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 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, 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 + 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 , 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