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Behaviour of headed stud connections for precast concrete panels under monotonic and cycled shear loading Neille, Donald Stewart 1977

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BEHAVIOUR OF HEADED STUD CONNECTIONS FOR PRECAST CONCRETE  PANELS UNDER MONOTONIC  AND CYCLED SHEAR LOADING  by DONALD STEWART NEILLE M.Sc.(Eng)(Witwatersrand)  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES Department o f C i v i l  Engineering  We accept t h i s t h e s i s as conforming t o the  required standard  The U n i v e r s i t y May, ®  of B r i t i s h  Columbia  1977  Donald Stewart N e i l l e , 1977  In p r e s e n t i n g the  requirements f o r an advanced degree a t the U n i v e r s i t y  of B r i t i s h Columbia, it  t h i s thesis i n p a r t i a l f u l f i l l m e n t of  freely  I agree t h a t the L i b r a r y s h a l l make  a v a i l a b l e f o r reference  and study.  I further  agree t h a t p e r m i s s i o n f o r e x t e n s i v e copying o f t h i s  thesis  f o r s c h o l a r l y purposes may be granted by the Head o f my Department or by h i s r e p r e s e n t a t i v e s .  I t i s understood  t h a t copying o r p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l gain  s h a l l not be allowed without my w r i t t e n  Department of C i v i l  Engineering  The U n i v e r s i t y of B r i t i s h Columbia 2075 Wesbrook M a l l , Vancouver, B.C. Canada. V6T 1W5 ii  permission.  Research  Supervisor:  Dr.  Richard  A.  Spencer  ABSTRACT  The  research  in  this  dissertation  of  predicting  the  buildings  under  test  and  data  connections  It via  a  tinct  earthquake  of  the  connections  of  an  precast  loads.  described  overall  objective  concrete Existing  procedures  of  panel laboratory  headed  stud  reviewed.  i s postulated to  stud  part  design  briefly  connection  that  shear  surrounding  loads  are  concrete  transmitted  by  three  dis-  mechanisms: friction  2.  bearing  3.  interaction on  aspects Friction  between of  end  f a c e p l a t e and  concrete  f a c e p l a t e on  concrete  of  between  l a b o r a t o r y models of  these  forces  comparison  with  particularly the  a  behaviour  1.  Tests  headed  forms  current  are  on  the  under  f a c e p l a t e on  and  designed  mechanisms between  studs  confirm  to  cycled  concrete  forces  loading. and  isolate  that  f a c e p l a t e and  remaining  concrete individual  a l l three  concrete  are  acting in a Bearing  interaction  of  between  exist. small  in  connection, the  end  studs  of and  s u r r o u n d i n g c o n c r e t e are shown t o be the main c o n t r i b u t i o n s t o the t o t a l l o a d c a r r i e d by a c o n n e c t i o n .  A simple analy-  t i c a l model i s p r e s e n t e d f o r the p r e d i c t i o n o f the u l t i m a t e shear l o a d c a p a c i t y o f a c o n n e c t i o n and a computer a l g o r i t h m i s proposed  f o r the p r e d i c t i o n o f the l o a d v e r s u s  b e h a v i o u r o f a c o n n e c t i o n under b o t h monotonic and  deflection cyclic  conditions.  E x i s t e n c e o f the t h r e e mechanisms whereby a connect i o n t r a n s f e r s a p p l i e d shear f o r c e s t o the s u r r o u n d i n g  con-  c r e t e c o n t r a d i c t s the shear f r i c t i o n e q u a t i o n which i s c u r r e n t l y used i n the d e s i g n o f c o n n e c t i o n s . t i o n s developed  The a n a l y t i c a l equa-  i n the i n v e s t i g a t i o n i n d i c a t e t h a t the s t r e n g t h  of a c o n n e c t i o n i s d i r e c t l y dependent upon t h e s t r e n g t h o f the s u r r o u n d i n g c o n c r e t e , as opposed t o the e x p r e s s i o n f o r shear f r i c t i o n , w h i c h does not c o n t a i n c o n c r e t e s t r e n g t h as a variable.  iv  TABLE OF CONTENTS  PAGE R i g h t s o f P u b l i c a t i o n , Copying  and Loan  i i  Abstract  i i i  L i s t of Tables  viii  L i s t of Figures  ix  Dedication  xvi  Acknowledgement  xvii  CHAPTER 1  INTRODUCTION  1  2  CURRENT INFORMATION ON HEADED STUD CONNECTIONS  7  3  2.1  Existing tion.  S t a t i c Load T e s t  2.2  C u r r e n t Design Procedure.  11  2.3  Recent C y c l i c T e s t s on Headed Stud Connections.  13  THE SHEAR LOAD-RESISTING COMPONENTS OF A CONNECTION  20  3.1  Probable Mechanisms.  20  3.2  L a b o r a t o r y Models of Mechanisms.  21  3.3  T e s t i n g Apparatus.  29  v  Informa-  7  CHAPTER 4  PAGE LABORATORY TESTS AND  Materials Tests.  4.2  T e s t s on F r i c t i o n  4.3  T e s t s on End B e a r i n g Specimens.  37  4.4  T e s t s on S i n g l e Studs i n Concrete.  46  Specimens.  37  T e n s i l e and Bending T e s t s on 60  ANALYTICAL MODELS  76  5.1 5.2  Approximations. Non-Linear Plane S t r e s s F i n i t e Element Program f o r P l a i n Concrete.  76  Non-Linear Plane Frame Program.  83  COMPARISON OF COMPUTER ANALYSES WITH LABORATORY RESULTS 6.1  7  35  Studs.  5.3 6  35  4.1  4.5  5  RESULTS  P r o p e r t i e s of S t e e l Anchor  77  93  Bars  and Studs.  93  6.2  Friction  95  6.3  End-Bearing Specimens.  95  6.4  S i n g l e Studs i n Concrete.  105  6.5  Complete  Headed Stud Connection.  112  SIMPLIFIED ANALYTICAL MODELS 7.1 U l t i m a t e Shear Load o f a Stud i n Concrete.  126  7.2 7.3  Specimen.  126  U l t i m a t e End-Bearing C a p a c i t y of Faceplate..  129  Comparison of Proposed Theory and Current Shear F r i c t i o n Theory w i t h L a b o r a t o r y Measurements.  129  vi  CHAPTER  PAGE 7.4 7.5  F r i c t i o n Between F a c e p l a t e and Concrete.  132  L o a d - D e f l e c t i o n Model f o r a Connection.  132  CONCLUSION 8.1 8.2  142  Confirmation of I n i t i a l tions . Future  Assump142  Research.  144 146  BIBLIOGRAPHY TRIANGULAR AND QUADRILATERAL PLANE STRESS/STRAIN F I N I T E ELEMENTS.  150  A. 1  Plane  Stress/Strain  Triangle.  150  A. 2  Plane  Stress/Strain  Quadrilateral.  163  A. 3  Patch  T e s t s on E l e m e n t s .  A. 4  E l e m e n t S t i f f n e s s and S t r a i n culation Subroutines.  A. 5  Performance Problems.  APPENDIX A  APPENDIX B  of Elements  173 Cal-  i n Selected  LOAD-DEFLECTION CURVE GENERATOR HEADED STUD CONNECTIONS  177 194  FOR 205  B. l  L i s t of V a r i a b l e s i n Subroutine Input/Output L i s t .  205  B. 2  Notes.  206  B. 3  Subroutine  APPENDIX C  STUDCO.  CONTROL OF AXIAL LOAD AND MOMENT INTERACTION FOR STUDS IN ITERATIVE CALCULATIONS  vii  207  220  LIST  OF  TABLES  TITLE  Summary  Concrete cimens  Tensile  of  Load  Data  Strengths  Strength  for  of  of  Connections  Laboratory  Steel  Spe-  Samples  Cumulative Rotations of Stud Bending Specimens S u b j e c t e d to C y c l i c Loading  Comparison Between Approximate Formula and R i g o r o u s D e r i v a t i o n o f S t u d Intera c t i o n Diagram  Trilinear Properties B a r s and Studs  of  Steel  Anchor  C o m p a r i s o n o f C a l c u l a t e d and M e a s u r e d Ultimate Strengths of Connections  Strain Energies Elements  from  Patch  Tests  on  C o m p a r i s o n o f S t r e s s e s and Deflections f o r P a r a b o l i c a l l y Loaded Square P l a t e  L I S T OF  FIGURES  TITLE Typical  precast concrete  panel  building.  T y p i c a l headed s t u d c o n n e c t i o n showing two common s t u d c o n f i g u r a t i o n s and t h e e a s e w i t h w h i c h m i n o r i m p e r f e c t i o n s and m i s a l i g n m e n t s may be accommodated. T y p i c a l push-out  specimen.  Shear l o a d - d e f l e c t i o n curves dia. x 4 i n . studs.  f o r 3/4  in.  Load-deflection curves to ultimate f o r 3/4 i n . d i a . x 4 i n . s t u d s . Details  of connections  Test r i g for c y c l i c tions.  tested.  loading of  connec-  Load-deflection  curve  - connection  Al.  Load-deflection  loops  - connection  A3.  Specimens f o r i n v e s t i g a t i o n of f r i c t i o n between f a c e p l a t e and c o n c r e t e . Test pieces for investigating end b e a r i n g .  faceplate  I s o l a t i o n o f b a r s and back o f f a c e p l a t e f r o m c o n c r e t e w i t h p l a s t i c foam. Machined crete .  studs  ready  f o r c a s t i n g i n con-  ix  TITLE  FIGURE  3.5  PAGE  Models f o r examination of i n t e r a c t i o n between stud and c o n c r e t e .  27  Specimens f o r t e s t i n g studs i n t e n s i o n and bending.  28  Apparatus f o r t e s t i n g s i n g l e studs i n c o n c r e t e , end b e a r i n g and f r i c t i o n specimens.  30  Measurement of displacements o f f r i c t i o n and end b e a r i n g specimens.  31  3.9  Set o f displacement t r a n s d u c e r s f o r meas u r i n g d e f l e c t e d shapes o f s t u d s .  31  3.10  Data a c q u i s i t i o n and r e c o r d i n g i n s t r u ments used i n t e s t s on c o n c r e t e .  33  3.11  Apparatus  34  3.12  Studs f i r m l y clamped t o c o n c r e t e b l o c k .  34  4.1  Bending  39  4.2  L o a d - d e f l e c t i o n curves f o r f r i c t i o n cimen F l .  spe-  4.3  L o a d - d e f l e c t i o n curves f o r f r i c t i o n cimen F2.  spe-  4.4  L o a d - d e f l e c t i o n curves f o r f r i c t i o n cimen F3.  spe-  4.5  L o a d - d e f l e c t i o n curves f o r end-bearing specimens B l & B2.  43  4.6  L o a d - d e f l e c t i o n curves f o r end-bearing specimens B3 & B4.  44  4.7  L o a d - d e f l e c t i o n curves f o r end-bearing specimens B5 & B6.  45  4.8  T y p i c a l c o n c r e t e f a i l u r e a t end o f f a c e plate.  47  4.9  L o a d - d e f l e c t i o n curve f o r stud S I .  48  3.6 3.7  3.8  f o r bending t e s t s on s t u d s .  t e s t on 1/2 i n . d i a . b a r s .  x  40 41 42  FIGURE  TITLE  PAGE  4.10  D e f l e c t e d shape of stud SI.  49  4.11  L o a d - d e f l e c t i o n curve f o r stud S2.  50  4.12  D e f l e c t e d shape of stud S2.  51  4.13  L o a d - d e f l e c t i o n curve f o r stud S3.  52  4.14  D e f l e c t e d shape o f stud S3.  53  4.15  L o a d - d e f l e c t i o n curve f o r stud S4.  54  4.16  D e f l e c t e d shape of stud S4.  55  4.17  L o a d - d e f l e c t i o n curve f o r stud S5.  56  4.18  D e f l e c t e d shape of stud S5.  57  4.19  L o a d - d e f l e c t i o n curve f o r stud S6.  58  4.20  D e f l e c t e d shape of stud S6.  59  4.21  S t r e s s - s t r a i n curves from t e n s i l e on studs.  4.22 4.23 4.24 4.25 4.26 4.27 4.28 4.29 4.30  Monotonic bending t e s t on stud SB1.  tests  specimen 65  C y c l i c bending t e s t on stud SB2.  specimen  C y c l i c bending t e s t on stud SB3.  specimen  66 67  Monotonic bending t e s t on stud SB4.  specimen 68  C y c l i c bending t e s t on stud SB5.  specimen  C y c l i c bending t e s t on stud SB6.  specimen  '  69 70  Monotonic bending t e s t on stud SB7.  specimen  C y c l i c bending t e s t on stud SB8.  specimen  C y c l i c bending t e s t on stud SB9.  specimen  xi  61  71 72 73  FIGURE 4.31 4.32  5.1 5.2 5.3 5.4 5.5  TITLE C y c l i c l o a d i n g caused t y p i c a l of studs i n f u s i o n welds.  PAGE fracture 74  Normalised load decrement versus normal i s e d d e f l e c t i o n from stud bending t e s t s . Q u a d r a t i c e q u a t i o n f i t t e d by l e a s t squares.  75  The approximation o f a stud i n c o n c r e t e by means o f a beam on Winkler s p r i n g s .  78  T r i a n g u l a r and q u a d r i l a t e r a l f i n i t e ments.  80  ele-  Concrete s t r e s s - s t r a i n model adapted work by Karsan & J i r s a .  from 82  B i a x i a l s t r e n g t h envelope f o r c o n c r e t e a f t e r Kupfer & G e r s t l e .  84  F i v e types o f l i n e members b u i l t plane frame model.  86  into  5.6  L o a d - d e f l e c t i o n model f o r c o n c r e t e s p r i n g s .  86  5.7  T r i l i n e a r h y s t e r e s i s loops used f o r both s t r e s s - s t r a i n and moment-curvature r e l a t i o n s h i p s f o r studs.  88  5.8  Stud c r o s s - s e c t i o n under u l t i m a t e  axial  f o r c e and bending moment.  90  5.9  Stud i n t e r a c t i o n diagram.  90  6.1  Bending t e s t on 1/2 i n . d i a . bars compared w i t h computer c a l c u l a t i o n s . Computer model of f r i c t i o n specimen with 6 i n . long anchor b a r s .  