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The ultimate load capacity of square shear plates with circular perforations : (parameter study) Martin, Anthony George 1985-12-31

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THE ULTIMATE LOAD CAPACITY OF SQUARE SHEAR PLATES WITH CIRCULAR PERFORATIONS  (PARAMETER STUDY)  by  ANTHONY GEORGE MARTIN B.ApSc,  University  A THESIS SUBMITTED  Of B r i t i s h  Columbia,  IN PARTIAL FULFILLMENT OF  THE REQUIREMENTS FOR THE DEGREE OF MASTER OF A P P L I E D SCIENCE in FACULTY OF GRADUATE STUDIES Department  of C i v i l  Engineering  We a c c e p t t h i s t h e s i s a s c o n f o r m i n g to the r e q u i r e d s t a n d a r d .  THE UNIVERSITY OF BRITISH COLUMBIA SEPTEMBER 1985 ©Anthony  1981  George  Martin,  1985  In  presenting  requirements  f o r the  COLUMBIA, I a g r e e for  reference  extensive granted  and  the  Representatives. this  thesis  written  thesis  in p a r t i a l  advanced degree at  that  copying by  this  the  Library  shall  study.  I further  of  thesis  Head  this of  my  gain  agree  that  shall  or  copying not  be  available  p u r p o s e s may by  his  or  or p u b l i c a t i o n allowed  the  BRITISH  permission  for scholarly  that  of  UNIVERSITY OF  make i t f r e e l y  Department  I t i s understood  for f i n a n c i a l  the  fulfillment  without  for be her of my  permission.  A n t h o n y G.  Department  of C I V I L ENGINEERING  THE UNIVERSITY OF BRITISH COLUMBIA 2070 Wesbrook P l a c e , V a n c o u v e r , Canada. V6T-1W5  D a t e : September,  1985  Martin,  P.  Eng.  ABSTRACT The  incremental  s t r u c t u r a l a n a l y s i s program NISA83 was  to  investigate various  of  square p l a t e s  shear  stress.  geometry plane  parameters a f f e c t i n g the u l t i m a t e  with c i r c u l a r  perforations  Both n o n l i n e a r  were t a k e n  capacities  material  into account  and b u c k l i n g  subjected  properties  in determining  capacities  of  capacity  to  and  used  uniform nonlinear  the ultimate i n perforated  shear  plates. The size  parameters  for  a concentric  constant  ratio  addition  various  most  buckling  parts.  was  parameters. elastic ultimate  The  width  studied  to  a shear p l a t e  and  ultimate the  of  for 0.2.  determine  to i t s  These  the a n a l y s i s in-plane  ultimate for  a In  the  original  and n o n l i n e a r  was  separated  capacity,  elastic  elastic-plastic  each  combination  were u s e d t o i d e n t i f y  r e s u l t s from p l a t e s  of v a r y i n g  material  of  buckling the  two  the importance of  both  contribute  to the reduced  with a c o n c e n t r i c a l l y located  s i z e showed e x c e l l e n t  experimental  and  analytical  c o r r e l a t i o n with other results  and t h e 3 - d i m e n s i o n a l b u c k l i n g  Variation size provided in  were  location  plate capacities.  The  capacity  to plate  two p a r a m e t e r s ,  determined  buckling  s t u d y were t h e h o l e  and the hole  diameter plates  this  capacity.  capacity  capacity  hole  shape t o r e s t o r e  the f i r s t  three  location,  doubler  in-plane  For into  of  effective  ultimate  investigated during  of  the center  some s i g n i f i c a n t  the u l t i m a t e  in-plane  for  both  capacity  Little  f o r a l l hole  i i  published  the  in-plane  a  standard  capacities.  l o c a t i o n of a hole results.  hole  of  change was locations.  found On t h e  other  hand, t h e  moving  the  compression buckling provides  hole  in-plane  thick  plates.  plate  and the  A  twice  p l a t e s t o be  doubler the  data  and  to provide  full  graphic  to a i d i n the  The  program  the  the  tedious  was  way  written  i n FORTRAN 77,  utilizing  graphics  package,  DI3000,  NISPLOT  generated  p l o t s of  When a c o m p l e t e  task  the  is  concentric  plate  than  as  part  of  available  narrow  and  as  the to  capacity. the  input  result,  this at  from a  independent  e l e m e n t mesh out  a  thesis. the  f r o m NISA83.  carried  size  recommended  in-plane  subroutines  a n a l y s i s was  9, fe.fs-  the  of c h e c k i n g  output  nodes and  generated.  elastic-plastic'  of d i s p l a y i n g t h e  n o d e s , e l e m e n t mesh, d e f l e c t e d s h a p e , and were  the  same t h i c k n e s s  to o b t a i n d e v i c e the  to  doubler  hole  developed  lab to process  diagonal  hole l o c a t i o n s .  its original  a convenient  after  more e f f e c t i v e  NISPLOT u s e d c o l o r g r a p h i c s  Engineering  check.  of  50%  that  optimum  p l a t e with  diameter  post-processor  concluded  the  by  ultimate  for a l l other  a n a l y s i s of  raised  tension  from t h e  i t was  p e r f o r a t e d p l a t e to  order  data  plate  Finally,  results  thin  In  Civil  the  l o w e r bound c a p a c i t y  wide and  restore  b u c k l i n g c a p a c i t y was  from  diagonal.  capacity  The showed  elastic  UBC  It  was  commercial graphics. for by  each  NISA83,  color stress f i l l  plots  TABLE OF CONTENTS Page ABSTRACT  i i  TABLE OF CONTENTS  iv  L I S T OF TABLES  v i i  L I S T OF FIGURES  viii  ACKNOWLEDGEMENTS  xi  NOMENCLATURE  x i i  CHAPTER 1.  2.  INTRODUCTION  1  1 .1  Background  1  1.2  Purpose  3  THEORETICAL BACKGROUND 2.1  3.  and Scope  5  The U l t i m a t e B e h a v i o r o f Shear  COMPUTER PROGRAMS 3.1  Plates  ,  5 9  NISA83  9  3.1.1  G e n e r a l Background  3.1.2 3.1.2.1 3.1.2.2 3.1.2.3 3.1.2.4 3.1.2.5  Time S t e p I n c r e m e n t C o n v e r g e n c e and D i v e r g e n c e Constant Load C o n t r o l Constant Arclength Control I t e r a t i o n Technique A p p l i e d I n c r e m e n t and I t e r a t i o n Algorithm  9 10 11 12 13  3.1.3  Nonlinear Material  16  3.1.4 3.1.4.1  Element L i b r a r y 2-Dimensional Element 3-Dimensional Element  3.1.4.2 3.2  9  14  18 Plane  Stress 19  Plate  Shell 21  NISPLOT  27  3.2.1  General Description  27  3.2.2  Stress F i l l  29  Routine  iv  4.  3.2.3  Visible  Surface P l o t t i n g  31  3.2.4  Metafiles  32  3.2.5  Flow  35  Charts  PLATE ANALYSIS 4.1  Variation  39 of Hole S i z e  40  4.1.1  P l a t e Geometry  4.1.2  Finite  Model  40  4.1.3 4.1.3.1 4.1.3.2 4.1.3.3  Results In-plane Y i e l d i n g 3-Dimensional E l a s t i c B u c k l i n g 3-Dimensional E l a s t i c - P l a s t i c Buckling .  44 44 47  4.2  Variation  Element  40  ...  51  of Hole L o c a t i o n  56  4.2.1  P l a t e Geometry  4.2.2  Finite  Model  57  4.2.3 4.2.3.1 4.2.3.2 4.2.3.3  Results In-plane Y i e l d i n g 3-Dimensional E l a s t i c B u c k l i n g 3-Dimensonal Elastic-Plastic Buckling  60 60 61  4.3  ...  65  Optimume D o u b l e r P l a t e  70  4.3.1  P l a t e Geometry  71  4.3.2  Finite  4.3.3 4.3.3.1  Results In-plane Y i e l d i n g  4.4 5.  Element  56  Convergence  Element  Model  71  w i t h Mesh R e f i n e m e n t  CONCLUSIONS  73 73 75 79  REFERENCES  83  APPENDIX A  D e r i v a t i o n o f C o n s i s t e n t S h e a r L o a d V e c t o r .... for the B i c u b i c I s o p a r a m e t r i c Element  APPENDIX B  ASCE S u g g e s t e d D e s i g n G u i d e s Web H o l e s  v  84  f o r Beams w i t h 88  APPENDIX C  Modification  APPENDIX D  Program  of NISA80 a t U.B.C  90  Listings  97  D.1  NISPLOT  97  D. 2  MESHGEN  112  APPENDIX E  Communications  Programs  118  E. 1  WORDSTAR O u t p u t on t h e MTS X e r o x  E.2  Transfer System  o f a VAX/VMS  File  9700  118  t o t h e UBC/MTS 119  vi  L I S T OF T a b l e No.  TABLES  Title  Page  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  3.1  Interpolation Element  Functions  3.2  Interpolation  Functions  4.1  F a i l u r e Mode C l a s s i f i c a t i o n  A.1  F o u r C u b i c Shape F u n c t i o n s a l o n g Boundary s=1, — 1 <r< 1  f o r B i l i n e a r Plane  No.  7  Stress 21  for Bicubic  Element  25 65  the B i c u b i c  vi i  Element 86  L I S T OF  FIGURES  Figure  No.  Title  2.1  Full  2.2  Perforated  3.1  Applied  I n c r e m e n t and  3.2  Initial  S t r e s s - S t r a i n Diagram  3.3  E l a s t i c - I d e a l - P l a s t i c M a t e r i a l Model Tension Test  Plate,  Ideal  Elastic-Plastic  PLate E l a s t i c - P l a s t i c  3.4  B i l i n e a r Plane Stress  3.5  Bicubic, Element  3.6 3.7  3.8  Page  Isoparametric,  Behavior  7  3ehavior  7  Iteration Algorithm  15  for Mild  Isoparametric  Steel  17  in Uniaxial 18  Element  Degenerated,  Plate  20 Shell 24  C o m p a r i s o n of t h e C l a s s i c a l F i n i t e Element F o r m u l a t i o n  Concept  and  Degeneration 26  S u b d i v i s i o n of t h e 16 Node I s o p a r a m e t r i c Element i n t o 16 S u b - R e g i o n s . E a c h S u b - R e g i o n i s F i l l e d w i t h a C o l o r A c c o r d i n g t o the S t r e s s L e v e l a t the Gauss I n t e g r a t i o n P o i n t i n the Sub-Region  30  L i s t i n g of the M e t a f i l e S o u r c e F i l e "mtr.log" t h a t R e s u l t e d i n the Compound P i c t u r e i n Figure  [3.9]  33  3.9  Compound P i c t u r e  3.10  Flow C h a r t  of  the  Frames P l o t t e d by  3.11  Flow C h a r t  of  the  Subroutine  using  a Metafile  34 NISPLOT  Interaction  in  35 the  Program NISPLOT Plate  36  4.1  Perforated  4.2  1/4  4.3 4.4  I n - p l a n e S t r e s s D i s t r i b u t i o n , 1/4 M o d e l . .' C o m p a r i s o n of the U l t i m a t e I n - p l a n e S h e a r 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 t h e F i n i t e E l e m e n t M e t h o d and the ASCE D e s i g n P r o p o s a l g i v e n by E q u a t i o n [4.2] 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  4.5  No.  Showing  P l a t e Model u s i n g  3x3  1/4  F.E.  Model  E l e m e n t Mesh  viii  41 43 45  46 49  No.  4.6  Elastic  4.7  Decreasing Perforated  4.8  Page  Title  Figure  Buckling Mode,Concentric  Hole  No.  50  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 P l a t e w i t h I n c r e a s i n g A p p l i e d Load  a 52  L o a d D e f l e c t i o n C u r v e of a S i m p l y S u p p o r t e d P e r f o r a t e d P l a t e with Various C o n c e n t r i c Hole S i z e s .  52  V a r i a t i o n of U l t i m a t e E l a s t i c - P l a s t i c B u c k l i n g C a p a c i t y of S i m p l y S u p p o r t e d P e r f o r a t e d P l a t e w i t h C o n c e n t r i c Hole S i z e  54  4.10  E-P  55  4.11  P l a t e Geometry and  4.9  Buckling  with  von-Mises S t r e s s Loading  used  i n the  t h e V a r i a t i o n of H o l e L o c a t i o n 4.12 4.13 4.14  . Finite  E l e m e n t M o d e l of H a l f  A n a l y s i s of  Parameter  the  56  Plate  58  F i n i t e E l e m e n t M o d e l of the T o t a l P l a t e Ultimate In-plane Capacity R e s u l t s for Various L o c a t i o n s N o r m a l i z e d t o the C o n c e n t r i c Hole Ultimate In-plane Capacity  4.15  Yield  at  90%  Ultimate  In-plane  4.16  Full  4.17  E l a s t i c Buckling Capacity Factors for Various L o c a t i o n s N o r m a t i z e d t o the C o n c e n t r i c Hole Capacity  62  P l a t e Model, von-Mises S t r e s s  Profile  of  the  Tension  4.19  Profile  of  the  Compression Diagonal  63  E-P  Buckling  at  90%  4.22  Geometry and L o a d i n g Doubler P l a t e  Ultimate  64 66 67  Capacity  of P e r f o r a t e d P l a t e  F i n i t e E l e m e n t Mesh of Doubler P l a t e  Hole  Diagonal  Ultimate E l a s t i c - P l a s t i c Capacity for Various L o c a t i o n s N o r m a l i z e d t o 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 Capacity  4.21  4.23  61  Load  4.18  4.20  59 Hole  Hole 68 69  with 72  Perforated Plate  with 72  ix  F i g u r e No. 4.24 4.25 4.26  A.1 B.1  Title  Page  No.  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 D o u b l e r P l a t e Area f o r V a r i o u s Doubler P l a t e Diameters  74  S p r e a d of Y i e l d Zones f o r S t a n d a r d Perforated Plates  75  and  Reinforced  C o n v e r g e n c e of t h e E l a s t i c B u c k l i n g L o a d w i t h 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 D/b =0.2. t/6=0.0l  78  Bicubic Isoparametric U n i f o r m Shear L o a d i n g  84  P l a t e S h e l l Element a l o n g One Edge  with  U l t i m a t e I n - p l a n e C a p a c i t y o f Shear Web w i t h a C i r c u l a r P e r f o r a t i o n as P r o p o s e d i n R e f e r e n c e [ 1 1 ] .  x  89  ACKNOWLEDGEMENTS The  author  Dr.  S.  F.  Stiemer  and  thesis He  for  his  gratitude  his valuable  advice  to  his  advisor  during  the  research  preparation. would  originally the  expresses  also  suggesting  like the  to  thank  t o p i c and  Dr.  P.  Osterrieder  for his valuable  help  for during  analysis. Much g r a t e f u l a p p r e c i a t i o n  Council  of  Canada  for their  i s due  t o the  financial  National  support  work.  xi  Research  throughout  this  NOMENCLATURE A list here.  of symbols u s e d  Conventional  repetitively  mathematical  in this  equivalent hole  length  Ai  cross sectional  area  of doubler  A  cross sectional  area  of c i r c u l a r  • width  operator  matrix  D  diameter  of c i r c u l a r p e r f o r a t i o n  D  diameter  of d o u b l e r  E  Young's modulus  d  F(u )  internal  H  equivalent hole  n  J  second  2  k  elastic  elastic  geometric  N(r,s)  element  P  load vector  AP  increment  MPa.)  matrix  using displacements  u  n  matrix  stiffness  shape  stresses  coefficient  matrix  functions  i n the load v e c t o r  Q{u )  out-of-balance  r,-  r a d i u s from t h e c e n t e r (polar coordinates)  R  d i s t a n c e from c e n t e r  S  factored ratio  n  200,000.  of the d e v i a t o r i c  stiffness  stiffness  Kg{?' ) n  Dt  height  buckling  K (u ) n  perforation  d  plate  (for steel,  invariant  incremental  r  {Dd~D)t  forces in configuration n  K{u ) n  plate  of p l a t e  B(u ) n  given  Description  A  b  is  i n d i c e s a r e not i n c l u d e d .  Symbol  h  thesis  load vector  f o r one t i m e P -  F(u n )  of the h o l e  t o node t  of p l a t e t o c e n t e r  of h o l e d i a m e t e r  xii  step  of p e r f o r a t i o n  t o p l a t e width  0.9D/6  Description  Symbol S(u")  stress  S .S..,S,  deviatoric  t  thickness  of p l a t e  t,  thickness  of doubler  r  matrix stresses  plate  elastic  buckling  stress  of f u l l  elastic  buckling  stress  of p e r f o r a t e d  exact  elastic  buckling  ultimate in-plane doubler p l a t e  stress  capacity  plate plate  of p e r f o r a t e d  of p e r f o r a t e d  plate plate  with  of f u l l  plate  r„  ultimate  elastic-plastic  buckling  stress  r„  ultimate plate  elastic-plastic  buckling  s t r e s s of  y  yield  T  or T  ,T  x y  y z  ,T  u u" Au  in-plane  in-plane  y  T  shear  z x  shear  s t r e s s of s t e e l ultimate  ultimate  stress  of f u l l  plate  s t r e s s of p e r f o r a t e d  equilibrium  displacement  vector  approximate e q u i l i b r i u m displacement increment i n approximate e q u i l i b r i u m vector  a  factor  A  load  v  Poisson's  trm  mean  cr„  yield  vector displacement  factor ratio  (0.3)  stress stress  °~zzi°'yy} 'zz  normal  1*  doubler  0  plate  stresses  exact  n  perforated  (for steel,  300. MPa.)  stresses plate  capacity  restoration  xiii  factor  1  INTRODUCTION  1.1  Background Flat  in  rectangular  modern  buildings  subjected  to  normally  a  out  a  of  then  t o as  This  components allow  reduce  or  v e r y common s t r u c t u r a l component  offshore  requires  often  pipes  structures  shear  shear p l a t e s  s u c h as  utility  the  plain  and  effectiveness  structure.  are  pure uniform  referred  Cost of  plates  stress. or  results  s h e a r webs of or  ducting  of  plates  s p a c e and  more h o l e s  girder  webs.  weight  These  These shear  h o l e s are  are  being  t o p a s s t h r o u g h the  o v e r a l l s t r u c t u r a l weight. reinforced circular  or  often  webs.  use  i n one  are  These  shear  optimal  and  cut  holes  web  and  plates  with  c a l l e d perforated  shear  plates. Early shear  elastic  plates  stress  was  function  in-plane done by  over  that  the  hole  s i z e r a t i o D/b  out the  an  the  of  plate  the  doubler  unreinforced  The  stability  investigated  the  finite  the  elastic  by  the  applied  the  load  f o r b o t h c l a m p e d and  of  perforated  finite  Airy's  results  show  r a p i d l y with  width).  the  hole.  He  He  the  carried  radial  shear  with  found that  d e c r e a s e d about stress  the  the  hole,  from  that  plates  was  CHEUNG [ 2 ] , They  used  plate. perforated  ANDERSON and  e l e m e n t method t o e s t a b l i s h the buckling  a His  very  to p l a t e  around  slender  ROCKEY,  square  s t r e s s d i s t r i b u t i o n produced  increases  very  of  plate.  is significantly  perforated of  who  increased  diameter  reinforcement  but  Djb,  domain o f  (hole  stress  first  [1],  i n v e s t i g a t i o n i n t o the  addition  the  WANG  stress concentrations  circumferential  of  stress analysis  the  simply  plates  and  shear  relationship the  hole  s u p p o r t e d boundary  1  size  between ratio,  conditions.  They  also  investigated  reinforcement jfc.  In  experimental  model, that  model  on  the test  stability  Using  calculated loading.  of  t h e y were a b l e  By  in-plane  both c i r c u l a r  experimental  experimental  shear  finite  results as  restricted approximation  size number  be  Their  of  D/b>  they  due t o  shear  Rayleigh-Ritz  applied  f o r both  s h e a r a n d moment  this  buckling  T h i s method was  shear p l a t e  combined  simply extended loading  perforations.  two p a r t  terms  and REDWOOD  The method was l a t e r  p l a t e s with  ratio,  and  e l e m e n t program  analytical  study c a r r i e d out c o n c u r r e n t l y ,  for perforated  the hole  would  the e l a s t i c - p l a s t i c  plate.  boundaries.  and r e c t a n g u l a r  work.  so  t h e p o t e n t i a l energy of  t o determine  beam webs w i t h  by UENOYA  URENOYA a n d REDWOOD c o r r e l a t e d t h e i r an  enough  plasticity  inserted into a  model o f a s q u a r e  s u p p o r t e d and clamped  for  their  material  plate  stress distribution  minimizing  of the p e r f o r a t e d  rectangular  Since  small  the material  T h e s e s t r e s s e s were t h e n  to  o u t an  elastic  ratio  d e a l t with  the e l a s t i c - p l a s t i c  one q u a r t e r  plates.  t o h a n d l e an  hole  coefficient,  carried  p l a t e nor the a n a l y t i c a l  a 2-dimensional  system  a  shear  slenderness  were f i r s t  expression.  to  able  effects  energy  capacity  work t h e y  the  yielding.  combined  buckling  analytical  only  between  shear b u c k l i n g  perforated  was  to material  The  this  chose t h e p l a t e  neither  [3].  to  study  they  subject  relationship  shape a n d t h e e l a s t i c  addition  analytical  the  solution small  became used  results  and with  provided  perforations. greater  in  past  excellent However, 0.7, t h e  Fourier  series  o f t h e d e f l e c t e d shape p r o d u c e d u n e x p e c t e d  results.  2  the  than  with  For  the  the  clamped p l a t e they  ultimate  elastic  d e c r e a s e d , as The  stability  in  in  and  with the  buckling  been  result  perforation  that  had  found  1.2  size  rather  than  a significant  load capacity  increasing hole  size  of  there  the  circular  effect  on  the  p l a t e . They a l s o  note  is a considerable  developed  i n the  plate  increase  before  the  i s reached.  P u r p o s e and Scope In  the  research  of  this  S t r u c t u r a l A n a l y s i s program, parameter  study  perforations. capacities; plastic hole  The  plate  capacities. buckling to  buckling  only  an  the  estimation  the  third  considers  i s obtained  establish  buckling  by  of  and  convergence  size  and  optimum  ultimate  in-plane  the  rate  elastic-  hole  accuracy  refining  loading  parameter,  the  the  a  circular  separate  major parameters,  a n a l y s i s of  shape,  with  three  elastic  Incremental  i s used t o c a r r y out plates  considers  two  Nonlinear  [7],  shear  yielding,  f o r the  Finally,  of  the  elastic  e l e m e n t model  and  exact  in  elastic  capacity.  Included implementation program  However,  NISA83  analysis  The  analysis  t h e s i s the  square  in-plane  location.  order  on  buckling,  doubler  The  i s increased  hole  REDWOOD show t h a t any  s h e a r p l a t e has  ultimate  capacity  i n c r e a s i n g the  capacity  from UENOYA and a  by  expected.  amount of p l a s t i c i t y  ultimate  that  it  as  part  of  this  thesis is  of a c o l o r g r a p h i c s was  originally  was  later  post  developed  extended  development  processor  called  t o c h e c k the  to provide  NISA83 o u t p u t .  NISPLOT i s u s e d t h r o u g h o u t  simple  of  plots  the  the model n o d e s and  post  the  element  3  NISPLOT.  input  data.  processing  research meshes  and  to  for  provide  for  data  checks. of  In a d d i t i o n ,  information The  operating  i t i s used t o i l l u s t r a t e  provided  program system.  by t h e f i n i t e  currently  r u n s on a VAX  I t i s written  g r a p h i c s p a c k a g e DI3000,  element  some  analysis.  11/730 under  i n FORTRAN  by P r e c i s i o n  and summarize  77, making  Visuals [8],  t h e EUNICE use o f t h e  2  THEORETICAL BACKGROUND  2.1  THE ULTIMATE BEHAVIOR OF SHEAR PLATES The  theoretical  uniform two  shear  loading  along  types of behavior.  imperfections the  behavior  stress the  material  ultimate  defines  k = 9.34  shear  shear  f o r simply  supported  ultimate  elastic  buckling  shear  stress,  slenderness point  level  the p l a t e  that  Equation  the m a t e r i a l  i s considered  slender.  