 , 0 . 0 , - 0 . 5 , - 0 . 5 , 0 . 0 , - 0 . 5 , 0 . 5 , - 0 . 5 / DATA F A C T / 1 . 0 , 9 . 0 , - 9 . 0 , 1 . 0 , 9 . 0 , - 9 . 0 , 1 . 0 , - 9 . 0 , 1 9 . 0 , 1 . 0 , - 9 . 0 , 9 . 0 , 8 1 . 0 , - 8 1 . 0 , 8 1 . 0 , - 8 1 . 0 / D A T A N P O I N T / 8 , 9 , 1 0 , 2 3 , 7 , 8 , 2 3 , 2 0 , 6 , 7 , 2 0 , 1 9 , 5 , 6 , 1 9 , 4 , 1 2 3 , 1 0 , 1 1 , 2 4 , 2 0 , 2 3 , 2 4 , 2 1 , 1 9 , 2 0 , 2 1 , 1 8 , 4 , 1 9 , 1 8 , 3 , 2 2 4 , 1 1 , 1 2 , 2 5 , 2 1 , 2 4 , 2 5 , 2 2 , 1 8 . 2 1 . 2 2 , 1 7 , 3 , 1 8 , 1 7 , 2 . 3 2 5 . 1 2 . 1 3 . 1 4 . 2 2 . 2 5 , 1 4 , 1 5 , 1 7 , 2 2 , 1 5 , 1 6 , 2 , 1 7 , 1 6 , 1 / DATA I C O L / 4 , 6 , 2 . 3 , 5 , 1 , 7 / R E T U R N END 1 1 1 APPENDIX D .2 MESHGEN c ********************************************************* c C PROGRAM TO GENNARATE NODE GRID C C T h i s v e r s i o n g e n e r a t e s a g r i d o f e l e m e n t s f o r a h o l e d p l a t e C A 1/4, 1/2 , o r f u l l p l a t e m o d e l c a n b e g e n e r a t e d i f t h e n o . C o f s i d e s s p e c i f i e d ( N S S ) i s 1 , 2 , o r 4 r e s p e c t f u l l y . C F i x e d b o u d a r i e s c a n b e s p e c i f i e d i f NSS i s n e g a t i v e . C C NSS = NO . OF S I D E S C NER = N O . OF E L E M E N T S R A D I A L L Y C N E A ( 4 ) = NO . OF E L E M E N T S PER A R C . C N N ( 4 ) = CORNER NODE N O . C c ********************************************************************** D I M E N S I O N N L ( 3 6 ) , R L ( 3 6 ) , A L ( 3 6 ) , A N G ( 5 ) , X Y ( 5 ) L O G I C A L F L A G COMMON N E A ( 4 ) , N N ( 4 ) , N S S , N E R , N N A , N N R , F L A G F L A G = . T R U E . INPUT=1 IOUT=7 P I = 3 . 1 4 1 5 9 2 6 5 4 D E G = P I / 1 8 0 . 0 NUMEL=0 OPEN ( U N I T = I N P U T , F I L E = ' N O D E . I N ' , S T A T U S = ' O L D ' ) OPEN ( U N I T = I O U T , F I L E = ' N O D E . O U T ' , S T A T U S = ' N E W ) READ ( I N P U T , 1 0 0 0 ) N S S , N E R , R 1 IF ( N S S . L T . O ) T H E N N S S = - 1 * N S S F L A G = . F A L S E . END IF DO 9 0 I S I D E = 1 , N S S READ ( I N P U T , 1 0 1 0 ) N E A ( I S I D E ) READ ( I N P U T , 1 0 2 0 ) A N G ( I S I D E ) , X Y ( I S I D E ) NUMEL = NUMEL + N E A ( I S I D E ) 9 0 C O N T I N U E IF ( N S S . E 0 . 4 ) T H E N NUMNO = NUMEL * 3 * ( N E R * 3 + 1 ) E L S E NUMNO = ( N U M E L * 3 + 1 ) * ( N E R * 3 + 1 ) END IF WRITE ( I 0 U T . 2 O O O ) NUMNO,NER A N G ( N S S + 1 ) = ANG( 1 ) + 9 0 . 0 * N S S IF ( N S S . E 0 . 2 ) T H E N X Y ( N S S + 1 ) = X Y ( 1 ) - 1 0 0 0 . 