6.2 6.3  Measured and computed l o a d - d e f l e c t i o n curves f o r f r i c t i o n specimen F2.  96 97 98  6.4  F i n i t e element g r i d of one quadrant of conc r e t e t e s t specimen used by Karsan & J i r s a .  100  6.5  Comparison of computed s t r e s s - s t r a i n curve w i t h t e s t by Karsan & J i r s a .  101  x n  FIGURE  6.6 6.7 6.8  TITLE 3/8 i n . end b e a r i n g f i n i t e elements.  PAGE  specimen modelled by 102  L o a d - d e f l e c t i o n curve f o r 3/8 i n . end b e a r i n g specimens.  103  Comparison of c a l c u l a t e d and measured maximum loads  f o r end b e a r i n g  specimens.  104  specimen.  106  6.9  Computer model of end b e a r i n g  6.10  Measured and computed l o a d - d e f l e c t i o n curves f o r end-bearing specimens B3 & B4. F i n i t e element g r i d of one h a l f of conc r e t e s l i c e c o n t a i n i n g stud shank.  108  6.12  L o a d - d e f l e c t i o n curve f o r concrete c o n t a i n i n g stud shank.  109  6.13  Computer model of stud i n c o n c r e t e .  6.14  Measured and computed curves f o r stud SI.  6.11  6.15 6.16 6.17 6.18 6.19 6.2 0 6.21  6.22  slice  107  I l l  load-deflection 113  D e f l e c t e d shape comparison and computed f o r c e s on stud SI w e l l a f t e r f i r s t y i e l d .  114  Measured and computed curves f o r stud S2.  115  load-deflection  D e f l e c t e d shape comparison and computed f o r c e s on stud S2 w e l l a f t e r f i r s t y i e l d .  116  Measured and computed curves f o r stud S3.  117  load-deflection  D e f l e c t e d shape comparison and computed f o r c e s on stud S3 w e l l a f t e r f i r s t y i e l d .  118  Measured and computed l o a d - d e f l e c t i o n curves f o r studs S4, S5 & S6.  119  D e f l e c t e d shape comparison and computed f o r c e s on studs S4, S5 & S6 w e l l a f t e r f i r s t yield.  120  Stud connection E2, due t o Spencer, simul a t e d by computer model.  122  xiii  FIGURE 6.23  TITLE Measured and computed  PAGE  load-deflection  curves f o r c o n n e c t i o n E2.  123  6.24  Stud a x i a l s t r a i n s f o r c o n n e c t i o n E2.  124  7.1  Approximation o f f o r c e s a c t i n g on a stud i n c o n c r e t e a t maximum l o a d . Approximate l o a d - d e f l e c t i o n model o f a connection.  128  Calculated load-deflection studs S4, S5 & S6.  136  7.2 7.3  12 8  curve f o r  7.4  C a l c u l a t e d l o a d - d e f l e c t i o n curve f o r c o n n e c t i o n E2.  137  7.5  C a l c u l a t e d l o a d - d e f l e c t i o n loops f o r stud bending specimen SB5.  138  Calculated load-deflection c o n n e c t i o n A3.  139  7.6  loops  7.7  L o a d - d e f l e c t i o n loops f o r c o n n e c t i o n E l measured by Spencer.  140  7.8  Calculated load-deflection nection E l .  141  A. 1  Complete c u b i c  A. 2  A t y p i c a l s i d e o f an element.  152  A. 3  Constrained cubic  159  A.4  Numerical i n t e g r a t i o n p o i n t s  A.5  Complete c u b i c q u a d r i l a t e r a l .  164  A. 6  Constrained cubic q u a d r i l a t e r a l .  164  A.7  Cantilever with parabolic  196  A.8  S t r a i n energy convergence f o r c a n t i l e v e r .  197  A. 9  Stresses i n c a n t i l e v e r at cross-section 12 i n . from support f o r 4 x 16 g r i d .  198  One quadrant o f p a r a b o l i c a l l y loaded square plate.  199  A.10  loops f o r con-  triangle.  152  triangle.  xiv  for triangle.  end load.  15 9  FIGURE A.11  TITLE Strain ly  energy  convergence  PAGE for parabolical-  loaded square p l a t e .  A.12  One h a l f  A.13  S t r a i n energy versus g r i d s i z e f o r s h o r t deep beam. S t r e s s e s i n s h o r t deep beam a t c r o s s - s e c t i o n 3 i n . from midspan f o r 8 x 8 g r i d . Results a r e compared w i t h an a p p r o x i m a t e s e r i e s s o l u tion.  A.14  of short  deep beam.  2 00 202  20 3  204  C.l  Stud i n t e r a c t i o n  diagram  224  C.2  Moment-curvature  diagram  224  xv  DEDICATED  TO MY PARENTS AND TO MY WIFE  ACKNOWLEDGEMENT OF THEIR MANY S A C R I F I C E S ON MY  BEHALF  xvi  ACKNOWLEDGEMENT  The his  advice  to  some  as  yet,  writer  i s indebted  and guidance  results  from  her enthusiastic  her  patient  The  with  and f o r h i s p e r m i s s i o n  h i s own  thanks  for  Bordignon  Spencer f o r  laboratory  tests  to refer which a r e ,  unpublished.  Special  Research  t o D r . R.A.  typing  support  of this  research  Council Masonry  connections  go t o t h e w r i t e r ' s  was  who  and encouragement,  Gaye, and  for  thesis.  financed  o f Canada. Ltd.  wife,  by t h e N a t i o n a l  Thanks supplied  f o rthe preliminary  xvii  a r e due t o M e s s r s . the concrete tests.  panels  1  CHAPTER  1.  INTRODUCTION  It  has been  conventional tion ing  skeleton  of multistorey purposes,  place and  economic  ency  t o depart  precast  the floors  panels  filled  have  from  buildings  altogether selves.  i n which  and t h e l o a d s  b u i l d i n g a r e shown  struction  i n R e f . 1.  of the r e l a t i v e l y  concrete  construction,  Social prefabri-  and t h e  use o f  panel  large frame  i s  knowledge  omitted  units  themlarge view i n  of precast  stage  of  to  panel  perspective  examples  precast  cladding  i n a single-storey  early  tend-  of construction i s  increased  i n the exploded  may  Because precast  units  dwell-  date.  by t h e p r e c a s t  of striking  found  type  and  greater  the conventional  and a number be  later  o r as p r e c a s t  carried  construc-  are cast i n  i s inevitable,  as p r e c a s t  frames,  Typical precast  1.1/  i n at a  i s a markedly  or conventional  t o use the  commercial  and frames  the conventional  There  America  indicated that  i n buildings, either  box-type  of  f o r both  i n the building industry  concrete  Fig.  i n North  of reinforced concrete  buildings  pressures  increasing.  panel  type  i n which  and t h e w a l l  cation  customary  con-  development  of the analysis  and  design  areas.  of  precast  While  continues  on  the  elastic  structural  loading,  little of  Moreover,  the  today  an  is  2  available tions tion  from  panel and  and  load  dwelling can  precast  panel  Of  has  2 3  be  the  cyclic  design  loads  structure  which  are  It  i s to  be  damage  under  these  and  of  such  connections  ture,  even  blast  under  on  in  precast  and  similar  i n worldwide  of  limited  on  full-scale  connec-  in  concrete Russia  structures,  These  with  and  the  respect  to  to  precast be  the  structural  engineer  is  concrete  expected  l a r g e r than  buildings,  during  any  of  the  the  other  structure will  overloads  during  a  severe  damage  attempted  is  deform cycles  by  large  that  design suffer  earthquake.  designing  inelastically of  in  lifespan  a  several  done  general.  expected  can  under  t e s t s were  production,  qualitative  is  informa-  precast  principally  use  that  that  which  cyclic  elements.  types  with  f o r mass  as  concern  may  few  loading.  intended  often  loads.  Limitation  tests  of  a  done,  regarded  particular  of  of  information  Some w o r k  been  structures  earthquake  a  ~  The  some  research  under  precast  connections  in  and  monotonic  behaviour,  loading  3 - 1 5  limited  behaviour  between  behaviour  monotonic  under  only  the  of  the  structures  results  the  is  exists  both  complication.  includes  or  about  v a r i e t y of  tests.  1 6  under  connections  under  fatigue  inelastic  elements  large  structures  lateral on  the  concerns  Japan,  and  i s known  added  mainly  buildings  quantitative information  concrete  conditions,  concrete  members  without  displacement  fracrever-  sals, for  and s t i l l  some  time  integrity  a s t o how m u c h  nor,  on t h e o t h e r  structural precast the  hand,  elements  There  yield  concrete structure  earthquake  As ned,  there  linear  t o ensure  loading. ments, been by  only  method,  under  which  various  that  the connected loads  funnelled integrity  have  load  programs,  been  used  i n a  through during  conditions  members.  - 2 6  Beams  Panels  are modelled  number o f d e g r e e s  the degrees  o f freedom  o f freedom  The  connections themselves  the  properties  o f which  employing  precast  by n o n - l i n e a r  a r e , as y e t , l a r g e l y  one o f t h e p r e r e q u i s i t e s  usual,  elements out so  ele-  have  as  a t c o n n e c t i o n s and j o i n t s are modelled  frame  earthquake  programs  finite  condensed  and non-  concrete  are modelled by  the  concrete  including  i n such  and columns  i s concer-  for linear  and p r e s t r e s s e d  p a n e l s , c o n n e c t i o n s and j o i n t s  Thus,  connections,  structures  Further extensions to include  2  large  of such  a r e a number o f computer  undertaken. "*  line  avail-  from  The l a t e r a l  of their  load-  i s paramount.  analyses of reinforced  structures  earthquake  i s no i n f o r m a t i o n  are p r i n c i p a l l y  f a r as t h e a n a l y s i s  displacement  under  instead.  and  the design of  i s required  whether  While  ensuring the d u c t i l i t y  requirements  ductility  capacities.  elements,  c o n n e c t i o n s , and maintenance  an  a  been  i n i t s infancy.  able  ultimate  structural  for similar  i s s t i l l  their  d e s i g n e r s have  of precast  connections ing  maintain  with  that remain. springs,  undetermined.  f o r increasing  the  5 quantitative concrete viour is  of connections that with  adopted  nections many  that w i l l  welded  in  recent  in  Fig.  under load ent  mechanisms  investigated  stud  conditions.  will  -  be more  univer-  enjoyed  headed  stud  of  con-  Locally,  denominator  have  It  i n the build-  the large variety  o n e common  stud  t h e beha-  y e t be i n v e s t i g a t e d .  connections  behaviour whereby  -  the  prominent  connection  favour i s shown  gives  mathematical  analysis,  concrete  may  carry  into From  procedures  and a d d i t i o n a l  aids  Three shear  differ-  loads are  mathematically.  are synthesized  load.  panels  l o a d i n g , and summarizes t h e  and modelled  an i n s i g h t under  the laboratory  f o rprecast  a connection  mechanisms  briefly  of the connections.  experimentally  connections  simplified  stud  describes  and c y c l e d shear  of these which  tions .  loading  types  reduce  A typical  o f headed  deflection  model  thus  investigation  monotonic  models  with  of precast  1.2.  This testing  connection  employing  years.  familiarity  different  possibly  headed  and d e s i g n  increased prefabrication  and w i l l  connections  fusion  the  under  industry, standard  sally  of the analysis  buildings i s a detailed  probable  ing  knowledge  into  the behaviour the data  a  The  computer  o f headed  thus  assembled,  are presented  as a i d s i n  i n the design  o f such  connec-  F i g . 1.2  Typical  showing two  headed s t u d  common  stud  connection  configurations  and the ease with which minor i m p e r f e c t i o n s and m i s a l i g n m e n t s  may  be  accommodated.  CHAPTER  CURRENT  2.1  INFORMATION  Existing Welded  use  headed  i n composite  this and in  static  R e f . 46.  tion,  which  direct shown few  t o most  cerned  with  were  of test,  used  b y many  Most  cases.  6  of these  Earlier  the l i m i t a t i o n that  data  tests  were  were  7  between  r e c o r d i n g was  a s i n F i g . 2.2  f o r instance.  became more  concerned  ultimate  cated  and r e s u l t s for results  McMackin tension  tests  were  recorded  1 2  field  may  be  studs  1  1  found  investiga-. i s the  of the  type  static,  with  a  load-deflection primarily  steel  and  terminated  conconcrete  a t a low  Investigators  to failure  conducted  on i n d i v i d u a l  i n this  capacities  by O l l g a a r d e t a l .  