has  a lower e l a s t i c  so  i t s ultimate  ideal  stocky  of  ROCKEY  [2]  [2.1].  yield  stress  expression, point,  to  the  equation  i n terms o f  T h e o r e t i c a l l y , at t h i s  at exactly  t h e same  load  buckles.  The u l t i m a t e  g o v e r n e d by t h e u l t i m a t e  Illustration  will  criteria  y  shear  simple  i s obtained.  [ 2 . 2 ] i s a l s o used  which a p l a t e  considered  (t/b),  vield  plate  material a  initial  buckling  cr /\/3.  y  f o r d e f i n i n g the balance or t r a n s i t i o n  transition  above  is T =  into  shear  s t r e s s by e q u a t i o n  square  the  plate  to  by t h e l e s s e r o f  t o the v o n - M i s p s  buckling  equating  the  subjected  p l a t e has no  i s governed  stress  By  [2.2],  shear  s t r e s s or the e l a s t i c  According  material  the e l a s t i c  capacity  shear  of the p l a t e .  plates  t h e b o u n d a r i e s c a n be s e p a r a t e d  I f the square  i t s ultimate  ultimate  of square  material  buckling capacity  the  effect  and s l e n d e r  to define  the slenderness  stocky,  o r below w h i c h  capacity  of a s t o c k y  shear  stress.  is  governed  of s l e n d e r n e s s  by  is  plate  yield  stress,  equation  [2.1].  on f a i l u r e  p l a t e s a r e shown i n f i g u r e  5  i t is  plate  A slender  s t r e s s than m a t e r i a l  ratio  mode  [2.1].  of  CT  cr  9.35TT JE  _  y  2  12(1-0.3 )  y/3 0.2614  In  reality is  initial  imperfections  the  to  a  the  The  other on  However,  takes  the  initially  effects  transition  the  of  Now  place  never p e r f e c t l y f l a t  on  the  plate  surface  over a  range  deviation  here  perfectly flat  or  transition  i s not  unperforated  imperfect  With the  between y i e l d i n g the  d e g r e e of  purpose  imperfections an  p l a t e s are  i n the  single point.  depending  (2.2)  b  never p e r f e c t l y u n i f o r m .  loading.  longer  E  however,  loading  2  eccentricities  in  and  buckling one  perfect  t o examine  the  be  the  no mode  values,  condition. effects  analysis will  uniform  will  is  failure  from t h e  p l a t e with  conditions  of  slenderness  p l a t e . The  the  introduction  from  of  and  shear  of  assume  loading.  subject  of  The  future  studies. The similar point  effect to  to  of  For occurs  s t r e s s e s and  is  the  transition the  buckling [2.1],  around  region,  plate stress. no  capacities.  between  the  change  reduction  is  of  a circular  imperfections.  intermediate  the  of  introducing  distinguish  yielding. yielding  that  of  hole.  the  i s no  elastic  slender  i n the  of  There  load  plate  The  buckling  c o n t r o l l e d by  the  the  elastic  buckling  solution existed  Therefore,  s o l u t i o n s f o r the  6  material  result  ultimate  ultimate  analytical  and  capacity.  zone I I , t h e  Unlike  distinct  of In  of  this this  capacity  elastic-plastic  stress in  f o r the  material  redistribution net  r e f e r r e d t o as  is  a  significant  causes a  path.  in a p l a t e  longer  buckling  plates,  This  hole  zone II  ultimate  equation ultimate  elastic-plastic  buckling  c a p a c i t i e s i n zone II must be  F i g . 2.1: F u l l P l a t e , I d e a l E l a s t i c - P l a s t i c Behavior Table  2.1:  Definition  of  F i g . 2.2: P e r f o r a t e d P l a t e , Elastic-Plastic Behavior Plate  objective  of  this  research  p l a t e c a p a c i t i e s i n each zone. varying  is  required  zone  I  is  S l e n d e r n e s s F a i l u r e Modes  material yielding elastic-plastic buckling elastic buckling  I II III  or  numerically.  FAILURE MODE  ZONE  The  determined  the  material  By  properties  i s to  restricting only  f o r each p a r a m e t e r c h a n g e . determined  displacements only.  The  by  one The  restricting  elastic  identify  buckling  7  the  the  the  displacements  finite  element  material model  capacity  ultimate  capacity to  of  model of  in-plane  zone I I I  is  evaluated then  applying  performing  matrix. by  by  Finally  a full  However, lateral  a small  a bifurcation  load  increment  a n a l y s i s on  the u l t i m a t e c a p a c i t y  three dimensional in  load  order  t o the  the r e s u l t i n g  and  stiffness  i n zone I I i s e s t a b l i s h e d  elastic-plastic  ultimate  to observe the o u t - o f - p l a n e  i s a p p l i e d a t or near  model  the c e n t e r  8  behavior  analysis. a  of t h e p l a t e .  small  3  COMPUTER PROGRAMS  3.1  NISA83 Nonlinear  finite  some t i m e and  are  this  contains  study,  program  well  associated  element p r o g r a m s have been a v a i l a b l e  easily  suited  accessible. several  f o r the  with perforated  3.1.1  General NISA83 on  the,  University  development  of  the  NISA83 supported  VAX  d i r e c t i o n of  Dr.  p r e v i o u s ten  variety  A  determine applying final the  included  the  same  programs,  ultimate  the  problem  are  r e f e r r e d t o as  Hafner,  the  list  of  i n Appendix  is  a  Universitat  Dr.  Ramm and  and  designed  computers.  A  Dr.  to  be  number  program o p e r a t i n g  on  a l l c h a n g e s made t o  the  NISA83  C.  it is  equilibrium load  i n one  configuration,  equilibrium NISA83,  Civil  program  Baustatik,  frame  to get  analysis  final  equilibrium  Columbia  is  Increment  non-linear the  main  comprehensive  Time S t e p a  problem  years.  of  U.B.C.  In  the  Analysis)  The  i n s t a n d a r d FORTRAN 77  were n e c e s s a r y  3.1.2  11/730.  the  modifications  s u b r o u t i n e s are  British  fur  a  VAX.  of  Instituts  is written by  instablity  Structural  the  under  over  make  plates.  Incremental  Department's,  Sattele  elastic-plastic  which  in  Background  Engineering  Stuttgart,  p r o g r a m , NISA83 u s e d  special features  shear  (Nonlinear  installed  The  for  in to a s e r i e s  the  step. or  to  structure  by  T h e r e maybe more t h a n  one  several  load  smaller  possible  position  configuration.  follows  time  not  always of  the  load  Like  paths  other  d e f l e c t i o n path  increments.  steps  9  leading  These  to  non-linear by  breaking  increments  Each beginning and is  time of  the  stresses used  the  the  load  may  equilibrium 3.1.2.1  checks The  of  the  be  C o n v e r g e n c e  NISA83  NISA83  the  time  required  And  has  by  either  several  is  the  forces  increment  load-displacement  control  increasing  vector  by  some  iterations  of  the  defined  new  the  astablished.  D i v e r g e n c e  each  iteration  made f o r c o n v e r g e n c e o r  configuration  what  is defined  step  At  displacements,  before convergence to  completes  criteria  a separate problem.  D e p e n d i n g on  incrementing each  as  equilibrium  time s t e p  configuration  are  the  a l l known.  Within  solution  As  by  i s considered  time s t e p  are  end  or  constant.  step  used  the  a  d i v e r g e n c e of  to define  converged or,  within  if  the  time  the  new  s o l u t i o n has  step,  solution. equilibrium  diverged  are  as  follows: Convergence -  i f the change i n the d i s p l a c e m e n t 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 v a l u e ( 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 b a l a n c e l o a d v e c t o r i s s t i l l l a r g e r than i n c r e m e n t a l a p p l i e d vector  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 c o n v e r g e n c e r e q u i r e m e n t 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 option  load  the  analysis  10  can  be  the  continued  from an  any  last  known e q u i l i b r i u m  iteration  NISA83  the  updates  displacements converge  by  whenever  any  the  analysis  and  the the  of  the  one  contains  solution  restart  the  last time  constant  control  used  user  specify  of  each time  time  step,  until  while  where  the  As  increments specified problem  level  load  are  the  reduced is  the  divergent Problems  last  stops.  the  to  Thus,  restart By  file  redefining NISA83  the  time  step  most common The  method a l l o w s  to define  i s held  the  constant  i t e r a t e s to the is  end  remain  load in  higher  new  small  within  magnitude.  i s unable case  the  condition the  configuration  established.  the  each  If  to converge  known e q u i l i b r i u m  load load  problems  time  step. the  ultimate  load,  the  a  level  is  ultimate  ultimate  for  increments along  approaches the  than  the  throughout  i n c r e m e n t method works w e l l  applied  this  i s the  s p e c i f i e d in constant  program  For of  is  program  fails  configuration.  used  level  solution  load  the  that  solution. average  load  equilibrium  restarting  element programs.  nonlinearities  path.  level  The  constant  the  Normally  the  increment  load  the  of  Control  in finite  step.  end  criteria  solution  i t s task  and  from t h i s l a s t  the  new  configuration.  parameters  load  the  the  the  complets  I f at  convergence  if a  above,  c o n v e r g e n c e or d i v e r g e n c e The  load  However,  criteria  step  the with  known e q u i l i b r i u m  3 . 1 . 2 . 2 C o n s t a n t Load  to  file  program s t o p s o r  i s continued  The  satisfies  stresses.  of  configuration.  load capacity  to is step  an  the  equilibrium  defined and  of  the  as  the  load  of  step. are  encountered  with  t h i s method when  approaches a b i f u r c a t i o n point,  11  or  the  behavior  the  load  becomes  highly  nonlinear.  In  the  these  increment  produce  caused  further reduction  still  a  higher  solution using  this  3.1.2.3  time with  of  the  special  Arclength the  Thus,  convergence  stiffness  load  in  turn  resulting  to the  in  equilibrium  c a p a c i t y becomes v e r y  difficult  the  f e a t u r e s of NISA83  a r c l e n g t h t o be  applied load vector  displacement so  Control  time s t e p c o n t r o l .  initial  step the  These d e f o r m a t i o n s  system  ultimate  i n the  method.  One  specify  t o the  changes  i n the  Constant Arclength  Constant  small  large deformations.  deformations.  close  cases  time  step  the  plus  incremental  T h i s method  u s e d as  a  The  load  level  n o r m a l of  the  incremental  i s equal  Riks-Wempner lets  the  reference.  i s assumed a  field.  load vector  i s the  user  In  variable  each along  i s scaled within displacement  t o the  defined  each  vector reference  arclength.  This the  l o a d time s t e p ,  the  converge  ultimate t o an  arclength the  i s extremely powerful  load d e f l e c t i o n path.  constant than  method  solution mean  that  able  to  explained  once t h e  This even  specified  equilibrium solution.  load  i f the  i n the follow  and  increment  allows  the  the  i s reduced program to  shear  load path  for  load  with  level  obtain  has  gone  p l a t e problem,  to the  12  plate  level the  until  to  follow  when u s i n g  load  However,  s t i f f n e s s matrix  buckling the  before,  i t i s impossible  c o n t r o l l e d method,  applied  obtained.  load  As  in i t s a b i l i t y  as  is  higher  program  constant  an  unknown, is  equilibrium  negative. the  to  the  equilibrium an  the  program  ultimate  This is  buckling  capacity  and t h e n on i n t o  Iteration  3.1.2.4  In  t h e two time  the post-buckling region.  Technique step control  conditions  are  specified.  iteration  the  solution  convergence  requirement  select the uses if  methods above, t h e r e q u i r e d end  Both  methods assume t h a t  will  will  improve  be s a t i s f i e d .  one o r a c o m b i n a t i o n  and  iteration  each  eventually  the  I n NISA83  of the M o d i f i e d  S t a n d a r d Newton-Raphson  with  Newton-Raphson  methods.  The f i r s t  l e s s CPU t i m e b u t t h e s e c o n d method c o n v e r g e s the  tangent  iteration.  stiffness  The d i f f e r e n c e  illustrated  by s t e p i n g  Associated  matrix  i s changing  with each  time  step  more  rapidly  between t h e two i t e r a t i o n  thougth a time  the user can and  method rapidly  with  each  techniques i s  iteration.  step are three  configurations.  (1) i s t h e 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 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 d i s p l a c e m e n t s i s known.  and a l l  (2) i s t h e 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 t h e l o a d p a t h . O n l y t h e end c o n d i t i o n ( l o a d l e v e l o r a r c 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 The  procedure  i n between  (1) and ( 2 ) .  t o g e t from c o n f i g u r a t i o n  (1) t o  (2)  i s as  follows: Step  1  calculate  the unbalanced  Q(u ) = P l  Step 2  2  = Ke(u l)  f o r t h e time  step. (3.1)  -  formulate the tangent c o n f i g u r a t i o n (1)  K(u l)  forces  +  stiffness  Kg(u l)  13  matrix i n  (3.2)  Step  3  solve f o r the incremental  Au=[^(u )]" P 1  Step  4  1  displacements  (3.3)  2  f i n d the t o t a l displacement c o n f i g u r a t i o n (n)  vector  f o r t h e new  u = u + Au n  Step  5  calculate  (-)  1  the i n t e r n a l  3  f o r c e s i n c o n f i g u r a t i o n (n)  (3.5)  F(u ) = J [B{u )] S{u ) dv n  n  4  T  n  Vol  Step  6  check f o r c o n v e r g e n c e [3.1.2.1]  or d i v e r g e n c e ,  Step  7  c a l c u l a t e the remaining time s t e p  unbalanced  section  forces for this  Q(u ) = P - F{u ) n  Step  8  2  (3.6)  n  form t h e new t a n g e n t s t i f f n e s m a t r i x K(u ) ( S t a n d a r d Newton-Raphson) o r u s e t h e o l d s t i f f n e s s m a t r i x K^u ) ( M o d i f i e d Newton-Raphson) n  1  Step  3.1.2.5  9  s o l v e f o r t h e next d i s p l a c e m e n t i n c r e m e n t and continue with step 4 t o 9 u n t i l l convergence.  Applied  Figure iteration provide the  [3.1]  I n c r e m e n t and I t e r a t i o n show  techniques  an e f f i c i e n t  how and  the  i n - p l a n e and e l a s t i c - p l a s t i c  provides beginning provides region  an of an  beyond  The  efficient  various  the r e s t a r t  algorithm.  Algorithm increment  option  This algorithm  method o f r a i s i n g  into  constant  procedure  combined  The a p p r o a c h  the load l e v e l path.  t o follow the load path  the p o s t - b u c k l i n g load step c o n t r o l  to  i s applied to a l l  buckling analyses.  the n o n l i n e a r p o r t i o n of the l o a d accurate  are  controls,  to  the  It  also  in  this  range. i s combined w i t h  14  the m o d i f i e d  MODIFIED NEWTON-RAPHSON WITH CONSTANT LOAD  1/  Deflection  RESTART  Deflection  Fig.  3.1: A p p l i e d  I n c r e m e n t and I t e r a t i o n  15  Algorithm  Newton-Raphson  iteration  technique  part  of  the  analysis.  load  i s s p e c i f i e d to approximately  80%  of  the  The  estimated  confined almost  load  ultimate  to small  step  up  to t h i s  control  continues  from  was  arclength  to  the  load  time  to  follow  the  load  is  able  to  follow  stop  the  b u c k l e s and  order  and  yield  to  plastic  criteria  NISA83  to  most p r o b l e m s w i l l  changed the  so  that  Standard  behave  equilibrium  constant  the  analysis  configuration.  f o r up  to  95%  order  of  2%  the  the  Newton-Raphson  i s r e s t a r t e d and  of  the  The  ultimate  of  the  current  control  control  fails  to converge,  the  constant  time step  control  i s invoked.  This  S t a n d a r d Newton-Raphson path the  f o r the true  beyond  Nonlinear In  known  in  yielding is  load.  adequate  step  step  with  3.1.3  be  material  level.  combined  plate  last  first  the  hole  Then NISA83  increments  load  After  the  found  using  specified  the  i s combined w i t h  procedure.  load,  Since  program parameters are  iteration  method  capacity.  areas around  elastically  Then t h e  level  f o r the  i s used.  Mathematically  a n a l y s i s . This  post-buckling  used  method  the  the  region.  Material the  deformations  r e a s o n a b l e model o f  the  i s then  l o a d - d e f l e c t i o n path u n t i l  i n t o the  identify  identify  r e s t of  iteration  The  plastic the the  point  at which e l a s t i c  start,  for various  von-Mises y i e l d strains,  elastic  plastic  criteria  can  [3.7]  16  is  stress states,  criteria,  well  behavior be  deformations  used  accepted  as  a by a  of s t e e l s .  expressed  by  equation  * /y/3 = J 9  C _x _ x  x =  1 / 2 2  —  (3.7)  = y/l/2 (5| + SJ + S?) + r%, + rj, + r*  (7,: m  5  - °"im  =  *  m  NISA83  also  p a r a m e t e r s E,  allows  <r , and v a l o n g w i t h y  For  mild  steel  the  stress-strain  hardening figure  curve  is  the  below t h i s material  the  usual  s t r a i n hardening  a r e between almost  occurs at s t r a i n  horizontal.  levels  model  i s used throughout  material  modulus Eh •  0.002 and  0.02, Most  above 0.02,  then strain  a s shown i n  c o n s i d e r e d were e x p e c t e d t o  l e v e l no s t r a i n h a r d e n i n g  t h i s m a t e r i a l model g i v e  figure  to s e l e c t  [ 3 . 2 ] . S i n c e a l l of the p l a t e s  plastic" of  i f the s t r a i n s  effects  have s t r a i n s Thus,  the user  commonly  known  the a n a l y s i s .  as  i s considered. "elastic  Under u n i a x i a l  the s t r e s s - s t r a i n  diagram  ideal tension  shown  [3.3].  400 -  o  0.000  0.005  0.010  0.015  0.020  0.025  Strain m m / m m Fig.  3.2:  Initial  Stress-Strain  Diagram f o r M i l d  17  Steel  in  400-  300  -  200  -  o co CO  100  0.000  0.005  0.010  0.025  0.020  0.015  Strain m m / m m Fig.3.3: S t r e s s - S t r a i n Diagram f o r E l a s t i c Ideal P l a s t i c M a t e r i a l Model i n U n i a x i a l T e n s i o n The  values  of the parameters used  to define  the  material  properties are; cr =  300.  y  E=  200,000.  E h=  0.0 MPa.  v=  3.1.4  Element  MPa.  0.30  Library  NISA83 c o n t a i n s an e x t e n s i v e from  truss,  elements plane  beam,  and, strain  degenerate problems  presented  for  curved  element  isoparametrics  the  two  forces  elements  here.  18  for  to,  4,8,9 o r  and p l a t e used  Elements  in  stress  or  a  family  of  16  nodes  bending. this  range  longitudinal  with plane  analysis  elements w i t h  membrane  library.  beam e l e m e n t s  2-dimensional  shell  involving of  and  4 and 8 node  plate  description  MPa.  A  research  for short is  3.1.4.1  2-Dimensional  For a  the  simple  two  plane  element  stress  reliable  element  is  perforated  plate  The  are  coordinates  for plane the  plate  capacities  T h i s 4 node i s o p a r a m e t r i c  documented a s b e i n g  with a doubler  well-conditioned  stress  problem  2-dimensional  plate,  and  [10],  domain  determine  and This  of  the  the  ultimate  yield capacity.  table  functions  of d o u b l e r  i s used.  to d i s c r e t i z e  element  in  analyses  element  results  used  in-plane plate  S t r e s s Element  dimensional  i s thoroughly  giving  given  Plane  i s d e r i v e d from [3.1].  used to  interpolation  As  of  local  the  interpolation  for a l l isoparametric  f o r mapping  the  the  the  r-s  element  from  element  displacement  elements  the  global  x-y  also  for  the  coordinates,  functions  and field,  equation  [3.8,3.9],  (3.8) (3.9)  In  order  formulation,  to maintain 2x2  T h i s order  stiffness  integral  a  constant.  coordinate element  energy  Gauss n u m e r i c a l  element.  is  the  of  integration  provided Some  constant  and  the e r r o r  numerical  integration  finite  i s used  element  with  this  element  the c o o r d i n a t e t r a n s f o r m a t i o n  matrix  higher  is refined  integration  the  e x a c t l y e v a l u a t e s the  order error  transformation matrix  mesh  bound of  the  i s introduced  if  the  i s not c o n s t a n t . However, i f t h e  trasformation  becomes n e g l i g i b l e .  is sufficient  to ensure  19  matrix Thus,  approches this  order  c o n v e r g e n c e as  a of the  mesh of  is refined the complete  [10].  and p r e s e v e t h e e n e r g y element  stiffness  bound.  A full  matrix i s given  in  derivation reference  Table  3.1: I n t e r p o l a t i o n F u n c t i o n s f o r B i l i n e a r P l a n e S t r e s s Element  Element Node No.  1 -r  1+s  1-s  1  1  1  1  -  1  -  2  -1  1  -  1  1  -  3  -1  -1  -  1  -  1  4  1  -1  1  -  -  1  3-Dimensional  Plate  a l l 3-dimensional order degenerated  that  inorder  to  analyses. due  to  and mesh i s a l s o  used  for  the  BATHE [12]  shell  in results  t o t h e nodes,  element However,  modelling some  in-  By a p p l y i n g t h e a p p r o p r i a t e the 3-dimensional  of the i n - p l a n e y i e l d  used  model  [3.2],  The t e r m  is  formulated.  element  is  based  from  functions,  called  "degenerated  on p l a t e  or  shell  the general 3-dimensional  21  used  "Lagrangian  cubic  formulation are given  element"  The c l a s s i c a l  is  capacities.  f o r t h e 16 node element  element  developed  i s used.  isoparametric plate  calculations.  interpolation  polynomials", table  analysis,  t o i n - p l a n e d i s p l a c e m e n t s and c a n be s u c c e s s f u l l y  the c a l c u l a t i o n The  element  f o r 3-dimensional  same e l e m e n t  conditions  restricted  shell  variations  plane ultimate capacity  Element  and some 2 - d i m e n s i o n a l  plate  results  eliminate  this  boundary  Shell  t h e 16 node, b i c u b i c reliable  changes,  in  4.0  1 +r  provides  for  *  s  For  found  I n t e r p c ) l a t i o n E' u n c t i o n  r  3.1.4.2  higher  Coordi nate  refers  f o r m u l a t i o n of theory. field  This  t o t h e way a  finite  theory  is  e q u a t i o n by making  some  restricting  ignoring  higher  equation  on  discretized  order  the  shell  surface.  