0 E L S E X Y ( N S S + 1 ) = X Y ( 1 ) . END IF N N ( 1 ) = N E A ( 1 ) * 3 IF ( N S S . N E . 1 ) T H E N DO 95 1 = 2 , N S S N N ( I ) = N N ( I - 1 ) + N E A ( I ) * 3 95 C O N T I N U E END IF DO 110 I S = 1 , N S S NNR=NER*3 N N A = N E A ( I S ) * 3 I I S = I S + 1 DO 100 I A N G = 1 , N N A I N O D E = N N ( I S ) - NNA + IANG ANGLE = ( I A N G - 1 ) * ( A N G ( 1 1 S ) - A N G ( I S ) ) / NNA + A N G ( I S ) I F ( I S . E O . 1 ) T H E N R 2 = X Y ( I S ) / S I N ( A N G L E * D E G ) E L S E IF ( I S . E 0 . 2 ) T H E N R2 = X Y ( I S ) / C O S ( A N G L E * D E G ) E L S E IF ( I S . E Q . 3 ) T H E N R2 = X Y ( I S ) / S I N ( A N G L E * D E G ) E L S E IF ( I S . E 0 . 4 ) T H E N R2 = X Y ( I S ) / C O S ( A N G L E * D E G ) END IF IF ( N S S . N E . 4 ) ANGLE = A N G L E - 4 5 . 0 0 C A L L P S P A C E ( N L . R L . A L , I N O D E , A N G L E , R 1 , R 2 , I O U T ) 100 C O N T I N U E 1 12 MESHGEN L i s t i n g 110 C O N T I N U E I F ( N S S . E 0 . 1 . O R . N S S . E 0 . 2 ) T H E N INODE = N N ( N S S ) + 1 I F ( N S S . E O . 1 ) T H E N R 2 = X Y ( 1 ) / S I N ( A N G ( 2 ) * D E G ) E L S E IF ( N S S . E Q . 2 ) T H E N R 2 = X Y ( 2 ) / C 0 S ( A N G ( 3 ) * D E G ) END IF ANGLE = A N G ( N S S + 1 ) - 4 5 . 0 0 C A L L P S P A C E ( N L . R L . A L , I N O D E , A N G L E , R 1 , R 2 , I O U T ) END IF C A L L LOAD ( N L . R L , A L , D E G . I O U T ) 1 0 0 0 FORMAT ( 2 I 5 . F 1 2 . 5 ) 1 0 1 0 FORMAT ( 1 5 ) 1 0 2 0 FORMAT ( 2 F 1 5 . 9 ) 2 0 0 0 FORMAT ( ' T i t l e ' , / , 2 1 4 , ' , 5 . 0 , ' 13 ' . 3 , ' , / . 3 ' 1, 2 , O , 0 , O , R e s t a r t ' , / , 4 ' 0 . 0 . 1 5 . 0 , ' , / , 5 ' 0 . 0 , 0 . 0 , 0 . 0 0 0 0 1 , , 3 0 0 0 0 0 . , ' , / , 6 ' , , 3 , 1, 0 , ' , / / ) STOP END S U B R O U T I N E P S P A C E i t * * * * * * * * : ) S U B R O U T I N E P S P A C E ( N L , R L , A L , I N O D E , A N G L E , R O , R 2 . I O U T ) D I M E N S I O N NL ( 1 ) , R L ( 1 ) , A L ( 1 ) L O G I C A L F L A G COMMON N E A ( 4 ) , N N ( 4 ) , N S S , N E R , N N A , N N R , F L A G POWER=1 .O/NNR R1=R0 J = I N O D E C 0 N S T = ( R 2 / R 1 ) * * P O W E R IF ( N S S . E Q . 1 . A N D . J . E 0 . 1 ) T H E N WRITE ( I O U T , 2 5 4 0 ) I N O D E , R 1 , A N G L E . A N D . J . E O . ( N N ( N S S ) + 1 ) ) T H E N I N O D E , R 1 . A N G L E . A N D . J . E O . 1) T H E N I N O D E , R 1 , A N G L E . A N D . J . E O . ( N N ( N S S ) + 1 ) ) T H E N I N O D E , R 1 , A N G L E I N O D E , R 1 , A N G L E T H E N + N N ( N S S ) E L S E I F ( N S S . E O . 