et al.  1 1  i n the  tests  of slip  with  in  investigators,  specimens'*' '  for  accepted  pertinent to this  deflection,  tors  developed  of the research  and r e l o a d i n g branches  the result  CONNECTIONS  information  connectors  on p u s h - o u t  i n F i g . 2.1.  i n some  STUD  of the investigations  type  test  curves  stud  test  summary  has been  unloading  with  A  One  shear  load  HEADED  c o n s t r u c t i o n and a r e w i d e l y  application. references  ON  2.  later  of the connec-  as t y p i c a l l y  indi-  i n F i g . 2.3.  sixty  combined  of various  shear  sizes,  and  and de-  8  Deflections  measured  between  concrete  and  steel  8  .„?|D.'Q|  n  o  --ii  CM  Section  Fig.2.1  Typical  A-A  push-out  specimen.  beam.  25  20 D.  a  15  Z)  -1—'  Q.  Spec. No.  10 +  Thiir limann 1 Viest 6A2 6B2 6A/4 6B/* 2B BGnjamin 8A 1B I  O  9/ *j y  c  °/ 0 f  _  0  0.02  +  +  0.04  0.06  (p.s.i.)  Quit (kip)  5080 ® 3870 0 A  24.2 32.0 32.5  3360 0 3260 A 3650 0 /4200 3650 •  21.2 22.5  0.08  D e f l e c t i o n (in.)  Fig. 2.2  Shear  l o a d - d e f l e c t i o n c u r v e s for /4 in. d i a . x 4 in. s t u d s . 3  7  30  Normal  o'  Light  weight  weight  concrete  concrete  f  10  23  0 0  0.1  Deflection  Load-deflection  0.3  0.2  curves  C U  (in.)  to u l t i m a t e  for  3// '4  J1 in. dia. x 4 i n . s t u d s  rived  an  interaction  u P' uc where  2.2  "PCI  Design  nections"  2 7  The encasing  Applied  ultimate  tension.  V u  Applied  ultimate  shear.  P' uc  Ultimate pullout strength c o n t r o l l e d by concrete.  V uc  Ultimate concrete of stud.  recently  under  Design  design  1  ultimate  and  the  pullout  capacity  tests  on  stud  moment.  "PCI  Manual  strength  i s determined 4  on  by  Design  the of  Con-  A  Area  o  ultimate  f  of  faceplate  can  a  stud  controlled  (2.2)  c  shear  concrete  concrete  of  from  (j) A o  0.85  a  and  stud  Procedure  w h e r e <J>  to  s e r i e s of  of  follows:  f' c  tion  shear  a  shear  p r o c e d u r e . c u r r e n t l y recommended  Handbook" i s as  completed  combined  P' uc  attached  (2.1)  P u  concrete  The  £ 1  vu  uc  Current The  v  +  Chadha connections  curve  cone  cylinder  shear  strength  be  determined  \  »  by  strength  of the  a  stud shear  fric-  concept  V uc  *  f  su  (2.3)  by  where <j>  0. 85 C r o s s - s e c t i o n a l area of stud shank Friction coefficient = Ultimate  su  The of concrete  concrete  stud  can be c a l c u l a t e d from 0.9  ^b  (2.4)  ^su  u l t i m a t e shear c a p a c i t y of a stud e x c l u s i v e of  can be c a l c u l a t e d from 0.75  V us  The and  s t r e n g t h of  u l t i m a t e t e n s i l e c a p a c i t y of a stud e x c l u s i v e  P' us The  tensile  0.9  A,  f  D  u l t i m a t e concrete  (2.5) S U  c a p a c i t y f o r combined  tension  shear l o a d i n g of headed studs welded t o a f a c e p l a t e  be determined from the i n t e r a c t i o n  7, u P' uc The  +  v  can  equation  7, £  u V uc  (2.6)  1  u l t i m a t e stud c a p a c i t y e x c l u s i v e of concrete  combined t e n s i o n and  shear l o a d i n g should  2 u P' us  +  V u V us  for  satisfy  2 £ 1  (2.7)  I t should be noted t h a t PCI l o a d f a c t o r of 4/3  additional  f o r the d e s i g n of t h i s type of c o n n e c t i o n  T h i s e f f e c t i v e l y reduces  2. 3  recommends an  the c a l c u l a t e d d e s i g n loads by  Recent C y c l i c t e s t s on Headed Stud  27  25%.  Connections  As p r e l i m i n a r y e x p l o r a t o r y r e s e a r c h f o r t h i s  investi-  g a t i o n , s i x headed stud connections of the type shown i n F i g . 1.2  were t e s t e d under q u a s i - s t a t i c monotonic and c y c l i c  loading.  shear  A comprehensive r e p o r t of t h i s r e s e a r c h has been  p u b l i s h e d by Spencer and N e i l l e ;  2 8  consequently,  only  salient  p o i n t s w i l l be i n c l u d e d here.  D e t a i l s of the s i x connections t e s t e d are shown i n F i g . 2.4. in.  Each c o n n e c t i o n was  cast i n a 2 x 4 f t . panel x 8  t h i c k and t e s t e d i n the r i g shown i n F i g . 2.5.  yoke was  The l o a d i n g  b o l t e d with high-strength f r i c t i o n g r i p b o l t s to a  5 x 12 x 5/8  i n . f a c e p l a t e which had been welded to the connec-  t i o n angle.  A d i s p l a c e m e n t - c o n t r o l l e d h y d r a u l i c jack a p p l i e d  a v e r t i c a l c y c l i c f o r c e t o the s t e e l p l a t e v i a the l o a d i n g yoke which prevented  n e i t h e r p u l l o u t of the studs nor  of the c o n n e c t i o n about a h o r i z o n t a l a x i s .  rotation  D e f l e c t i o n s of  the c o n n e c t i o n were measured between the top of the 5/8 i n . p l a t e and  the top of the c o n c r e t e p a n e l .  Connection A.1  was  loaded m o n o t o n i c a l l y t o  I t s l o a d - d e f l e c t i o n curve i s shown i n F i g . 2.6. c o n n e c t i o n s were s u b j e c t e d t o c y c l i c  failure.  The  remaining  loading at frequencies  14  Details  Connection  angles  of  studs,  connection  and p a n e l r e i n f o r c e m e n t .  CM  A'1 5'  L  A2, A 3  m  5-  / 3WLx12 Long #  X  B 1 3^3 x tfLx12'Long #  3  f f  B 2 3'x2'x #L 10 Long 3  #  X  B3  s u y  c  = 60 000p.s.i. =  A 600p.s.i.  • Denotes  no. U  grade  bar.  60  MHO.*  S^WLxlo'LongJ^  Fig.2.4  Details  of  connections  tested.  15  30  20  10 -H  0 0  0.1  0.2  .0.3  0.4  0  D e f l e c t i on (i n.)  Fig. 2.6  L o a d - d e f l e c t i o n curve - c o n n e c t ion A t  r a n g i n g from 0.01  to 0.02  Hz.,  with a few c y c l e s i n the  t i c range f o l l o w e d by c y c l e s of i n c r e a s i n g amplitude failure.  up  The  i s g i v e n i n Table  omitted, were found  l o a d i n g , c o n n e c t i o n A.1  y i e l d i n g , equal almost  I.  but i n which the 4/3  connections  tended  s t r e n g t h envelope amplitude.  The limit.  degrade.  lity  sustained a high load a f t e r  to the maximum l o a d at f i r s t  yield, for remain-  t o some s o r t of r a p i d l y degrading  cyclic  yield  l o a d - d e f l e c t i o n loops e x h i b i t e d a  For loads w i t h i n t h i s l i m i t , the  loops  f o r loads o u t s i d e the l i m i t they  Both the y i e l d s t r e n g t h envelopes  l i m i t s were a r b i t r a r i l y chosen i n F i g .  P r i o r to t h i s research, Al-Yousef cyclic  load  under c y c l i c l o a d i n g w i t h g r a d u a l l y i n c r e a s e d  remained s t a b l e , and to  cal-  to be c o n s e r v a t i v e . Under mono-  a d e f l e c t i o n up to 14 times t h a t at f i r s t y i e l d . The  stability  and  d e s i g n u l t i m a t e loads f o r these c o n n e c t i o n s ,  c u l a t e d by PCI d e s i g n procedures f a c t o r was  A.3  A summary of the u l t i m a t e loads  d e s i g n loads f o r the connections  ing  to  T y p i c a l l o a d - d e f l e c t i o n loops f o r c o n n e c t i o n  are shown i n F i g . 2.7.  tonic  elas-  48  and  stabi-  2.7.  performed r e v e r s e d  l o a d i n g t e s t s on a number of push-out type  Many of the l o a d - s l i p curves  the  continued  from h i s r e s u l t s bear  specimens. striking  resemblances t o the l o a d - d e f l e c t i o n curves of the above i n vestigation.  1  Fig.2.7  1  +  Load-deflection  30  l o o p s - c o n n e c t i o n A3  19  TABLE I SUMMARY OF LOAD DATA FOR CONNECTIONS  Design U l t i m a t e S t r e n g t h Connection  *  Concrete (kip)  Maximum Load  Steel (kip)  Up (kip)  Down (kip)  Mode of Failure  Al  27. 2  27.2  35. 4  -  Concrete  A2  27.2  27.2  31.5  30.2  Stud  A3  27.2  27.2  29.3  27.0  Stud  Bl  26.4  27.2  31. 8  28.0  Concrete  B2  25.5  26.9  30.0  32. 0  Stud  B3  25.5  26. 9  23.4*  24.0*  Stud  Test results error.  unreliable  because  of connection  fabrication  20  CHAPTER  THE  SHEAR L O A D - R E S I S T I N G  3 .1  in  the  the  Mechanisms  From  exploratory cyclic  detailed  stud to  chapter,  stud  two  routes  of  the  series  effects  of  that  of  monotonic  i n concrete variations  common of  each  to  complex  and  by  that  After the  a  The  were  of  and  cyclic  shear  and  the  a  research  i s not  angle  Firstly, failed  the  loads more  tests  of  on  the and  face-  subject  in this  tests  concrete  i n compression  headed  load  stud  preliminary cyclic  exam-  further,  i s the  included  more  scru-  an  shear  examination  reinforcement,  latter  was  several pertinent physical  a l l specimens.  connection  was  with  a much  typical  second  panels,  of  further  first  which  briefly  behaviour  The  i n panel  investigation  there  that  mechanisms  CONNECTION  described  evident  emerged:  concrete.  Observations cated  A  tests,  warranted.  research  arrangements.  separate  was  OF  transmitted externally-applied  surrounding  connections  plate  of  was  various  connection  the  i t was  connections  investigation  extensive  a  the  previous  ination  COMPONENTS  Probable  headed  tiny,  3.  of  discourse.  indi-  features at  and  the  end  spalled  off  as  the  was  concluded  adjacent initial  load  what  of  finally  plates,  mill  in  the  transfer  between  of  observations  forces  were  concrete  by  the  from  the  the  the  the  significantly  of  and  surfaces  the  the  to  the each  entire  from  an  to ob-  striations  and  behind  face-  the  indicated  that  connection  to  possible  concrete.  postulated a  or  It  the  tests,  i t was  shear,  f a c e p l a t e s and  via  studs,  concrete,  concrete  i t was  the  faceplate. Apart and  direction  commenced.  f a c e p l a t e on  In  concrete  studs  the  transmitted three  of  contributed  behind  on  yielding  connection.  between  scale  and  bearing  away  of  happened  aligned  these  each  out  interaction  deposited  end  broke  pulled  had  the  a maximum  probably  stiffness  connection  vious  that  concrete  faceplate  see  reached  friction  As  a  external the  result shear  surrounding  mechanisms:  3.1.1  F r i c t i o n between concrete.  3.1.2  Bearing  3.1.3  I n t e r a c t i o n between s t u d s , f a c e p l a t e and c o n c r e t e by b e a r i n g o f t h e s t u d s on c o n c r e t e and b e n d i n g o f t h e studs.  3.2  Laboratory Four  Each  set  shear each 8  was  of  sets  end  of  of  to  components  specimen  that  cylinders  28  days  in a  was  were  for  surrounding  concrete.  Mechanisms  isolate  A l l specimens  least  f a c e p l a t e and  f a c e p l a t e on  strengths. at  of  laboratory test  force-resisting  i n . concrete  of  Models  designed  concrete  inside  and  specimens  some  aspect  tabulated cast,  cast  six  of  the  above. 4  room.  devised. three Along  i n . diameter  f o r measurement  cylinders  curing  were  were  of  properly  with by  cylinder cured  3.2.1 for  F r i c t i o n Specimens:  These specimens were  the i n v e s t i g a t i o n of f r i c t i o n  designed  between the i n s i d e of a  f a c e p l a t e and the adjacent c o n c r e t e .  D e t a i l s of the three  specimens made are g i v e n i n F i g . 3.1.  Each f a c e p l a t e was  h e l d up a g a i n s t the c o n c r e t e by two 1/2 i n . diameter  steel  bars which were welded t o the f a c e p l a t e a t one end and t o an angle s e c t i o n embedded i n the c o n c r e t e a t the o t h e r .  These  s t e e l bars had a d i f f e r e n t l e n g t h i n each specimen and were isolated  from the surrounding  foam p o s s e s s i n g l i t t l e bending  concrete with r i g i d  bearing strength.  plastic  The t e n s i l e and  p r o p e r t i e s of the bars were measured from a s e t of  t e n s i l e and bending  t e s t s , the r e s u l t s of which appear i n  the f o l l o w i n g c h a p t e r .  3.2.2  End b e a r i n g Specimens:  S i x t e s t p i e c e s were c o n s t r u c -  ted  of 1/2 i n . t h i c k f a c e p l a t e s w i t h t h i c k n e s s e s of 1/4,  3/8  and  1/2 i n . b e a r i n g on the c o n c r e t e , as shown i n F i g . 3.2.  Again the f a c e p l a t e s were h e l d i n p l a c e by two 1/2 i n . d i a meter s t e e l bars p o s s e s s i n g the same p r o p e r t i e s as the bars used f o r the f r i c t i o n  tests.  