formulation  approach.  No  the  domain.  behavior  is  dimensional  the  field  structural  system  model. A schematic  small it  in this more  to  a  formulation  system  than  is  is  to  the  a  based  equations.  Therefore,  i s discretized  directly  The  classical used  to  2-dimensional  on  the  any by a  then  domain.  assumptions a r e  equation  and  2-dimensional  2-dimensional  general  field  comparison  displacements  A structural  or displacement  3-dimensional  Instead  (ie.  terms) t o reduce  u s i n g the elements  degenerate  reduce  assumptions  full  3-  3-dimensional finite  element  o f t h e two f o r m u l a t i o n s i s shown i n  f i g u r e [3.6] Geometrically  nonlinear  deflection  theory  displacement  terms t h a t a r e n e g l e c t e d  allowed  the user  Lagrangian Lagrangian brief  local  large  either  deflection  Updated Lagrangian  coordinate  At  stresses,  known  in  throughout  this  second  in linear  formulation.  order  analyses.  Here  large strain NISA83  or  the  Total Updated  f o r a l l buckling analysis  Formulation  the  beginning  called  strains  the  of each time  step  and  a  the  element  i s updated  A l l information  displacements  used  A t t h e end o f t h e t i m e  22  element  incremental  system",  configuration.  and i t i s  on t h e  element  and t h e e q u i l i b r i u m  configuration  the increment.  with  i s based  the " c o r o t a t i o n a l  known e q u i l i b r i u m  about  the  using  t h e Updated L a g r a n g i a n  system t r a c k i n g  system,  last  are solved  follows.  coordinate  the  consider  f o r m u l a t i o n i s used  deformations.  to  to apply  description The  which  problems  as  a  are  reference  s t e p , when a new  equilibrium strains  configuration  and  stresses  configuration, With system  ready  i s established,  a r e updated or transformed  t o s t a r t t h e next  t h i s large  deflection  s h a r e s t h e same r i g i d  stresses  or  point  element.  Again  element matrix  integration  thickness thickness  direction  integration  reference  if  the  The Simpson provides  r a t i o of the p l a t e  this  new  corotational  to  t h u s any the  last  of the element. i n t h e r - s p l a n e and a  of i n t e g r a t i o n  integral  i s a constant.  t h e element  are with  system,  increment.  through the thickness  t h i s order  stiffness  theory  configuration  4 x 4 Gauss n u m e r i c a l Simpson  time  to  body modes a s t h e e l e m e n t ,  strains calculated  known d e f o r m e d e q u i l i b r i u m A  the coordinate  i s used w i t h  exactly  coordinate  numerical  23  than 1 0 .  this the  transformation  integration  reliable results  i s greater  evaluates  5  i f the  i n the  width  to  10 In Fig.  local  space  3.5: B i c u b i c ,  (projection Isoparametric,  on  to  the  plane  Degenerated,  24  Plate  t=0) Shell  Element  Table Elem Node No. 1  3.2:  Interpotation  s  1  1  for Bicubic  Element.  I n t e r p o l a t i o n F u n c t i o n * 256  coc r d . r  Functions  Fact  1+r  3r+1  3r-1  1-r  1 +s 3s+1  3s-1  1 -s  1  1  1  1  -  1  1  1  -  2  -1  1  -  1  1  1  1  1  1  -  3  -1  1  -  1  1  1  -  1  1  1  4  1  1  1  1  1  -  . -  1  1  1  5  1/3  9  1  1  -  1  1  1  1  -  6  -1/3  -9  1  -  1  1  1  1  1  -  7  -1  1/3  9  -  . 1  .1  1  1  1  -  <  8  -1  "1/3  -9  -  1  1  1  1  -  1  9  "1/3  -1  -9  -  1  1  -  1  1  10  1/3  -1  9  1  -  1  -  1  1  1 1  1  -1/3  -9  1  1  -  -  1  12  1  1/3  9  1  1  -  1  -  13  1/3  1/3  81  1  -  1  1  -  14  -1/3  1/3  -81  -  1  1  1  -  15  -1/3  "1/3  81  -  1  1  -  1  16  1/3  -1/3  -81  -  1  -  1  -  <  <  •  1  25  -  <  <  -  -  CLASSICAL  CONCEPT  DEGENERATION  Assumptions  Analytical reduction _3_dim—— 2 dim over thickness  Assumptions  D i s c r e t i z a tion 3 dim  point  numerical ^  over  volume  Discretiza tion 2 dim — p o i n t numerical j* over  surface  Dis-  place mentL*. solution  3.6:  C o m p a r i s o n of t h e C l a s s i c a l C o n c e p t and D e g e n e r a t i o n F i n i t e Element F o r m u l a t i o n  26  NISPLOT  3.2  3.2.1  General As  commonly  p r o g r a m use  c a n  t h e  be  U . B . C ,  c o n c i s e  f o r m .  r u n n i n g  c o l o r  g r a p h i c  d i m e n s i o n a l t o  can  a n y be  p r o c e s s  i s  w r i t t e n o r  was  element  d e v e l o p e d  Department from  c h e c k i n g  a n a l y s i s  o f  t h e  a  d a t a  used  element  s t r e s s e s  i n t o  i n p u t  was  t o  C i v i l  NISA83  i t  s t a n d a r d  FORTRAN  77  DI3000  NISPLOT  r u n s  under  t h e  I t  on  t h e  amount  of  s t o r e d  on  VAX  i s  EUNICE  o p e r a t i n g  d i s k .  On  t h e However  t h e  n a t i v e  4.1  a s  a  mesh,  3 -  ( d i s p l a y e d  t o  s t o r a g e  i n  system, o t h e r a l l  V i s u a l s  i n o n l y  h a n d , i n p u t  o f  t h e  system  o f  one  o b j e c t  S i n c e  t h e  p r o c e s s  t h e  EUNICE  NISA83 a n d  o n l y  o u t p u t  27  i s  o p e r a t i n g  b e c a u s e  t h e  o f  f i l e s  g r a p h i c s  t h e  C i v i l  a r e  a s  s t o r a g e  c a n  a be  E n g i n e e r i n g  c o n v e r t i n g  under  UNIX  r e q u i r e s  l i b r a r y  i s  a  V A X  system  l i b r a r y  l i b r a r y runs  makes  C o l u m b i a  w h i c h  a n d  d i s k .  a n d  i s  [ 8 ] ,  s u b r o u t i n e  o n l y  r e q u i r e d  c o m m e r c i a l  B r i t i s h  EUNICE  so  p r o g r a m  l a n g u a g e  t h e  form  a r e  T h e  DI3000  system  c u r r e n t l y  m o d u l a r t h a t  e n v i r o n m e n t ,  VAX-VMS  11/730  i n  U n i v e r s i t y  V A X . T h e  d i s k  p r o g r a m i n g  P r e c i s i o n  n e s s e s a r y  l a r g e  t h e  by  a n d  e f f i c i e n t l y .  s u b r o u t i n e s  EUNICE  i s t h e  r e s t r i c t i o n s  on  g e n e r a l  m o d i f i c a t i o n s  a n d  p a c k a g e ,  t o  v e r y  q u i c k l y  g r a p h i c s  s y s t e m .  f o r  element  o u t  s t a n d a r d  e m u l a t o r .  a  subsequent  c a l l s  D e p a r t m e n t  t h e  o u t p u t  a n  f i n i t e  NISPLOT  p r o d u c i n g o r  i n  i n  o p p o s e d  u s e d  A f t e r  s h a p e s ,  w r i t t e n  11/730  o f t h e  p o s t - p r o c e s s o r ,  c a r r i e d  s o f t w a r e  was  a n a l y s i s .  c h a n g e s  t o  p r o g r a m  by  c o l o r s ) .  NISPLOT t h a t  T h e  NISPLOT  a n  g e n e r a t e d  c a p a b i l i t i e s t o  d e f l e c t e d  seven  o u t p u t  l e n g t h y .  g r a p h i c s  b e f o r e  up  known,  v e r y  c o l o r  E n g i n e e r i n g , more  Description  s t o r e d t h e  VMS  t o  t o on  t h e t h e  VAX-VMS EUNICE  compat i b l e . Like  any  subroutines device  other  program t h a t makes c a l l s  of DI3000, t h e  driver  At  the C i v i l  to  a Seiko  to c r e a t e  Engineering  linked  the  device  Graphics  t o any  other  Corp.  dependent  l a b the  Graphics  the d e v i c e  driver  input  data  displacement, "plot.str" three  of  and  the  input one  specified  Unfortunately, is  given  validity  when of a  following  of  the  user  prompted,  file  When g i v e n  files  exist the  The  are  The  can  files. and  The  prompt  must-be as  but  the  user  that  program  user  one, the  if  the  two  or  geometry  also  for  t o see it  not  and all file  i f the  does  correct f i l e  does  a  geometry,  "plot.dis",  program c h e c k s  sure  the  output  supply  when r e q u e s t e d ,  linked  DI3000.  prompts the  "plot.geo",  user  be  image c o u l d by  NISA83  module.  IRIS-1200, and  The  supported  default  stress files  respectively.  must a l w a y s be files  files.  executable  Inc.  WX4636R p l o t t e r .  graphics  to the c o r r e c t  p r o g r a m can  NISPLOT i s s e q u e n t i a l l y s t r u c t u r e d and the  the  o b j e c t module i s l i n k e d  D-SCAN GR-1104, S i l i c o n  Watanabye I n s t r u m e n t  to  not. type  check  the  plot  the  type.  a l l three  files  as  input,  NISPLOT w i l l  frames: 1)  t h e mesh nodes and  node  numbers  2)  t h e e l e m e n t mesh and e l e m e n t numbers w i t h e l e m e n t g r o u p i n a new color.  3)  a solid color stress f i l l isoparametric plate s h e l l  4)  3 - d i m e n s i o n a l p l o t of t h e d e f l e c t e d shape w i t h 2 c o l o r s , one f o r t h e n e a r and one f o r t h e f a r s i d e , of t h e 16 node i s o p a r a m e t r i c p l a t e s h e l l element.  28  of t h e 16 element  each  node  5)  The  3 - d i m e n s i o n a l d e f l e c t e d shape w i t h a seven c o l o r s o l i d f i l l of t h e 16 node i s o p a r a m e t r i c plate s h e l l element  program  mesh and  the  elements  title  element  or  3.2.2  plate  with  level  First maximum range  calculated  and is  level.  Next  follows.  Each  16  Each  shown i n f i g u r e cubic  contained  within fills  these  with  4x4  other  deflected  color  sub-region  for  of  the  routines  shape  stress  Gauss  16  of  i s consistent  the  4x4  with  fill,  the  are  16  node  integration.  the  of  von-Mises  l e v e l s and  searched  points.  to determine  one  s u b - r e g i o n s are  Calculation functions at  a sub-region  the  sub-region  area.  of  the  29  defined  in  gauss  routine  to  each  elements into  integration  these p o i n t s  given  this  i s subdivided  Gauss  by  25  nodes  16  integration  as  using  [3.2],  the  as  point.  i s done  table  selects  the  Then  assigned  p r o c e e d s through a l l the  contains  node  von-Mises  stresses.  a color  element  16  the  Gauss i n t e g r a t i o n are  the  all with  stress  von-Mises s t r e s s  the  work f o r any  supported  isoparametric  interpolation the  and  six  region  [3.9].  examining  color  at  routine node  b o u n d a r i e s of  the  that  element  into  the  sub-regions. The  each  minimum v a l u e s  divided  only  element  Routine  color  a l l the  shape w i t h  nodes,  The  stresses,  element  fills  a  and  NISA83 l i b r a r y .  are  shell  Fill  routine  elements  and  element  routines  element  deflected  Stress This  universal  the  dependent,  isoparametric  f o r p l o t t i n g the  i n the  plotting  fill,  stress  are  contained  involving solid  routines  By  point correct  6  <5>  1—Q 3  o—t-  I  +  + I 6  17  19  1 3  1 4  7  O  21  20  Oi  +  -I-  7  16  22  2  15  4  - - T -  + 8  O  <V1 1 5  1 6  -4.-4--  ^ 23  25  +  +  6"  I  10  o  +  +  11  1  9  14  12  13  4>  10 O #  Global  Nodes  S u b d i v i d e d E l e m e n t Nodes (25 n o d e s , 16 r e g i o n s )  -f- Gauss  Integration  Points  Fig 3.7: S u b d i v i s i o n o f t h e 16 Node I s o p a r a m e t r i c E l e m e n t 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 A c c o r d i n g t o the S t r e s s L e v e l a t the Gauss I n t e g r a t i o n P o i n t i n the Sub-Region.  30  3.2.3  Visible The  Surface  Plotting  s u b r o u t i n e VISBLE  improve  the  quality  distinguishing plane  and  given  surface  detect  i f the  The stress element  any  surfaces that are v i s i b l e By  the v i e w i n g  t o p or bottom  r o u t i n e and  calculates  of  the  solid  The  the the  solid  fill  the v i e w i n g  two  the  dark light  t o p and The  fill  bottom  routine  more s o p h i s t i c a t e d of t h i s  f u t u r e work  then  easily  from  both  the v i e w i n g  into  subroutine  By  calling  using a d i f f e r e n t  F o r complex  algorithm i s required, JENSSEN  i n which  by  calling  fill  pattern  convex  this  top  results.  surfaces, but  bottom  visible  Similarly,  top  negative  all  illustration  then  the  filling  the  u,v  point(1)-  if i t is plane.  of  passed  a l l the v i s i b l e  simple  the  coordinates  filling  distinguished.  the  before  is positive  and  to  viewer.  converted The  any  able  surface are  result  w e l l f o r the  problem.  thesis.  in this  then  plane,  surfaces a clearer works  is  i t i s p o s s i b l e to create p l o t s  r o u t i n e t w i c e and  s h a p e s of t h e p l a t e  scope  b l u e and blue,  surfaces are  stress  on  of  v e c t o r s formed by  I f the  r o u t i n e t w i c e , once  with  surfaces with  for  on  normal  routine just  system.  of two  viewing  to the  x,y,z  by  i n the  is called  global  to  plot  subroutine  a sub-region  viewing  bottom s u r f a c e i s v i s i b l e  surfaces  the  local  point(3)-point(2).  is visible  fill  These p o i n t s a r e  the c r o s s product  p o i n t ( 2 ) and surface  the  the  surface i s v i s i b l e  is filled.  subroutine.  coordinates  the  s u r f a c e s u b r o u t i n e VISBLE  sub-region  this  examining  plane  t h r e e or more p o i n t s t h a t l i e on to  r o u t i n e that attempts  three-dimensional  that are not.  in  hidden fill  of  between  those  i s a simple  deflected however,  i s beyond  a the  [6] p r o v i d e s a good r e f e r e n c e f o r  area.  31  3.2.4  Metafiles A metafile i s a device  on  the  host  reference. and  creates  stored  and  requests that  i s that  which i s s t o r e d  file  they  for future  c a n be r e c a l l e d  access  a Metafile  called  translator  to  the  NISA83  be c r e a t e d ,  "NISAPLOT.MFL".  NISPLOT A l l the  frames drawn by NISPLOT on t h e c u r r e n t d e v i c e a r e t h e n file,  frames,  as w e l l as a p p e a r i n g  or p i c t u r e s ,  on t h e v i e w i n g  device.  a r e s t o r e d i n the standard  the-DI3000 g r a p h i c s p a c k a g e .  closes  When NISPLOT f i n i s h e s  format  running i t  the f i l e .  If  a  modified. reason  metafile  is  NISPLOT p l o t s  for this will  manipulations  detailed Users'  description Manual  explanation To  recall the  invoked  (ie.  the t i t l e  the  only  become a p p a r e n t  that  c a n be  from  pictures  i n the f i r s t  done  [8] under t h e c h a p t e r  pictures  from  the  By d i v i d i n g  further  with  the  sequence  is  frame.  The  d i s c u s s i o n of translator.  c a n be f o u n d  A  i n t h e DI30.00  "Metafiles".  Only  a  brief  i s outlined..  the m e t a f i l e ,  DI3000 m e t a f i l e t r a n s l a t o r mtrans.seiko).  plotting  after  of the t r a n s l a t o r  to define a viewing  attributes.  requested,  of p o s s i b l e m a n i p u l a t i o n s  device,  or  permanent  t o r u n NISPLOT o r h a v i n g  opens a new f i l e  in this  These p l o t  user  file  by anyone u s i n g t h e DI3000 m e t a f i l e  t h e user  subsequent  the  plot  files.  When  the  as a  The i d e a of t h e M e t a f i l e s  having  output  of  and may be kept  manipulated  without  independent  NISAPLOT.MFL, t o a new  linked  t o the device  The m e t a f i l e t r a n s l a t o r  allows  is the  p o r t a n d window and t h e n  r e q u e s t any o f  Metafile  these  be  drawn  in  the screen of the viewing  f o u r v i e w i n g p o r t s and r e q u e s t i n g t h a t  32  viewing  d e v i c e i n t o two  different  picture  be  drawn  i n each p o r t ,  since  the t i t l e  will  not appear  a compound p i c t u r e i s  was o n l y  drawn on t h e f i r s t  subsequent  compound p i c t u r e t h e u s e r  port  f o r the f u l l  device  screen  drawn.  This  picture  i n s t e a d of i n each  One  last  causes the t i t l e  and t h e n  However,  p i c t u r e generated, i t  i n any o f t h e view p o r t s .  the  generated.  To p l a c e  the t i t l e  can d e f i n e a select  t o a p p e a r under  new  on  view  p i c t u r e one t o be the t o t a l  compound  sub-picture.  f e a t u r e 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 t o d e f i n e t h e m e t a f i l e , v i e w p o r t s , windows and even t h e order  o f p i c t u r e s drawn c a n be s e t up i n a l o g  Metafile  translator  command  source.  in  sequence. T h i s  i s a c t i v a t e d , the user  The t r a n s l a t o r source  file  then  assigns  execute these  to display a series  SET  MF  SET  W  1  ( - 1 . 0  1.0  SET  W  2  ( - 1 . 0  1.0  - 0 . 9 5  SET  V  1  ( - 1 . 0  0 . 0  0 . 0 5  SET  V  2  (  1.0  0 . 0 5  1.0)  SET  V  3  ( - 1 . 0  0 . 0  - 0 . 9  0.05)  SET  V  4  (  1.0  - 0 . 9  0.05)  SET  V  5  ( - 1 . 0  1.0  - 1 . 0  SET  DEFAULT  SET  DEFAULT  0 . 0 0 . 0 W  - 1 . 0  file  as a  commands  of  results.  1.0) 0.95) 1.0)  1.0)  2  BOX ON  P  2  V  1  NOEJECT  DRAW  P  3  V  2  NOEJECT  DRAW  P  4  V  3  NOEJECT  DRAW  P  5  V  4  NOEJECT  SET  the  NISAPLOT.MFL  DRAW  DRAW  this  When  c a n be s e t up t o draw p i c t u r e s f r o m  s e v e r a l M e t a f i l e s i n sequence  1  will  file.  BOX O F F P  1  V  5  W  1  Fig. 3.8: L i s t i n g o f t h e M e t a f i l e S o u r c e F i l e "mtr.log" t h a t R e s u l t e d i n t h e Compound P i c t u r e i n F i g u r e [ 3 . 9 ]  33  Fig.  3.9:  Compond  Picture  using  a  METAFILE  3.2.5 F l o w  Charts  Figure the  [3.10] shows a f l o w c h a r t  d i f f e r e n t frames p l o t t e d . F i g u r e  flow chart  (  o f NISPLOT showing how  o f NISPLOT w i t h  respect  [ 3 . 1 1 ] shows a more  subroutines  detailed  interact.  START)  PLOT U N D E F L E C T E D S H A P E  DEFINE INPUT  IN 3 - D T H E N D E F L E C T E D  FILES  S H A P E WITH O N E C O L O R SOLID FILL  PLOT A N D NUMBER N O D E S  PLOT A N D NUMBER E L E M E N T S IF (3dstr)  NO  PLOT U N D E F L E C T E D S H A P E IN 3 - D T H E N D E F L E C T E D S H A P E WITH SOLID C O L O R  PLOT E L E M E N T S WITH  S T R E S S FILL  SOLID C O L O R S T R E S S FILL  Fig.  3.10 Flow C h a r t  of the Frames P l o t t e d  35  to  By NISPLOT  ( START )  J^d5plt),  /SETUP/  /READNO/  /READNO/  VIEW  MAXMIN  ADDDIS  T  r  DRAY  PLTELE (  VIEW  PLTNOD  - NODNUM  PLTHED  PLTHED  ADDDIS  /READEL/  FILLEL  •  PLTELE  ELENUM  VISBLE  < E N D )  >  DRAY ELENUM  PLTELE  r  PLTHED  ELESTR ( DATIN ELESTR  1 *  PLTELE  SHAPE  -  PLTELE  VISBLE DRAY  LEGEND  (eof.ne.  NO  LEGEND  Fig.  3.11:  Flow C h a r t  of t h e S u b r o u t i n e P r o g r a m NISPLOT 36  Interaction  i n the  Descriptions  of  subroutines  in figure  [3.10]  Description  Subroutine SETUP  asks the o p e r a t o r f o r the input t h e g e o m e t r y , d i s p l a c e m e n t s and these f i l e s  READNO  reads  MAXMIN  d e t e r m i n e s t h e maximum and t h e node c o o r d i n a t e s  VIEW  s e t s t h e frame v i e w i n g a t t r i b u t e s window, view p o r t e t c .  PLTNOD  marks t h e  NODNUM  numbers the  PLTHED  p l o t s the heading at  the  READEL  reads  node  PLTELE  s e t s t h e 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 e a c h e l e m e n t and c o l l e c t s t h e node numbers f o r e a c h l i n e one a r r a y , r e a d y f o r DRAY  the  x,y,z  coordinates  node p o i n t w i t h  i n the  node p o i n t  element  line  by  the  nodes  minimum g l o b a l d i m e n s i o n s i e . 2-D  or  3-D  of  view,  "+"  marks b o t t o m of  the  page  numbers in  DRAY  draws t h e  ELENUM  numbers t h e  READST  reads  ELESTR  s e p a r a t e s t h e 16 node i s o p a r a m e t r i c e l e m e n t i n t o 16 s u b - r e g i o n s w i t h 25 n o d e s , t h e n f i l l s e a c h of t h e subr 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 a t the Gauss p o i n t i n the s u b - r e g i o n  DATAIN  initializes  SHAPE  c o n v e r t s t h e 16 node i s o p a r a m e t r i c e l e m e n t t o a 25 e l e m e n t u s i n g t h e e l e m e n t shape f u n c t i o n s f o r interpolation  VISBLE  c h e c k s t o see i f t h e s u b - r e g i o n in the v i e w i n g plane  i n the  given  of  f i l e names c o n t a i n i n g s t r e s s e s , and opens  elements at element  PLTELE their  mid  point  stresses  v a r i a b l e s used  i n ELESTR  37  t o be  node  plotted is visible  Subroutine  Description  LEGEND  writes the s t r e s s of t h e frame  legend  i n t h e upper  ADDDIS  s c a l e s and a d d s t h e n o d e a l d i s p l a c e m e n t s o r i g i n a l nodal coordinates  FILLEL  s e p a r a t e s t h e 16 node i s o p a r a m e t r i c e l e m e n t i n t o 9 s u b r e g i o n and f i l l s e a c h s u b - r e g i o n w i t h one o f two c o l o r s d e p e n d i n g on t h e r e s p o n s e from V I S B L E  38  right-hand corner t o the  4  PLATE  ANALYSIS  Numerical square of  analysis  p l a t e s with c i r c u l a r  t h r e e p a r a m e t e r s on  For  was  the  first  two  in-plane, e l a s t i c  buckling  capacities  was  plate) only  p l a t e t h i c k n e s s was ) was  ratio  ratio,  as d e s c r i b e d  the  standard  1000  mm  which  the  x  was  1/98.7,  (t/b  plate, 10 mm.  very  the  and  the  the  effects  shear  hole  plates.  location)  ultimate  For  the  elastic-plastic  third  parameter  in-plane ultimate  given  by  The done  the  concentric 0.90b.  (the  capacities  hole  geometries  and  balanced  slenderness  o u t s i d e dimensions f o r  the a n a l y s i s ,  equation  balance  [2.2].  ratio  This  dealt with  were  f o r the  1000 of  full  slenderness  the  to  combined  mm  x  1/100,  plate  ensured  of that  material  and  modes.  standard The  the h o l e was  of  the  first  square  plate  diameter  of  location  moved  parameter, described  t h e h o l e was  parameter,  about  the  models were u s e d d u r i n g  hole  the  surface. the  study  size,  above  was  with  v a r i e d from center Many of  the  a  0.156  of a  0.26  different hole  size  location.  In o r d e r geometry  ultimate  thickness  This provided a plate slenderness  hole. For  diameter  hole  used throughout  investigation  on  ideal  i n s e c t i o n [ 2 . 