1 WR ITE ( I O U T , 2 5 6 0 ) E L S E I F ( N S S . E O . 2 WR ITE ( I O U T , 2 5 4 0 ) E L S E I F ( N S S . E O . 1 WR ITE ( I O U T , 2 5 4 0 ) E L S E WRITE ( I O U T , 2 5 0 0 ) END IF DO 2 0 0 1=1 ,NNR I F ( N S S . E O . 4 ) INODE= INODE E L S E INODE= INODE + N N ( N S S ) + 1 END IF R 1 = R 1 * C 0 N S T I F ( I . E O . N N R ) T H E N N L ( J ) = I N O D E R L ( J ) = R 1 A L ( J ) = A N G L E END IF C C S I M P L Y S U P P O R T E D BOUNDARY C I F ( F L A G ) T H E N C C ONE QUARTER P L A T E C I F ( N S S . E Q . 1 . A N D . J . E Q . 1 . WRITE ( I O U T , 2 5 4 0 ) INODE E L S E I F ( N S S . E O . 1 . A N D . J WRITE ( I O U T , 2 5 5 0 ) INODE E L S E IF ( N S S . E O . 1 . A N D . J WR ITE ( I O U T , 2 5 6 0 ) INODE E L S E IF ( N S S . E Q . 1 . A N D . d WRITE ( I O U T , 2 5 7 0 ) INODE . A N D . I . N E . N N R ) T H E N R 1 . A N G L E E 0 . 1 . A N D . I . E O . N N R ) T H E N R 1 . A N G L E E O . ( N N ( N S S ) + 1 ) .AND R 1 . A N G L E E O . ( N N ( N S S ) + 1 ) .AND R 1 . A N G L E I . N E . N N R ) THEN I . E O . N N R ) THEN 113 MESHGEN L i s t i n g E L S E I F ( N S S . E Q . 1 . A N D . I . E Q . N N R ) T H E N W R I T E ( I 0 U T . 2 5 3 0 ) I N O D E , R 1 , A N G L E O N E H A L F P L A T E E L S E I F W R I T E E L S E I F W R I T E E L S E I F W R I T E E L S E I F W R I T E E L S E I F W R I T E F U L L P L A T E I . N E . N N R ) T H E N ( N S S . E 0 . 2 . A N D . J . E Q . 1 . A N D . ( I O U T . 2 5 4 0 ) I N O D E , R 1 . A N G L E ( N S S . E O . 2 . A N D . J . E Q . 1 . A N D . I . E Q . N N R ) T H E N ( I 0 U T . 2 5 5 O ) I N O D E . R 1 . A N G L E ( N S S . E Q . 2 . A N D . J . E Q . ( N N ( N S S ) + 1 ) . A N D . I . N E . N N R ) T H E N ( I O U T . 2 5 4 0 ) I N O D E , R 1 . A N G L E ( N S S . E Q . 2 . A N D . J . E Q . ( N N ( N S S ) + 1 ) . A N D . I . E Q . N N R ) T H E N ( I 0 U T . 2 5 8 O ) I N O D E . R 1 . A N G L E ( N S S . E O . 2 . A N D . I . E Q . N N R ) T H E N ( I O U T . 2 5 3 0 ) I N O D E . R 1 . A N G L E E L S E I F ( N S S . E Q . 4 . A N D . I F ( J . E Q . ( N N ( 2 ) + 1 ) ) W R I T E ( I 0 U T . 2 5 1 O ) E L S E I F ( J . G T . ( N N ( 2 ) + 1 ) W R I T E ( I 0 U T . 2 5 2 O ) I N O D E E L S E W R I T E ( I 0 U T . 2 5 3 O ) I N O D E E N D I F I . E Q . N N R ) T H E N T H E N I N O D E , R 1 . A N G L E . A N D . J . L E . R 1 , A N G L E R 1 , A N G L E ( N N ( 3 ) + 1 ) ) T H E N I N T E R N A L E L S E W R I T E ( I O U T . 