The bars and the back o f each  f a c e p l a t e were i s o l a t e d from the c o n c r e t e w i t h r i g i d foam, except  f o r approximately  plastic  1/8 i n . of the lower edge and  the e n t i r e end of the f a c e p l a t e which were i n c o n t a c t w i t h the c o n c r e t e .  A t y p i c a l specimen, ready  for casting i n  c o n c r e t e , i s shown i n F i g . 3.3.  3.2.3  S i n g l e Studs i n Concrete:  The studs f o r these  labo-  r a t o r y models were machined from b r i g h t bar s t e e l , as opposed  r 1  7/  s/g  '/? d i a . b a r  1//  Zx2Y/  4  L  faceplate  CM  Plastic  foam  4;in.  4,6 or 8 a n c h o r bar length  Sect ion  A-A A  Fig.3.1 friction  Specimens between  for  faceplate  investigation and  of  concrete. CO  F i g . 3.2  Test  pieces for investigating  f a c e p l a t e end  bearing.  to  commercial studs which have a cold-formed  of  the shank.  same as those  The  head a t one  t e n s i l e p r o p e r t i e s of the s t e e l were the  f o r the bars used i n the f r i c t i o n  specimens.  Three specimens were c a s t w i t h 6 i n . studs of 1/2, 3/4  i n . diameter  ( F i g . 3.4)  a l l w i t h studs of 5/8 at  a 1/2  5/8  and  and a f u r t h e r t h r e e were c a s t ,  i n . diameter  and a s p e c i a l  the f a c e p l a t e , as shown i n F i g . 3.5.  a l l of 5/8  end  The  stiffener  f a c e p l a t e s were  i n . p l a t e and were i s o l a t e d from the concrete  i n . gap.  During  p l a t e s were supported  by  t e s t i n g , the lower ends o f the f a c e -  by l o a d c e l l s  i n s t e a d of the u s u a l  second s t u d .  A 1/8 to  i n . wide s l i t extended from the top of the stud  the top of the c o n c r e t e over the whole l e n g t h of the stud.  T h i s was  p r o v i d e d to a l l o w f o r l a t e r access to the stud by  means of displacement  t r a n s d u c e r probes so t h a t the deformed  shape of the stud c o u l d be measured d u r i n g t e s t i n g . studs had  s m a l l dimples  drilled  at 5/8  along t h e i r top s u r f a c e s to f a c i l i t a t e the probes as shown i n F i g .  3.2.4  i n . between c e n t r e s a c c u r a t e l o c a t i o n of  3.4.  Studs i n Tension and Bending.  Three t e n s i l e  specimens were assembled by f u s i o n welding 5/8 Fig.  i n . diameter 3.6.  w i t h two plates. below.  test  three 6 i n . by  commercial studs, head t o end,  Nine bending  The  as shown i n  specimens were c o n s t r u c t e d , each  of the same studs fusion-welded  to 5/8  i n . face-  The method of t e s t i n g these specimens i s d i s c u s s e d  26  F i g . 3.3  Isolation  faceplate  from  Fig. 3.4  of  bars  and  back  of  c o n c r e t e w i t h p l a s t i c foam.  Machined  studs ready  in c o n c r e t e .  for  casting  1  15  7 /  '/8 s l i t  for  -probes •••00  \ \ \ \  c CD  R  V of  probes  5/g f a c e p l a t e  Details Section  A-A  Fig.3.5  Models  for  interaction between  stud  examination and  of  concrete.  of  studs  19 7  i 1  '  u  Tensile  2  1I  specimens  P  Fig.3.6 studs  Specimens in  tension  f o r testing  and  bending.  3. 3  Testing A  testing  different shown  the  of concrete  3.7.  application  beam  Each  machine,  directly was  used  bolted  on  concrete  i t proved  instead,  of i t s span,  equal  to two-thirds  load  An  T h e beam a reaction  of the load  tested  to the head  of the apply  applied  on  by  loading a  loaded  on t h e  by  face-  to  rested was  as  firmly  intermediate  one e n d o f w h i c h  producing  was  too d i f f i c u l t  faceplate.  three  constructed  of the loading  t o be  the  b e d a n d was  downward  to the faceplate.  third  was  specimen  the t e s t i n g  of the size  t o each  t o accommodate  specimens  of a vertical  Because  testing loads  r i g designed  i n position  plate.  pad  types  i n Fig.  clamped  Apparatus  bearing a t one-  faceplate  the testing  machine.  Deflections mens w e r e m e a s u r e d between  a clamp  faceplate, for  these  3.9.  also  be  tion  curve  plotter.  i n Fig. were  by means  One  seen  i n two  tiers  f o r each  figure. stud  The d e f l e c t i o n  was  and the  Load-deflection directly  of each  of a battery  of the load  i n this  3.8.  on  single  cells  a n X-Y  stud  plotted  a  continuously was  trans-  shown i n  mentioned  testing,  plot  plotter.  i n concrete  frame  previously  During  for this  curves  of displacement  i n the s p e c i a l  speci-  transducer  of the concrete  plotted  shape  and end b e a r i n g  of a displacement  the bottom  deflected  mounted  Fig.  near  specimens  measured  ducers  by means  a s shown  The was  of the f r i c t i o n  read  can  load-deflecon from  an  X-Y  the  Fig. 3.7  Apparatus  for  in c o n c r e t e , end bearing and  testing  single  friction  studs  specimens. o  Fig. 3.8  Measurement  of  of f r i c t i o n  and  displacements end  bearing  specimens.  F i g . 3.9  Set  transducers deflected  of for  shapes  displacement measuring of  studs.  transducer all  load  closest  and  electronic  most via  be  to  a  A  plot  printed at  voltages  returned  the  will  were  which  shown  the  read  by  loads  i n the  typewriter  means  them  and  the any  of  an  were and  a l -  deflections  foreground  of at  intervals  readings  processed  d e f l e c t e d shape  on  regular  These  a l l measured  typewriter  of  At  system.  mini-computer  terminal  3.10.  faceplate.  acquisition  immediately the  the  displacement  data  transmitted  to  stud  in Fig.  could  stage  of  also  the  test.  i The tests in  on  Fig.  spacer  apparatus,  commercial 2.5.  placed  block  of  which  the  set  studs  of  the  studs  shown  D e f l e c t i o n s were device  quasi-static  and  cyclic  the loads  hydraulic jack  faceplate.  continuously  on  to  and and  bend  top  used  was  firmly  3.12.  The  was  of  through  X-Y  plotter.  r i g shown a  steel  clamped span  between  loading were  a yoke  to  a  over 1/2,  the  faceplate.  a p p l i e d by  L o a d - d e f l e c t i o n curves an  the  v a r i e d between  each  a  of  bending  f a c e p l a t e had  measured  were  f o r the  adaptation  in Fig.  in.  clamping  free  an  were  the  the  was  studs  1-1/2  controlled  as  3.11,  studs,  between  concrete  and  or  Each  in Fig.  1  top  of  Monotonic  displacementbolted  again  to  plotted  Fig. 3.10  Data  instruments  acquisition  used  in  tests  concrete.  and on  recording studs  in  F i g . 3.11 f o r bending  Apparatus tests  on studs.  F i g . 3.12  Studs  firmly  c l a m p e d to c o n c r e t e  block.  35  CHAPTER  4. f  LABORATORY  4 .1  Materials Tests  on  specimens  are  tation  tests  of  4.1.1 was  the  diameter ders  by  were  upper  platen.  on  fixed  are  4.1.2  from  machined  aging strain  specimens.  Each  concrete  of  studs.  were  per  the  Strains over  a  a  of  on  measure-  six 4 in.  these  spherical  minute.  tensile  steel were  gauge  directly  strengths  presen-  cylin-  machine  seat  on  strength  Results  of  the test  these  II.  standard of  to  strength  l o a d i n g f o r each  p.s.i.  laboratory  laboratory testing  p l a t e n and of  the prior  Measurements  standard  i n Table  samples  cylinder  compressive  rate  Three  extensometercurves  a  1800  presented  Steel:  machined  d e s c r i b e d below  the  lower  The  approximately  tests  and  collectively  made w i t h a  RESULTS  materials constituting  8 in. cylinders.  had  was  the  average  which  AND  Tests  Concrete:  ment  TESTS  used  test  specimens  f o r the  anchor  measured  by  means  length of  2  i n . and  plotted  on  an  X-Y  were  of  bars an  aver-  stress-  recorder.  TABLE I I CONCRETE STRENGTHS OF LABORATORY SPECIMENS  Principal Dimension  Cylinder Strength  (p.s.i.)  Specific Weight (p.c.f.)  Specimen Description  Specimen Number  F r i c t i o n between  Fl  4 i n . span  6192  210  153. 3  back o f f a c e p l a t e  F2  6 i n . span  6165  502  155. 0  F3  8 i n . span  5663  239  154.1  Bl  1/4 i n . f a c e p l a t e  4593  121  151.1  B2  1/4 i n . f a c e p l a t e  4810  305  153. 3  B3  3/8 i n . f a c e p l a t e  4926  146  153. 3  B4  3/8 i n . f a c e p l a t e  4933  86  152.8  B5  1/2 i n . f a c e p l a t e  4812  96  151. 8  B6  1/2 i n . f a c e p l a t e  4955  216  153.7  SI  1/2 i n . d i a . stud  7704  165  154. 3  S2  5/8 i n . d i a . stud  6775  243  153.3  S3  3/4 i n . d i a . stud  6910  . 223  153. 9  S4  5/8 i n . d i a . stud with s t i f f e n e r  5800  349  154.0  and concrete  End b e a r i n g of f a c e p l a t e on concrete  I n t e r a c t i o n between stud  and  surrounding concrete  Mean  Standard Deviation  S5  II  II  5211  427  154. 7  S6  II  II  4674  257  154. 8  All  specimens e x h i b i t e d a w e l l - d e f i n e d y i e l d p o i n t and  t u r e o c c u r r e d a t s t r a i n s of about  20%.  frac-  Relevant p r o p e r t i e s  measured i n these t e s t s are p r e s e n t e d i n Table I I I .  Bending  t e s t s were conducted  bars f o r l a t e r comparison  on two  1/2  i n . diameter  w i t h a computer model. Load  versus  d e f l e c t i o n curves from these t e s t s appear i n F i g . 4.1.  4. 2  T e s t s on F r i c t i o n  Specimens  Each f r i c t i o n specimen was  put through s e v e r a l l o a d -  i n g and u n l o a d i n g loops as shown i n the curves i n F i g s . 4.2  t o 4.4.  The  l o a d i n g i n one d i r e c t i o n o n l y . completed  load-deflection  t e s t i n g apparatus p e r m i t t e d  Consequently,  each loop  was  by removing the l o a d e n t i r e l y and moving the f a c e -  p l a t e back t o zero displacement by means of a s m a l l h y d r a u l i c j a c k a p p l i e d between the bottom of the f a c e p l a t e and the bed of the t e s t i n g machine.  The  j a c k was  then removed and r e -  l o a d i n g commenced v i a the t e s t i n g apparatus. a p p l i e d by the jack were not measured.  Reverse  Each t e s t was  ducted s l o w l y and took approximately h a l f - a n - h o u r t o  4.3  loads concomplete.  T e s t s on End B e a r i n g Specimens D e f l e c t i o n s of these specimens were m o n o t o n i c a l l y  i n c r e a s e d t o a maximum of about was  completely removed.  0.4  i n . , a f t e r which the l o a d  A l l l o a d - d e f l e c t i o n curves showed  a marked drop i n l o a d a t a d e f l e c t i o n of between 0.05 0.12  i n . as shown i n F i g s . 4.5  t o 4.7.  and  Simultaneously w i t h  t h i s drop i n l o a d the c o n c r e t e at the end of the  faceplate  TABLE I I I TENSILE STRENGTH OF STEEL SAMPLES  Specimen Number  Y i e l d Stress (p. s . i . )  1  67 800  74 000  29. 71 x 1 0  6  2  67 900  73 500  30.08 x 1 0  6  3  68 300  74 000  30. 03 x 1 0  6  Ultimate Stress (p.s. i . )  E l a s t i c Modulus (p. s. i . )  z  A  = = = =  z A >© —  A  AW  if  Jl  4?  p  1 \ 3=—T  I!  5/2 in. -*l  "  n>  J  ////  ! - II  0  0.1  1  0.2  0.3  Deflection A  F i g . 4.1  Bending  test  0.4  Q5  1  0.6  (in.)  on/2 in. dia. bars.  0.7  0  0.1  0.3  0.2  0.5  O.U  Def l e e t ion (in.)  Fig.4.4  L o a d - d e f I ection curves  for  friction  specimen  F3.  10 B e a - ing t h i c k n e s s = 3/ in. 8  8  /  II ^  ^  B4  1  Bh  I  6  2  B3  ^  1  f  0  r-—ZTZ.-  0  01  0.2  0.3  ^ti  J 0.5  0M  D e f I e c t i o n (in.)  g. 4.6  L o a d - d e f I e c t ion curves  for  end-bearing  specimens  B 3 & B4.  0  0.1  0.2  0.3  0.4  0.5  D e f l e c t i on (in.)  4.7  L o a d - d e f l e c t ion c u r v e s  for end-bearing  specimens  B5&B6.  broke away, as shown f o r a t y p i c a l specimen i n F i g . 4.8. A f t e r t h i s concrete f a i l u r e ,  the l o a d g r a d u a l l y b u i l t up  a g a i n , p r o b a b l y because o f f r i c t i o n between the lower end of the f a c e p l a t e and the remaining took approximately  4. 4  concrete.  Each t e s t  twenty minutes t o complete.  T e s t s on S i n g l e Studs  i n Concrete  R e s u l t s o f these t e s t s are presented i n F i g s . 4.9 t o 4.20.  