1 ] . The  analyses  failure  s e l e c t e d so t h a t t h e  c l o s e to the  c l o s e t o the  resulting  buckling  or  b u c k l i n g , and  were d e t e r m i n e d .  width  and  (hole size  of  standard  determined. The  to  of  p e r f o r a t i o n s to e s t a b l i s h  parameters  of a d o u b l e r  on a s e r i e s  the u l t i m a t e c a p a c i t i e s  ultimate  shape  performed  to determine on  the  significance  the c a p a c i t y of  capacities  were c a l c u l a t e d  the  of n o n l i n e a r m a t e r i a l  perforated  plate,  f o r each v a r i a t i o n  39  of  three the  two  major p a r a m e t e r s . plane y i e l d  capacity,  elastic-plastic yield the  capacity hole  capacity plane  the  elastic  buckling  buckling  capacity  on t h e p l a t e c a p a c i t y . was u s e d  ultimate  4.1  capacity  The u l t i m a t e  due t o t h e p r e s e n c e  elastic-plastic  together  buckling  The  first  parameter  investigated  ultimate  plate  capacity  and  elastic-plastic  diameters  from  4.1.1  Plate The  f o r the u l t i m a t e  square  By by the  i n section  Finite  loading  out-ofFinally,  calculation  for i t s effect a  of  on  centrally  in-plane,  6=1000 mm  diameters.  D/b,  varied  the  located  elastic  buckling, for  hole  r=l0 mm,  was  The  ratio  from 0.15,  of  hole  0.2,  0.3,  constant  to the  [3.1.3], Model g e o m e t r y and l o a d i n g , a n d  boundary c o n d i t i o n s ,  t o be a n a l y s e d .  (see f i g u r e  and  p a r a m e t e r s were h e l d  symmetry o f t h e p l a t e  appropriate  p l a t e needed  plate  Element  observing  applying  hole  thickness,  0.4,...to 0.9. A l l m a t e r i a l  4.1.2  with a  c a p a c i t i e s were o b t a i n e d  plate,  concentric  to plate  specifications  buckling  Geometry  with  diameter  elastic  0.156 t o 0.96.  standard  analyzed  y i e l d i n g around  i n the e l a s t i c  was t h e s i z e o f  buckling  in-plane  capacity.  Size  Results  The  of the p e r f o r a t i o n .  V a r i a t i o n of Hole  perforation.  and t h e u l t i m a t e  of m a t e r i a l  t o e s t a b l i s h t h e change  two f a c t o r s were c o n s i d e r e d  the ultimate i n -  were c a l c u l a t e d .  i n d i c a t e the i n f l u e n c e  stiffness  the  F o r e a c h model c o n f i g u r a t i o n ,  only  The two d i a g o n a l s  [ 4 . 1 ] ) a r e two a x e s o f symmetry  and g e o m e t r y . F o r t h e s y m m e t r i c  40  buckling  one q u a r t e r  of  of the square f o r the  modes t h e r e  plate will  Fig.  4.1:  Perforated  Plate  Showing  1/4  F.E.  Model  be  no  rotation  perpendicular  about  to  these  axes,  each other  and  these  two  axes  for a l l deformations  due  remain to  this  loading. As a  3x3  shown  grid  i n f i g u r e [4.2]  was  used  e l e m e n t mesh was The the  used  f o r a l l the  conditions  load  outer  restricting  edge the  and  the  displacement  two field  element  mesh was  used  plate  ultimate  in-plane  capacity.  was  quarter  separated  Nodes  were  spacing  concentration gradients  the  concentric  then p l a c e d by of  were t h e  plate. hole  desired.  axes to  f o r the  slices along  equation  by  highest  and  same 9  of  suit  3-dimensional  imposed a l o n g  symmetry.  Then,  i n - p l a n e movement o n l y a n a l y s i s of  lines  [4.1].  nodes a r o u n d t h e  This  deg.  This  the  a  radial  material yielding  was  by the  plate lines.  proportional  provided  p e r f o r a t i o n where  the  perforated  s e c t i o n of  spaced  using  in  analyses.  For  10 e q u a l l y  these  element  model were v a r i e d t o  p l a t e model made o f a 90  i n t o nine  given  of  boundary c o n d i t i o n s were  same  The  node p l a t e s h e l l  f o r one  calculation  buckl_ing, d i s p l a c e m e n t plate  16  t o model a q u a r t e r  boundary  ultimate  the  a  dense  the  stress  most  severe.  (4.1)  42  4.2:  1/4  Plate  Model  using  3x3  Element  Mesh  The  consistent  applied  along  e l e m e n t mesh for  the  load  coordinates, resulting  defined  was d e v e l o p e d  were used w i t h  ultimate  capacity  in-plane  and t h e u l t i m a t e  calculated 4.1.3.1  capacity  capacity,  hole  element  global  element  load  load  global  and  then  vector.  size  design  results proposal  of  appendix  hole  had a d i r e c t  of the f i n i t e show  sizes.  effect  [4.4]  and  between d e c r e a s i n g  buckling  capacity  was  size.  on t h e u l t i m a t e i n c r e a s i n g hole  decrease  in ultimate  e l e m e n t work a n d  The d e s i g n  and a c i r c u l a r the  buckling  an e x c e l l e n t c o r r e l a t i o n  B and was b a s e d on t h e  of  elastic  Yielding  of the p e r f o r a t e d p l a t e . With  The  figure  traction  i n A p p e n d i x A. The  ultimate  of the hole  f o u n d t o be a c o r r e s p o n d i n g  results  of  consistent  elastic-plastic  f o r each v a r i a t i o n In-plane  The  plate  vector  f o r the element  the  model c o n s i s t e n t  load  Results  The  range  i n terms  was  distorted  stress  formulation  t o c a l c u l a t e each element  a  consistent  a n d i s documented  [A.8]  stress  such  model t h e u n i f o r m  A general  was added t o t h e t o t a l  4.1.3.  was  vector,  equations  coordinates this  to c o r r e c t l y  shear  Using  t h e use o f t h e e x a c t  the p l a t e boundary.  consistent  f o r a uniform  plate boundaries.  required  each element  along  load vector  element  show a l m o s t  [4.2] i s  ASCE p r o p o s a l  perforation.  finite  equation  This  a straight  entire  derived  in  square  with  illustrated  line  c a p a c i t y and i n c r e a s i n g h o l e  ASCE [11]  the  along  been  there  capacity.  assuming a  equation  work have  size  the for  in-plane  the in  correspondence  size.  (4.2)  >/4/35 - 2 5 + 1 2  44  1.2 1.1-  Hole S i z e R a t i o  D/b  F i g . 4.4: C o m p a r i s o n of t h e U l t i m a t e I n - p l a n e 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 t h e F i n i t e E l e m e n t Method and t h e ASCE D e s i g n P r o p o s a l g i v e n by Equation [4.2]  46  4.1.3.2  3-Dimensional E l a s t i c  The hole  variation  size  yield  capacity size  When  elastic  different  For the smaller  buckling  from  hole  that  sizes  was a l m o s t p r o p o r t i o n a l t o t h e h o l e  holes  straight  of t h e u l t i m a t e  was s i g n i f i c a n t l y  capacity.  Buckling  the e l a s t i c  capacity  diameter  became  of the  with  in-plane  the decrease  size.  in  F o r t h e medium  b u c k l i n g c a p a c i t y was much lower t h a n t h e  line correlation  the hole  capacity  found  i n the in-plane  became g r e a t e r  significabtly  than  lower  yield  0.2 6  than  a  capacity.  the  buckling  straight  line  correlation. The simply figure  elastic  buckling  coefficients  plate  boundaries  supported [4.5],  UENOYA, capacity equation  and  compared  with  f o r both are  other  illustrated  theoretical  REDWOOD [ 3 ] a n d MARCO [ 9 ] . The u l t i m a t e i s related  to  the e l a s t i c  t h e clamped a n d  results  elastic  buckling  in by  buckling  coefficient  by  [4.3]. (4.3)  UENOYA element determine surface  and  stress  REDWOOD used a c o m b i n a t i o n a n a l y s i s and a R a y l e i g h - R i t z  the e l a s t i c was  of  buckling c o e f f i c i e n t .  discretized  the f i n i t e  The p e r f o r a t e d element  using  stresses  were then s u b s t i t u t e d i n t o t h e minimum p o t e n t i a l e n e r g y  represented  throughout  elastic  plate  the s t r e s s d i s t r i b u t i o n  The  An  method  to  provided  loads.  elements.  method  stress  and  (CST)  energy  finite  constant  expression,  triangle  by  in-plane  t h e domain.  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  deflected by t h e f i r s t  shape eight  in  the energy  terms of a  47  analysis  the  buckling  expression  Fourier  The  series.  was The  resulting solution smaller  values for a f u l l  holes.  0.4  their  the  current  finite  using  model  analysis. plate the  of The  shell  work.  MARCO  model was made up o f 16 Despite  using  applied  Also,  a  higher  softening  buckling A  the  l o a d would typical  illustrated  stiffness  work.  throughout  A  the  isoparametric,  than  i n the model, the  current  i s t h e u s e of a d i f e r e n t a n a l y s i s i s done.  load t o the  matrix,  buckling  plate  thus  a  mode f o r t h e q u a r t e r  before  If the  load.  slightly  lower  tend  a r e more e v e n l y  This  out-of-plane  movement o f t h e p l a t e  displacements  occur  i n these  model  is  t o be c o n c e n t r a t e d i n  the tension diagonal,  the compression d i a g o n a l the boundary.  plate  t h e c o m p r e s s i o n d i a g o n a l , on  side, the displacements  across  were  result.  of the p l a t e . Along  because a l o n g  than  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  side, the displacements  tension  lower  the b u k l i n g  i n figure [4.6], A along  right-hand  the c e n t e r  load  a n a l y s i s was done he w o u l d g e t l o w e r b u c k l i n g  i f a lower o r d e r  have  used  fewer e l e m e n t s  f o rthis  initial  greater  by MARCO  bicubic,  were s l i g h t l y  before  buckling  i n the current  p e r f o r a t e d p l a t e was  increament  bifurcation  hand  the  work f o r t h e  becomes  determined  t h e same program a s u s e d  he o b t a i n e d  load  D/b,  classical  o f MARCO.  coefficients  One p o s s i b l e e x p l a n a t i o n  initial  the  buckling  the  the current  size,  e l e m e n t work o r t h a t  elements.  results  as t h e h o l e  with  a n a l y s i s g i v e s a much h i g h e r  elastic  obtained  p l a t e and a g r e e w i t h  However,  elastic  The  full  showed a good c o r r e l a t i o n  in  areas.  48  distributed.  the plate  tension this  on t h e l e f t This i s  i s subjected  to  tends t o r e s t r a i n the area.  Thus,  lower  16  Hole Size Ratio D/b  Fig.  4.5: V a r i a t i o n o f 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 Concentric Hole Size  49  with  Fig.  4.6:  Elastic  Buckling  Mode,  Concentric  Hole  4.1.3.3  3-Dimensional E l a s t i c - P l a s t i c  The  combined  studied  by  capacity  determining  of  nonlinear  the  path  was d e t e r m i n e d each  behavior  at various  equilibrium  bifurcation  analysis  load.  the  When  stiffness is  matrix  reduced.  strength  along  material  elastic-plastic  the  material  y e i l d i n g has lowered  as the current  plates the  rapid very  small  plateau increase  determinant  point the  defines  only  starts  the  load  to  the  yield,  post  capacity.  capacity  is  the buckling  path.  the  capacity buckling Finally,  reached  load  a  buckling  and the b u c k l i n g  hole  when  t o the  of  four  same  perforated  s i z e s a r e shown i n f i g u r e [ 4 . 8 ] . the l a t e r a l  90% o f t h e u l t i m a t e stiffness.  a small  The  increase  applied  have l o w e r e d  the buckling  of t h e s t i f f n e s s the u l t i m a t e  matrix  Finally, capacity becomes  51  is a  are very  increases  l o a d . There i s  to increase  elastic-plastic  plate  load there deflection  loading.  While  displacements  in applied  as d i s p l a c e m e n t s c o n t i n u e in  load-  established  determine  d e f l e c t i o n paths  i n the p l a t e  nonlinearities the  buckling  behaves e l a s t i c a l l y  q u i c k l y with  the  level.  load  At approximately change  long  load  different  material  small.  a  resulting  with  was  the buckling  ultimate  The  to  softened  increase  typical  the l o a d - d e f l e c t i o n  t h e p l a t e move o u t - o f - p l a n e  of the p l a t e  of  The p l a t e b u c k l i n g  first  the  value  i n the  were  buckling  significance  configuration  is effectivly  As  [4.7],  was p e r f o r m e d  plate  The  i s displayed  stages  and geometry  elastic-plastic  plate.  shown i n f i g u r e  new  material  the ultimate  perforated  material  deflection  As  e f f e c t s of n o n l i n e a r  Buckling  with  only a  when  sufficiently  negative.  bukling  the  capacity  This of  150  140Stable  130-  Unstable •  120  110-  10090-  Legend Load Path Buckling Load  80-  70 4  ig.  6  8  I  10  i  14  12  18  16  Lateral Deflection mm 4.7: D e c r e a s 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 C a p a c i t y 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.  of  200  Full Pate  D/b=0.15 D/b=0.2  D/b=0.3  D/b=0.5  0  2  4  6  8  i  10  1  i  12  1  i  1  14  i  16  i  - I — ' — l — ' — I — ' — l —  20  22  24  Lateral Deflection 18m m F i g . 4.8: Load D e f l e c t i o n C u r v e o f a S i m p l y S u p p o r t e d Perforated Plate with Various Concentric Hole S i z e s  52  26  28  Although figure  the  [4.8]clearly  changes  with  becomes  longer  deflection holes  curves  hole  there  before  the  indicates  size.  that  load  the p l a t e  Thus,  plates  with  Figure  [4.8]also clearly  slope  and w i l l  larger holes shows t h a t  results  of  slope  carry  more  the ultimate  i s s u b s t a n t i a l l y reduced with  plate  steeper  be  capacity  larger  the  still  will  load-  for  in  a  plateau  the  i s that  Also,  plateau  the  of  distortion  i s reached. is stiffer  of t h e  increase  T h i s means  be more o u t - o f - p l a n e  load.  buckling  size  a t t h e same t i m e t h e  ultimate  characteristics,  the length  As t h e h o l e  becomes s t e e p e r .  will  similar  demonstrates that  and  path  have  more  ductile.  elastic-plastic increasing  hole  size. The results  by  UENOYA  capacities in  the present f o r the  The r e s u l t s  from l o a d - d e f l e c t i o n c u r v e s  the  ultimate  along  with  l o a d was d e f i n e d  of the present  similar  numerical  elastic-plastic  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  figure [4.9].  ultimate  study  t o those  as the h i g h e s t  study  buckling illustrated  were  obtained  i n f i g u r e [ 4 . 8 ] . The  load  level  obtained  on  l o a d - d e f l e c t i o n curve. The  hole  two s e t s o f r e s u l t s  size,  lower  but the c u r r e n t  values  explanations relatively calculate  show good agreement  stiff the  T h e r e may  discrepancies.  CST e l e m e n t ,  stress  smaller  work o f t h e a u t h o r g i v e s s u b s t a n t i a l l y  f o r the l a r g e r h o l e s . f o r these  f o r the  Firstly,  u s e d by UENOYA  distribution  may  stress field.  have  underestimation  of the t r u e  under  estimated  the R a l y l e i g h - R i t z minimization  higher  plate capacity.  Secondly,  two  possible  t h e use of and  the  REDWOOD,  resulted  I f the s t r e s s e s  by r e s t r i c t i n g  53  be  to  in  were  would p r o d u c e the  an  a  displacement  1.2  1.1  T  0  1  0.1  1  0.2  1  0.3  1  0.4  1  1  0.5  1  0.6  Hole Size Ratio  0.7  1  0.8  1—  0.9  D/b  F i g . 4.9: V a r i a t i o n of U l t i m a t e E l a s t i c - P l a s t i c B u c k l i n g C a p a c i t y of S i m p l y S u p p o r t e d P e r f o r a t e d P l a t e w i t h C o n c e n t r i c H o l e S i z e .  54  Fig.  4.10:  E-P  Buckling  with  von-Mises  Stress  function assumes  t o the f i r s t that  eight  the r e s u l t i n g  displacement  represented  by a c o m b i n a t i o n  sufficient  for  the  compared.  However,  hole  increases,  to  size  accurately  full  the  plate  the  represent  The  second parameter  various  locations  ultimate  in-plane,  capacities 4.2.1  were Plate  displacement  accurately  assumption  mode t o  which  was  i t was the  f u n c t i o n may n o t be a b l e  buckling  mode.  The  result  capacity.  Location studied  The c e n t e r on t h e p l a t e elastic  may be  UENOYA  t e r m s become more dominate a s  the b u c k l i n g  V a r i a t i o n of Hole  plate capacity.  buckling  the lowest  '  field  series,  o f t h e s e modes. T h i s  i f higher  would be t o o v e r - p r e d i c t 4.2  terms of t h e F o u r i e r  was t h e l o c a t i o n o f a h o l e  of a standard surface.  buckling,  hole  was p l a c e d  F o r each l o c a t i o n  and e l a s t i c - p l a s t i c  on in the  buckling  determined. Geometry  b/2  b/2  •b/2  Fig.  b/2  4.11: P l a t e Geometry a n d L o a d i n g u s e d i n t h e A n a l y s i s o f the V a r i a t i o n o f H o l e L o c a t i o n P a r a m e t e r  56  A  standard  various  hole with a diameter  locations  shown  in figure  location, deg.,  [4.11].  ANG.  and  5/6=0.15 and  one  quarter  i n the p l a t e .  of  the  The  R/b,  of  The  0.2b  parameters used  The  hole  p l a t e area  only  due  centered  at  loading  are  geometry and  were v a r i e d as  0.3.  was  to d e f i n e the  follows;  ANG=45 t o  needed t o be  to the  symmetry  hole 135  considered in loading  in and  geometry.  4.2.2  Finite  for  Element Model  Two  b a s i c e l e m e n t meshes were r e q u i r e d t o do  the  eccentric  located along  By  along  required.  However,  there  no  was  The  half  model  method.  half The  coordinate corner Each  of  made up The  the  and  this if  was CPU  time  full  the p l a t e ,  a similar  then  3x3  grid. and  spacing  by  hole  a x i s of  center  symmetry  a full  f o r the  half-plate a  p l a t e model possible  the  l i n e s were t h e n t h e model by  used  by  the  same  of  a  e s t a b l i s h e d to  nine  in a half  radial  lines.  57  each  sections.  and  full  model  respectively.  f o r the c e n t e r was  polar  isoparametric  of t h e model was  system e a c h q u a r t e r  e q u a l l y spaced  diagonal  it  into quarter  elements  i n each q u a r t e r used  was  required.  as  origin  using  This resulted thirty-six  about  analysis.  s e l e c t e d as  modeled  was  boundary  model  l o c a t e d on  p l a t e models were g e n e r a t e d  manner a s was  ten  not  whenever  dividing  was  polar coordinate  sections  used  Radial  nodal  was  symmetry and  system.  of e i g h t e e n  i s an  o n l y a one  the h o l e  c e n t e r was  in a  there  axis,  hole  section  elements  in  less  When t h e  analysis  a p p l y i n g the a p p r o p r i a t e d i s p l a c e m e n t  a x i s of  significantly The  locations.  a p l a t e diagonal  this diagonal. conditions  hole  the  determined  h o l e models. divided  Ten  into  nodes were  With nine then  CC  Fig.  4.12:  Finite  Element  Model  of  Half  the  Plate  Fig.  4.13:  Finite  Element  Model  of  the  Total  Plate  placed by  along  equation  each r a d i a l  half  [4.13]. with  p r o p o r t i o n a l s p a c i n g as  examples o f t h e r e s u l t i n g  and f u l l  finite  models a r e i l l u s t r a t e d  These e l e m e n t meshes were u s e d  some  distortion  4.2.3  Results  4.2.3.1  In-plane  Results in  with  figure  through-out  [4.14].  As t h e 0.2b d i a m e t e r  capacity.  one l o c a t i o n  o t h e r models.  reached to  small  area  material In interest, type  located  had the h o l e  the  i n the  near  started  around  Once s u f f i c i e n t  lower  in-plane  h a d more than  yielding  and t h e o u t e r  a  2% and  had o c c u r r e d  plate  this  c o n f i g u r a t i o n was  this  of f a i l u r e .  a l l sides.  material failure  shear  Most  and  the  This  plate  to the outer  considered a  local  mode  webs w o u l d n o t l i k e l y  shear  the  plate  an u l t i m a t e p l a t e c a p a c i t y  local  in reality  the  was r e s t r i c t e d  Therefore  theory  with  mechanism was s i m i l a r  hole  r a t h e r than  the other  between  boundary,  The f a i l u r e  boundary  ANG.=90,  i n t h i s model a s  the  failure  72/6 =0.3,  c a p a c i t y than  between  on  the  t h e p l a t e boundary  the hole  c o n f i g u r a t i o n s but y i e l d i n g  but  ultimate  parameters,  a significantly  i t s ultimate capacity.  boundary.  this  only  d e f i n e d by  boundary  the other  the analyses,  in capacity.  Yielding  hole  fact  location  produced  models.  inner  In  model  D/b =0.2,  and  h o l e was moved over  yield  The  [4.12]  a n a l y s i s are i l l u s t r a t e d  change  a 4% change  for  Yielding  s u r f a c e t h e r e was l i t t l e  had  mesh  t o accommodate t h e p l a t e g e o m e t r y .  from t h e i n - p l a n e y i e l d i n g  This  element  in figure  plate  change.  given  [4.1].  Typical the  line  limit.  may  be  of  experience  webs have a f l a n g e o r s t i f f e n e r  would  provide  6 0  a  mechanism  for  '6 ID 1  •  *'  *  •>  /  0.962 .•••«•..  •-.0.980  .•0.980  N  .^"6.989 /  '•0.989  ".'0595  -  10  R/b=0.3  b  /  S-  1.03  0.995' '  \,  v  ^ R / b . = O.I5  \  2  b  /2  .  F i g . 4.14: 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 R e s u l t s f o r V a r i o u s Hole L o c a t i o n s Normalize t o the C o n c e n t r i c Hole Ultimate In-plane Capacity redistribution forces  into  through plate  o f f o r c e s . The a r e a s  t h e boundary  the  stiffeners  at regions  4.2.3.2  stiffeners. and t h e n  ultimate  the  elastic  or  f o r c e s would  significant  of the p l a t e .  tension diagonal  back  travel  into  the  when  changes  the hole  I f the hole  to the compression  in  the  location was  moved  diagonal  the  b u c k l i n g c a p a c i t y c o u l d be i n c r e a s e d by a s much a s 50%.  A number o f r e s u l t s illustrated factor  would t r a n s f e r  Buckling  b u c k l i n g c a p a c i t y occur  away f r o m t h e c e n t e r  from  These  be t r a n s f e r r e d  the in-plane capacity,  elastic  strain  strain.  3-Dimensional E l a s t i c  Unlike  moves  of lower  of h i g h  of  in figure  f o r the e l a s t i c  [4.17].  