2 5 0 0 ) I N O D E , R 1 , A N G L E E N D I F F I X E D B O U N D A R Y E L S E O N E Q U A R T E R P L A T E I F ( N S S . E Q . 1 . A N D . J . E Q . 1 . A N D . I . N E . N N R ) T H E N W R I T E ( I 0 U T , 2 6 4 O ) I N O D E , R 1 , A N G L E E L S E I F ( N S S . E Q . 1 . A N D . J . E Q W R I T E ( I 0 U T . 2 6 5 0 ) I N O D E , R 1 E L S E I F ( N S S . E Q . 1 . A N D . J . E Q W R I T E ( I 0 U T . 2 6 G 0 ) I N O D E , R 1 E L S E I F ( N S S . E Q . 1 . A N D . J . E Q W R I T E ( I 0 U T . 2 6 7 O ) I N O D E , R 1 E L S E I F ( N S S . E Q . 1 . A N D . I . E Q . N N R ) T H E N W R I T E ( I 0 U T . 2 6 3 0 ) I N O D E , R 1 , A N G L E 1 . A N D . I . E Q . N N R ) T H E N A N G L E ( N N ( N S S ) + 1 ) . A N D . I . N E . N N R ) T H E N A N G L E ( N N ( N S S ) + 1 ) . A N D . I . E Q . N N R ) T H E N A N G L E O N E H A L F P L A T E E L S E I F W R I T E E L S E I F W R I T E E L S E I F W R I T E E L S E I F W R I T E E L S E I F W R I T E F U L L P L A T E ( N S S . E Q . 2 . A N D . J . ( I 0 U T . 2 6 4 O ) I N O D E . ( N S S . E Q . 2 . A N D . J . ( I 0 U T . 2 6 5 O ) I N O D E , ( N S S . E Q . 2 . A N D . J . ( I 0 U T . 2 6 4 0 ) I N O D E . ( N S S . E Q . 2 . A N D . J . ( I 0 U T . 2 G 8 O ) I N O D E , ( N S S . E Q . 2 . A N D . I . ( I 0 U T . 2 G 3 O ) I N O D E , E Q . 1 . A N D . I . N E . N N R ) T H E N R 1 . A N G L E E Q . 1 . A N D . I . E Q . N N R ) T H E N R 1 . A N G L E E Q . ( N N ( N S S ) + 1 ) . A N D . I . N E . N N R ) T H E N R 1 . A N G L E E Q . ( N N f N S S ) + 1 ) . A N D . I . E Q . N N R ) T H E N R 1 . A N G L E E Q . N N R ) T H E N R 1 . A N G L E E L S E I F ( N S S . E Q . 4 . A N D . I . E Q . N N R ) T H E N I F ( J . E Q . ( N N ( 2 ) + 1 ) ) T H E N W R I T E ( I O U T . 2 6 1 0 ) I N O D E , R 1 , A N G L E E L S E I F ( J . G T . ( N N ( 2 ) + 1 ) . A N D . J . L E . W R I T E ( I O U T . 2 6 2 0 ) I N O D E , R 1 , A N G L E E L S E W R I T E ( I 0 U T . 2 6 3 O ) I N O D E , R 1 , A N G L E E N D I F ( N N ( 3 ) + 1 ) ) T H E N 114 MESHGEN L i s t i n g c C I N T E R N A L C E L S E WRITE ( I O U T . 2 5 0 0 ) I N O D E , R 1 , A N G L E END I F END I F 2 0 0 C O N T I N U E C c S IMPLY S U P P R T E D BOUNDARY L 2 5 0 0 FORMAT ( 1 4 . ' , 0 , 0 , 0 , 0 , 0 , 1, 0 0 0 , , 2 ( F8 3 , . ' ) . ' 0 , 1 ,2 2 5 1 0 FORMAT ( 1 4 , ' , 1 . 