The l o a d on each stud was reduced  t o zero  sev-  e r a l times t o produce l o a d i n g and u n l o a d i n g branches i n the l o a d - d e f l e c t i o n curve which was recorded d i r e c t l y on an X-Y p l o t t e r .  The l o a d t r a n s d u c e r s , p r e v i o u s l y shown  i n F i g . 3.7, gave an i n d i r e c t measurement o f the t e n s i o n i n the s t u d .  Measurements o f stud t e n s i o n were made a t  i n t e r v a l s throughout  each t e s t and are i n c l u d e d i n the l o a d -  d e f l e c t i o n curves.  Tension measurements f o r s t u d S2 a r e  omitted because i t was d i s c o v e r e d a f t e r the t e s t t h a t measurements from one o f the l o a d c e l l s were e r r a t i c , due  t o a poor e l e c t r i c a l c o n n e c t i o n .  probably  Included w i t h each  l o a d - d e f l e c t i o n curve are f o u r d e f l e c t e d shapes o f the stud which were measured a t d i f f e r e n t stages o f the t e s t . should be noted  It  t h a t the s c a l e s on the d e f l e c t i o n axes o f  these graphs become p r o g r e s s i v e l y c o a r s e r w i t h i n c r e a s e d deflection.  T e s t s on studs SI t o S3 i n d i c a t e d t h a t the 1/2 i n . i s o l a t i o n gap between f a c e p l a t e and concrete was probably  47  F i g . 4.8 f a i l u r e at  Typical end  of  concrete faceplate.  J  1 r  l.b  /  "* — ~  '  F  -^ ^ — —  . — ^  Si  1  •  •  CL  i< i *  /  1  i/ 1  •l •/ ll  te  11  / *  i'  /  *'  1  \\ \\  V? in.  ns ion  V)  f<?'  c  IF' ' 1 / / / ' /"  dia. stud  0  <  c o  I;  t(  •  £—Stud  axial  I '/  ;/  1 •  F " 1  1.  ]  i  I  a  i  1  a  il  I  ii  /I  /<  \ (  /'/  1 /I  n  I. u  n  /  ^ ...  -i—»  I o-  0  0.05  0.1  0.15  0.2  D e f I ec t ion (in.)  Fig. 4.9  Load-deflection  curve  for  stud  S1.  0.25  Distance  from  concrete  f a c e (in.)  0.  —I 2  °*-o=fco=»o—O-KD4 6  L o a d = 2.0 kip.  0-04-0-0-0400 ^  2  4  6  L o a d = 4.0 k i p .  7  Q-o—Qj-e-o—Q40.  ^—1  lO?  2  4  6  L o a d = 5.06 k i p .  2  4  L o a d = 5.32 k i p  Fig.4.10  Deflected  s h a p e of s t u d  S1  51  Distance  from concrete  4  face  (in.)  6  L o a d = 2.0 k i p .  CT-  10  /  2  4  6  L o a d = 6.04 k i p .  c o  0 0 / 2  (_>  CD  4  6  L oad = 7.5 k i p .  Q  L o a d = 8.0 k i p .  •08-O-  I  F i g . 4.12  Deflected  shape of s t u d  S2.  Distance  from  concrete  4  f a c e (in.)  6  4  6  L o a d = 12.65 k i p .  QH3—0-„Q4=Q~ O  2  4  L o a d = 13.78 k i  F i g . 4.14  Deflected  shape  of s t u d  S  Distance  from  concrete  f a c e (in.)  410=0-0^.©.  0  4  6  0014}:©=.©-©4 _ 0  4  ' 6  0024  =0—©4©—  ^2  4  6  ^{<-0—Q—,Q^Q_ ©  0  /  2  4 L o a d =13.31 k i  Fig.4.16  Deflected  shape  of  stud  S4  57  Distance  from concrete  f a c e (in.)  _^0=0—Of-O-O—O-fO-  '2  4  6  p L o a d =2.0 k i p .  0}-0=0-Of-0-O—040— /2  4  6  P L o a d = 6.97kip.  c o  d^°^°^04-o-o-o4 o— 0 / 2 4 6  -i—'  L o a d =11.0kip. Q  0  p  2  4  6  L o a d = 13.13 kip.  F i g . 4.18  Deflected  s h a p e of s t u d  S5.  F i g . 4.19  Load-deflection  curve  for  stud  S6.  10  Distance  from  •A  U  p  2  concrete  f a c e (in.)  0-fO—  6  L o a d = 1.94 kip.  o-o-o  L o a d = 6.78 kip.  003+ .0-  1 ^Q-o-itt— u  •o-fo-  4  6  L o a d = 11.18 k i p .  2  U  L o a d =12 .21 k i p  F i g . 4.20  Deflected  shape  of s t u d  S6.  unrealistic.  Studs S4 t o S6 were thus manufactured  s t i f f e n e r i n the 1/2  i n . gap, F i g . 3.5.  The  with a  stiffener  probably formed a s t r e s s - r a i s e r because studs S4 t o S6 f r a c t u r e d , whereas studs SI t o S3 remained ger d e f l e c t i o n s . approximately corded.  Stud S5 was  intact for lar-  a c c i d e n t a l l y preloaded with  8 k i p . , d u r i n g which no measurements were r e -  The p r e l o a d was  the u s u a l manner.  removed and t e s t i n g commenced i n  The e f f e c t s of the p r e l o a d are r e f l e c t e d  i n the d e f l e c t e d shapes, which are d i f f e r e n t from those of studs S4 and S6, and i n the l o a d - d e f l e c t i o n curve which exhibits a slight stiffening  4.5  T e n s i l e and Bending  4.5.1  Tensile Tests:  i n the f i r s t  T e s t s on  load  branch.  Studs  S t r e s s versus s t r a i n curves  these t e s t s appear i n F i g . 4.21.  from  S t r a i n s were measured  over a gauge l e n g t h of 2 i n . s i t u a t e d c e n t r a l l y on the shank of the middle  stud.  The welds i n a l l t h r e e specimens r e -  mained i n t a c t w h i l e f r a c t u r e o c c u r r e d i n one shanks.  of the s t u d  In each case necking of the stud shank o c c u r r e d  o u t s i d e of the extensometer  gauge l e n g t h and,  as a r e s u l t ,  the u n l o a d i n g p o r t i o n of the s t r e s s - s t r a i n curve c o u l d not be measured.  The  r e s u l t s f o r the t h r e e specimens show  g r e a t e r v a r i a b i l i t y than i s normally expected t e s t s on s t e e l . of alignment each t e s t  for tensile  T h i s i s a t t r i b u t a b l e t o the probable  lack  and c o n c e n t r i c i t y of the t h r e e studs comprising  specimen.  70  >— o — © — o  60  __ —0—- . _ 0  —  —  0-^A^Q'  50  1  40 30  1/  1  o 7 •" G  o  Spec:imen ST 1  A  ST 2  1  ST3  / o /  / 0 -  !  0  1  0.25  I  1  0.5  0.75 Strain  4.21  1.0  1-  —1 1.25  1.5  (%)  S t r e s s - s t r a i n curves from tensile  tests  on  studs.  4.5.2 ted  Bending T e s t s : with  tions  spans o f  of the  l o a d was  4.25  4.28.  loaded  Hz.,  No  to zero,  as  had  Deflec-  increased  been  reached,  shown i n F i g s .  fractures occurred  i n the  4.22,  monotonically-  s p e c i m e n o f e a c h g r o u p was  cyclic  with  fracture  an  loading at  occurred,  but  cycles  the  F i g s . 4.23,  amplitude  before  the  i n the  cyclic  of the  curve  c  possesses  cated  in. Figs.  cycle.  4.25  constant  tests,  with  plotted  on  respect  the  until final  to c y c l i c  for three 4.27  to  five  and  4.30.  cyclically-loaded and  load-  speci-  faceplate,  4.31.  be  which i n c r e a s e s with  the  Each monotonic l o a d - d e f l e c t i o n  and  transition 4.28.  a s s o c i a t e d maximum d e f l e c t i o n  malized  of  order  l o a d - c a r r y i n g c a p a c i t y may  a distinctive 4.22,  the  The  subjected  specimen i n F i g .  i n a l l of the  to  4.29.  f u s i o n welds between s t u d  of  subjected  i n each c y c l e  and  f r a c t u r e s i n the  A degradation  amplitude  versus  kept  of  i n c r e a s e , F i g s . 4.24,  shown f o r a t y p i c a l  noticed  4.26  similarly  was  next  Without exception, mens o c c u r r e d  frequencies  i n c r e a s e i n amplitude  member o f e a c h g r o u p was  as  i n . , F i g . 3.6.  a large deflection  reduced  second  quasi-static  ing,  1-1/2  specimens.  The  .01  1 and  s p e c i m e n s were t e s -  s p e c i m e n o f e a c h g r o u p were  until  when t h e and  1/2,  first  monotonically  Groups o f t h r e e  The  point load  i n any  (D,P) do  indi-  decrement  c y c l e was  nor-  t o P and D f o r t h e same s p a n , and o o g r a p h shown i n F i g . 4.32. A parabola was  fitted  t o the data p o i n t s by the method o f l e a s t squares 2*  AP  2.43  + 8.22  D D.  + 93.5  D  x 10  -3  (4.1) where AP  =  Load decrement  D  =  Maximum d e f l e c t i o n i n the same cycle.  =  C o o r d i n a t e s of t r a n s i t i o n p o i n t on monotonic l o a d - d e f l e c t i o n curve of specimen w i t h same span.  i n a cycle.  E q u a t i o n 4.1 i s i n c o r p o r a t e d i n the l o a d - d e f l e c t i o n model of a c o n n e c t i o n i n Chapter 7.  The cumulative r o t a t i o n o f a stud bending specimen under c y c l i c  l o a d i n g i s an approximate i n d i r e c t measure of i t  t o t a l energy d i s s i p a t i o n b e f o r e  fracture:  Sd  0  =  up t o fracture  where  0  =  (4.2)  Cumulative r o t a t i o n t o f r a c t u r e .  6d =  F a c e p l a t e d e f l e c t i o n increment.  L  Bending span as d e f i n e d i n F i g . 3.6  =  T h i s may be used as an i n d i c a t o r o f stud f r a c t u r e i n s t r u c t u r a l a n a l y s i s programs.  The c u m u l a t i v e r o t a t i o n s of  the l a b o r a t o r y specimens are l i s t e d  i n Table IV.  TABLE IV CUMULATIVE ROTATIONS OF STUD BENDING SPECIMENS SUBJECTED TO CYCLIC LOADING  Stud Bending Specimen Number  Cumulative R o t a t i o n to F r a c t u r e  SB2  3.136  SB3  2. 984  SB5  3.404  SB6  5.338  SB8  3 . 066  SB9  4.814 Mean  3.790  CTi  F i g . 4.23  Cyclic  b e n d i n g t e s t on s t u d  specimen  SB2.  20  (D ,£) 0  15  S p a n 1 in. 10  / 0  0.05  0.1  0.15  0.2  Deflection  F i g . 4.25  M o n o t o n i c bending  0.2 5  0.3  0.3 5  (in.)  test  on s t u d  specimen  SB4. 00  F i g . 4.26  C y c l i c bending  t e s t on s t u d  specimen  SB5.  4.27  Cyclic  bending  test  on s t u d  specimen  SB6  Span  ' I  1/2 in. 1  2  / 0  0.1  0.2  0.3 Deflection  Fig.4.28  Monotonic  0.4  0.5  0.6  0.7  (in.)  bending test  on  stud  specimen  SB7.  12  12 H 0.25  1  0.2  1  0.15  1  0.1  1  1  0.05  0  Deflection  Fig.4.29  Cyclic  bending  1  test  0.05  1  1  1  H  0.1  0.15  0.2  0.25  (in.)  on s t u d  specimen  SB8.  F i g . 4.30  Cyclic  bending  t e s t on s t u d  specimen  SB9.  4.31  Cyclic loading  caused  typical  f r a c t u r e of studs in f u s i o n welds.  < 0_  c  c  CD  £ i_  u  CD "O T3 CO O  •D CD  LO  ro  .,0.5^  1.0  Normalised  F i g . 4.32  Normalised  f r o m s t u d bending  load  tests.  deflection  decrement  Quadratic  versus  equation  normalised fitted  by  deflection  least  squares.  CHAPTER  ANALYTICAL  Two the  computer  programs  l a b o r a t o r y models  tical  model  of  these  programs  a  and  to  complete  and  the  5.  MODELS  were  written  subsequently  connection.  approximations  for analysis provide  A  an  analy-  description  involved  is  of  of  presented  below.  5 .1  Approximations Ideally,  large  assemblage  sent-day  large  a  stud  of  computer  of  of  degrees  non-linear tant. sional which too  necessary  expensive,  finite  to  or  elements.  the  be  can  of  In a  iterative  represented  finite  this  the  connection  for  in non-linear  analyses,  With  the  this  scope  i n mind,  of a  provide  large  number  i s most  s o l u t i o n s and concrete  the  impor-  three-dimenand  are  a  Pre-  investigation  relationships  beyond  by  elements.  probably  accommodate  involved.  material behaviour  constitutive are  facilities  memory  freedom  Unfortunately  can  three-dimensional  computing  sufficient  i n concrete  as  steel, yet  either  three-dimensional simpler  idealization  is  77 considered  which i s modelled along  Winkler s p r i n g s , tic  2 9  as d e p i c t e d i n F i g . 5.1.  A linear elas-  form o f t h i s a n a l y s i s has p r e v i o u s l y been used f o r dowel  j o i n t s i n concrete The  the l i n e s of a beam on  pavements  s p r i n g s have n o n - l i n e a r  simulate  concrete  49  and channel shear  connectors.  50  l o a d - d e f l e c t i o n p r o p e r t i e s and  around a stud while  the beam - which r e p r e -  sents a stud - i n c o r p o r a t e s the e l a s t o - p l a s t i c behaviour o f steel.  The p r o p e r t i e s of each concrete  from a n o n - l i n e a r plane slice  of concrete  s p r i n g are determined  s t r e s s f i n i t e element a n a l y s i s of a  which i s a t r i g h t angles t o the stud.  T h i s approximation ignores any d i r e c t or shear s t r e s s t r a n s f e r from one concrete  slice  s t a t e of s t r e s s i n c o n c r e t e  i s ignored  s t a t e i n the plane  of each s l i c e  p a r t i c u l a r l y conservative Richart et a l .  3 0  and Balmer  t e s t s on concrete  t o another.  The t r i a x i a l  and only a b i a x i a l  i s considered.  i n the case of t r i a x i a l 31  T h i s may be compression.  have shown by means o f t r i a x i a l  under c o n f i n i n g f l u i d p r e s s u r e  t h a t con-  c r e t e s t r e n g t h s may be i n c r e a s e d between 4.1 and 7.0 times the unconfined recorded  strength.  On the other hand, Kupfer e t a l .  