the  elastic  concentric  hole  so t h a t d i r e c t  locations.  If  the hole  buckling capacities  A l l the c a p a c i t i e s are given  b u c k l i n g c a p a c i t y of  a  c o m p a r i s o n s c a n be  plate made  as a  with  a  between  was l o c a t e d i n t h e t e n s i o n d i a g o n a l  61  are  the  62  Fig.  4.16:  Full  Plate,  von-Mises  Stress  factor  ranged  However, 1.52.  along  This  buckling  from  0.992 t o  1.017.  the compression  This  diagonal  means t h a t t h e r e was  a 52%  c a p a c i t y by moving the h o l e  thecompresion  is essentially the  factor  increase in  from t h e  constant.  increased the  to  elastic  tension diagonal  to  diagonal.  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 f o r V a r i o u s H o l e Hole L o c a t i o n s Normalize to the C o n c e n t r i c Hole C a p a c i t y . The figure the  elastic  [4.18] and  tension  buckling  mode  [4.19] i l l u s t r a t e s ,  d i a g o n a l and profile  along  similar  shape t o a s i m p l e  compression  diagonal,  complicated  s h a p e s . The  hole  boundary then  at  the  sine  edges o f  p l a t e the d i s p l a c e m e n t s  the  eigenvector  different  diagonal  wave. appeared  displacements  the  plate  shown  in  profiles  respective.  t e n s i o n d i a g o n a l was  the p r o f i l e  the  the  the  compression  displacement in  of  very  However,  The  smooth,  across  t o be made up  the  o f more  were more p r o n o u n c e d a t  the p l a t e .  were v e r y  s m a l l and  showed t h a t some were a c t u a l l y  areas.  64  At  the  corners  examination negative  of  in  the of of  these  4.2.3.3  3-Dimensional E l a s t i c - P l a s t i c  The  results  plastic  are  perforated  related  a  the The  typical  produced either  buckling  as  of change  elastic-plastic  an  of  ultimate  o f t h e two  i f there  buckling  f a c t o r s were c o n s i d e r  hole  i s moved t o c o m p r e s s i o n d i a g o n a l ,  was  so  would  e x p e r i e n c e an Examples failure  fail  in-plane of  Model  models  4.1:  failure  that  geometry lower However,  yield  capacity  Thus  the  The would  than  if  if  the  as  the buckling  no  undergo  when  capacity that  plate  the would  (zone I ) . a  zone  [4.1].  F a i l u r e Mode  Character i st i c s ANG. R/b D/b  or  failure.  underwhen zone I and  in table  buckling  significantly,  buckling.  yield  modes, a r e g i v e n Table  increased  than the i n - p l a n e  without  and  alone.  the  plate  location is  little  II)  capacity  buckling  higher  ultimate  t h e p l a t e would  elastic  much  was  was  material  elastic-plastic  capacity  The  elastic  (zone  The  concentrically  f o r each hole  capacity,  nonlinear  the  the  elastic-  [4.20].  capacity.  in  showed t h a t  buckling  figure  of  plastic  plate capacity  results  effects  a factor  elastic  magnitude  i n the e l a s t i c  combined  are d e t a i l e d i n  plate ultimate  to  capacity.  capacity expressed  elastic-plastic  change  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  buckling  capacities  Buckling  Classification  MPa.  MPa.  MPa.  Failure Class  A  0.2  0.3  45  136.9  133.5  128.3  zone I I  2C  0.2  0.0  -  138.4  131.3  126.2  zone I I  E  0.2  0.3  135  136.9  207.5  138.8  zone I  65  II  Fig.  4.18:  Profile  of  the  Tension  Diagonal  :  Profile  of  the  Compression  Diagonal  *6  b/2  •  -j.  b  /  2  F i g 4.20: U l t i m a t e E l a s t i c - P l a s t i c C a p a c i t y f o r V a r i o u s Hole L o c a t i o n s Normalized t o the C o n c e n t r i c Hole U l t i m a t e E l a s t i c - P l a s t i c Buckling Capacity Since  the  eccentric  hole  concentric failure  elastic always  h o l e and,  were  found  capacity  was  capacity.  However,  could  not  decreased, the  in  never  occur.  higher  value figure  this  that  this  analysis.  of  the a  plate  plate  buckling  The  lower  ratio  of  buckling  the  the  in-plane failure  plate  b u c k l i n g c a p a c i t y w o u l d be r e d u c e d , would r e m a i n u n c h a n g e d ,  a  (zone I I I )  elastic  than  with  with  does n o t mean t h a t a zone I I I  I f the slenderness  the e l a s t i c  forcing  were while  the p l a t e  mode.  concentrically f o r a l l other  [4.20],  than  significantly  a zone I I I f a i l u r e The  c a p a c i t y of  no examples o f e l a s t i c  in-plane capacity  into  buckling  a l l but  perforated plate provides  a lower  hole  reference  locations.  one h o l e  location  68  With  p r o d u c e d an  bound to  ultimate  capacity hole  factor  l o c a t i o n that  same  one  that  section  yield  yield  elastic  had  buckling  are  the  location,  buckling  lowest  capacity  capacity  prevented,  capacity  capacity.  capacity  little  for hole  had  minimum u l t i m a t e  any l o c a t i o n o f a g i v e n  increased on  size  local  in-plane equal to  hole.  the c o n c e n t r i c elastic-plastic  of the Moving  in-plane  i s given  will  If  the e l a s t i c  the  in  in-plane the hole buckling capacity.  buckling  capacity  by t h e c o n c e n t r i c  configuration.  4.3  Optimum D o u b l e r The  final  effectiveness perforation.  Plate  parameter of  the  investigated  addition  The o b j e c t i v e  of the doubler  around  restoring  some o r a l l o f t h e p l a t e s  of t h i s  doubler p l a t e  the hole,  in  of a doubler  stresses  goal  ultimate  elastic-plastic  hole  t h e web  ultimate  capacities.  effect  occur  of f o r c e s .  combination  buckling  will  with concentric  in  material  be a p p r o x i m a t e l y  The  the  discussed  around  the  The  was  local  indicated that  was g o v e r n e d by a  of t h e p l a t e  Thereforethe  will  of a p l a t e  away from t h e c e n t e r and  then  analysis  and e l a s t i c  As  mode o f f a i l u r e  and s t i f f e n e r s  hole.  capacity  was due t o a  f o r the r e d i s t r i b u t i o n  f o r any h o l e  The  this  ultimate  capacity.  low v a l u e  the flanges  failures  in-plane  hole  this  a mechanism  capacity,  of the c o n c e n t r i c  had a low i n - p l a n e  since  material  that  i n a lower  I t i s u n l i k e l y that  reality, provide  than  resulted  [4.2.3.1]  failure.  the  greater  limiting  s t u d y was t o e v a l u a t e  the  study  plate  was  the  around  the  p l a t e was t o l o w e r t h e  the m a t e r i a l  in-plane  yield  yielding  and  capacity.  The  t h e e f f e c t i v e n e s s of  shapes.  70  various  4.3.1  Plate The  standard  circular A  Geometry  perforation  doubler  plate  perforated  plate  the  doubler  2.425Z>and  4.3.2  the  Like  symmetry  around was  models  the  of  8x5  an  [3.1.4.2], into  mesh of The  eight  plate  each  line  plane  stress  model was  was  each r a d i a l  line  2.425D.  The  and  set  by  to  and  varied  to  the  thickness  of  1 .3D  from  to  node was  same  line.  [4.23]  b o u n d a r y . The  then  radii  of  located  section plate  lines. node  four  1.3£J  along  consisted  quarter  first next  and  only.  in  radial The  of  symmetry  movements  d i v i d i n g the spaced  axes  displacement  a x e s of  in figure  by  each r a d i a l hole  these  perforated  The  elements d i s c u s s e d  nine e q u a l l y  the  The  in-plane  were s e t a c o n s t a n t last  study.  attached  required.  shown  generated  on  at  the  concentrically  along  model  nodes were t h e n p l a c e d  of  1.50t.  to  diagonals.  plate  sections  diameter  model was  were a p p l i e d  one-quarter  was  diameter  f o r the  d i s p l a c e m e n t s were r e s t r i c t e d The  on  The  part  diameter  Model  one-quarter  were a g a i n  The  0.25*  from  this  shape  hole.  varied.  other  the  circular  the  Element  boundary c o n d i t i o n s all  the  0.26  with a concentric  used t h r o u g h o u t  thickness  the  only  was  in  plate  Finite  plate  square p l a t e  placed  nodes  1.75D, the  Six  outer  on  2.0D plate  boundary. The for  the  plates  plate  internal only  doubler This  doubler  plate  allowed  was  element  the  first  r i n g s . For element  s i z e s two the  .modeled by  or  same model t o  w i t h m i n i m a l c h a n g e s between  the  ring  three be  specifying a thicker smallest  was  thickened.  element used  models.  71  diameter  rings  For  were  throughout  the  plate doubler larger  thickened. analysis  T  X\  \  Dd  \\200 \  E (Ty  JL  v  = 200 000 M P a 300 M P a = 0.3  t = 10  500  Fig.  500  4 . 2 2 : Geometry and L o a d i n g of P e r f o r a t e d with Doubler Plate  Fig.  4.23:  F i n i t e E l e m e n t Mesh of P e r f o r a t e d with Doubler Plate 72  Plate  Plate  4.3.3  Results  4.3.3.1  In-plane  The  doubler p l a t e  sectional than  a  areas,  As  significant The equation  [4.24]  i n the c a p a c i t y of  a plate  d  the doubler p l a t e  optimum d o u b l e r  plate  thickness *  the  into  t h e body of t h e p l a t e .  the  inner  plate  perforation  1> (r  the a d d i t i o n  perforation  boundary,  I n s t e a d , a second y i e l d boundary.  This  of  In  D = 2.0D  1.0. S o l v i n g f o r an  d  (4.4) a t the  inner  t h e h o l e and up  Once t h e y i e l d i n g h a d e x t e n d e d t o the outer  pattern.  but  load  plate  plate  h a d been  significant  Again, y i e l d i n g started into  spread  rapidly  73  from t h e  the  attaned.  t h e r e was a  i t d i d not extend  from  edges,  at the  the  plate.  zone d e v e l o p e d a t t h e o u t e r d o u b l e r  yielding  and  d  the parameters  yielding started  of a doubler  to the y i e l d  1.0, t h e  arid p r o p a g a t e d a r o u n d  boundary  modification  is  y  became u n s t a b l e and t h e u l t i m a t e  With  factor  capacity.  by  ~ f)  v  plate,  boundary of the p e r f o r a t i o n  given  t =t.  d  doubler  is  of almost  d  s i z e a s , D =2.0D,  t h e r e was a  plate  unperforated  t > gives  diameters.  factor.  a doubler plate  Uy = fy + Without  was i n c r e a s e d  factor  of the  nondimensional plate  restoration  restoration  plot  a  doubler  restoration  i t i s shown t h a t  cross  was more e f f e c t i v e  vs.  with doubler  to i t s original  A /A=1 »0 h a s a c a p a c i t y for  $  t h e same  [4.24] a  factor  ratio,  given  plate  In f i g u r e  I f the capacity  was r e s t o r e d  figure  doubler  plate.  diameter  increase  [4.4].  thin  i s shown f o r v a r i o u s  plate  capacity  showed t h a t ,  restoration  area  the doubler  plate  thick  capacity plate  analysis  a wide,  narrow,  effective doubler  Yielding  edge  plate  of the  doubler Once  plate  the  boundary A  t o the  yielding there  p l a t e b o u n d a r y as had  extended  the  load  inward t o the  inner  was  no  further  increase  in plate  comparison  of  the  ultimate  yield  standard -is g i v e n  and  the  in f i g u r e  two  reinforced perforated  level  increased. perforation  capacity.  patterns,  p l a t e s as  for  described  the above  [4.25],  0.5  1.0  Doubler Plate Area A /A A  Fig.  4.24:  E f f e c t i v e Capacity Area f o r Various  R e s t o r a t i o n F a c t o r vs D o u b l e r Doubler P l a t e Diameters  74  Plate  doubler plate reinforcement D = 400 j t = 20 d  without  reinforcement  d  Fig.  4.1  4.25: S p r e a d of Y i e l d Zones f o r S t a n d a r d 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  C o n v e r g e n c e w i t h Mesh R e f i n e m e n t An  approximate  rigorously reduced,  proven the  solution that,  method  as  will  method the  finite From  and  show an  element  finite  of  the  the  system s t r a i n  eigenvalue provide load.  an The  energy (l/n) . p  asymptotic  theory  element  upper bound v a l u e theory  where  also  f o r the  a s y m p t o t i c a l l y converge  n  i s the  strain  been  finite  to the e x a c t  75  The exact  i n any  The  energy  bound v a l u e shown t h a t  stiffness  number o f e l e m e n t s  is  the  matrix  system's e l a s t i c  s t a t e s t h a t the  size  requirements.  shown t h a t t h e  element  be  solution.  these  i t has  can  solution.  the  p r o v i d e s a lower  finite  it  of bound on  a l l of  Furthermore  of t h e  exact  some s o r t  be  if  or element  convergence to  i t can  solution  energy.  analysis  will  the  formulation s a t i s f i e s  element  finite  step size  render  a p p r o x i m a t e method s h o u l d p r o v i d e solution  i s enhanced  an will  buckling  element solution one  for  strain like  direction  and  p  depends  on  continuum  problem  formulate  the  under  the  of  interpolation  P r o o f s of The  the element a r e a  has  been  element  consistent uniform  shear  shear  Exact  was  calculated  i s used.  sufficient  to  fourth,  and  third  fifth  element  suficient Instead cases,  and  for  however,  Jacobian matrix Since exact  of  the  on  the  by  the  boundaries  the  subjected  time  to  is a  constant. of  be  of t h e  Jacobian  coordinate  will  this  integral  the  have terms t o  the  require  may  be  other  terms i n  integration  of o r d e r  nine  integral t o use  must be  be  or  ten.  for  all  a  reduced  sufficient  i f the d e t e r m i n a n t  the element  76  to  with  stiffness  integration  stiffness  must  integral,  shown i t i s s u f f i c i e n t  the  numerical  stiffness  would  the  if  element  i n the J a c o b i a n m a t r i x  integral  integration  exact  analysis.  integration  the determinant  the  applied  and  the  element  the  evaluate a polynomial  [ 1 0 ] has  made  and a p p l i e d .  exactly integrating  evaluate  each  s i x t h power. C o m b i n i n g  stiffness  BATHE  integration, exactly  power and  be  satisfied  the element  i n the determinant  to f u l l y of  Along  accuracy  t r a n s f o r m a t i o n m a t r i x . Terms second  throughout  automatically  exactly evaluate  a l l term  to  between e l e m e n t s and  r e q u i r e s a l o t of CPU The  used  i s assumed.  vector  integration  integration  including  load  the  be  between e l e m e n t s  satisfied  formulation.  functions  of  t h e s e p r o p e r t i e s can  domain as w e l l as  boundaries  equation  c o n s i s t e n t l o a d must  l o a d i n g between e l e m e n t s was  finite  the  differential  c o n s i s t e n t load requirement  element The  the  conditions.  system  integration The  and  element.  certain  over  the g o v e r n i n g  stiffness  of  integral  to the  is  not  realistic  order  a lower  integration  asymptotically convergence the o r d e r By with  order the  finite  will  be  value.  supported  perforated  element  the  calculated using  mesh  the  accuracy  plate  the  1/4  may  still  and  not  was v a r i e d  c a p a c i t y g i v e by e a c h mesh.  algorithm.  A  was  of elements  new  through exact  repeated.  of each l e a s t  solution,  and the r e l a t i v e  the best value  established.  The  a  simply  to  permutations the  for  set  up  results  of t h i s  and  squares f i t  the  and  this  cumulative  f i t a s s o c i a t e d w i t h an assumed elastic  the  of r e l a t i v e  assumed  r e c o r d i n g of  to  buckling  calculated  was  A  perforated  A Log-Log p l o t  f o r the exact  The  6x6.  elastic  was t h e n  was  0.26.  the p o i n t s u s i n g a l e a s t  By k e e p i n g  work  of  1x1  of  i n one d i r e c t i o n  solution  squares  error  rate  buckling  The  of  from  assume f o r t h e e x a c t  buckling  passed  elastic  model  buckling capacity  of the p l a t e  line  the  was done on e a c h o f t h e s e  elastic  number  convergence  with a hole diameter  plate  v a l u e was t h e n  vs  of  plate  capacity  was  lower  error  by NISA83 were e s t i m a t e d . standard  analysis  approximate A  the  for  bifurcation  error  a  However, t h e r a t e of  t h a t t h e r e was an a s y m p t o t i c  performed  process  solution  g o v e r n e d by t h e i n t e g r a t i o n  refinement,  capacities  straight  Using  of the element.  mesh  error  i s used.  element  converge t o the exact  assuming  plate.  integration  exact  buckling capacity  work a r e shown  in  figure  [4.26]. The  s l o p e of the l i n e  convergents rate  rate,  for this  in figure  i s approximately  [4.26],  representing  -3. T h e r e f o r e ,  the convergence  e l e m e n t and model i s n t o t h e power -3 or  77  the  (1/n) . 3  Also with of  given  e a c h mesh. 5%.  I t shows t h a t  A more r e f i n e d g r i d  solution  for  capacities, not  by t h e f i g u r e i s t h e r e l a t i v e  the ultimate  but would  present  mesh a l r e a d y  consistent depending  element on  capacities results  hole  that  would have p r o v i d e d elastic  11/730.  size  express  with  accurate  and e l a s t i c - p l a s t i c  buckling  a n d CPU t i m e which was  The f u l l  from a l l the a n a l y s e s  in  ultimate  and  n o t due t o c h a n g e s  varying  and l o c a t i o n , the  same  error  a more  p l a t e model w i t h  u s e d t h e c a p a c i t y o f t h e VAX. mesh,  associated  t h e 3x3 e l e m e n t mesh has an  r e q u i r e more s t o r a g e  a v a i l a b l e on t h e VAX  error  degrees  the  of  using  error.  ultimate Therefore,  compared. Any c h a n g e s  c a p a c i t i e s was a t t r i b u t e d t o t h e p a r a m e t e r s i n the modeling  studied  technique.  0.001  Number Of Elements In One Direction n F i g . 4.26: C o n v e r g e n c s o f t h e E l a s t i c B u c k l i n g Load w i t h Mesh R e f i n e m e n t f o r a C o n c e n t r i c a l l y H o l e d P l a t e w i t h Z>/6 = 0.2, 1/6 = 0.01  78  a  distortion  calculated  relative  were d i r e c t l y  By  the  5  CONCLUSIONS The  behavior  perforation different failure width). in-plane  at  failure  stocky  material  in-plane  yield by  intermediate  so  the  buckling the  of  The  The  capacity  capacity.  ultimate  failure.  For  i s much l o w e r ultimate  No  are  i s governed  Therefore,  numerical  t h a n the  / the  elastic  therefore  Finally,  yield  for  capacity  and  magnitude.  the  and  The  buckling  elastic-plastic  p l a t e s with  methods a r e  the  material  is  solution e x i s t s for  of p e r f o r a t e d  three  by  p l a t e s the  yielding  by  of  is limited  same  both m a t e r i a l  analytical  capacity  the  one  ( thickness  capacity.  of  circular  determines  capacity  in-plane  a  by  t/b  slender  p l a t e s , both the  mode i s a f u n c t i o n of  which  capacity  buckling  capacity  with  described  ratio,  ultimate  plate  elastic  buckling  ultimate  be  parameter  capacity. the  plate  slenderness  capacity. the  shear  l o a d can  p l a t e s the  slender  elastic  failure  plate  yield  capacity  controlled  square  modes.  mode i s the For  a  i t s ultimate  buckling  the  of  determining  this  required  type  to  of  estimate  these c a p a c i t i e s . The on  program NISA83 was  perforated  plates.  program c a l c u l a t e d t h e buckling  capacity  c a p a c i t y . The the  of  control  deflection  each  ultimate the  option  time s t e p c o n t r o l and  efficiency step  For  and  restart  u s e d t o c a r r y out  in-plane  ultimate of  the  p a t h of  the  to  be  plate  well  yield  capacity,  buckling  a b i l i t y to  "constant  suited  i n t o the  post  79  to  the  elastic  methods c o n t r i b u t e d The  study  configuration  elastic-plastic  p r o g r a m and  iteration  these c a l c u l a t i o n s . proved  parameter  a parameter  change to  the  arclength"  time  follow  buckling  the  region.  load-  Some the  minor  Civil  E n g i n e e r i n g VAX  capacity a  1/4  was  sufficient  plate.  time  c h a n g e s were r e q u i r e d  larger  If  The  became  would be  plotting  system  it  was  output  files  graphical  found  d i s p l a c e m e n t s or any  finite  similar NISA83  memory  modeling  used  the  storage  were  CPU  became  attempted  developed  r a n under  f o r the  a  t h e EUNICE o p e r a t i n g  completely compatible with The  both  information that data  checks  output  the  VMS  i s presented in a  and  post-processing  s t r e s s e s make a g r a p h i c s p r o g r a m a n e c e s s i t y  element  first  p r o g r a m . I t i s reccommended  results  straight  line  decreasing ASCE  parameter  of a c o n c e n t r i c  The  design  the  storage  that  NISPLOT  to i n c l u d e a l l the elements  for  (or a  in  the  library.  variation  the  and  the program  p r o g r a m ) be e x t e n d e d  The  a  speed  p l a t e model was  n o n l i n e a r problem  t o be  for  and  NISPLOT, was  from NISA83. form  computer  on  required  program,  from NISA83. A l t h o u g h ,  running  the n o n l i n e a r problem  full  large  a much l a r g e r  computer  The  to handle  However, when t h e  requirement  critical.  11/730.  t o g e t NISA83  plate  hole  correlation capacity.  tended  in  the  study  was  the  size.  f o r the u l t i m a t e  Suggested  rules  investigated  in-plane y i e l d  between The  capacity  increasing  results  were a l s o  Design G u i d e l i n e s . to o v e r e s t i m a t e the  hole  I t was  showed  size  and  correlated  found  that  in-plane capacity  to  these of  the  plate. The parameter results. elastic  elastic  buckling  variation For  holes  buckling  were larger  capacity  capacities  in  agreement  than was  0.4  calculated with  of t h e  reduced  80  for  other  plate  each  published  width,  significantly  below  the a  straight account  line i n the  Finally,  correlation.  