1 , 1 , 0 . 0 , 1, 0 0 0 , , 2 ( F 8 3 . . ' ) , ' 0 . 1 , 2 2 5 2 0 FORMAT ( 1 4 . ' , 0 . 1 , 1 . 0 . 0 , 1, 0 0 0 , . 2( F8 3 , . ' ) . ' 0 , 1 , 2 2 5 3 0 FORMAT ( 1 4 , ' , 0 , 0 . 1 , 0 . 0 . 1. 0 0 0 . , 2 ( F8 3 , . ' ) , ' 0 , 1 , 2 2 5 4 0 FORMAT ( 1 4 . ' , 0 , 1 . 0 , 1 , 0 , 1. 0 0 0 , , 2 ( F 8 3 . . ' ) . ' 0 . 1 , 2 2 5 5 0 FORMAT ( 1 4 , ' , 0 , 1 , 1 , 1 , 0 , 1, 0 oo - . , 2 ( F 8 3 . . ' ) . ' 0 . 1 . 2 2 5 6 0 FORMAT ( 1 4 , ' , 1 , 0 , 0 , 0 , 1 . 1. 0 0 0 , , 2( F8 3 , , ' ) , ' 0 . 1 , 2 2 5 7 0 FORMAT ( 1 4 , ' . 1 , 0 , 1 , 0 , 1 . 1. 0 0 0 , , 2 ( F 8 3 . . ' ) , ' 0 , 1 . 2 2 5 8 0 /•* FORMAT ( 1 4 , ' , 1 , 1 . 1 . 1 , 0 , 1, 0 0 0 , , 2 ( F 8 3 . . ' ) , ' 0 . 1 , 2 c C L A M P E D BOUNDARY 2 6 1 0 FORMAT ( 1 4 , ' , 1 , 1 , 1 , 1 . 1 . i , 0 0 0 . , 2 ( F8 3, , ' ) , ' 0 , 1 , 2 2 6 2 0 FORMAT ( 1 4 . ' , 0 , 1 , 1 . 1 , 1 , 1, 0 0 0 , , 2 ( F 8 3 . , ' ) . ' 0 , 1 , 2 2 6 3 0 FORMAT ( 1 4 , ' , 0 , 0 . 1 , 1 , 1 . 1 , 0 0 0 , , 2 ( F 8 3 , , ' ) , ' 0 . 1 , 2 2 6 4 0 FORMAT ( 1 4 . ' , 0 . 1 . 0 . 1 , 0 , 1 , 0 0 0 , , 2 ( F 8 3 . . ' ) , ' 0 . 1 , 2 2 6 5 0 FORMAT ( 1 4 , ' . 0 , 1 , 1 . 1 . 1 , 1, 0 0 0 . , 2 ( F 8 3 , . ' ) . ' 0 . 1 , 2 2 6 6 0 FORMAT ( 1 4 , ' , 1 , 0 . 0 , 0 , 1 , 1 , 0 0 0 . , 2 ( F 8 3 , . ' ) . ' 0 , 1 , 2 2 6 7 0 FORMAT ( 1 4 , ' , 1 , 0 , 1 , 1 , 1 , 1 , 0 0 0 , , 2 ( F 8 3 , . ' ) , ' 0 , 1 .2 2 6 8 0 FORMAT ( 1 4 , ' , 1 , 1 , 1 , 1 . 1 . 1 • 0 00. . . 2 ( F 8 3 . , ' ) , ' 0 , 1 , 2 RETURN c END Q ************ S U B R 0 U T I N E L 0 A D * * * * * * * * * * * C S U B R O U T I N E LOAD ( N L , R L , A L . D E G , I O U T ) D I M E N S I O N N L ( 1 ) , R L ( 1 ) , A L ( 1 ) , D I S ( 3 6 ) , P ( 3 0 ) 1 , S U M ( 4 ) , P E L E ( 4 ) , I F I R S T ( 1 6 ) , I L A S T ( 16) L O G I C A L F L A G COMMON N E A ( 4 ) , N N ( 4 ) , N S S . N E R , N N A , N N R , F L A G C C P R I N T E L E M E N T NODE NUMBER ING C DO 2 9 0 I R = 1 , N E R WRITE ( I O U T , 2 0 0 0 ) N N ( N S S ) / 3 C A L L E L E N O ( I F I R S T . I L A S T , I R , N N , N S S ) WRITE ( I O U T . 2 0 1 0 ) ( I F I R S T ( I ) ,I = 1 . 16) IF ( N E A ( N S S ) . N E . 1 ) 1 WRITE ( I O U T , 2 0 2 0 ) N N ( N S S ) / 3 , ( I L A S T ( I ) , I = 1, 16 ) 2 9 0 C O N T I N U E C C P R I N T NUMBER OF LOAD P O I N T S C I F ( N S S . E 0 . 4 ) T H E N NUMLP = 2 * N N ( N S S ) + NSS + 1 E L S E NUMLP = 2 * ( N N ( N S S ) + 1) END IF WRITE ( I O U T , 2 0 3 0 ) NUMLP T H I C K = 1 0 . 