i n c r e a s e s o f up t o only 27% above unconfined  strengths  32  concrete  i n b i a x i a l compressive t e s t s u s i n g s t e e l brush  p l a t e n s t o a v o i d any p o s s i b i l i t y o f a c o n f i n i n g s t r e s s i n the third direction.  5.2  v  Non-Linear Plane S t r e s s F i n i t e Element Program f o r P l a i n Concrete The  concrete  c o n s t i t u t i v e r e l a t i o n s h i p s included i n  78  Stud in  encased concrete. Typical by  slice  finite  properties  modelled  elements of  gives  springs.  •/////////>  Computer  model  of s t u d  F i g . 5.1 concrete  in c o n c r e t e  The by m e a n s  approximation  of a beam  of a s t u d i n  on W i n k l e r  springs.  the program are e s s e n t i a l l y the same as those used by Darwin and Pecknold; thus o n l y a summary w i l l be g i v e n here. Work i n 33  t h i s f i e l d has a l s o been d e s c r i b e d by Cervenka and P h i l l i p s and Z i e n k i e w i c z ,  and Gerstle *' 31  35  36  Numerical s o l u t i o n i s achieved by means of the d i s placement  method u s i n g an i n c r e m e n t a l i t e r a t i v e s o l u t i o n i n  which the e f f e c t s of n o n - l i n e a r m a t e r i a l behaviour are  inclu-  ded as r e s i d u a l l o a d terms i n a Newton-Raphson p r o c e s s . f i n i t e elements  37  The  employed are t r i a n g u l a r or q u a d r i l a t e r a l w i t h  corner nodes o n l y and t h r e e degrees of freedom per node, as shown i n F i g . 5.2.  Displacements  along an element  edge are  compelled t o be l i n e a r i n a d i r e c t i o n p a r a l l e l t o the edge, and c u b i c a t r i g h t angles t o the edge. s t r e s s element element,  T h i s a l l o w s the plane  to be compatibly combined w i t h a p l a t e  w i t h c u b i c displacements along i t s edges,  p l a t e and s h e l l a p p l i c a t i o n s . a more complete  For readers who  d e s c r i p t i o n of these elements  bending  i n folded  are i n t e r e s t e d  i s provided i n  Appendix A.  Increments  of e i t h e r nodal displacements or loads are  a p p l i e d , a l l o w i n g f o r the s i m u l a t i o n of e i t h e r l o a d - or d i s p l a c e m e n t - c o n t r o l l e d l a b o r a t o r y experiments.  During the i n -  cremental s o l u t i o n , a r e c o r d of s t r a i n s and a s s o c i a t e d c i p a l s t r e s s e s i s kept f o r the c e n t r o i d o n l y of each Cumulative  prin-  element.  b i a x i a l s t r a i n s are s t o r e d i n the form of "equiva-  X  5.2  Triangular  and  right  angles  to  edge.  quadrilateral  finite  elements. CO  o  lent uniaxial e.  strains" =  defined  33  }  IU  by:  Aa.  /  (5.1)  1  '  V  E. l  A l l load increments i n which Aa.  =  i n c r e m e n t a l change i n s t r e s s direction.  E.  =  tangent e l a s t i c modulus i n i direction.  i  =  1,2 r e f e r s directions  in i th  In each increment the strain i s calculated Ae . xu  =  stress  change i n e q u i v a l e n t  uniaxial  from:  a.  - a.  1  new  to p r i n c i p a l 1 and 2.  1  E. l  (5.2)  . ,  old  S t r e s s increments are  related  to s t r a i n increments  by: dada, d 12 T  / E ^ 1-v  Symmetrical  0  de.  0  de,  h(E +E -2/E E ; 1  2  1  2  dY 12 (5.3)  where  th E^  =  tangent e l a s t i c modulus i n i direction. Each d i r e c t i o n has a s t r e s s - s t r a i n model as shown in Fig. 5.3.  v  =  Poisson r a t i o . 0.2 i n t e n s i o n - t e n s i o n and comp r e s s i o n - c o m p r e s s i o n and has a s t r e s s - d e p e n d e n t value i n tensioncompression and u n i a x i a l compress i o n . 33  S =0.145S^ + 0.130S p  e  o  Fig. 5.3  Concrete  stress-strain  model  adapted  f r o m work  by K a r s a n & J i r s a . CO CO  Values o f E^ and  which are p o s i t i v e and g r e a t e r  than a s p e c i f i e d minimum p o s i t i v e v a l u e are a c c e p t a b l e i n the program.  Values l e s s than t h i s minimum value are r e -  p l a c e d by the minimum p o s i t i v e v a l u e , even on the unloading  curve of the s t r e s s - s t r a i n diagram  ensure t h a t the assembled tive-definite.  of F i g . 5.3, t o  s t i f f n e s s m a t r i x i s always  posi-  E r r o r s i n t r o d u c e d by t h i s are c o r r e c t e d i n  the c o m p i l a t i o n o f the a s s o c i a t e d r e s i d u a l l o a d v e c t o r which i s compiled from the s t r e s s e s e x i s t i n g i n the element. Thus the square r o o t s used i n e q u a t i o n  (5.3) are always  C r a c k i n g i s modelled by r e d u c i n g the a s s o c i a t e d  real.  elastic  modulus t o the minimum p o s i t i v e v a l u e and again any s t r e s s e s a c r o s s the c r a c k are reduced t o zero i n each i t e r a t i o n by the subsequent  r e s i d u a l load vector.  The f o l l o w i n g s i x  c r a c k c o n f i g u r a t i o n s are i n c l u d e d i n the program l o g i c : no c r a c k , one c r a c k , f i r s t  crack c l o s e d , f i r s t  crack c l o s e d and  second crack open, both c r a c k s c l o s e d , both c r a c k s open.  The b i a x i a l s t r e n g t h envelope of Kupfer and G e r s t l e Fig.  3 9  5.4, i s imposed upon the p r i n c i p a l s t r e s s e s by s u i t a b l y  s c a l i n g the s t r e s s e s i n the c o n c r e t e model o f F i g . 5.3 up or down.  5. 3  Non-Linear  Plane Frame Program  T h i s program was w r i t t e n t o p r o v i d e the approximate "beam on Winkler s p r i n g s " model p r e v i o u s l y mentioned. Again the displacement method i s used with an i t e r a t i v e tal  s o l u t i o n of the Newton-Raphson  37  incremen-  type and e i t h e r nodal  1.4 /  Normalised  principal  stress  R = 1/,/ c 1  F i g . 5.4  Biaxial  strength  envelope  T  for c o n c r e t e a f t e r Kupfer & G e r s t l e CO  displacements in  or nodal  loads may be a p p l i e d .  t h i s program are l i n e members;  types  and  however, f i v e  different  These members  (type 1 i n  are i n c l u d e d :  5.3.1 Fig.  A l l elements  L i n e a r l y E l a s t i c Members:  5.5), are i n c l u d e d f o r the s i m u l a t i o n o f f a c e p l a t e s other members which always remain l i n e a r l y e l a s t i c .  They may c a r r y a x i a l  5.3.2  l o a d s , shear f o r c e s and bending moments.  Faceplate F r i c t i o n Simulator:  l y e l a s t i c , but c a r r i e s  T h i s member i s l i n e a r -  a x i a l compressive f o r c e s o n l y ,  (type  2 i n F i g . 5.5). When such a member i s i n compression, a f r i c tion  l o a d equal t o the c o e f f i c i e n t  o f f r i c t i o n times the  f o r c e i n the f r i c t i o n s i m u l a t o r i s a p p l i e d a t the j o i n t and in  the d i r e c t i o n  specified,  as demonstrated i n F i g . 5.5.  5.3.3  Bond S p r i n g : The bond s p r i n g i s l i n e a r l y e l a s t i c and  carries  a x i a l f o r c e s o n l y , but ceases t o operate  an a x i a l l o a d o f more than a s p e c i f i e d posed,  as soon as  c r i t i c a l value  i s im-  (type 3 i n F i g . 5.5).  5.3.4  Concrete S p r i n g :  T h i s i s a n o n - l i n e a r member t h a t  carries  compressive a x i a l  f o r c e s only and i s used t o simulate  f a c e p l a t e end-bearing crete  (type 4a), and s t u d s . b e a r i n g  (type 4 i n F i g . 5.5).  on con-  The l o a d - d e f l e c t i o n curve f o r  t h i s s p r i n g has the same a l g o r i t h m as the concrete model shown i n F i g . 5.3.  The s c a l e s along  the axes are a l t e r e d  86  © Denotes  fixed  joint  o Denotes  pinned  joint  Five  types  F i g . 5.5 members  built into  plane  of  frame  line model.  max  o  max. Deflection  Fig. 5.6  D  L o a d - d e f I e c t i on m o d e l for c o n c r e t e  springs.  to  reflect  Fig.  load  5.6.  versus  d e f l e c t i o n , however,  The p r o p e r t i e s  D o  the  load-deflection  finite  element  5.3.5  analysis  Non-Linear  which  may  have  diagrams  linear was  curve,  along  stresses  t h e member  derived  as  described.  line  elements  and a s i m i l a r  which  i s at r i effect  element  within  i s linear  to i t .  integrated  the length  each  with  The three  a t the centre  bending  moments  o f moments  element Gauss  element.  of the element  and s t r a i n s  and a r e c o r d  forces  of this  vary  Axial and a integralinearly  and c u r v a t u r e s  for  i s stored.  and b e n d i n g  are governed  moments  by a f a i l u r e  a t any envelope  cross-section which  may  be  follows:  With pression  3 7  whereas  integration point  an e l e m e n t  stress  The B a u s c h i n g e r  a t r i g h t angles  along  i s stored,  Axial of  function  integration points  of axial  each  algorithm  5.7.  i s numerically  record  across  already  s t r e s s - s t r a i n curve  i n Fig.  and c u b i c  are constant  point  a plane  are non-linear  t h e same  displacement  forces  tion  from  define  i n the model.  matrix  quadrature  .which max'  The s t r e s s - s t r a i n and moment-curva-  have  a s shown  i t s axis  stiffness  These  curve.  not included  The  v i a the program  Studs:  both  and P  are obtained  a non-linear  moment-curvature ture  curve,  , D max  a s shown i n  reference  or tension  to Fig.  i s given  5.8,  by:  the area  i n bending  com-  F i g . 5.7  Trilinear hysteresis  loops  used  s t r e s s - st r a i n and moment - c u r v a t u r e r e l a t i o n s h i p s  for for  both studs. 00  co  A = 2r'  I  " \  whe re r u  s i n 2 e  o  (5.4)  ~ o e  r a d i u s of stud arcsin a r depth o f a x i a l compression o r t e n s i o n zone  0  2a  The bending moment l e v e r arm i s given by: 3 2y = 2r~  cos 0  (5.5)  0  3A  The u l t i m a t e  a x i a l force acting at a cross-section i  P = (irr - 2A) f su where f su  =  The u l t i m a t e  ultimate  (5.6) steel  strength ^  bending moment a c t i n g a t the cross-  section i s : M = 2Ay f  (5.7)  su  When no bending moment i s present,  the u l t i m a t e  axia  f o r c e i s g i v e n by: P , = irr f pi su  S i m i l a r l y , when no a x i a l f o r c e i s p r e s e n t  (5.8)  the u l t i -  mate bending moment i s g i v e n by: M . = 4r pi  z  f r  SU  (5.9)  Bending  Stud  F i g . 5.8 ultimate  axial  cross-section  force and  0-2  Axial  0.4  0.6  bending  0.8  under  moment.  1.0  M M Pi  Fig. 5.9  Stud i n t e r a c t i o n  diagram  From equations 3 M  (5.6)  to  (5.9)  fl  (5.10) Pi sin2 8  Q  +  6  (5.11)  Q  Pi By an  taking  i n t e r a c t i o n diagram of the  tained.  From i n s p e c t i o n  is d i f f i c u l t M/M  d i f f e r e n t v a l u e s of 8^  t o get  form shown i n F i g . 5.9  of equations  +  1 P1J P  M  interaction  M  The  approximate  a n c  ^  parabolic  (5.12)  = 1  Pi  With f u r t h e r  be  r e f e r e n c e t o F i g . 5.9,  of M and' P r e s u l t i n g i n a p o i n t  on the  seen i n Table  any  V.  combination  diagram below  the  line +  Pi i s admissible.  M M  For  a combination of M and  between the  curve, the  s t r e s s - s t r a i n and  down by  (5.13)  = 1  Pi  in a point  scaled  P  curve  g i v e s very good r e s u l t s , however, as can  straight  ob-  (5.11) i t p  f o r the  2  (5.10) and  is  a d i r e c t r e l a t i o n s h i p between / p ^  ^ f o r computation purposes.  relationship  TT/,  between 0 and  s t r a i g h t l i n e and  the  P which r e s u l t s interaction  moment-curvature diagrams  linear interpolation  to conform to the  a c t i o n diagram, as d e s c r i b e d i n Appendix  C.  are inter-  TABLE  V  COMPARISON BETWEEN APPROXIMATE FORMULA AND RIGOROUS DERIVATION OF STUD INTERACTION DIAGRAM  M M . Pl  p  V  % Difference  Rigorous equations (5.10) & (5.11)  Approximate equation (5.12)  0  1. 000  1. 000  0.00  0.1  0. 991  0. 990  -0.10  0.2  0. 963  0.960  -0.31  0.3  0. 916  0.910  -0.66  0.4  0. 851  0.840  -1.29  0.5  0.765  0. 750  -1. 96  0.6  0. 660  0.640  -3. 03  0.7  0.533  0.510  -4.32  0.8  0. 384  0.360  -6.25  0.9  0. 208  0.190  -8. 65  1.0  0.000  0.000  0.00  93  CHAPTER 6. COMPARISON OF COMPUTER ANALYSES WITH LABORATORY RESULTS  6.1  P r o p e r t i e s of S t e e l Anchor Bars and Studs C o o r d i n a t e s o f the t r i l i n e a r  a l g o r i t h m of F i g . 