This  design  web  the  of any  ultimate  reduction  where b u c k l i n g  elastic-plastic  were c a l c u l a t e d f o r e a c h h o l e  size.  other  current  published  capacity The  of  the  ultimate  either  the of  buckling  load  The  with  the  material  in-plane  The  a concentric  of  material  yielding  variation  hole  the  plate  in-plane  calculation in hole  capacity  reduces  plate  was  was hole  the  and  hole  of  the  elastic  found to  for  each  of a  plate  l o c a t i o n except too  close  possibility the  of  close to  the  that  local  near boundary  could  buckling  increase  plate tension  concentric any  location.  capacity.  l o a d was  of  plate  variation in  capacity  located  the  values.  a good a p p r o x i m a t i o n  i n any was  than  capacity.  is l i t t l e  The  others.  the  calculated  there  to the  lower  buckling  a n a l y s i s was  hole  that by  was  load  l o c a t i o n y i e l d e d some u n e x p e c t e d  moved from t h e The  hole  seemed t o p r o v i d e  between t h e  hole  buckling  the  a hole  capacities  showed  capacity  location.  p l a t e there  buckling  diagonal.  hole  buckling  elastic  capacity  into  occur.  overestimated  ultimate  i n the  I f the  elastic was  the  yield  with  edge.  boundary  The  around  p l a t e even w i t h  plate  reduce the  the  taken  r e s u l t s were compared  slightly  or  be  could  results  r e s u l t s i n d i c a t e that  plate capacity  the  The  buckling  yield  thus lowering  ultimate  a perforated to  elastic-plastic  second parameter  location. the  The  p l a t e have been  in-plane  Yielding  The  work.  should  of  hole the  diagonal  produced  hole  by  results. t o 50%  to the  the  locations.  81  up  for  if  each The the  compression  lowest  elastic  The be  ultimate  elastic-plastic  a combination  away  from  of t h e f i r s t  the center  would be g o v e r n e d buckling  load  capacity.  the c o n c e n t r i c  capacity  location  also  An  gave  The  elastic  plate passing  relative  the  convergence  made  plate  plate  find  yield elastic  that  capacity  this  buckling  hole provides  load  to  the  a lower  of a  plates  line  establish  from  3  same  was  doubler  *«//t = 1.0, and in-plane  this  the  yield  accuracy  i n a l l the for  the  the  one  82  quarter  elements.  Log-Log  The e r r o r  By  plot  of  ( n ) ,  i n the  work, was e s t i m a t e d  t o be 5%.  and  analyses.  i n one d i r e c t i o n ,  was d e t e r m i n e d .  throughout  of c i r c u l a r  1x1 t o 6x6  through  v s number o f e l e m e n t s  (1/n)  reinforcement  the hole.  was c a l c u l a t e d  straight  of  as  dimensions,  plate  best  used  the in-plane  to  and t h i c k n e s s  a mesh r a n g i n g  rate  element g r i d , plot  of a d o u b l e r  using  error  i f the e l a s t i c  elastic-plastic  o f t h e e l e m e n t mesh u s e d  buckling  model  capacity  the lowest  elastic-plastic  perforated  was  rate  than  with a concentric  a s i t would have w i t h o u t  convergence  higher  ultimate  The d o u b l e r  a  attempt  capacity,  hole provided  The d i a m e t e r  were v a r i e d .  capacity  yield  plate  size.  effectiveness  •0/6=2.0,  By moving t h e h o l e  the ultimate  not a s u r p r i s e  f o r the u l t i m a t e  investigated.  plates  was  had t h e l o w e s t  t h e same h o l e The  also  it  Thus t h e p l a t e  bound v a l u e with  of the p l a t e  by t h e i n - p l a n e  buckling  c a p a c i t i e s appeared to  two c a p a c i t i e s .  became s i g n i f i c a n t l y  Since  capacity.  buckling  from  a  3x3 this  REFERENCE 1  Wang, C h u - K i a . " 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", P r e s e n t e d a t a m e e t i n g of t h e 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 , O h i o O c t . 2-3, 1945.  2  Rockey, K. C , A n d e r s o n , R. G., and Cheung, Y. K. "The B e h a v i o r of Square s h e a r Webs H a v i n g A C i r c u l a r H o l e " , Symp. on T h i n W a l l e d 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 C o l l e d g e of Swansea, C r o s b y 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 S h e a r B u c k l i n g of Square P l a t e s w i t h C i r c u l a r H o l e s " , Computers and S t r u c t u r e s , V o l . 8 , pp. 291-300, Pergamon P r e s s L t d . , 1978.  4  Uenoya, M. and Redwood, R. G. " B u c k l i n g of Webs W i t h O p e n i n g s " , Computers and S t r u c t u r e s , V o l . 9 , pp. 191-199, Pergamon P r e s s L t d . , 1978.  5  Redwood, R. G., and Uenoya, M. " C r i t i c a l L o a d s f o r Webs w i t h H o l e s " , J o u r n a l of t h e S t r u c t u r a l D i v i s i o n , ASCE, V o l . 1 0 5 , No. 105, pp. 2053-2067, O c t . 1979.  6  J a n s s e n , T. L. "A S i m p l e E f f i c i e n t H i d d e n L i n e A l g o r i t h m " , Computers and S t r u c t u r e s , V o l . 1 7 , pp. 563-571, Pergamon P r e s s L t d . , 1983.  7  H'afner, L., Ramm, E . , S a t t e l e , J . M., and S t e g m u l l e r , H. "NISA80 Proqrammdokumentation Programmsystem", B e r i c h t des Instituts fur Baustik, Universitat Stuttgart, 1981.  8  P r e c i s i o n V i s u a l s I n c . "DI3000 U s e r s G u i d e " , P r e c i s i o n V i s u a l s , 6260 L o o k o u t Road, B o u l d e r , C o l o r a d o , 80301 USA, March 1984.  9  Marco, Renzo " B u c k l i n g of P l a t e s w i t h C i r c u l a r B.Ap.Sc. T h e s i s , U n i v e r s i t y o f M a i t o b a , 1984.  10  Bath^ K l a u s - J u r g e n " F i n i t e E l e m e n t P r o c e d u r e s i n E n g i n e e r i n g A n a l y s i s " , P r e n t i c e H a l l I n c . , Englewood C l i f f s , N. J . 1982  11  Subcommitty on Beams w i t h Web O p e n i n g s " S u g g e s t e d D e s i g n G u i d e s f o r Beams w i t h Web H o l e s " , ASCE, J o u r n a l of t h e S t r u c t u r e s D i v . , V o l . 9 7 , pp.2707-2728, Nov. 1971.  12  Bathe^ K l a u s - J u r g e n and B o l o u r c h i , S a i d "A G e o m e t r i c and M a t e r i a l N o n l i n e a r P l a t e S h e l l E l e m e n t " , Computers and S t r u c t u r e s , V o l . 1 1 , pp. 23-48, Pergamon P r e s s L t d . , 1980.  83  Holes",  APPENDIX A D e r i v a t i o n of t h e C o n s i s t e n t S h e a r Load V e c t o r f o r the B i c u b i c Isoparametric Element. Using  a  higher  isoparametric modeled  more  element  with  elements  a  small  means  that  dependent  shown  that  much  by  on  element  gives number each  excellent  Work  done  element  distortion  as  the  vector  he  does  i n order  of  reccommend  to minimize  using  to  lower  any e r r o r s  problem  using  fewer  and  hence, [12]  has  not a f f e c t e d  as  by BATHE  i s  the  bicubic  a  distorted  isoparametric many  the  However,  i s more  loads.  as  results  of elements.  element  the a p p l i e d  such  the bicubic  Nevertheless, load  order  order  element caused  elements. consistent  by  the element  distortion. The vector shear  following for  is  the bicubic  applied  along  terms  of the element  used  to determine  the d e r i v a t i o n of isoparametric  one b o u n d a r y . coordinates.  the c o n s i s t e n t  the  consistent  element,  The The  load  with  vector  resulting  shear  vector  a  load uniform  i s defined  equations  f o r any  in  c a n be  distorted  element.  Fig.  A.1:  Bicubic  Isoparametric P l a t e S h e l l Element S h e a r L o a d i n g a l o n g One E d g e  84  with  Uniform  The  definition  load vector P  of  the  i s g i v e by  i t h term of  equation  Pi = JJ  the  element  consistent  [A.1].  q(x,y)N (r,s)dxdy  (A.l)  i  Area  where <l{ iy)  =  x  applied traction  Pi =  the  r,s  the  reduced to the  element  A  surface  vector  shape f u n c t i o n f o r node x  isoparametric  local  the  i ' t e r m of c o n s i s t e n t l o a d  ^ ( ^ 3 ) =  For  on  coordinate limits  of  element  t h i s can  s y s t e m and  -1  to  the  be  converted  volume  into  integration  1.  +1+1 P  = J J  {  q (z, y) 7Y, ( r , s) detJ dr ds  (A.2)  -1-1 where  If  detJ = <j(r,s)  determinant matrix  is  everywhere e l s e ,  uniform  equation  of t h e  along  [A.2]  along  can  be  Jacobian  transformation  the edge  s=1  reduced  and  zero  to.  +1 Pi = q j  i = 2,6,5,1  Ni detJ dr  (4.3)  -1 and  ^  dN-  .  dr  3  1 through  4,  j=2,6,5,l  or c o n v e r t i n g  the  i n d i c e s to  gives the  +1 Pk = q f Nkiy^x^dr  x  2  k = 1,2,3,4  x  3  85  line  integral,  Table  A . I : Four  TV* * 1 6  Ni  C u b i c Shape F u n c t i o n s a l o n g Boundary s=1, -1<r<1  fact.  (r+1)  N  3  Multiplying  (1-r)  1  1  1  1  —  1  1  9  1  1  —  1  1  1  1  1  -  -9  2  (3r-1 )  -  1  N  (3r+1)  t h e B i c u b i c Element  these  out and t a k i n g t h e d e r i v a t i v e s  ^=(^)(-3r  3  gives:  r2 + 3 r - l )  +  (4.4)  ^4=(^)(9r  3  + 9r -r-l) 2  ffi.(±)(-^  +  /-9\ ,  dN  2  9  )  .  9  -«r = ( » ) (-  Expanding  ttr+1  r  2  +  2  r  + ) 3  (4.5)  e q u a t i o n [A.3]  . - l  f -£  - l  9  +X3 Nt  -1  dr  +  Xl  JNt 9Jl±dr  -1  86  (4.6)  E a c h of  the  f o u r terms  N  = A  k  Thus, the  k  (a r* + b r k  k  ^  = B,(eir2  integration  f o r these  +1  i s of  the  form  dr  k  where  [A.6]  in equation  2  /ir +  +  +  c r+d) k  ri  t e r m s need o n l y be  done once,  +1  N -g~r dr = A Bi j k  (a r  k  k  3  + b r + c r + d ) for2 + fir + g ) dr 2  k  k  k  t  -l 2A B k  =  Substituting equations simple  the  \  { h € l  ^  and  element with uniform  x  shear  k  P  3  = 0.300Z! =  -  -0.0875X!  +  {A.7)  d k e i  f o r a ,b ,c d ,ei,fi  values  k  k  [A.6]  equation  kt  g  k  results  t  in  f o r the c o n s i s t e n t l o a d v e c t o r f o r s t r e s s a l o n g one  -0.7125Z! +  =  k9l  3  -0.500ZJ + 0.7125x  2  4  + (<**e< + c fi + b )  fl)  =  P  P  a  [A.5] into  expressions  P  ~ "  5  corresponding  [A.4]  linear  t  0.0x  2  1.0125x  2  +  2  + 0.300x  -  0.300i  1.0125z  O.O13  +  2  -  +  3  +  3  -  87  0.0875x  0.300x  3  four an  side.  0.7125x  0.7125x  from  +  '  4  4  U.8) 4  0.500x  4  ,  Appendix B ASCE Suggested The  ultimate  g i v e n by e q u a t i o n  Design Guides i n - p l a n e shear  r  a  )  '  (  *  m  )  A = 0A5D  [ B . 1 ]gives:  S = 0.90(D/6) i s made i n e q u a t i o n s  a n d t h e two e q u a t i o n s  expression, equation shear  §  t h e above i n t o e q u a t i o n  the substitution  [B.4]  (  hole H = 0.9D  If  [ 1 1 ] i s g i v e n by:  »- ( ' "f )v r E »  — *  Substituting  s t r e s s o f a p e r f o r a t e d web a s  [ 1 8 ] i n Reference  f  For a " c i r c u l a r  f o r Beams with Web Holes  [B.7],  a r e combined and s i m p l i f i e d , a s i n g l e i s obtained  f o r theultimate in-plane  s t r e s s f o r a s q u a r e p l a t e p e r f o r a t e d by a  a = 3.o(i-l)  fy  »  T  [ B . 3 ] and  88  hole.  (B.6)  _^(l-S)(l/S-l) \Jl + 3(l/S-  circular  l)  2  Hole Size Roitio D/b Fig.  B . 1 : U l t i m a t e I n - p l a n e C a p a c i t y o f S h e a r Web w i t h a C i r c u l a r P e r f o r a t i o n as Proposed i n Reference [ 1 1 ]  89  APPENDIX C Modification  o f NISA80 a t U.B.C.  NISA80 The f o l l o w i n g i s a summary of t h e c h a n g e s t h a t were made t o O c t o b e r 83 v e r i o n of NISA80, c a l l e d NISA83 on t h e C i v i l E n g i n e e r i n g VAX 11/730, a t U.B.C. the  PROGRAM NISA83C Change  file  names t o s u i t SLF( 1 ) =' DRA2 S L F ( 2 ) = ' DRA2 S L F ( 3 ) = ' DRA2 S L F ( 4 ) = ' DRA2 S L F ( 5 ) = ' DRA2 S L F ( 6 ) ; ' DRA2 S L F ( 7 ) : ' DRA2  from  the disk  [SCRATCH [SCRATCH tSCRATCH [SCRATCH [SCRATCH [SCRATCH [SCRATCH  names a t U.  ]NISA20, SCR* ]NISA21, SCR' ]NISA22, SCR' ]NISA2 3, SCR' ]NISA24 SCR' ]NISA25 SCR' ]NISA26 SCR'  F1='DRA2:[SCRATCH]NS01.SCR' F2='DRA2:[SCRATCH]NS 0 2.SCR' F3='DRA2:[SCRATCH]NS03.SCR' F8='DRA2:[SCRATCH]NS08.SCR' F9='DRA2:[SCRATCH]NS09.SCR' to  SLF(1)='NISA20.SCR* SLF(2)='NISA21.SCR' SLF(3)='NISA22.SCR* SLF(4)='NISA23.SCR' SLF(5)='NISA24.SCR' SLF(6)='NISA25.SCR' SLF(7)='NISA26.SCR' C F1='NS01.SCR' F2='NS02.SCR' F3='NS03.SCR* F8='NS08.SCR' F9='NS09.SCR'  SUBROUTINE INPUT Changes have been made t o t h e c y l i n d r i c a l n o d a l i n p u t r o u t i n e s so t h a t t h e u s e r c a n s e l e c t t h e n o r m a l a x i s . NAXIS=0  y-z plane i s s p e c i f i e d x i s t h e nomal a x i s  NAXIS=1  same a s NAXIS=0  NAXIS=2  x-y p l a n e i s s p e c i f i e d z i s t h e nomal a x i s  from  20  to  20  in polar  coordinates  in polar  coordinates  READ(INP,1000) - - - - ,Z(N),KN,IT WRITE(IOUT,2002) - - - - ,Z(N),KN,IT 1000 FORMAT( ----,15,12) 2001 FORMAT( - - - - ,5X,2HIT/) 2002 FORMAT( - - - - , I 5 , 2 X , I 5 ) READ(INP,1000) - - - - ,Z(N),KN,IT,NAXIS WRITE(IOUT,2002) - - - - ,Z(N),KN,IT,NAXIS  90  1000 FORMAT( 2001 FORMAT( 2002 FORMAT( from C C C  CYLINDRICAL 50  to  - - - - ,15,12,12) ,5X,2HIT,3X,5HNAXIS/) ,I 5,2X,I 5,2X15)  C C C  DUM = Z(N) * RAD Z(N) = Y(N) * SIN(DUM) Y(N) = Y(N) * COS(DUM) CYLINDRICAL  50  COORDINATES  COORDINATES  CONTINUE IF (NAXIS.EQ.2) THEN DUM = Z(N) * RAD Z(N) = X(N) X(N) = Y(N) * COS(DUM) Y(N) = Y(N) * SIN(DUM) ELSE DUM = Z(N) * RAD Z(N) = Y(N) * SIN(DUM) Y(N) = Y(N) * COS(DUM) END IF  from  DX = (X(N)-X(NOLD)) / XNUM  to  I F (NAXIS.EQ.2) THEN DZ = (Z(N)-Z(NOLD)) / XNUM ELSE DX = (X(N)-X(NOLD)) / XNUM END I F  from  C C C  CYLINDRICAL 60  to  C C C  ROLD = Y(NOLD) / COS(DUMOLD) RNEW = Y(N) / COS(DUM) CYLINDRICAL  60  COORDINATES  COORDINATES  I F (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 I F  SUBROUTINE MAIN Change t h e open s t a t e m e n t s f o r f i l e s NPLOT a n d NPLOT1 from u n f o r m a t e d t o f o r m a t e d . The r e a s o n f o r t h i s change i s t h e EUNIC FORTRAN 77 c o m p i l e r gave an e r r o r when i t r e a d from an u n f o r m a t e d f i l e . T h i s p r o b l e m seems t o be s p e c i f i c t o t h e U. B. C. VAX and i s r e l a t e d t o r u n n i n g VMS w i t h a EUNIC e m u l a t o r r a t h e r t h a n n a t i v e UNIX. from  I F ( I P G . E Q . I ) OPEN(UNIT=NPLOT,- - - ,FORM=UFM) IF(MODEX.NE.O) OPEN(UNIT=NPLOT1,- - - ,FORM=UFM)  to  I F ( I P G . E Q . I ) OPEN(UNIT=NPLOT,- - - ,FORM=FM) IF(MODEX.NE.O) OPEN(UNIT=NPLOT1,- - - ,FORM=FM)  91  SUBROUTINE D3INP Change t h e w r i t e from  WRITE  to 2010  from  from u n f o r m a t e d  t o formated.  (NPLOT)NUMEP,MN  WRITE (NPLOT,2010)NUMEP,MN FORMAT(2I5)  SUBROUTINE PLN Change t h e w r i t e  to  statements  statements  WRITE WRITE WRITE  from u n f o r m a t e d  t o formated.  (NPLOT) HED (NPLOT) NUMNP,NUMEG (NPLOT) X1,Y1,Z1  WRITE (NPLOT,2000) HED WRITE (NPLOT,2010) NUMNP,NUMEG WRITE (NPLOT,2020) X1,Y1,Z1 2000 FORMAT (A72) 2010 FORMAT (215) 2020 FORMAT (1P,3E15.6)  SUBROUTINE D3PLOT Changes where made t o a l l w r i t e formated output.  statements  from u n f o r m a t e d t o  f rom  WRITE WRITE WRITE WRITE WRITE WRITE WRITE WRITE WRITE  NPLOT) N , 1 3 , ( I N O D E ( I ) , I = 1 , 1 3 ) NPLOT) N,9,(INODE(I),1=1,9) PLOT) N,5,(INODE(I),1=1,5) NPLOT) N,4,(INODE(I = 1,4) NPLOT) N,4,(INODE(I = 1,4) NPLOT) N,4,(INODE(I = 1,4) NPLOT) N,4,(INODE(I = 1,4) = 1,3) NPLOT) N,3,(INODE(I = 1 ,3) NPLOT) N,3,(INODE(I  to  WRITE WRITE WRITE WRITE WRITE WRITE WRITE WRITE WRITE FORMAT FORMAT FORMAT FORMAT FORMAT  NPLOT ,2000) N,13 ,(INODE(I),1=1,13) NPLOT ,2010) N,9, ( I N O D E ( I ) , 1 = 1 , 9 ) NPLOT ,2015) N,5, ( I N O D E ( I ) , 1 = 1 , 5 ) NPLOT ,2020) N,4, ( I N O D E ( I ) , 1 = 1 , 4 ) NPLOT ,2020) N,4, ( I N O D E ( I ) , 1 = 1 , 4 ) NPLOT ,2020) N,4, (INODE(I),1=1,4) NPLOT ,2020) N,4, (INODE(I),1=1,4) NPLOT ,2030) N,3, ( I N O D E ( I ) , 1 = 1 , 3 ) NPLOT ,2030) N,3, ( I N O D E ( I ) , 1 = 1 , 3 ) (215, 2 X , 1 3 ( 1 5 ) ) (215, 2 X , 9 ( I 5 ) ) (215, 2 X , 5 ( I 5 ) ) (215, 2 X , 4 ( I 5 ) ) (215, 2 X , 3 ( I 5 ) )  2000 2010 2015 2020 2030 Note;  in line 3 WIRTE(PLOT)N,- - PLOT i s n o t a t y p i n g T h i s i s how i t a p p e a r e d i n t h e o r i g i n a l v e r t i o n .  SUBROUTINE PLOTGEO Change t h e w r i t e f rom  WRITE  statement  from u n f o r m a t e d  t o formated.  (NPLOT) N , I E P , ( I N O D E ( I ) , I = 1 , I E P )  92  error,  WRITE  to 2000  SUBROUTINE WRITE Change t h e w r i t e from to  (NPLOT,) N,IEP, (I NODE(I),I = 1 , 1 E P ) (215,2X,<IEP>,15)  FORMAT  s t a t e m e n t s from u n f o r m a t e d  to formated.  WRITE (NPLOT)PHED WRITE (NPLOT)DSI,DS2,DS3 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 o f t h e c h a n g e s t h a t were made t o the June 84 v e r i o n of NISA80, c a l l e d NISA84 on t h e C i v i l E n g i n e e r i n g VAX 11/730, a t 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( 2010 FORMAT(  German t e x t German t e x t  to  2000 FORMAT( 2010 FORMAT(  English English  SUBROUTINE from  to  text text  ) ) ) )  OPENRF 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' NISA.RN1' "SCRATCH" NISA.RN2' "SCRATCH" NISA.RN3' "SCRATCH" NISA.RN4' "SCRATCH" NISA.RN5' "SCRATCH" NISA.RN6' "SCRATCH" NISA.RN7'  'SCRATCH: NISA.RN1' 'SCRATCH: NISA.RN2' 'SCRATCH: NISA.RN3' :'SCRATCH: NISA.RN4' ••' SCRATCH:NISA.RN5' •'SCRATCH: NISA.RN6' •'SCRATCH: NISA.RN7'  93  C's have been p l a c e d i n t h e f i r s t column o f t h e s e c o n d v e r s i o n of OPENRF so t h a t i t i s n o t c o m p i l e d by t h e U. B. C. Vax 11/730 T h i s second v e r s i o n i s f o r the Cray computer. SUBROUTINE FOPEN from  CHARACTER  NAME*40  to  CHARACTER  NAME*(*)  SUBROUTINE HEDIN M o d i f i e d the output heading t o acknowledge that the work i s b e i n g done a t t h e U. B. C. s i t e on t h e C i v i l E n g i n e e r i n g Vax 11/730. SUBROUTINE INPUT Changes have been made t o t h e c y l i n d r i c a l n o d a l i n p u t r o u t i n e s so t h a t t h e u s e r c a n s e l e c t t h e n o r m a l a x i s . NAXIS=0  y-z plane i s s p e c i f i e d x i s t h e nomal a x i s  NAXIS=1  same a s NAXIS=0  NAXIS=2  x-y p l a n e i s s p e c i f i e d z i s t h e nomal a x i s  from  20  to  20  coordinates  i n polar  coordinates  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) 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  CYLINDRICAL 50  to  i n polar  C C  c 50  COORDINATES  DUM = Z(N) * RAD Z(N) = Y(N) * SIN(DUM) Y(N) = Y(N) * COS(DUM) CYLINDRICAL  COORDINATES  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 I F from  DX = (X(N)-X(NOLD))  to  I F (NAXIS.EQ.2) THEN DZ = (Z(N)-Z(NOLD)) / XNUM ELSE DX = (X(N)-X(NOLD)) / XNUM END I F  from  C C C  CYLINDRICAL COORDINATES 60  to  / XNUM  C C C  ROLD = Y(NOLD) / COS(DUMOLD) RNEW = Y(N) / COS(DUM) CYLINDRICAL COORDINATES  60  I F (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 I F  SUBROUTINE DKTM T h e r e was a c o m p i l e t i m e e r r o r b a c a u s e o f t h e m u l t i p l e d e c l a r a t i o n o f t h e 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 o f t h e c h a n g e s t h a t were made t o t h e u p d a t e o f t h e 3D-PLATE SHELL ELEMENT i n s t a l l e d S e p t . 17 85. C i v i l E n g i n e e r i n g VAX 11/730, a t U.B.C.  on  TRANSFER F I L E D3DMAIN.FOR When t h e T r a n s f e r f i l e D3DMAIN.FOR was r e a d f r o m t h e IBM d i s k e t t e u s i n g t h e program KERMIT a non ASCII c h a r a c t e r was f o u n d on l i n e No. 1117. The c h a r a c t e r was e d i t e d from t h e f i l e u s i n g t h e PC e d i t o r , EDLIN, a n d r e p l a c e d by ?. Line  1112 t o  1119  SUBROUTINE D3LSS C  Q  (A,G,GI,IT)  ******************************************************  c  *  C C C C  * * * *  * TRANSFORM STRESS AND STRAIN LOCAL-GLOBAL LOCAL 3-DIRECTION IS ZERO ? ? ? A  ... VECTOR TO BE TRANSFORMED  95  * * * *  TRANSFER F I L E D3DINP.FOR The same p r o b l e m o f a non ASCII c h a r a c t e r o c c u r e d i n l i n e No. 393 of t h e t r a n s f e r program D 3 D I N P . f o r . A g a i n t h e c h a c t e r was e d i t e d from t h e f i l e b e f o r e t r a n s f e r o f t h e f i l e was c o m p l e t e d t o t h e VAX. Line  390 t o 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 a s 1 (commplete t h i c k n e s s i n t e g r a t i o n ) the program s t o p p e d b e c a u s e o f an e r r o r i n t h e v a r r i a b l e a r r a y d i m e s i o n . Which a r r a y was n e v e r d e t e r m i n e d , however, t h e v a r r i a b l e NBO i s n e v e r a p p e a r s i n t h e p a r r a m e t e r s t r i n g i n t h e s u b r o u t i n e D3DISD. from C 120 CALL D3DISD 1  (DISD,DDISD,B,ALFN,EDIS,DC,DCA(1,1,N),NC(1,N), HTET(1,1,N),IEL,MN,NBO,ND,IFORM,HHI)  120 CALL D3DISD 1  (DISD,DDISD,B,ALFN,EDIS,DC,DCA(1,1,N),NC(1,N), HTET(1,1,N),IEL,MN,ND,I FORM,HHI)  C to C C  96  APPENDIX D P r o g r a m  Listings  APPENDIX D . l NISPLOT Q  *********************************************************  C C C C C C  U N I X  N I S A 8 3 P L O T  c  C C C  THIS PROGRAM USES "DI-3000" TO PLOT THE F I N I T E ELEMENT GRID AND THE D E F L E C T E D SHAPE FROM "NISA83" IN A 3-D FORM.  c p  c c c c c c c c c  V E R S I O N  C R E A T E  ME T A F I L E S  INPUT AND OUTPUT F I L E S . GEO. INPUT F I L E (FORMATED) DISP. INPUT F I L E (FORMATED) STRESS INPUT F I L E (FORMATED) OUTPUT PLOTED TO OUTPUT SCRATCH F I L E FULL PLATE GEO. DEVICE TYPE  Q  OUTPUT  'INF I L E ' 'INF I L E ' 'INFILE ' 'NOUT' 'I SCR' ' ' 'MDEV' 'NDEV'  = = = = = = = =  1 4 3 6 7 8 0 1  (FROM (FROM (FROM  USER) USER) USER)  (METAFILES) (GRAPHICS)  *****************************************************************************  IMPLICIT REAL*4(A-H.O-Z) COMMON / PLT / I N O D E ( 1 0 0 , 1 3 , 5 ) , I E L ( 1 0 0 , 5 ) , N ( 1 0 0 , 5 ) . NMAX(5) COMMON / MAX / R M I N ( 3 ) , R A V E ( 3 ) , RMAX(3), RATIO 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 ) COMMON / SVIEW / D ( 3 ) , U ( 3 ) COMMON / HEAD / PHEAD INTEGER NUMP, NUMEG LOGICAL D3STR, D3PLT, METST, PHEAD CHARACTER*45 HED. PHED CHARACTER EOF DIMENSION S T R E S S ( 1 6 , 3 0 , 5 ) , 1X(500), Y(500), Z(500), DX(500), DY(500). DZ(500). 2 R X ( 5 0 0 ) , R Y ( 5 0 0 ) . R Z ( 5 0 0 ) . 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 J B E G I N CALL J D I N I T (NDEV) CALL JDEVON (NDEV) IF ( M E T S T ) T H E N CALL JFSOPN ( 3 , 0 , 0 , ' N I S A P L O T . M F L ' ) CALL J D I N I T (MDEV) CALL UDEVON (MDEV) END I F CALL JASPEK ( 1 .RATIO) IF ( R A T I O . L T . 