0 C C C A L C U L A T E THE D I S T A N C E BETWEEN P O I N T S C DO 3 0 0 I = 1 , N N ( N S S ) NN5=NN(NSS )+1 11=1+1 IF ( I I . E 0 . N N 5 . A N D . N S S . E O . 4 ) 11=1 X 1 = R L ( I ) * C O S ( A L ( I ) * D E G ) 115 MESHGEN L i s t i n g Y 1 = R L ( I ) * S I N ( A L ( I ) * D E G ) X2 = R L ( I I ) * C O S ( A L ( I I ) * D E G ) Y2 = R L ( I I ) * S I N ( A L ( 1 1 ) * D E G ) D I S ( I ) = SORT ( (X2-X1 ) * ( X 2 - X 1 ) + ( Y 2 - Y 1 ) * ( Y 2 - Y 1 ) ) 3 0 0 C O N T I N U E C C FOR E A C H S I D E C A L C U L A T E THE C O N S I S T E N T LOAD VECTOR C DO 3 3 0 I S I D E = 1 .NSS ID IR = 1 I D I R 2 = 2 DMULT = 1 . 0 S U M ( I S I D E ) = 0 . 0 DO 3 0 5 1 = 1 , 3 0 P ( I ) = 0 . 0 3 0 5 C O N T I N U E DO 3 2 0 I E L = 1 , N E A ( I S I D E ) ID = N N ( I S I D E ) - N E A ( I S I 0 E ) * 3 + ( I E L - 1 ) * 3 + 1 A = D I S ( I D ) B = D I S ( I D + 1 ) + A C = D I S ( I D + 2 ) + B C A L L C O N S T L ( P E L E . A . B . C ) DO 31.0 d=1 .4 NP = ( I E L - 1 ) * 3 + J P ( N P ) = P ( N P ) + P E L E ( J ) * T H I C K 3 1 0 C O N T I N U E 3 2 0 C O N T I N U E C C S E T THE C O R R E C T S I G N AND O I R E C T I O N FOR E A C H S I D E , C T H E N C H E C K THE SUM OF THE LOAD V E C T O R . C I F ( N S S . E 0 . 4 ) T H E N I F ( I S I D E . E O . 2 . O R . I S I D E . E O . 4 ) ID IR=2 I F ( I S I D E . E Q . 2 . O R . I S I D E . E O . 3 ) DMULT = -1 E L S E D M U L T = 1 . 0 / S O R T ( 2 . 0 ) END I F C C P R I N T THE NODAL LOADS C DO 3 2 5 1= 1 , ( N E A ( I S I D E ) * 3 + 1 ) P ( I ) = P ( I ) * D M U L T S U M ( I S I D E ) = S U M ( I S I D E ) + P ( I ) K = N N ( I S I D E ) - N E A ( I S I D E ) * 3 + I I F ( K . E 0 . N N 5 . A N D . N S S . E 0 . 4 ) K=1 WRITE ( I 0 U T . 2 0 4 0 ) N L ( K ) . I D I R . P ( I ) I F ( N S S . E O . 1 ) T H E N P ( I ) = - 1 . 0 * P ( I ) WRITE ( I 0 U T . 2 0 4 0 ) N L ( K ) , I D I R 2 , P ( I ) END I F 3 2 5 C O N T I N U E 3 3 0 C O N T I N U E C C P R I N T L A T E R A L LOADS C C DO 3 4 0 1= 1 , N N ( N S S ) , 3 C I F ( N S S . N E . 4 ) T H E N C I F ( I . E O . 1 ) T H E N C WRITE ( I O U T . 2 0 5 0 ) I ,1+1,1+2 C E L S E C WRITE ( I 0 U T . 2 O 6 O ) 1,1+1.1+2 C END IF C E L S E C WRITE ( I 0 U T . 2 O 6 O ) 1,1+1,1+2 C END IF C 3 4 0 C O N T I N U E C I F ( N S S . N E . 4 ) WRITE ( I O U T . 2 0 7 0 ) N N ( N S S ) + 1 WR ITE ( I O U T . 2 0 8 0 ) WRITE ( I O U T . 2 0 9 0 ) ( I . S U M ( I ) . I = 1 , 4 ) 1 16 MESHGEN L i s t i n g 2 0 0 0 FORMAT ( 1' 7 , ' . 