5.7,  which i s used f o r m o d e l l i n g both the s t r e s s - s t r a i n and moment-curvature p r o p e r t i e s o f s t e e l anchor bars and studs, are  r e c o r d e d i n Table V I .  The c o o r d i n a t e s of the t r i l i n e a r  s t r e s s - s t r a i n curves were chosen t o approximate curves measured i n t h e l a b o r a t o r y as c l o s e l y as p o s s i b l e . A momentc u r v a t u r e diagram was c a l c u l a t e d f o r each diameter of bar or stud on the u s u a l assumption t h a t s t r a i n s v a r y l i n e a r l y w i t h d i s t a n c e from the n e u t r a l a x i s o f a c r o s s - s e c t i o n . A s e t o f p r o g r e s s i v e l y l a r g e r c u r v a t u r e s was chosen, and from the s t r a i n s a t each c u r v a t u r e , a s t r e s s diagram a c r o s s the c r o s s s e c t i o n was c o n s t r u c t e d w i t h the use o f the c o r r e s p o n d i n g l a b o r a t o r y s t r e s s - s t r a i n curve.  The bending moment a t a  c r o s s - s e c t i o n was c a l c u l a t e d from t h i s s t r e s s diagram and trilinear  moment-curvature c o o r d i n a t e s were chosen t o r e p -  r e s e n t the r e s u l t i n g moment-curvature diagram as c l o s e l y as possible.  TABLE VI TRILINEAR PROPERTIES OF STEEL ANCHOR BARS AND STUDS  Diameter (in.)  1* 2  Coordinates (See F i g . 5.7)  Axial Strain  Stress (P- s. i . )  Curvature  (X! , y i )  0. 00217  65 000  0.00986  880  (x  /Y2 )  0.00251  70 000  0.03500  1 350  (x ,Ya)  0.20000  90 000  3.25000  1 600  (X! , y i )  0.00184  55 000  0.00840  750  (x  0.00215  60 000  0.03500  1 170  (x rYs)  0.20000  80 000  3.25000  1 400  (Xi  rYl)  0.00217  65 000  0.00690  1 560  (X  2  ,Y2 )  0.00251  70 000  0.02770  2 600  (X  3  ,ys)  0.20000  90 000  2.75000  3 100  (Xi , y i )  0.00217  65 000  0.00578  2 690  (X2 ,Y2 )  0.00251  70 000  0.02310  4 160  (X  0.20000  90 000  2.50000  4 960  2  3  2  2  ,y ) 2  3  5* 8  3* 4  3  rY3)  Anchor bars and machined studs Commercial studs  (in.r  1  Moment (lb. i n . )  Relevant p r o p e r t i e s  from Table VI were used f o r the  comparison between measured and c a l c u l a t e d  load-deflection  curves f o r a 1/2-in.diameter s t e e l c a n t i l e v e r bar w i t h an end l o a d , F i g . 6.1.  The measured and c a l c u l a t e d  results  show good agreement. 6.2  F r i c t i o n Specimen A computer model of the f r i c t i o n  long anchor bars i s shown i n F i g . 6.2.  specimen w i t h 6 i n . Loads from the o r i -  g i n a l computer c a l c u l a t i o n s bore no resemblance  t o the mea-  sured l o a d - d e f l e c t i o n curve f o r t h i s specimen and never exceeded 1 k i p . The secondary e f f e c t of a x i a l s h o r t e n i n g anchor bars due to bending was computer program siderably  subsequently i n c l u d e d  of the  i n the  and the c a l c u l a t e d loads were t h e r e a f t e r  improved.  con-  The c a l c u l a t e d curves shown i n F i g . 6.3  were a l l c a l c u l a t e d a t a c o n s t a n t c o e f f i c i e n t of f r i c t i o n of 0.3.  These curves show some d i s t i n c t d i f f e r e n c e s  laboratory  curves which are p r o b a b l y p r i m a r i l y due to a v a r y -  i n g c o e f f i c i e n t of f r i c t i o n  i n the l a b o r a t o r y  appears t h a t the r e a l c o e f f i c i e n t of f r i c t i o n than 0.3  a t s h o r t d e f l e c t i o n s and d u r i n g  specimen.  and an i n c r e a s e d  It  i s much h i g h e r  initial  load  and t h a t i t decreases to below 0.3 w i t h i n c r e a s e d  6. 3  from the  cycles,  deflection  number of c y c l e s .  End-Bearing Specimens The n o n - l i n e a r  described  plane s t r e s s f i n i t e element  i n Chapter 5 was  end-bearing specimens.  program  used f o r computer a n a l y s i s of the  The program was  first  t e s t e d on a  computer model of a c y c l i c a l l y - l o a d e d c o n c r e t e t e s t s p e c i -  300+  -Q  Q_ (0  o  0  0.1  0.2 End  Fig. 6.1  Bending  test  0.3 deflection  0.U  0.5  0.6  0.7  (in.)  on ^2 in. dia. bars compared  with  computer  calculations.  Member for  w i t h high axial stiffness  displacement  CO  6 at  -©  1/2 d i a . a n c h o r  \"  control  = 6  (T)  /7  .0.69  4  ©~  IP  barsfi Faceplate  CD  © Denotes  f i x e d joint  o Denotes  pinned joint  Friction -o  s i m u l a t o r s (T) ©-  — ~®——9  1  "  1 2. 1/  Fig. 6.2  C o m p u t e r m o d e l of f r i c t i o n  specimen  w i t h 6 in. long anchor  bars.  5  —0  U  3 2 Q.  o  0 -1 -2 -3 Deflection  Fig. 6.3  (in.)  Measured and computed l o a d - d e f l e c t i o n c u r v e s for friction  s p e c i m e n F2. CO CO  99 men  t e s t e d by  The  computer  trol  Karsan  where a l l t h e  effect  pressive versus  Jirsa  calculations  were done u s i n g d i s p l a c e m e n t the  a t the  strain  a distance  similar  t o t h a t used  K a r s a n and  Jirsa) The  along  Finite  is  had  edge o f  the  Hawkins ' ' 51  areas  52  and  f o r long  may  and  readily  where r e s t r a i n t  be  adapted  effects  are run  specifying  the  i n the  proved cal  t o be  three  average,  a finer  finite  11%  too  times  1/4,  0.85  element g r i d  the  to  as  model  3/8  and  3/8  finite  stiff,  by in  concrete  1/2  as  while  in.  end-  i n . specimen i n of a n a l y s i s  areas  the  y-direction  along  Work  the  by  end-bearing surrounding  of  element  control the  by  concrete  models  shown i n t h e t h e maximum  shown i n F i g . . 6.8.  d i d not  (over  ensure  under d i s p l a c e m e n t  too  com-  results.  T h i s type  o f F i g . 6.7, high,  average  of the  t o compact  faceplate. • A l l three  l o a d - d e f l e c t i o n curve  were, on on  displacements  about  by  i m p o s e d by  E a c h a n a l y s i s was  simulate  results  i n F i g . 6.8.  concrete.  step under the  value  narrow end-bearing  as d e p i c t e d  to  centre-line  original  laboratory  analyses.  con-  model  The of  6.4.  a c t u a l experiment  to t h a t f o r the  f o r the  concrete, 5 3  stress  element models of the  were u s e d only  the  t o be m u l t i p l i e d  specimens, s i m i l a r  valid  i n the  maximum n o r m a l i s e d  a g r e e m e n t between computed  6.6,  the v e r t i c a l  i s compared w i t h  m o d e l o f F i g . 5.3  Fig.  the  y-direction  horizontal centre-line  average  bearing  edge o f  o f a heavy t e s t i n g machine p l a t e n .  stress  6.5.  top  same amount i n t h e  the  Fig.  w h i c h i s shown i n F i g .  3 8  nodes a l o n g  were d i s p l a c e d t h e the  and  improve r e s u l t s  typiloads  A single appre-  run  Nodes  along  top  edge a l l have s a m e y - di s p l a c e m e n t .  Reinfor cement s i m u l a t e d with line  elements.  T h i c k n e s s = 3 in. f ' = 5 000 p.s.i. c  Fig. 6.4  F i n i t e e l e m e n t grid of one quadrant  of 3ft  concrete  test  specimen  u s e d by K a r s a n & J i r s a :  1  1  J  T e s t A C 3 - 1 0 , f = 5 010 p.s.i. , Ref. 38 c  0.  0.5 Normalised  1.0 strain  1.5  2.0  S= /r e  o  Fig. 6.5  C o m p a r i s o n of computed stress-  s t r a i n curve w i t h t e s t by  Karsan & J i r s a .  102  Fig. 6.6 specimen modelled  3/gjp  e n c  by f i n i t e  j  bearing elements.  103  S~  H  •01  .02  Faceplate  Fig.6.7 for %  1  -03  -OU  d e f I ect ion (in.)  Load - d e f l e c t i o n in. end  1  bearing  curve  specimens.  h  -05  104  Q £  -o o  £ E X  CO  £ -a o  —*  u fd u_  0  0.1  0.2  Faceplate  Fig. 6.8 maximum  0.3 thickness  0.4  0.5  0.6  (in.)  Comparison of c a l c u l a t e d and measured loads  for  end  bearing  specimens.  ciably,  l e a d i n g t o the c o n c l u s i o n  t h a t the program d i d not  model the shear-type f a i l u r e of the concrete p l a t e too w e l l .  under the f a c e -  The computer models do appear t o c o n f i r m ,  however, t h a t the r e l a t i o n s h i p between f a c e p l a t e and  thickness  f a c t o r e d maximum l o a d i s l i n e a r , F i g . 6.8.  A b e t t e r i d e a l i s a t i o n of the end-bearing specimen was  the plane frame computer model of F i g . 6.9.  viously described bearing  f r i c t i o n model was combined w i t h an end-  s p r i n g modelled by the l o a d - d e f l e c t i o n curves of  F i g . 5.6. the  The p r e -  The maximum l o a d o f the s p r i n g was taken from  laboratory  curve o f F i g . 6.8, and the d e f l e c t i o n at maxi-  mum l o a d was taken t o be the average o f those from the l a b o r a t o r y curves o f F i g s . 4.5 t o 4.7, namely 0.025 i n .  Similar-  l y the d e f l e c t i o n a t f a i l u r e o f t h i s s p r i n g was 0.09 i n . The l o a d - d e f l e c t i o n curve of t h i s model bears a r e a s o n a b l e resemblance t o the l a b o r a t o r y curves as shown i n F i g . 6.10.  6. 4  S i n g l e Studs i n Concrete Each s i n g l e stud i n concrete  on Winkler s p r i n g s , as d e s c r i b e d  was modelled by a beam  i n Chapter 5.  The f i n i t e  element g r i d o f F i g . 6.11 was used t o determine the l o a d d e f l e c t i o n c h a r a c t e r i s t i c s , F i g . 6.12, of a s l i c e of concrete of u n i t t h i c k n e s s .  Note t h a t the Y-load recorded  6.12 i s f o r only one h a l f o f the s l i c e .  This analysis  c a t e s t h a t t h e maximum l o a d c a p a c i t y of concrete stud i s :  i n Fig.  under a  indi-  106  Member for  w i t h high axial stiffness  displacement  CO  control  (T^)-  6 at 1 = y  -©  S  -0.69*  ©-  1" '/2 d i a . a n c h o r  r~ bars( 5  V  Faceplate  to i  ©Denotes  f i x e d joint  o Denotes  pinned joint  Friction  simulator  End-bearing  Fig.6.9  (T)  (^)  concrete  spring(4a)-  Computer model of end bearing  specimen.  D e f I e ct ion (in.)  Fig. 6.10 deflection  Measured and c o m p u t e d curves  for  end-bearing  specimens  load-  B3 & B4.  108  AU  support  nodes  f i x e d against  |Y  rotation.  T h i c k n e s s =1 in. fc = 4 800  Stud by  represented  linear  elastic  Typical  — elements  support  for  nodes  on  centre-line.  Fig.6.11 of  —  Finite  concrete  element  slice  grid  containing  of  one  stud  half shank.  p.s.i.  Vertical crete side  crack on  of  in c o n -  compression s ud  °\  shank  ©  Stud  L 0 deflection  Fig. 6.12 concrete  shank  cracks  away  from c o n c r e t e  on  tension  1  -001 of  side.  +  j  -002 stud  centre  -003 relative  L o a d - d e f l e c t ion slice  -004  containing  to  point A  curve  for  stud  shank.  -005 (in.)  where R  =  maximum r e a c t i o n p e r u n i t l e n g t h of stud  d  =  stud  f^ =  diameter.  concrete c y l i n d e r strength.  and t h i s maximum r e a c t i o n occurs a t a d e f l e c t i o n o f 0.004 i n .  These p r o p e r t i e s were used t o g e t h e r w i t h the  l o a d - d e f l e c t i o n curves of F i g . 5.6 t o r e p r e s e n t the concrete s p r i n g s used i n t h e plane c o n c r e t e , F i g . 6.13.  frame computer model of a s t u d i n  I n i t i a l computer runs r e s u l t e d i n c a l -  c u l a t e d l o a d - d e f l e c t i o n curves which were t o o s t i f f  and which  had maximum loads which were f a r t o o low i n comparison w i t h the l a b o r a t o r y curves.  By t r i a l  and e r r o r , i t was d e t e r -  mined t h a t the r e a c t i o n p e r u n i t l e n g t h o f stud should be increased to R = 5.0 d f ^  (6.2)  w i t h a d e f l e c t i o n a t maximum l o a d o f .05 i n . and the d e f l e c t i o n a t complete f a i l u r e of the s p r i n g was s e t a t .25 i n .  The much h i g h e r v a l u e s are not t o o s u r p r i s i n g  because  i t must be remembered t h a t the f i n i t e element a n a l y s i s i s only t r u e f o r a plane s t r e s s c o n d i t i o n .  The concrete immed-  i a t e l y under a stud can be h e a v i l y c o n f i n e d by surrounding c o n c r e t e and i s t h e r e f o r e probably  under a t r i a x i a l  compres-  Member for  Concrete  with  high a x i a l  displacement  c o n t r o l (T)  springs Stud  shank(?)