1) THEN CALL JVSPAC (-1.0, 1.0, -RAT 10,' RAT 10 ) ELSE I F (RATIO.GT. 1 ) THEN CALL JVSPAC (-1.0/RATIO, 1.0/RATIO, - 1 . 0 , ELSE RATI0=1.0 END I F CALL J S E T D B ( 0 )  97  1.0)  NISPLOT L i s t i n g READ IF  IN  AND PLOT  (NPLATE  FULL  .NE.O)  PLATE  AND ELEMENT  MODEL.  THEN  NVIEW=0 NPLOT=NPLATE CALL  READNO  ( X , Y , Z , H E D , N U M P , N U M E G .  CALL  MAXMIN  ( X . Y . Z . N U M P )  CALL  VIEW  (NVIEW)  CALL  READEL  (NUMEG.NPLOT,ISCR)  CALL  JOPEN  CALL  PLTELE  ( X , Y , Z . N U M E G , N V I E W )  CALL  PLTHED  (HED)  CALL  JCLOSE  CALL  JPAUSE  CALL  JFRAME  END  ISTOP)  1,NPLOT.I SCR  (NDEV)  IF  READ  IN  FULL  MESH  NODE  POINTS  AND  ELEMENTS.  NVIEW=1 NPLOT=NPGEO CALL  READNO  ( 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 )  CALL  MAXMIN  ( X , Y , Z , N U M P )  CALL  VIEW  (NVIEW)  CALL IF  A  PLOT  (METST)  SUBROUTINE  PLOT  T H E NODE  POINTS.  (SUBROUTINE  PLOTNO )  THEN  CALL  JOPEN  CALL  PLTHED  CALL  TO  ( H E D )  JCLOSE  CALL  JPAUSE  CALL  (NDEV)  JFRAME  PHEAD=.FALSE. END  IF  CALL  JOPEN  CALL  PLTNOD  CALL  PLTHED  CALL  ( X , Y . Z , N U M P , N O U T ) ( H E D )  JCLOSE  CALL  JPAUSE  READ CALL  (NDEV)  ELEMENT READEL JFRAME  CALL  JOPEN  C A L L . P L T E L E  PLOTEL)  SCR)  ( H E D )  JPAUSE  D3STR  WITH  IF  (SUBROUTINE  JCLOSE  CALL IF  ELEMENT  ( X , Y , Z , N U M E G . N V I E W )  PLTHED  CALL  AND PLOT  (NUMEG,NPLOT,I  CALL  CALL  DATA  A  (NDEV)  .TRUE.  STRESS  READ  COLOR  IN  THE  STRESS  F I L E  AND PLOT  THE UNDEFLECTED  SHAPE  F I L L  THEN  (D3STR)  NPLOT=NPSTR CALL  VIEW  (NVIEW)  CALL  READST  (STRESS,NMAX,STRMAX,NUMEG,NPLOT,I  CALL CALL  JOPEN  CALL  ELESTR  CALL  PLTELE  CALL  PLTHED  CALL  (X ,  Y . Z . S T R E S S , S T R M A X , N U M E G . 2 )  ( X , Y , Z , N U M E G . 2 ) ( H E D )  JCLOSE  CALL  LEGEND  (STRMAX)  CALL  JPAUSE  (NDEV)  END IF  SCR)  JFRAME  IF  D3PLT  .TRUE.  THAN  PLOT  THE DEFLECTED  AND THE ORIGINAL  98  SHAPE  IN  3-D.  NISPLOT IF  ( D 3 P L T )  Listing  THEN  NPLOT=NPDIS CALL IF  READNO  ( 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  ( I S T O P . L T . O ) DO  39  GOTO  SCR,I  STOP)  5 0  I T E R = 1 . 1 0  NVIEW=2 CALL  VIEW  CALL  ADDDIS  CALL  (NVIEW) ( X , Y , Z .  D X . D Y . D Z .  R X , R Y , R Z , N U M P , - 1)  JFRAME  CALL  JOPEN  CALL  PLTELE  CALL  PLTHED ( H E D )  CALL  ( X . Y , Z . N U M E G . 3 )  JCLOSE  CALL  JPAUSE  (NDEV)  CALL  ADDDIS  ( X , Y , Z .  CALL  JOPEN  D X . D Y . D Z ,  CALL  F I L L E L  CALL  F I L L E L  ( R X , R Y , R Z , N U M E G , 2 )  CALL  P L T E L E  ( R X , R Y . R Z , N U M E G , 2 )  CALL  1 )  JCLOSE  WRITE  ( N O U T , 1 1 0 0 )  READ IF  ( R X . R Y , R Z , N U M E G ,  R X , R Y , R Z . N U M P , 1 )  ( 5 ,' (A1 ) ' )EOF  ( E O F . E C ' S '  .OR.  E O F . E C ' s ' )  GOTO  5 0  c  C  IF  C  A  D3STR  AND D3PLT  STRESS  IF  COLOR  (D3STR) IF  T H E DEFLECTED  .OR.  E O F . E C ' C )  JOPEN  CALL  P L T E L E  CALL  ELESTR  CALL  ELESTR  ( R X , R Y , R Z , S T R E S S , S T R M A X , N U M E G . 2 )  CALL  P L T E L E  ( R X , R Y . R Z , N U M E G , 2 )  CALL  PLTHED ( H E D )  ( X , Y , Z . N U M E G , 3 ) ( R X , R Y , R Z , S T R E S S , S T R M A X , N U M E G  , 1 )  JCLOSE  CALL  LEGEND  WRITE  (STRMAX)  ( N O U T , 1 1 0 0 )  READ END  WITH  THEN  CALL  IF  SHAPE  JFRAME  CALL  END  PLOT  THEN  ( E O F . E C ' C CALL  ARE TRUE  F I L L  ( 5 ,' (A1 ) • • )EOF  ( E O F . E O . ' S '  . O R .  E O F . E O . ' S ' )  GOTO  5 0  I F  I F  CONTINUE  39  CONTINUE  50  END  I F  CLOSE IF  ROUTINE  (METST)  THEN  CALL  JDEVOF  (MDEV)  CALL  JDEND  (MDEV)  END  1100  PLOT  I F  CALL  JDEVOF  (NDEV)  CALL  JDEND  (NDEV)  CALL  JEND  CLOSE  (UNIT=NPGEO)  CLOSE  (UNIT=ISCR)  F O R M A T ( / , '  <RETURN>  TO CONTINUE,  S<RETURN>  TO  S T O P . ' )  STOP END Q  C  c  *******************************  S  U  B  R  O  U  T  ************  I  N  E  V  I  E  *********************************  W  **************************************************************************+* SUBROUTINE  VIEW  (NVIEW)  I M P L I C I T  REAL*4  COMMON  /  MAX /  R M I N ( 3 ) ,  COMMON  /  SVIEW  /  INTEGER  ( A - H . O - Z ) R A V E ( 3 ) ,  R M A X ( 3 ) ,  RATIO  D ( 3 ) . U ( 3 )  NVIEW  CALL  JRIGHT  CALL  JVUPNT  ( . T R U E . ) ( R A V E ( 1 ) , R A V E ( 2 ) . R A V E ( 3 ) )  99  NISPLOT IF  (NVIEW.EO.1) D( 1 ) = - 3 . 0 D(2)=-3.0 D ( 3 ) = -1 . 0 U( 1 ) = - 1 . 0 U(2)=-1.0 U(3) = 4.0  Listing  THEN  UMIN = RMIN( 1 ) - R A V E ( 1 ) UMAX = R M A X ( 1 ) - R A V E ( 1 ) VMIN=RMIN(2)-RAVE(2) VMAX=RMAX(2)-RAVE(2) CALL CALL CALL  JNORML(0.0.0.0.-1.0) JUPVEC(0.0,1.0.0.0) JWINDO (UMIN,UMAX,VMIN.VMAX)  CALL CALL CALL  JNORML(0.0,0.0.-1.0) JUPVEC(1.0,1.0,0.0) JWINDO (UMIN,UMAX,VMIN,VMAX)  CALL JPERSP (-10.0) ELSE IF ( N V I E W . E O . O ) THEN UMIN = RMIN( 1 ) - R A V E ( 1 ) * 0 . 7 UMAX=RMAX( 1 ) - R A V E ( 1 ) * 0 . 7 VMIN=RMIN(2)-RAVE(2)*0.7 VMAX=RMAX(2)-RAVE(2)*0.7  CALL JPERSP (-10.0) ELSE IF ( N V I E W . E O . 2 ) THEN DUM=RMAX(1)-RMIN(1) UMIN=-0.65*DUM UMAX= 0 . 6 5 * D U M VMIN=-0.65*DUM  VMAX= O . G 5 * D U M DIST=(RMAX(1)-RMIN(1))*0.90  c  100  110  1000 1010 1020  CONTINUE  WRITE ( G . 1 0 1 0 ) (D(I),I=1,3) READ ( 5 , 1 0 0 0 , E R R = 1 0 0 ) BX.BY.BZ IF (BX.EO.0.0 .AND. B Y . E O . 0 . 0 .AND. B Z . E O . 0 . 0 ) ELSE D ( 1 ) = BX D ( 2 ) = BY D(3)= BZ CONTINUE 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 ) BX.BY.BZ IF (BX.EO.0.0 ELSE U(1)=BX U(2)=BY U(3)=BZ END I F END I F  FORMAT FORMAT FORMAT  .AND.  BY.EO.0.0  (3G12.6) (/,' NORMAL V E C T O R (' UP V E C T O R X . Y . Z  .AND.  THEN  BZ.EO.0.0)  THEN  X.Y.Z ?',3F10.3) ?',3F10.3)  c  C A L L JNORML (D(1 ) , D ( 2 ) , D ( 3 ) ) CALL JUPVEC (U(1 ) , U ( 2 ) , U ( 3 ) ) C A L L JWINDO (UMIN,UMAX,VMIN,VMAX) CALL JVUPLN (DIST) CALL JPERSP (DIST*-3.0) ENDIF CALL JWCLIP ( .TRUE . ) RETURN END  c **************************** C  S U B R O U T I N E  S E T U P  Q  *****************************************************************************  SUBROUTINE SETUP (NPGEO.NPDIS,NPSTR,NPLATE,I 1 D3STR.D3PLT.METST) IMPLICIT REAL*4 (A-H.O-Z) C H A R A C T E R CHAR LOGICAL D 3 S T R , D 3 P L T , M E T S T , S T A T CHARACTER * 20 I N F I L E  100  SCR,  NISPLOT c c c  GEOMETRY  AND  SCRATCH  Listing  FILE  WRITE(6,2000) CALL IFILE (INFILE,STAT) IF ( .NOT. STAT ) GOTO 100  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 ' ) IF (INFILE .EO. '2f.geo') THEN OPEN  )  (UNIT =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 ELSE  NPLATE  NPLATE=0 END IF REWIND I S C R REWIND NPGEO DISPLACEMENT CONTINUE WRITE(6,2010)  1 10  READ ( 5 , ' ( A 1 ) ' , E R R = 1 1 0 ) CHAR IF (CHAR.EO.'y' .OR. C H A R . E O . ' Y ' ) THEN D3PLT=.TRUE . WRITE(6,2020) CALL IFILE (INFILE,STAT) IF (.NOT. S T A T ) GOTO 120 OPEN (UNIT=NPDIS.FILE=INFILE.STATUS='o1d') REWIND N P D I S ELSE D3PLT=.FALSE. END IF  120  C C C  STRESS  READ ( 5 , ' ( A 1 ) ' , E R R = 1 3 0 ) CHAR IF (CHAR.EQ.'y' .OR. C H A R . E O . ' Y ' ) D3STR=.TRUE. WRITE(6,2040) CALL IFILE (INFILE,STAT) IF (.NOT. S T A T ) GOTO 140  140  OPEN (UNIT=NPSTR REWIND N P S T R ELSE D3STR=.FALSE . END IF  C C C  WRITE (6,2050) READ ( 5 , ' (A1 ) ' ) IF (CHAR.EO.'Y' METST=.TRUE. ELSE  2050  Q  THEN  , F I L E =I N F I L E , STATUS= ' o l d ' )  METAFILE  2000 2010 2020 2030 2040  C  FILE  CONTINUE WRITE(6,2030)  130  Q  FILE  CHAR .OR.  CHAR.EQ.'y')  THEN  METST=.FALSE. END IF FORMAT ( 'GEOMETRIC INPUT F I L E NAME?'.$) FORMAT ( / / . ' D O YOU H A V E A D I S P L A C E M E N T F I L E Y/N ? FORMAT ( ' D I S P L A C E M E N T INPUT F I L E NAME? '.$) FORMAT ( / / , ' D O YOU H A V E A S T R E S S F I L E Y / N ? '.$) FORMAT ( ' S T R E S S INPUT F I L E NAME? ',$)  FORMAT RETURN END  (//,'DO  YOU  WANT  TO  CREATE  ***************  S U B R O U T I N E  A  METAFILE  Y/N?  '.$)  '.$)  *******************************************  I F I L E  *****************************************************************S  SUBROUTINE I FILE(INFILE.STAT) CHARACTER*20 INFILE LOGICAL STAT  1 0 1  ***********  *******  NISPLOT READ(5,'(A20)')  Listing  INFILE  INQUIRE (FILE=INFILE,EXIST=STAT) IF ( S T A T . E O . . F A L S E . ) THEN WRITE(6,*) ' **** E R R O R **** WRITE(6,*) ' F I L E DOES NOT E X I S T '  WRITE(6,*) END IF RETURN END  '  TRY A G A I N  '  * * * * * * * * * * * * * * * * * * * * * * * * * * * * * , * * * * , * * « * , » * * * * * * * * * * * * * * * * * * * * * * * « *  c  C  S U B R O U T I N E  Q  220 240 2000 2010 2020  c  C c  R E A D N O  *******************************************************************  SUBROUTINE 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 ) IMPLICIT R E A L M (A-H.O-Z) INTEGER NUMP,NUMEG,ICORD,NPLOT.ISCR 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) THEN READ(NPLOT.2010) NUMP,NUMEG WRITE(ISCR,2010) NUMP,NUMEG END IF 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 STOP) 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 ) CONTINUE CONTINUE FORMAT ( A 4 5 ) FORMAT ( 2 1 5 ) FORMAT (1P.3E15.6) RETURN END  ************************************************************* S U B R O U T I N E  R E A D E L  ****************************************************************** SUBROUTINE  READEL  ( N U M E G . N P L O T , I SCR )  IMPLICIT REAL*4(A-H,0-Z) COMMON / P L T / I N 0 D E ( 1 0 0 , 1 3 . 5 ) , INTEGER NUMEG,NPLOT,ISCR  IEL(100,5),  N(100,5),  NMAX(5)  c  C C  READ NODES FOR E L E M N T  OF E L E M E N T GROUP N U M .  AND S T O R E  IN  INODE(300,13,5)  c  DO  4 2 0 NUM=1,NUMEG READ(NPLOT,2000) NMAX(NUM),NUMEL WRITE(ISCR,2000) NMAX(NUM),NUMEL DO 4 1 0 L O O P = 1 , N M A X ( N U M ) READ(NPLOT,2010) I EL(LOOP,NUM),N(LOOP,NUM), 1 ( INODE(LOOP,J,NUM),J=1,N(LOOP,NUM)) WRITE(ISCR,2010) I EL(LOOP,NUM),N(LOOP,NUM), 1 (INODE(LOOP,J,NUM),J=1,N(LOOP,NUM ) ) 410 CONTINUE 420 CONTINUE WRITE (ISCR,2020) 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 I N NODE AND E L E M . RETURN END c  C C  DATA.  **'./)  ****************************************************************** S U B R O U T I N E  R E A D S T  ****************************************************************** SUBROUTINE  READST  CHARACTER*45 DIMENSION  (STRESS,NMAX,STRMAX,NUMEG,NPLOT.I  SHED  STRESS( 1 6 , 3 0 , 5 ) , N M A X ( 5 ) ,STRMAX(2)  STRMAX(1)=10000.0  STRMAX(2)=-10.0 READ (NPLOT.1020)SHED DO 7 2 0 N U M = 1 , N U M E G DO 7 1 0 I E L = 1 , N M A X ( N U M ) / 5  102  SCR)  NISPLOT READ  WRITE DO  IF  1  700  1  ( S T R E S S ( I , I E L , N U M ) , I = 1 , 16)  (ISCR. 1000)  ( S T R E S S ( I , I E L . N U M ) , I = 1, 1 G )  1 = 1 , 1G  ( S T R E S S ( I . I EL,NUM) .LT.STRMAX( 1) ) STRMAXf1)=STRESS(I,IEL.NUM)  IF  700  (NPLOT, 1000)  Listing  ( STRESS(I.I EL,NUM).GT.STRMAX(2) ) STRMAX(2)=STRESS(I.IEL.NUM)  CONTINUE  710  CONTINUE  720  CONTINUE  1000  FORMAT  1020  IF  ((STRMAX(2)-STRMAX(1))  FORMAT  (4(4(2X,  (2X.A45)  .LT.  1PE 1 2 . 5 )  /))  0.001  )  STRMAX(2)=STRMAX(2)+1.0  RETURN END C  S U B R O U T I N E SUBROUTINE IMPLICIT  M A X M I N  MAXMIN  REAL*4  COMMON  /  MAX  DATA  /  1.0,1.0,1.0  DIMENSION R  /  (X,Y,Z,NUMP)  (A-H.O-Z)  RMIN(3),  RAVE(3),  RMAX(3),  X(500),Y(500),Z(500),R(3)  RATIO  /  RMIN(1)=X(1)  RMIN(2)=Y(1 ) RMIN(3)=Z( 1) RMAX(1 )=X( 1 )  RMAX(2)=Y( 1 ) R M A X ( 3 ) = Z( 1 ) DO  510  IF IF IF IF  IF IF  I=2,NUMP (X(I) .LT .RMIN(1)) ( X ( I ) . G T .RMAX( 1 ) ) ( Y ( I ) . L T .RMIN(2)) ( Y ( I ) . G T .RMAX(2) ) (Z(I ) .LT .RMIN(3)) ( Z ( I ) . G T .RMAX(3) )  CONTINUE D E L T X = R M A X ( 1 -) R M I N ( 1 ) DELTY =RMAX(2) -RMIN(2) DELTZ =RMAX(3) -RMIN(3) IF  ((DELTY/RATIO) DELT=DELTY  .GT.  RMIN(1) = X(I RMAX(1) = X(I  RMIN(2) = Y(I RMAX(2) = Y(I RMIN(3) = Z(I RMAX(3) = Z(I  DELTX)  THEN  R(1)=1.0/RATIO ELSE DELT=DELTX R(2)=RATI0 ENDIF  DO  520  C  C  c c c c c c  520 1=1,3 R A V E ( I ) = ( R M A X ( I ) + R M I N ( I ) )/2  RMAX(I)=RAVE(I) RMIN(I)=RAVE(I ) CONTINUE RETURN END  .0  + R(I)*DE LT*0.59 R(I)*DELT*0.59  A * * * * * * * * * * * * * * . * . * * * * * * * * * * * * * * * * * * * * . * * * * * * * * . * * S U B R O U T I N E  A D D D I S  SUBROUTINE A D D D I S ( X , Y , Z , D X . D Y , D Z , R X . R Y , RZ , N U M P . N C A S E ) IMPLICIT REAL*4 (A-H.O-Z) COMMON / MAX / R M I N ( 3 ) , RAVE(3), RMAX(3), RATIO DIMENSION X ( 5 0 0 ) , Y ( 5 0 0 ) , Z ( 5 0 0 ) , NCASE  = -2  NO  NCASE  = -1  ADD ONLY  NCASE  =  SCALE  NCASE  =  0  SCALING THE  1 DO B O T H  IS  DONE.  CONST.  TO  Z  DISPLACEMENTS.  Z-DISPLACEMENT  THE  ABOVE.  (DZ)  AND  ADD  1DX(500),DY(500),DZ(500), RX(500),RY(500),RZ(500) CONST = NCASE * 0 . 0 6 5 * (RMAX( 1)-RMIN( 1 ) ) IF (NCASE . E O . -1) THEN C 0 N S T = 1 6 . 0 * CONST DMULT=0.0 ELSE IF (NCASE .EQ. -2) THEN  103  TO O R I G I N A L  COODINATES.  NISPLOT L i s t i n g CONST=0.0 DMULT=0.0 ELSE S E A R C H T H R O U G H T H E Z D I S P L A C E M E N T S AND T H E S C A L E SO T H A T T H E Y H A V E A P P R O X . 20% SCREEN WINDOW. RMAX(3)=DZ( 1) RMIN(3)=DZ( 1) DO 6 0 0 I = 2 , N U M P  IF (DZ(I).GT.RMAX(3)) IF (DZ(I).LT,RMIN(3 ) ) CONTINUE IF (RMAX(3).LE,RMIN(3)) DMULT=1 . 0 ELSE  600  DMULT= END IF END IF  DO  Q  *  RMAX(3 ) = DZ( I ) RMIN(3)=DZ(I ) THEN  (RMAX(1)-RMIN(1 ) )  /(RMAX(3)-RMIN(3))  6 1 0 1 = 1 ,NUMP RX(I ) = X ( I ) + D X ( I ) RY(I)=Y(I)+DY(I)  RZ(I)=Z(I) CONTINUE RETURN END  610  + DZ(I)*DMULT  +  CONST  ***********************  C C  0.25  THEM  S U B R O U T I N E  P L T H E D  ***************************************************************************** SUBROUTINE  '  IMPLICIT  COMMON  COMMON  /  /  PLTHED  (HED)  REAL*4(A-H.O-Z)  MAX  HEAD  CHARACTER*45 CHARACTER*1  /  /  RMIN(3), PHEAD  HED.  FLAG  RAVE(3),  RMAX(3),  RATIO  NEWHED  L O G I C A L PHEAD DIMENSION WX(4),WY(4),WZ(4) c  C C  LOCATE SO T H E IF IF  THE VIEWPLANE AND T H E S E T T H E T E X T ATTRIBUTES H E A D I N G W I L L A P P E A R A T T H E T O P OF T H E P A G E .  (PHEAD .EO. . F A L S E . ) RETURN (RATIO . L E . 1.0) THEN CALL J C O N V W (-1 . 0 , R A T 1 0 , W X ( 1 ) , W Y ( 1 ) , W Z ( 1 ) ) C A L L JCONVW ( O . O . R A T I 0 , W X ( 2 ) , W Y ( 2 ) , W Z ( 2 ) ) C A L L JCONVW ( 0 . 0 , 0 . 0 , W X ( 3 ) , W Y ( 3 ) , W Z ( 3 ) ) C A L L JCONVW ( 1 . 0 , - R A T I 0 , W X ( 4 ) , W Y ( 4 ) , W Z ( 4 ) )  ELSE CALL CALL CALL CALL END IF  JCONVW JCONVW JCONVW JCONVW  (- 1 , 0 / R A T I O , 1 . 0 , W X ( 1 ) , W Y ( 1 ) , W Z ( 1 ) ) (0.0, 1.0,WX(2),WY(2),WZ(2) ) (0.0,0.0,WX(3),WY(3),WZ(3) ) (1,0/RATIO,-1.0,WX(4),WY(4).WZ(4))  CXBASE=WX(2)-WX(1) CYBASE=WY(2)-WY(1) CZBASE=WZ(2)-WZ(1) CXPLAN=WX(1)-WX(3) CYPLAN=WY(1)-WY(3) C Z P L A N = WZ( 1 ) - W Z ( 3 ) CXSIZE=0.065*RATI0*SQRT(CXBASE*CXBASE+CYBASE*CYBASE+CZBASE*CZBASE) CYSIZE=0.055*SQRT(CXPLAN*CXPLAN+CYPLAN*CYPLAN+CZPLAN*CZPLAN) CALL JUPDAT WRITE ( 6 , 2 0 0 0 ) READ ( 5 , 1 0 0 0 ) FLAG IF (FLAG.EO.'Y' .OR FLAG.EO.'y') WRITE (6,2001) . READ ( 5 , 1 0 0 1 ) NEWHED END IF CALL JBASE (CXBASE.CYBASE, CZBASE) CALL JPLANE(CXPLAN,CYPLAN, CZPLAN) CALL CALL CALL  0  THEN  JSIZE (CXSIZE.CYSIZE) JCOLOR (0) JJUST (2,3)  1 04  NISPLOT CALL CALL  Listing  J3MOVE ( W X ( 2 ) , W Y ( 2 ) . W Z ( 2 ) ) JFONT (18)  CALL J F A T T R ( 1 , 1 . 0 . 1 . 3 , 1 6 3 8 3 ) IF (FLAG.EO.'Y' .OR. F L A G . E O . ' y ' ) CALL JFSTRG (NEWHED) ELSE CALL JFSTRG (HED) END IF  C C C  WRITE  FOOTNOTE  CXSIZE=CXSIZE/2  AT  BOTTOM  OF  THEN  PAGE  .0  CYSIZE=CYSIZE/2 .0 CALL JLWIDE(SOOO) CALL ddUST (3,1) CALL JCOLOR (2) CALL JSIZE (CXSIZE.CYSIZE) CALL CALL 1000 1001  2000 2001  J3M0VE (WX(4),WY(4),WZ(4)) JFONT (1)  CALL J3STRG ( ' U . FORMAT (A1 ) FORMAT ( A 4 5 ) FORMAT FORMAT  (/,' (/,'  1'B RETURN END  B.  C.  CIVIL  ENGINEERING')  D o y o u w a n t a new t i t l e ? E n t e r new T i t l e ' , / ,  1  2  3  y/n') 4  E' )  * * * * * » * * * * * * * * * * * * * * * * * * * * * * * * * * * * ^  c  C  S U B R O U T I N E  P L T N O D  C »*****«».»*•.******•«*..««**««*«****»**.»*«.*************•**.*.,*•*  720  SUBROUTINE PLTNOD (X.Y.Z.NUMP) IMPLICIT REAL*4(A-H.O-Z) CHARACTER*1 YES COMMON / MAX / R M I N ( 3 ) , R A V E ( 3 ) . INTEGER NUMP DIMENSION X(500),Y(500),Z(500) C A L L JCMARK (2) CALL JCOLOR (0) DO 7 2 0 1 = 1 , N U M P CALL J3MARK (X(I ) .Y(I ) . Z ( I ) ) CONTINUE  RMAX(3) ,  RATIO  c  C  NUMBER  THE  NODE  POINTS.  c  1000 2000  CALL JUPDAT WRITE(6,2000) READ ( 5 , 1 0 0 0 ) YES IF (YES.EO.'Y' . O R . Y E S . E O . ' y ' ) C A L L NODNUM FORMAT ( A 1 ) FORMAT (/,' NODE N U M B E R I N G ? y/n ',$) RETURN END  ****************************************************************  c  C c  (X.Y.Z.NUMP)  S U B R O U T I N E  P L T E L E  ****************************************************************  SUBROUTINE P L T E L E (X,Y,Z.NUMEG,NWRITE) IMPLICIT REAL*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(100.5), 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 NUMEG DIMENSION X(500).Y(500),Z(500) c  C C 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 P L O T O U T S I D E OF T H E E L E M E N T I N S O L I D LINE, AND P L O T ANY I N T E R N A L L I N E S W I T H D A S H E D L I N E S . LOOP OVER E L E M E N T G R O U P S .  c  ICOLOR = 1 IF (NWRITE . E O . 3) IC0L0R=2 DO 8 1 0 N U M = 1 , N U M E G IF (NWRITE .EO. 1) I C 0 L 0 R = N U M IF (ICOLOR G E . 3) I COLOR = ICOLOR+1 CALL JCOLOR(ICOLOR)  105  NMAX(5)  NISPLOT IELO=0 OO 8 0 0 K = 1 , N M A X ( N U M ) IF (IELO.EO.IEL(K.NUM)) ISTYL=3 ELSE ISTYL=0  Listing  THEN  IELO=IEL(K,NUM) END I F IF  800 810  ( N W R I T E . E 0 . 1 .OR. N W R I T E . E 0 . 2 CALL JLSTYL (ISTYL) C A L L ORAY (X.Y,Z,K,NUM) END IF CONTINUE CONTINUE  .OR.  ISTYL.EQ.O)  THEN  c  C  WRITE  ELEMENT  NO.  IN  THE  MIDLE  OF  THE  ELEMENT.  c  CALL ELENUM RETURN END  Q  C  (X , Y , Z , N U M E G , N W R I T E )  ****************************************^  S U B R O U T I N E  Q  D R A Y  ****************************  S U B R O U T I N E DRAY (X,Y,Z,K,NUM) IMPLICIT REAL*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 ) . INTEGER NRAY.K.NUM DIMENSION X(500),Y(500),Z(500), 1 XARRAY(12),YARRAY(12),ZARRAY(12) NRAY=N(K,NUM)-1 DO 3 0 0 J = 2 , N ( K , N U M )  N(100,5),  NMAX(5)  XARRAY(J- 1) = X(INODE(K,J.NUM)) YARRAY(J- 1 ) = Y(IN0DE(K.J.NUM))  300  ZARRAY(J-I) = Z(INODE(K,J,NUM)) CONTINUE CALL J3M0VE(X(IN0DE(K, 1.NUM)),Y(INODE(K, 1 Z(INOD E(K, 1 .NUM))) CALL J3P0LY(XARRAY,YARRAY,ZARRAY,NRAY) RETURN END  Q  C  N O D N U M *****************************************************************************  S U B R O U T I N E NODNUM (X.Y.Z,NUMP) IMPLICIT REAL*4(A-H,0-Z) COMMON / MAX / R M I N ( 3 ) , R A V E ( 3 ) , R M A X ( 3 ) , RATIO 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 ) IF (NUMP.GE.200) RETURN C A L L JCONVW (0.0.0.0,WX(1),WY(1),WZ(1)) C A L L JCONVW ( 0 . 5 , 0 . 0 . W X ( 2 ) , W Y ( 2 ) ,WZ(2)) CXBASE=WX(2)-WX(1) CYBASE=WY(2)-WY(1) CZBASE=WZ(2)-WZ(1) CALL JBASE (CXBASE.CYBASE.CZBASE) CALL JPATH (1) XSIZE=0.013*(RMAX(1)-RMIN(1)) YSIZE=0.013*(RMAX(2)-RMIN(2)) CALL JSIZE(XSIZE,YSIZE) CALL JCOLOR (1) CALL JJUST (3,3) DO 7 0 0 1 = 1 , N U M P WRITE(CHAR, ' ( 13) ' ) I CALL J3M0VE(X(I),Y(I),Z(I))  700  ),  *****************************************************************************  S U B R O U T I N E  Q  C  1.NUM)  CALL J3STRG(CHAR) CONTINUE RETURN END  106  NISPLOT L i s t i n g C c  S U B R O U T I N E  E L E N U M  «««**«**.«.*****.**.*.****^^  SUBROUTINE ELENUM ( X , Y , Z . N U M E G , N W R I T E ) IMPLICIT REAL*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 ) , COMMON / MAX / R M I N ( 3 ) , R A V E ( 3 ) , R M A X ( 3 ) , RATIO C H A R A C T E R * 3 CHAR DIMENSION X ( 5 0 0 ) . Y ( 5 0 0 ) , Z ( 5 0 0 ) . W X ( 3 ) , W Y ( 3 ) , W Z ( 3) IF (NWRITE.EO.2 .OR. NWRITE.EO.O) THEN RETURN END I F C  NMAX(5)  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,WX(2),WY(2),WZ(2)) C A L L JCONVW (0.0,0.5,WX(3),WY(3),WZ(3)) CXBASE=WX(2)-WX(1) CYBASE=WY(2)-WY(1) CZBASE=WZ(2)-WZ(1) CXPLAN=WX(3)-WX(1) CYPLAN=WY(3)-WY(1) IF  C  (CXPLAN.EO.O  .AND. CYPLAN.EO.O)  RETURN  CZPLAN=0.0 CALL JBASE (CXBASE.CYBASE.CZBASE) CALL JPLANE(CXPLAN,CYPLAN,CZPLAN) CALL JPATH ( 1 ) CALL JJUST (2,2) XSIZE=0.022*(RMAX(1)-RMIN(1)) YSIZE=0.022*(RMAX(2)-RMIN(2)) CALL JSIZE(XSIZE,YSIZE) IC0L0R=2 DO  8 4 0 NUM=1,NUMEG IF (NWRITE ,E0. 1)  ICOLOR=NUM  IF (ICOLOR . G E . 3) ICOLOR=ICOLOR+1 CALL JCOLOR (ICOLOR) IELO=0 DO 8 3 0 K = 1 , N M A X ( N U M ) IF (IELO.NE.IEL(K,NUM)) IELO=IEL(K,NUM) RELX=0.0 RELY=0.0 RELZ=0.0 NN=N(K,NUM)-1 DO 8 2 0 1 = 1 , N N  THEN  RELX=RELX+X(INODE(K,I.NUM)) RELY=RELY+Y(INODE(K,I,NUM)) RELZ=RELZ+Z(INODE(K,I,NUM)) CONTINUE  820  RELX=RELX/NN RELY=RELY/NN RELZ=RELZ/NN  830 840  c  C c  WRITE(CHAR.'(13)') IEL(K.NUM) CALL J3M0VE (RELX,RELY,RELZ) CALL J3STRG (CHAR) END IF CONTINUE CONTINUE RETURN END  »*«***»*******•****************.***»*^^ S U B R O U T I N E F *********************».***^^  I  L  L  E  L  E  SUBROUTINE F I L L E L (X,Y,Z,NUMEG,NSURF) IMPLICIT REAL*4(A-H.O-Z) LOGICAL VISBLE. OK COMMON / P L T / I N O D E ( 1 0 0 , 1 3 . 5 ) , I E L C 1 0 0 . 5 ) , N( 1 0 0 . 5 ) . NMAX(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 ) D A T A NOD / 1, 2, 3, 5, 16. 13. 11, 14, 15, 1 2 3  2, 3, 1 3 , 1 6 ,  IF  4. 6 . 1 5 . 1 4 . 1 4 . 1 5 . 5 . 1 5 , 1 4 . 1 1 , 9. 8,  12. 13, 16, (NSURF.EO.1) THEN  16,  13.  12.  10.  9.  6, 7,  8/  107  NISPLOT  Listing  IC0L0R=4 ELSE IC0L0R=6 ENDIF INTEN =16384  CALL JPINTR ( 1) CALL JCOLOR (ICOLOR) CALL JPIDEX (ICOLOR,INTEN) DO 9 5 0 N U M = 1 , N U M E G  C C C  COLLECT DO  900  930 940 950  C C  C C C C  FOR  ONE  ELEMENT  IN  TO ONE  STRING.  9 1 0 1=2,3 INUMB(1 + 11) = I N O D E ( L E L + 1 , I , N U M ) INUMB(I+13)=IN0DE(LEL+2,I,NUM) CONTINUE START  FILLING  ELEMENT  9 3 0 1=1,9 DO 9 2 0 J = 1 , 4 DX(J)=X(INUMB(NOD(I, J) ) ) DY(J)=Y(INUMB(NOD(I,J) ) ) DZ(J)=Z(INUMB(NOD(I,J) ) ) CONTINUE OK = V I S B L E (DX,DY,DZ,NSURF) IF (OK) CALL J3PLGN (DX.DY.DZ,4) CONTINUE CONTINUE CONTINUE CALL JPINTR (0) RETURN END **************************  S U B R O U T I N E  E L E S T R  * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *'* * * * * * * * * * SUBROUTINE LOGICAL  C C C C C  NUMBERS  940 LEL=1.NMAX(NUM) ,5 DO 9 0 0 1 = 1 , 1 2 INUMB(I)=INODE(LEL,I,NUM) CONTINUE  IMPLICIT  C C C  NODE  DO  920  Q  THE  DO  910 C C C  ALL  ELESTR  (X,Y,Z,STRESS,STRMAX,NUMEG,NSURF)  REAL*4(A-H,0-Z)  VISBLE,  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 / S T R / 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 ) DIMENSION X(1).Y(1),Z(1),ELX(4),ELY(4).ELZ(4) DIMENSION XNEW(25),YNEW(25),ZNEW(25) DIMENSION S T R E S S ( 1 6 , 3 0 , 5 ) , STRMAX(2) CALL DATIN IC0LD=O INTEN =16384 IF (NSURF.EO.1) THEN IVALUE=8 ELSE IVALUE= 1 END IF CALL JPINTR (IVALUE) LOOP DO  OVER  ALL  ELEM.  