1 2 , ' , 3 , 0 , 0 , 0 , 1 6 , , 4 , 4 , 5 , 0 , 1 , 1 , 1 , 1 , ' , / , 2 ' 1 , 7 . 7 0 E - 0 5 . 0 . 0 . ' , / , 3 ' 2 0 0 0 0 0 . 0 . 0 . 3 . 1 . 2 . 3 0 0 . 0 , 0 . 0 , ' ) 2 0 1 0 FORMAT ( 1' 1 , 1 6 , 3 3 4 , 1 , 0 , 0 , 0 , 1 0 . 0 , ' , / , 1 6 1 4 ) 2 0 2 0 FORMAT ( 1 1 3 , ' , 1 6 , 3 3 4 , 1 , 3 , 0 , 0 , 1 0 . 0 , ' , / , 1 6 1 4 ) 2 0 3 0 FORMAT ( 1 1 4 , ' , 1 , 3 , ' , / , 2 ' 1 , 3 ' , / , 3 ' 0 . 0 , 0 . 0 ' , / , 4 ' 1 . 0 . 1 . 0 . ' . / . 5 ' 2 . 0 , 2 . 0 , ' ) 2 0 4 0 FORMAT ( 1 4 , ' , ' , 1 2 , ' , 1 , ' , F 1 0 . 4 ) 2 0 5 0 FORMAT ( 1 4 , ' , 3 , 1, 0 . 0 4 ' , / , + 1 4 , ' , 3 . 1 , 0 . 1 2 ' , / , + 1 4 , ' , 3 , 1 , 0 . 1 2 ' ) 2 0 6 0 FORMAT ( 1 4 , ' , 3 , 1, 0 . 0 8 ' , / , + 1 4 , ' . 3 , 1 , 0 . 1 2 ' , / , + 1 4 , ' , 3 , 1 , 0 . 1 2 ' ) 2 0 7 0 FORMAT ( 1 4 , ' , 3 , 1, 0 . 0 4 ' ) 2 0 8 0 FORMAT ( ' 1 , 1, 1 , 0 , 4 , , , 1 . 0 , ' , / , 1 ' 1 ' ) 2 0 9 0 FORMAT 1 ( 4 ( / , ' SUM OF THE F O R C E S FOR S I D E ' , 1 2 , ' I S ' , F 1 5 . 6 . ' s q m m ' RETURN END C c * * * * * * * * * * * * * * * S U B R O U T I N E C O N S T L * * * * * * * * * * C SUBROUT INE C O N S T L ( P E L E , A , B , C ) D I M E N S I O N P E L E ( 4 ) C C C A L C U L A T E THE C O N S I S T E N T LOAD FOR THE C U B I C SHAPE F U N C T I O N C C . . • • C 0 A B C C P E L E ( 1 ) = 0 . 7 1 2 5 * A - 0 . 3 0 0 0 * B + 0 . 0 8 7 5 * C P E L E ( 2 ) = 1 . 0 1 2 5 * B - 0 . 3 0 0 0 * C P E L E ( 3 ) = - 1 . 0 1 2 5 * A + 0 . 7 1 2 5 * C P E L E ( 4 ) = 0 . 3 0 0 0 * A - 0 . 7 1 2 5 * B + 0 . 5 0 0 0 * C RETURN END C Q * * * * * * * * * * * * * * 5 (j g R • u j i N E E L E N 0 * * * * * * * * * * * * * * C S U B R O U T I N E ELENO ( I F I R S T , I L A S T , I R , N N , N S S ) D I M E N S I O N I F I R S T ( 1 6 ) , I L A S T ( 1 6 ) , I M U L T ( 1 6 ) , J M U L T ( 1 6 ) , 1 I A D D ( 1 6 ) . J A D D ( 1 6 ) , N N ( 4 ) DATA IMULT / 3 , 0 , 0 , 3 , 2 , 1 , 0 , 0 . 1 . 2 , 3 , 3 , 2 , 1 , 1 , 2 / DATA IADD / 4 , 4 , 1 , 1 , 4 , 4 , 3 , 2 , 1 , 1 , 2 , 3 , 3 , 3 , 2 , 2 / DATA JMULT / 2 , - 1 , 0 , 3 , 1 , 0 , 0 , 0 , 1 , 2 , 3 . 3 . 2 , 1 , 1 . 2 / DATA vJADD / 0 , 0 , 3 , 3 , 0 , 0 , 1 , 2 , 3 , 3 , 2 , 1 , 1 , - 1 , 2 , 2 / NO=NN(NSS ) I F ( N S S . N E . 4 ) N0=N0+1 I S T A R T = N O * ( I R - 1 ) * 3 J S T A R T = N O * ( I R - 1 ) * 3 + N 0 + 1 DO 5 0 0 1 = 1 , 1 6 I F I R S T ( I ) = I M U L T ( I ) * N O + I S T A R T + I A D D ( I ) IF ( N S S . E O . 4 ) T H E N I L A S T ( I ) = J M U L T ( I ) * N O + J S T A R T - J A D D ( I ) E L S E I L A S T ( I ) = I F I R S T ( I ) + N N ( N S S ) - 3 END IF 5 0 0 C O N T I N U E RETURN END 117 APPENDIX E Computer Communications APPENDIX E . l WORDSTAR Output on the MTS Zerox 9700 The f o l l o w i n g commands w i l l t r a n s f e r a WORDSTAR f i l e on the IBM PC to MTS and then p r i n t the f i l e on the Xerox 9700 l a z e r p r i n t e r . Require: WORDSTAR DISKETTE WINDOW DISKETTE o b t a i n a b l e from the UBC book s t o r e I or G account on the UBC MTS system In WORDSTAR p r i n t the f i l e to a d i s k f i l e . P "filename" y " f i l e p r i n t " RETURN RETURN Y RETURN X // e x i t WORDSTAR when p r i n t i n g i s f i n i s h e d // // change to WINDOW d i s k e t t e // A>WSCLRBIT " f i l e p r i n t " " f i l e c l e a r " // get the a t t e n t i o n of the smart switch with kermit // A>KERMIT SET BAUD 4800 CONN * 6 // MTS on port 6, VAX VMS on port 5 // c t r l ] C EXIT // using the same d i s k e t t e run WINDOW // A>WINDOW G // or I depending on the l o c a t i o n MTS account // SIG " c c i d " "password" CREATE " f i l e n e w " %T PC " f i l e c l e a r " MTS " - f i l e t e m p " ASCII RUN PC:WPPRINT SCARDS="-filetemp" SPRINT="filenew" PAR=WORDSTAR SET PROUTE=CNTR SET DELIVERY=CIVL // or CNTR //) CON *PRINT* PORTRAIT ONESIDE COPY " f i l e n e w " *PRINT* SIG %EXIT Other commonly used MTS comands a r e : DIS *PRINT* // d i s p l a y s p r i n t s t a t i s t i c s // REL *PRINT* // r e l e a s e p r i n t to p r i n t e r // SYS QUE USER // show que or time of p r i n t i n g // " f i l e t y p e " f i l e names p r o v i d e d by the user // // enclose comments 118 APPENDIX E.2 T r a n s f e r of a VAX-VMS F i l e to the UBC/MTS System The f o l l o w i n g commands w i l l t r a n s f e r a f i l e from the C i v i l E n g ineering VAX 11/730 to the UBC/MTS system. RETURN "VAXid" // sign on to the VAX // "password" SD "defaultDIR" ALLOC TXAO // d i a l up UBC (228-1401) on modem // KERMIT SET SPEED 1 2 0 0 CONN G (I) SIG " c c i d " // sign on to the UBC MTS system // "password" RUN *KERMIT RECEIVE "filename" c t r l P // c t r l ] C on the IBM PC // SEND "filename" CONN EXIT SIG // s i g n o f f the MTS system // EXIT LO // sign o f f the VAX VMS system // " " e n c l o s e user i d ' s , filenames, d i r e c t o r i e s and passwords // // en c l o s e comments 119

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