GROUPS.  AND  EACH ELEMENT  IN  THE  GROUP.  8 9 0 NUM=1.NUMEG DO 8 8 0 NE L E = 1 , N M A X ( N U M ) / 5 LEL= (NELE-1)*5 + 1 C A L L S H A P E TO G E N E R A T E A 25 SAME O U T E R BOUNDARIES A S T H E 16 N O D E I S P A R A M E T R I C CALL  SHAPE(X,Y,Z,  NODED E L E M E N T  WITH  THE  ELEM.  XNEW,YNEW,ZNEW,LEL,NUM)  F I L L T H E 16 S E C T I O N S OF T H E E L E M E N T THE S T R E S S L E V E L IN EACH S E C T I O N .  WITH  108  THE  COLOR  ASSOCIATED  WITH  NISPLOT L i s t i n g DO 1  C C  START  C  C  Q  /  ELEMENT  8 5 0 J= 1 . 4 ELX(J)=XNEW(NPOINT(J, IAREA) ) ELY(J)=YNEW(NPOINT(J,IAREA)) ELZ(J)=ZNEW(NPOINT(J,IAREA)) CONTINUE OK = V I S B L E ( E L X , E L Y . E L Z , N S U R F ) IF (OK) CALL J3PLGN (ELX,ELY,ELZ,4) CONTINUE  860 C  880 890  FILLING  +0.05)  DO  850  C  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 ) (STRMAX(2)-STRMAX(1)) + 1 IF (ICNEW.NE.ICOLD) THEN IF (ICNEW.GT.7) ICNEW=7 IF (ICNEW.LT.1) ICNEW=1 ICOLD=ICNEW CALL JCOLOR (ICOL(ICNEW)) CALL JPIDEX (ICOL(ICNEW).INTEN) END IF  END  LOOP  OVER  CONTINUE CONTINUE CALL JPINTR RETURN END  ELEMENTS  IN  GROUP,  AND END  LOOP  OVER  ALL  GROUPS.  (0)  ***************************  C  S U B R O U T I N E  S H A P E  Q  *****************************************************************************  SUBROUTINE SHAPE( X , Y , Z , XNEW,YNEW,ZNEW, LEL.NUM) COMMON / S T R / 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 / I N O D E ( 1 0 0 , 1 3 , 5 ) . I E L ( 1 0 0 . 5 ) , N( 1 0 0 , 5 ) . NMAX(5) D I M E N S I O N X ( 1 ) , Y ( 1 ) , Z ( 1)  • 901 C C C  • ' 900  910 C C C C  DIMENSION XNEW(25), YNEW(25). DIMENSION A ( 8 ) , S F ( 16) DO 9 0 1 1 = 1 , 2 5 XNEW(I)=0.0 YNEW(I)=0.O ZNEW(l)=0.0 CONTINUE COLLECT  ALL  THE  NODE  NUMBERS  ZNEW(25),  FOR  ONE  INUMB(16)  ELEMENT  IN  TO O N E  STRING.  DO  900 1=1,12 INUMB(I)=INODE(LEL,I,NUM) CONTINUE DO 9 10 1 = 2 , 3  INUMB(1+11)=INODE(LEL+1,I,NUM) INUMB(I+13) = INODE(LE L+ 2,I,NUM) CONTINUE CALCULATE DO  C C  940  FIND  NEW  C O O R D I N A T E S OF  A 25  NODED  ELEMENT.  THE  LOCAL  COORDINATES R  ICOORD=1,25 THE  SHAPE  FUNCTIONS GIVE  R=RS(1.ICOORD) S=RS(2,ICOORD) 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 1  SF(J)=A(NF(1,J)) * A(NF(2,J)) * A(NF(3.J)) * A(NF(4,J)) * A(NF(5,J))*A(NF(6,J))*FACT(J)/256.0  109  AND  S.  NISPLOT 920  Listing  CONTINUE DO  930 1=1,16 XNEW(ICOORD)=XNEW(ICOORD) YNEW(ICOORD)=YNEW(ICOORD) ZNEW(ICOORD ) = ZNEW(ICOORD) CONTINUE CONTINUE RETURN END  930 940  Q  + S F ( I )*X(INUMB(I )) + SF(I)*Y(INUMB(I )) + SF ( I ) * Z(INUMB(I))  ****************************************************  C  S U B R O U T I N E  L E G E N D  Q  *****************************************************************************  SUBROUTINE LEGEND (STRMAX) COMMON / S T R / 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 ) DIMENSION  STRMAX(2),RELX(4),RELY(4),RELZ(4)  CHARACTER*17 CHAR DATA RELX / - 0 . 0 1 , -0.10, 0.0, DATA RELY / - 0 . 0 2 5 , 0.0. 0.05, DATA R E L Z / 0 . 0 , 0.0, 0.0. 0.0 CALL JRESET CALL CALL CALL CALL CALL  JRIGHT JVPORT JVUPNT JNORML dUPVEC  0.10 / 0.0 / /  ( .TRUE . ) ( ( ( ( (  .45,1.0,-1.0,1.0) 0.0,0.0,0.0) 0.0,0.0,-1.0) 0.0,1.0,0.0) -.275, .275. -1.0,  1.0)  CALL CALL CALL  dWINDO dPERSP JWCLIP  CALL  dOPEN  CALL CALL CALL CALL CALL CALL CALL PNTX  JPINTR (1) JSIZE (0.04,0.04) JdUST (1,2) JCOLOR (0) J3M0VE ( - 0 . 2 7 5 , 0 . 8 7 , 0 . 0 ) JHSTRG ( ' [ B U N D ] S T R E S S * [ B L C ] T [ E L C ]  (-1.0) ( .FALSE . )  C  M[BLC]PA*MM' )  dSIZE ( 0 . 0 2 5 . 0 . 0 2 5 ) = -0.160 DO 1 0 0 1 = 1 , 7 PNTY = 0 . 8 7 - 1 * 0 . 0 7 5 51 52 IF  = STRMAX(1) + = STRMAX(1) + (I.E0.7) THEN  (1-1)*(STRMAX(2)-STRMAX(1)) /6.0 ( I ) * ( S T R M A X ( 2 ) - S T R M A X ( 1) ) / 6 . 0 -  WRITE(CHAR,1010) ELSE  100  C 1000 1010  c C  0.1  S1  WRITE(CHAR,1000) S1,S2 END I F CALL dPIDEX ( I C O L ( I ) , 1500) CALL d3M0VE (PNTX,PNTY,O.0) CALL dR3PGN (RELX,RELY,RELZ,4) CALL d3STRG (CHAR) CONTINUE CALL dPINTR (0) CALL dCLOSE FORMAT FORMAT RETURN END  (F7.1,' (F7.1)  T0',F7.1)  ******************************** LOGICAL  FUNCTION  VISBLE  Q  ****************************************************************  LOGICAL FUNCTION VISBLE (ELX.ELY,ELZ,NSURF) DIMENSION 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 C C C C C  S E E I F T H E P L A N E D E F I N E D BY T H E F O U R P A S S E D P O I N T S V I S I B L E UNDER T H E CURRENT V I E W I N G TRANSFORMATION NSURF  =1 =2  FILL FILL  UNDERSIDE TOPSIDE  (-Z) (+Z)  110  IS  NISPLOT DO  C  100 1 = 1 , 4 C A L L UCONWV CONTINUE  100  Listing  (ELX(I ) , E L Y ( I ) , E L Z ( I ) ,VX(I ) .VY(I ) )  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 ) = -DDX1*DDY2 + DDX2*DDY 1 IF ( N S U R F . E O . 2 ) V A L U E ( I ) = V A L U E ( I ) CONTINUE  110 C  VISBLE  C  = VALUE(1)  . G T . 0 . 0 0 .OR.  * -1.0  VALUE(2)  .GT. 0.00  RETURN END  Q  *****************************************************  C  S U B R O U T I N E  Q  D A T I N  **************************************************************************** SUBROUTINE DATIN COMMON ( 6, 7 . 1, 6 D A T A NF/ / S T 1 R, 2 /, 3 ,N5 F, 6 1 2,3,4,5,6,7, 2 2,3,4,6,7,8, 3 1,2,3,6,7,8, 4 1.2,4,5.6,8.  ) 1, R 2 ., 52 .56) ,, 7N, P O I 1N.T3(, 44 ,. 15 6. 6 ) ,. F7 A , 2S ,( 4 , CT(16),ICOL(7) 2,3,4,5,6,8, 1.3,4,6,7,8, 1,2,3,5,7,8, 1.2.4.5.7,8.  2,3.4,5.7,8, 1,2.4.6.7.8, 1,2,3,5.6,8, 1,3.4.5.7.8.  5  1,3,4.5,6,8^/ D A T A 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 FACT / 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 , 81.0,-81.0, 81.0,-81.0 / DATA NPOINT / 8 , 9 , 1 0 , 2 3 , 7,8,23,20, 6,7,20,19, 5,6,19,4, 1 23,10,11,24, 20,23,24,21, 19,20,21,18, 4,19,18,3, 2 24,11,12,25, 21,24,25,22, 18.21.22,17, 3,18,17,2. 3 25.12.13.14. 22.25,14,15, 17,22,15,16, 2,17,16,1 / DATA ICOL / 4 , 6 , 2 . 3 , 5 , 1 , 7 / RETURN END  11 1  APPENDIX D . 2 MESHGEN *********************************************************  c  c  C C C C C C C C C C C C  PROGRAM  TO  GENNARATE  NODE  GRID  T h i s v e r s i o n generates a g r i d of elements f o r a holed plate A 1/4, 1/2, o r f u l l p l a t e model c a n be g e n e r a t e d i f the no. o f s i d e s s p e c i f i e d (NSS) is 1,2,or 4 respectfully. 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 is negative. NSS NER NEA(4) NN(4)  = = = =  N O . OF N O . OF N O . OF CORNER  SIDES ELEMENTS ELEMENTS NODE N O .  RADIALLY PER ARC.  **********************************************************************  c  DIMENSION LOGICAL  90  95  100  NL(36),RL(36),AL(36),ANG(5),XY(5)  FLAG  COMMON NEA(4),NN(4),NSS,NER,NNA,NNR,FLAG FLAG=.TRUE. INPUT=1 IOUT=7 PI=3. 141592654 DEG=PI/180.0 NUMEL=0 OPEN ( U N I T = I N P U T , F I L E = ' N O D E . I N ' ,STATUS= ' OLD' ) OPEN (UNIT= 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 ) NSS,NER,R1 IF (NSS.LT.O) THEN NSS=-1*NSS FLAG=.FALSE. 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 ) CONTINUE IF (NSS.E0.4) THEN NUMNO = N U M E L * 3 * (NER*3+1) ELSE NUMNO = ( N U M E L * 3 + 1 ) * (NER*3+1) END IF WRITE ( I 0 U T . 2 O O O ) NUMNO,NER ANG(NSS+1) = ANG( 1 ) + 9 0 . 0 * N S S IF (NSS.E0.2) THEN XY(NSS+1) = X Y ( 1 ) - 1 0 0 0 . 0 ELSE XY(NSS+1)=XY(1). END IF NN(1)= NEA(1)*3 IF (NSS.NE.1) THEN DO 9 5 1=2,NSS NN(I)= NN(I-1) + NEA(I)*3 CONTINUE END IF DO 1 1 0 I S = 1 , N S S NNR=NER*3 NNA=NEA(IS)*3 IIS=IS+1 DO 1 0 0 I A N G = 1 , N N A I N O D E = N N ( I S ) - NNA + I A N G A N G L E = ( I A N G - 1 ) * ( A N G ( 1 1 S ) - A N G ( I S ) ) / NNA + A N G ( I S ) IF (IS.EO.1)THEN R2=XY(IS) / SIN(ANGLE*DEG) E L S E IF (IS.E0.2)THEN R2 = X Y ( I S ) / COS(ANGLE*DEG) E L S E IF (IS.EQ.3)THEN R2 = X Y ( I S ) / S I N ( A N G L E * D E G ) E L S E IF (IS.E0.4)THEN R2 = X Y ( I S ) / COS(ANGLE*DEG) END IF IF ( N S S . N E . 4 ) ANGLE = A N G L E - 4 5 . 0 0 CALL PSPACE (NL.RL.AL,INODE,ANGLE,R1,R2,IOUT) CONTINUE  1 12  MESHGEN 110  1000 1010 1020 2000  Listing  CONTINUE IF ( N S S . E 0 . 1 .OR. N S S . E 0 . 2 ) THEN INODE = N N ( N S S ) + 1 IF ( N S S . E O . 1) THEN R2=XY(1) / SIN(ANG(2)*DEG) ELSE IF (NSS.EQ.2)THEN R2=XY(2) / C0S(ANG(3)*DEG) END IF  2 3 4 5  6  ANGLE = ANG(NSS+1) - 4 5 . 0 0 CALL PSPACE (NL.RL.AL,INODE,ANGLE,R1,R2,IOUT) END IF C A L L LOAD (NL.RL,AL,DEG.IOUT) FORMAT (2I5.F12.5) FORMAT ( 1 5 ) FORMAT (2F15.9) FORMAT ( ' T i t l e ' , / , 1 4 , ' , 5 . 0 , ' 13 ' . 3, ' , / . ' 1, 2 , O , 0 , O , R e s t a r t ' , / , ' 0. 0 . 1 5 . 0 , ' , / , '  0 . 0 , 0 . 0 , 0.00001 ,  ' , STOP END  ,  3,  1,  0,',//)  ,300000.  S U B R O U T I N E  ,' , / ,  P S P A C E  it********:)  SUBROUTINE PSPACE (NL,RL,AL,INODE,ANGLE,RO,R2.IOUT) DIMENSION NL( 1) ,RL( 1 ) ,AL( 1 ) LOGICAL FLAG 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=INODE C0NST=(R2/R1)**POWER IF (NSS.EQ.1 .AND. J . E 0 . 1 ) THEN WRITE ( I O U T , 2 5 4 0 ) I N O D E , R 1 , A N G L E  ELSE IF ( N S S . E O . 1 .AND. J . E O . (NN(NSS) + 1 ) ) THEN WRITE (IOUT,2560) INODE,R1.ANGLE ELSE IF ( N S S . E O . 2 . A N D . J . E O . 1) T H E N WRITE (IOUT,2540) INODE,R1,ANGLE ELSE IF ( N S S . E O . 1 .AND. J . E O . ( N N ( N S S ) + 1 ) ) THEN WRITE (IOUT,2540) INODE,R1,ANGLE ELSE  WRITE (IOUT,2500) INODE,R1,ANGLE END IF DO 2 0 0 1 = 1 , N N R IF ( N S S . E O . 4 ) THEN INODE=INODE + NN(NSS) ELSE INODE=INODE END IF R1=R1*C0NST IF ( I . E O . N N R )  + NN(NSS)  + 1  THEN  NL(J)=INODE RL(J)=R 1 AL(J)=ANGLE END IF C C C  SIMPLY  C C C  ONE Q U A R T E R  IF  SUPPORTED (FLAG)  BOUNDARY  THEN PLATE  IF  (NSS.EQ. 1 .AND. J . E Q . 1 WRITE ( I O U T , 2 5 4 0 ) INODE ELSE IF ( N S S . E O . 1 . A N D . J WRITE ( I O U T , 2 5 5 0 ) INODE ELSE IF ( N S S . E O . 1 . A N D . J WRITE ELSE IF  WRITE  (IOUT,2560) INODE (NSS.EQ.1 .AND. d (IOUT,2570)  INODE  I .NE.NNR ) THEN .. A N D . R1.ANGLE E0.1 . A N D . I . E O . N N R ) THEN R1.ANGLE I.NE.NNR) E O . ( N N ( N S S ) + 1 ) .AND  THEN  I.EO.NNR)  THEN  R1.ANGLE E O . ( N N ( N S S ) + 1 ) .AND R1.ANGLE  113  MESHGEN ELSE  IF  ( N S S . E Q . 1  WRITE ONE  HALF  I.EQ.NNR)  THEN  INODE,R1,ANGLE  PLATE  ELSE  ( N S S . E 0 . 2  IF  ELSE  ( N S S . E O . 2  IF  ( N S S . E Q . 2  IF  ELSE  .AND.  ( N S S . E O . 2  IF  I .NE.NNR)  THEN  J . E Q . 1  .AND.  I.EQ.NNR)  THEN  J . E Q . ( N N ( N S S ) + 1 )  I.NE.NNR)  THEN  J . E Q . ( N N ( N S S  )+1 )  .AND.  I . E Q . N N R )  THEN  INODE.R1.ANGLE .AND.  ( I O U T . 2 5 3 0 )  .AND.  INODE,R1.ANGLE .AND.  ( I 0 U T . 2 5 8 O )  WRITE  .AND.  INODE.R1.ANGLE  ( N S S . E Q . 2  IF  WRITE ELSE  .AND.  ( I O U T . 2 5 4 0 )  WRITE  J . E Q . 1  INODE,R1.ANGLE  ( I 0 U T . 2 5 5 O )  WRITE ELSE  .AND.  ( I O U T . 2 5 4 0 )  WRITE  FULL  .AND.  ( I 0 U T . 2 5 3 0 )  Listing  I.EQ.NNR)  THEN  INODE.R1.ANGLE  PLATE ELSE  IF  ( N S S . E Q . 4  IF  .AND.  ( J . E Q . ( N N ( 2 ) + 1 ) ) WRITE ELSE  I.EQ.NNR)  ( I 0 U T . 2 5 1 O ) IF(  WRITE  THEN  THEN INODE,R1.ANGLE  J . G T . ( N N ( 2 ) + 1 )  .AND.  J . L E . (NN(3)+1)  ( I 0 U T . 2 5 2 O )  INODE  R1,ANGLE  ( I 0 U T . 2 5 3 O )  INODE  R1  )  THEN  ELSE WRITE END  ,ANGLE  IF  INTERNAL ELSE WRITE END FIXED  ( I O U T . 2 5 0 0 )  INODE,R1,ANGLE  IF  BOUNDARY  ELSE ONE  QUARTER IF  PLATE  ( N S S . E Q . 1 WRITE  ELSE  IF  ( N S S . E Q . 1  WRITE ELSE  ( I 0 U T . 2 6 5 0 )  IF  ( N S S . E Q . 1  WRITE ELSE  ( I 0 U T . 2 6 G 0 )  IF  ( N S S . E Q . 1  WRITE ELSE  ( I 0 U T . 2 6 7 O )  IF  ( N S S . E Q . 1  WRITE ONE  HALF  ( I 0 U T . 2 6 3 0 )  J . E Q . 1  .AND.  I.NE.NNR)  THEN  INODE,R1,ANGLE .AND.  J . E Q  INODE,R1 .AND.  J . E Q  INODE,R1 .AND.  J . E Q  INODE,R1 .AND.  1  .AND.  I.EQ.NNR)  THEN  ANGLE (NN(NSS)+1)  .AND.  I . N E . N N R )  THEN  .AND.  I . E Q . N N R )  THEN  ANGLE (NN(NSS)+1) ANGLE  I . E Q . N N R )  THEN  INODE,R1,ANGLE  PLATE  ELSE  IF  ( N S S . E Q . 2  WRITE ELSE  ( I 0 U T . 2 6 4 O )  IF  ( N S S . E Q . 2 ( I 0 U T . 2 6 5 O )  WRITE ELSE  ( N S S . E Q . 2  IF  ( I 0 U T . 2 6 4 0 )  WRITE ELSE  ( N S S . E Q . 2  IF  ( I 0 U T . 2 G 8 O )  WRITE ELSE  ( N S S . E Q . 2  IF  ( I 0 U T . 2 G 3 O )  WRITE FULL  .AND.  ( I 0 U T , 2 6 4 O )  .AND.  J . EQ.1  .AND.  I.NE.NNR)  THEN  I.EQ.NNR)  THEN  INODE. R1.ANGLE .AND.  J . EQ.1  .AND.  INODE, R1.ANGLE .AND.  J . E Q . ( N N ( N S S )  1 )  .AND.  I . N E . N N R )  THEN  J . E Q . ( N N f N S S ) + 1)  .AND.  I.EQ.NNR)  THEN  +  INODE. R1.ANGLE .AND.  INODE, R1.ANGLE .AND.  I. EQ.NNR)  I N O D E , R1  THEN  .ANGLE  PLATE ELSE  IF IF  ( N S S . E Q . 4  .AND.  ( J . E Q . ( N N ( 2 ) + 1 ) ) WRITE ELSE WRITE  ( I O U T . 2 6 1 0 ) IF(  I . E Q . N N R )  THEN  THEN INODE,R1,ANGLE  J . G T . ( N N ( 2 ) + 1 )  .AND.  J . L E . (NN(3)+1)  ( I O U T . 2 6 2 0 )  INODE,R1,ANGLE  ( I 0 U T . 2 6 3 O )  INODE,R1,ANGLE  ELSE WRITE END  IF  114  )  THEN  MESHGEN  Listing  c C C  INTERNAL ELSE WRITE END  END  C  200  IF  L 2500 2510 2520 2530 2540 2550 2560 2570 2580  FORMAT FORMAT FORMAT FORMAT FORMAT FORMAT FORMAT FORMAT FORMAT  SUPPRTED  (14. (14, (14. (14, (14. (14, (14, (14, (14,  /•*  CLAMPED  c 2610 2620 2630 2640 2650 2660 2670 2680  c  FORMAT FORMAT FORMAT FORMAT FORMAT FORMAT FORMAT FORMAT  C C C  C C C  C C C  BOUNDARY  1, 1, 1, 1. 1. 1, 1. 1. ' , 1 , 1 . 1 . 1 , 0 , 1,  ' ' ' ' ' ' ' '  , , , , , , , .  0 1 0 0 0 0 1 1  , . . , , , , ,  0 1 1 0 1 1 0 0  , , , . . , , ,  0 1 1 1 0 1 0 1  , , . , , , , ,  0 0 0 0 1 1 0 0  , . . . , , , ,  0 0 0 0 0 0 1 1  , , , . , , . .  , . , . . , ,  1 1 1 1 1 0 1  . , , , . , ,  1 1 1 0 1 1 1  . , . , , , ,  0 00, 0 00, 0 00, 0 00. 0 00, 0 oo . 0 00, 0 00, 0 00,  , 2( F8 ,2(F8 . 2( F 8 , 2( F8 ,2(F8 ,2(F8 , 2( F 8 ,2(F8 ,2(F8  3, 3. 3, 3, 3. 3. 3, 3. 3.  . . . . . . , . .  ' ). ' ' ), ' ').' '),' ' ). ' ' ). ' ' ), ' '),' ' ),'  0 , 1,2 0.1, 2 0,1 , 2 0,1, 2 0.1, 2 0 . 1. 2 0.1, 2 0,1 . 2 0 . 1, 2  0 0 0 0 0 0 0 0  , 2( F8 , 2(F8 ,2(F8 ,2(F8 ,2(F8 ,2(F8 ,2(F8 . 2(F8  3, 3. 3, 3. 3, 3, 3, 3.  , , , . . . . ,  ' ' ' ' ' '  0,1 , 2 0,1, 2 0.1 , 2 0.1 , 2 0.1, 2 0,1, 2 0 , 1.2 0,1, 2  -  BOUNDARY  (14, (14. (14, (14. (14, (14, (14, (14,  ' ' ' ' ' ' '  , , , , . , ,  1 0 0 0 0 1 1  , , , . , , ,  1 1 0 1 1 0 0  , , . . , . ,  1 1 1 0 1 0 1  i, 1, 1 , 1 , 1, 1 , 1 ,  ',1,1,1,1.1.  1  •  00. 00, 00, 00, 00. 00. 00, 00..  ) ) ) ) ) )  ,' .' ,' ,' .' .' ' ),' ' ) , '  RETURN END  Q  C  INODE,R1,ANGLE  CONTINUE SIMPLY  c  (IOUT.2500)  IF  ************  S  U B  R 0 U T  I  N E  L  0 A D  ********* * *  S U B R O U T I N E LOAD (NL,RL,AL.DEG,IOUT) DIMENSION NL(1),RL(1),AL(1),DIS(36),P(30) 1 , S U M ( 4 ) , P E L E ( 4 ) , I F I R S T ( 1 6 ) , I L A S T ( 16) LOGICAL FLAG COMMON N E A ( 4 ) , N N ( 4 ) , N S S . N E R , N N A , N N R , F L A G PRINT  ELEMENT  NODE  NUMBERING  DO  2 9 0 IR=1,NER WRITE ( I O U T , 2 0 0 0 ) NN(NSS)/3 CALL ELENO ( 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 . 1 6 ) IF (NEA(NSS).NE.1) 1 W R I T E ( I O U T , 2 0 2 0 ) N N ( N S S ) / 3 , ( I L A S T ( I ) , I = 1, 1 6 ) 290 CONTINUE PRINT  NUMBER  OF LOAD  POINTS  IF  (NSS.E0.4) THEN NUMLP = 2 * N N ( N S S ) + NSS + 1 ELSE N U M L P = 2 * ( N N ( N S S ) + 1) END IF W R I T E ( I O U T , 2 0 3 0 ) NUMLP THICK=10.0 CALCULATE DO  THE DISTANCE  BETWEEN  300 I=1,NN(NSS) NN5=NN(NSS)+1 11=1+1 IF ( I I . E 0 . N N 5 .AND. NSS.EO.4) X1=RL(I) * C O S ( A L ( I ) *DEG)  POINTS  11=1  115  MESHGEN  C C C  300  Y1=RL(I) * SIN(AL(I)*DEG) X2 = R L ( I I ) * COS(AL(II )*DEG) Y2 = R L ( I I ) * SIN(AL(11)*DEG) D I S ( I ) = SORT( (X2-X1 )*(X2-X1 ) CONTINUE FOR E A C H DO  305  SIDE  CALCULATE  Listing  + (Y2-Y 1 ) *(Y2-Y 1 )  THE CONSISTENT  LOAD  VECTOR  3 3 0 ISIDE=1.NSS IDIR = 1 IDIR2 = 2 DMULT = 1 . 0 SUM(ISIDE) =0.0 DO 3 0 5 1 = 1 , 3 0 P(I)=0.0 CONTINUE DO  320 IEL=1,NEA(ISIDE) ID = N N ( I S I D E ) - NEA(ISI0E)*3 A = DIS(ID) B = DIS(ID+1)+A  + (IEL-1)*3  + 1  C = DIS(ID+2)+B CALL CONSTL (PELE.A.B.C) DO  310 320 C C C C  S E T T H E C O R R E C T S I G N AND O I R E C T I O N F O R E A C H T H E N C H E C K T H E SUM OF T H E LOAD V E C T O R . IF  ( N S S . E 0 . 4 ) THEN IF ( I S I D E . E O . 2 .OR. I S I D E . E O . 4 ) IF ( I S I D E . E Q . 2 .OR. I S I D E . E O . 3 ) ELSE DMULT=1.0/SORT(2.0) END I F  C C C  PRINT DO  325 330 C C C C C C C C C C C C C C C  3 1 . 0 d=1 . 4  NP = ( I E L - 1 ) * 3 + J P(NP) = P(NP) + PELE(J)*THICK CONTINUE CONTINUE  T H E NODAL  IDIR=2 D M U L T = -1  LOADS  325 1=1,(NEA(ISIDE)*3+1) P(I) = P(I)*DMULT SUM(ISIDE) = SUM(ISIDE)+P(I) K = NN(ISIDE)-NEA(ISIDE)*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 ) IF ( N S S . E O . 1 ) THEN P(I) = - 1 . 0 * P(I)  WRITE END I F CONTINUE CONTINUE PRINT  (I0UT.2040)  LATERAL  NL(K),IDIR2,P(I)  LOADS  DO 3 4 0 1 = 1 , N N ( N S S ) , 3 IF ( N S S . N E . 4 ) THEN IF ( I . E O . 1 ) THEN WRITE ( I O U T . 2 0 5 0 ) I,1+1,1+2 ELSE WRITE ( I 0 U T . 2 O 6 O ) 1,1+1.1+2 END I F ELSE WRITE ( I 0 U T . 2 O 6 O ) 1,1+1,1+2 END I F 3 4 0 CONTINUE IF ( N S S . N E . 4 ) WRITE ( I O U T . 2 0 7 0 ) NN(NSS)+1 WRITE (IOUT.2080) WRITE ( I O U T . 2 0 9 0 ) (I.SUM(I).I=1,4)  1 16  SIDE,  )  MESHGEN L i s t i n g 2000  FORMAT  (  1'  7, ' . 1 2 , ' , 3 , 0 , 0 , 0 , 16,  2'  1,7.70E-05.0.0.',/,  3' 2010  200000.0.0.3.1.2.300.0,0.0,')  FORMAT 1'  2020  (  1, 16,334, 1 , 0 , 0 , 0 , 1 0 . 0 , ' , / ,  FORMAT 113,  2030  1614)  (  ',16,334,1,3,0,0,10.0,',/,1614)  FORMAT  (  114, ' , 1 , 3 , 2'  ' , / ,  1,3',/,  3'  0.0,  4'  1.0.  1.0.'./.  5'  2 . 0 ,  2 . 0 , ' )  0 . 0 ' , / ,  2040  FORMAT  ( 1 4 , ','  2050  FORMAT  ( 1 4 , ',  +  14,  ',  3.  +  14,  ',  3,  2060  , 4 , 4 , 5 , 0 , 1 , 1 , 1 , 1 , ' , / ,  FORMAT  , 1 2 , ', 3,  1,  1,',F10.4) 0.04',/,  1 , 0 . 1 2 ' , / , 1 , 0 . 1 2 ' )  ( 1 4 , ',  3,  1,  0.08',/,  + 1 4 , ' . 3 , 1 , 0 . 1 2 ' , / , +  ',  14,  3,  1 , 0 . 1 2 ' )  2070  FORMAT  (14,  2080  FORMAT  ( ' 1 ,  1  '  2090  ' , 3 , 1,  1,  0.04')  1 , 0 , 4 ,  , , 1 . 0 , ' , / ,  1' )  FORMAT 1  SUM  ( 4 ( / , '  OF  THE  FORCES  FOR  SIDE',12,'  IS',F15.6.'  s qmm'  RETURN END C c  * * * * * * * * * * * * * * *  S  U  B  R  O  U  T  I  N  E  C O N S T L  * * * * * * * * * *  C SUBROUTINE  DIMENSION  C C  CONSTL  CALCULATE  C C  .  C  THE  CONSISTENT  .  0  C  (PELE,A,B,C)  PELE(4)  LOAD  FOR  •  A = 0.7125*A  PELE(2)  =  PELE(3)  =-1.0125*A  PELE(4)  = 0.3000*A  -  -  CUBIC  SHAPE  FUNCTION  •  B  PELE( 1)  THE  C  0.3000*B  +  0.0875*C  1.0125*B  -  0.3000*C  +  0.7125*C  +  0.5000*C  0.7125*B  RETURN END C Q  * * * * * * * * * * * * * *  5  (j  g  R •  u  j  i  N  E  E  L  E  N  0  * * * * * * * * * * * * * *  C SUBROUTINE  1  DIMENSION  IADD(16).  DATA  IMULT  DATA  ELENO  DATA  JMULT  vJADD  NO=NN(NSS) IF  ILAST(16),  JADD(16),NN(4)  IADD  DATA  ( IFIRST,ILAST,IR,NN,NSS )  IFIRST(16), /  IMULT(16),  JMULT(16),  3 , 0 , 0 , 3 , 2 , 1 , 0 , 0 . 1 . 2 , 3 , 3 , 2 , 1 , 1 , 2  /  4 , 4 , 1 , 1 , 4 , 4 , 3 , 2 , 1 , 1 , 2 , 3 , 3 , 3 , 2 , 2  /  0,  /  (NSS.NE.4)  2 , - 1 , 0 , 3 , 1 , 0 , 0 , 0 , 1 , 2 , 3 . 3 . 2 , 1 , 1 . 2 0 , 3 , 3 , 0 , 0 , 1 , 2 , 3 , 3 , 2 , 1 , 1 , - 1 , 2 , 2  N0=N0+1  ISTART=NO*(IR-1)*3  JSTART=NO*(IR-1)*3+N0+1  DO  500  1=1,16  IFIRST(I)= IF  THEN  +  ILAST(I)  = JMULT(I)*NO  ILAST(I)  =  ELSE  500  IMULT(I)*NO  (NSS.EO.4)  END  IF  ISTART +  +  JSTART  IADD(I) -  JADD(I)  IFIRST(I)+NN(NSS)-3  CONTINUE  RETURN END  117  / / / /  APPENDIX APPENDIX  E Computer C o m m u n i c a t i o n s  E . l WORDSTAR O u t p u t  on t h e MTS Z e r o x  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 t h e IBM PC t o MTS a n d t h e n p r i n t t h e f i l e on t h e X e r o x 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 t h e UBC book I o r G a c c o u n t on t h e UBC MTS s y s t e m  In WORDSTAR p r i n t P "filename" y "fileprint" RETURN RETURN Y RETURN X  the f i l e  to a disk  store  file.  / / 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 t o WINDOW d i s k e t t e / /  A>WSCLRBIT "fileprint" "fileclear" // g e t t h e a t t e n t i o n o f t h e smart s w i t c h w i t h k e r m i t / / A>KERMIT SET BAUD 4800 CONN *6 / / MTS on p o r t 6, VAX VMS on p o r t 5 / / ctrl]C EXIT // u s i n g t h e same d i s k e t t e r u n WINDOW / / A>WINDOW G / / o r I d e p e n d i n g on t h e l o c a t i o n MTS a c c o u n t / / SIG "ccid" "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 S C A R D S = " - f i l e t e m p " S P R I N T = " f i l e n e w " PAR=WORDSTAR SET PROUTE=CNTR SET DELIVERY=CIVL / / o r CNTR //) CON *PRINT* PORTRAIT ONESIDE COPY " f i l e n e w " *PRINT* SIG %EXIT Other DIS REL SYS  commonly used MTS comands a r e : *PRINT* // displays p r i n t s t a t i s t i c s // *PRINT* // release p r i n t t o p r i n t e r // QUE USER / / show que o r t i m e o f p r i n t i n g / /  "filetype" // //  f i l e names p r o v i d e d e n c l o s e comments  118  by t h e u s e r  APPENDIX E . 2 T r a n s f e r o f a VAX-VMS F i l e  t o t h e UBC/MTS  The f o l l o w i n g commands will transfer a f i l e E n g i n e e r i n g VAX 11/730 t o t h e UBC/MTS s y s t e m . RETURN "VAXid" "password" SD " d e f a u l t D I R " ALLOC TXAO  " //  " //  the  Civil  / / s i g n on t o t h e VAX / /  // d i a l KERMIT SET SPEED 1 2 0 0 CONN G (I) SIG " c c i d " "password" RUN *KERMIT RECEIVE " f i l e n a m e " ctrlP SEND " f i l e n a m e " CONN EXIT SIG EXIT LO  from  System  up UBC (228-1401) on modem / /  / / s i g n on t o t h e UBC MTS s y s t e m / /  // c t r l ] C  on t h e IBM PC / /  / / s i g n o f f t h e MTS s y s t e m / / / / s i g n o f f t h e VAX VMS s y s t e m / /  e n c l o s e user i d ' s , filenames, passwords e n c l o s e comments  119  d i r e c t o r i e s and  

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