-  Faceplate © D e n o t es  fixed  o Denotes  p i n n e d joint  F i g . 6.13  stiffness  Computer  (T)-  joint  model  of s t u d  in c o n c r e t e .  112 s i v e s t r e s s c o n d i t i o n which can r e s u l t i n a f a r higher load c a p a c i t y than the plane s t r e s s c o n d i t i o n would i n d i c a t e , as mentioned  i n Chapter 5.  Equation  (6.2) and the d e f l e c t i o n s o f .05 in.and  .25 in. were used i n a l l of the computations  of the load-de-  f l e c t i o n curves o f F i g s . 6.14, 6.16, 6.18 and 6.20. measured and c a l c u l a t e d  The  l o a d - d e f l e c t i o n curves o f the studs  i n c o n c r e t e compare very f a v o u r a b l y , as do most of the def l e c t e d shapes  of F i g s . 6.15, 6.17, 6.19 and 6.21. T h i s lends  credence t o the shear f o r c e , bending moment and r e a c t i o n  dia-  grams which were e x t r a c t e d from the c a l c u l a t i o n s a t a p o i n t i n each l o a d - d e f l e c t i o n curve w e l l a f t e r f i r s t y i e l d . maximum p o i n t s i n each bending moment diagram two p l a s t i c hinges have formed  indicate  The that  i n the stud shank, one a t the  f a c e p l a t e and one a t some d i s t a n c e i n t o the c o n c r e t e . shear f o r c e diagram passes through zero a t the second  The plastic  hinge and most o f the c o n c r e t e r e a c t i o n t o the stud o c c u r s between the two p l a s t i c h i n g e s .  6. 5  Complete Headed Stud For comparison  a complete  Connection  of a plane frame computer model w i t h  c o n n e c t i o n , the l a b o r a t o r y r e s u l t s from c o n n e c t i o n  E2, i n v e s t i g a t e d by S p e n c e r ^ were chosen because r e c e n t experiments have been monitored  in his  w i t h c o n n e c t i o n s , s t r a i n s i n the studs by means of s t r a i n gauges.  A p a i r of  these gauges were s i t u a t e d near the head o f the stud and  6  +  3  +  CL  T3 O  0  0.1  0.15  D e f I e c t ion (i n.)  Fig. 6.14  Measured  and  c o m p u t e d l o a d - d e f l e c t i o n c u r v e s for stud  S1. CO  114  Distance  Fig.6.15  from  concrete  0  0-5  Distance  from  face  1-0 back  (in.)  1.5 of  2-0  f a c e p l a t e (in.)  D e f l e c t e d s h a p e comparison  computed forces on s t u d  S1 w e l l a f t e r f i r s t  and yield.  10  +  0  0.05  0-1  0-15  Deflection  Fig. 6.16  Measured  and c o m p u t e d  0-2  0-25  (in.)  load-deflection  c u r v e s for s t u d S2. M  116  Distance  from  concrete  Distance  F i g . 6.17  from  back  Deflected  c o m p u t e d forces on s t u d  face  of  shape  (in.)  f a c e p l a t e (in.)  comparison  S2 w e l l after f i r s t  and  yield.  L o a d (kip.)  /LIT  Distance  from  concrete  0  0.5  Distance  F i g . 6.19  from  f a c e (in.)  1.0 back  of  1-5  f a c e p l a t e (in.)  D e f l e c t e d shape  c o m p u t e d forces on s t u d  2.0  c o m p a r i s o n and  S3 well a f t e r f i r s t  yield.  CL  _£  O  0.1 D e f I e c t i o n (in.)  F i g . 6.20  Measured  and c o m p u t e d l o a d - d e f l e c t i o n  curves  for studs S4, S5 & S6.  Distance  from  concrete  Distance  from  Fig. 6.21  D e f l e c t e d shape  forces on  studs  S4,  face  (in.)  concrete  face  comparison and  S5 & S6  well after  (in.)  computed first  yield.  121 another p a i r i n the r e g i o n from the f a c e p l a t e . of the stud  of the p l a s t i c hinge f u r t h e s t away  One o f each p a i r was cemented on the top  shank and the other on the bottom.  D e t a i l s o f specimen E2 and i t s a s s o c i a t e d  computer model  appear i n F i g . 6.22. Measured and computed l o a d - d e f l e c t i o n are  curves  shown i n F i g . 6.23, t o g e t h e r w i t h the c a l c u l a t e d i n d i v i d u a l  loads c a r r i e d by studs, f a c e p l a t e  f r i c t i o n and f a c e p l a t e end-  bearing .  The  laboratory  t e s t and the computer c a l c u l a t i o n s  indicate  t h a t both the upper and lower studs y i e l d s u b s t a n t i a l l y i n a x i a l tension,  as i n d i c a t e d i n F i g . 6.24. In the l a b o r a t o r y  test this  t e n s i l e y i e l d i n g occurs s i m u l t a n e o u s l y w i t h the peak load on the c o n n e c t i o n , whereas the c a l c u l a t i o n s i n d i c a t e t h a t  axial tensile  y i e l d i n g occurs a f t e r the peak l o a d i s reached and a t a much greater  d e f l e c t i o n o f the f a c e p l a t e .  There are no  laboratory  measurements nor any r e s u l t s from the computer c a l c u l a t i o n s which explain  t h i s d i f f e r e n c e . One p o s s i b i l i t y  i s t h a t i n the a c t u a l  specimen the c o n c r e t e being compressed under each stud c a r r y very h i g h compressive s t r e s s e s  appears t o  because i t i s c o n f i n e d  by  s u r r o u n d i n g c o n c r e t e . Expansion of the c o n c r e t e , i n a d i r e c t i o n p a r a l l e l t o the studs, i s p r o b a b l y r e s t r a i n e d by the f a c e p l a t e , thus a p p l y i n g  t e n s i l e f o r c e s t o the studs, which may e x p l a i n the  h i g h a x i a l s t r a i n s i n the studs a t an e a r l y stage i n the t e s t . On the  other hand, the computer model does not c o n t a i n  and  i t i s a x i a l shortening  this effect  of the stud due t o bending alone  produces the a x i a l t e n s i l e s t r a i n s at a l a t e r stage.  that  End  b e a r i n g s p r i n g (4a  1/2'n. dia. x 6 i n . s t u d s  .IV/ Friction s i m u l a t o r ^ ) — ^  2 x 2 x 7 ^ in. a n g l e —  CXJ  oo  f = 6 050 p.s.i. r  Faceplate(T^  CNJ  Displacement ©Denotes Details  c o n t r o l member(T^)  f i x e d joint. Computer  of s p e c i m e n E2  oDenotes  pinned joint  model  45  Fig. 6.22  Stud c o n n e c t i o n E2, due to S p e n c e r ,  s i m u l a t e d by  computer  model. to to  30  25  20  15  10  0 0  0.05  0-1 Deflection  6.23  Measured and  computed  0-15  0.2  (in.)  l o a d - d e f l e c t i o n curves  for c o n n e c t i o n  _45  E2  125 The in  important p o i n t t o note i s t h a t both studs are  a x i a l t e n s i o n when the maximum shear load on the connec-  t i o n i s reached.  T h i s i n d i c a t e s t h a t the common p r a c t i c e  of c o n s i d e r i n g moment e q u i l i b r i u m about some p o i n t i n the connection,  w h i l e i g n o r i n g the h o r i z o n t a l i n t e r a c t i o n f o r c e s  between f a c e p l a t e and c o n c r e t e ,  i s meaningless.  Such a c a l -  c u l a t i o n would i n d i c a t e t h a t the t o p stud i n F i g . 6.22 i s i n t e n s i o n and the bottom stud  i n e q u a l and o p p o s i t e  While the studs are y i e l d i n g i n t e n s i o n ,  compression.  i t must be  remembered t h a t p l a s t i c hinges have formed i n each stud as w e l l .  The formation  shank  of each p l a s t i c hinge f u r t h e s t away,  from the f a c e p l a t e i s confirmed by the s t r a i n s measured f o r connection  E2.  In the computer c a l c u l a t i o n s the stud  have reached the y i e l d  s u r f a c e o f F i g . 5.9.  forces  The computer  c a l c u l a t i o n s i n d i c a t e t h a t the t e n s i l e f o r c e s i n the studs are about 30% of t h e i r f u l l y i e l d value present  w i t h no bending moment  and the moments a t the p l a s t i c hinges are t h e r e f o r e  about 90% o f the f u l l p l a s t i c moment with" no a x i a l f o r c e s sent,  i n accordance w i t h F i g . 5.9.  pre-  126  CHAPTER 7. SIMPLIFIED ANALYTICAL MODELS  7.1  U l t i m a t e Shear Load o f a Stud i n Concrete Common f e a t u r e s of the computed shear f o r c e ,  bending  moment and r e a c t i o n diagrams f o r s i n g l e studs i n c o n c r e t e i n the p r e v i o u s chapter suggest tion  the simple a n a l y t i c a l  o f the f o r c e s on a stud shown i n F i g . 7.1.  approximaThe r e a c -  t i o n diagrams o f F i g s . 6.15, 6.17, 6.19 and 6.21 can r e a d i l y be approximated  by a r e c t a n g u l a r diagram, between the two  p l a s t i c hinges, o f h e i g h t k f ^ , where k i s a  constant.Measure-  ments o f the areas o f the r e a c t i o n diagrams o f the p r e v i o u s c h a p t e r , between the two p l a s t i c between 4.3 0 and 4.50.  hinges, y i e l d e d v a l u e s f o r k  Thus t h e t o t a l r e a c t i o n a p p l i e d by  the c o n c r e t e i s g i v e n by: C = 4.4 f'dL c where f ^ =  (7.1)  concrete c y l i n d e r strength  d  =  stud shank diameter  L  =  d i s t a n c e between p l a s t i c  hinges.  127 For v e r t i c a l e q u i l i b r i u m o f the s h o r t l e n g t h o f stud between the p l a s t i c  hinges V  where V  u  = C  (7.2)  = u l t i m a t e shear l o a d on stud,  u  For e q u i l i b r i u m of moments about the p l a s t i c hinge at A V L = u  + 2M , pl  2  where M ^ = F u l l y p l a s t i c moment o f stud ^ shank which may be c a l c u l a t e d from equation (5.9). i.e.  From equations  V  r 2  2  i  M  = ^ + —-P-±.  u  (7.1),  (7.3)  L  (7.2) and (7.3) M <->  = /rriT^d  L  7  4  c Once L i s c a l c u l a t e d , tions  (7.1) and (7.2).  i s easily  found from equa-  The d e s i g n u l t i m a t e shear l o a d , V^,  would be g i v e n by V  u  = d>V  u  where cj) = customary c a p a c i t y r e d u c t i o n factor. For studs  subjected  t o combined shear and t e n s i l e  loads the f u l l y p l a s t i c moment would be reduced a c c o r d i n g t o the i n t e r a c t i o n diagram o f F i g . 5.9.  IVu M  pi  A  i  .7.1 a stud  n  c n  M  u  u  Pi  A p p r o x i m a t i o n of f o r c e s in c o n c r e t e  at m a x i m u m  acting load.  //////  F i g . 7.2 deflection  Approximate model  of a  load-  connection.  where P P  M  7.2  = T e n s i l e l o a d on s t u d . u s  = Ultimate t e n s i l e capacity of stud w i t h no bending moment present. = Reduced p l a s t i c moment of stud shank.  U l t i m a t e End-bearing  C a p a c i t y of F a c e p l a t e  For a f a c e p l a t e w i t h t h i c k n e s s much l e s s than the f a c e p l a t e width, the s t r a i g h t l i n e of F i g . 6.8 may be used. i.e. where  P . = 1.2A, f ub f c  (7.7)  = U l t i m a t e end-bearing c a p a c i t y of f a c e p l a t e . = Bearing area o f f a c e p l a t e .  The  d e s i g n u l t i m a t e b e a r i n g c a p a c i t y o f the f a c e p l a t e  would be g i v e n by  P  7. 3  ub  * ub  =  P  Comparison o f Proposed  ( 7  '  8 )  Theory and Current Shear  F r i c t i o n Theory w i t h L a b o r a t o r y Measurements. Comparisons are made i n Table V I I with f o u r t e e n conn e c t i o n s , a l l o f which are o f the type shown i n F i g . 1.2. The studs o f the c o n n e c t i o n s of s e r i e s A were p a r a l l e l t o the s i d e f a c e s o f the concrete panels i n which they were c a s t ( F i g . 2.4) and the remainder sides.  were at 45 degrees  For comparison purposes,  t o the p a n e l  no c a p a c i t y r e d u c t i o n was  TABLE V I I COMPARISON OF CALCULATED AND MEASURED ULTIMATE STRENGTHS OF CONNECTIONS 6 in.dia. Concrete Cylinder Strength (p. s. i . )  Faceplate Angle Size (in. )  Al  4600  A2  (in.)  Ultimate Strength Shear F r i c t . Eqn. 2.3 (kip.)  Ultimate Strength Proposed Theory (kip.)  4x3x3/8  5/8  33.2  35. 6  4600  4x3x3/8  5/8  33.2  35. 6  A3  4600  4x3x3/8  5/8  33.2  Bl  4600  3x3x3/8  5/8  B2  4600  3x2x3/8  B3  4600  CI  Specimen  Stud Diameter  Measured Maximum Load Up  Down  (kip.) 35.4  (kip.) *  31. 5  30.2  35.6  ' 29.3  27.0  33.2  33.6  31. 8  28.0  5/8  33.2  31. 5  30.0  32. 0  3x2x3/8  5/8  33.2  31.5  23.4**  24.0**  6450  3x3x3/8  5/8  33.2  42. 0  48.5  44.0  C2  6450  3x3x3/8  5/8  33.2  42.0  44.4  39.8  C3  6450  3x3x3/8  5/8  33.2  42. 0  41.9  44.8  C4  6450  3x3x3/8  5/8  33.2  42.0  41.3  C5  6450  3x3x3/8  5/8  33.2  42. 0  38.2 *  C6  6450  3x3x3/8  5/8  33.2  42.0  40.3  42.5  El  5850  2x2x1/4  1/2  21.2  23.4  24.4  E2  5850  2x2x1/4  1/2  21.2  23. 4  29.4  24.9 *  A & B S e r i e s from Ref. 28. C & E S e r i e s due t o Spencer, Ref. 45 * Monotonic Test ** Test R e s u l t s U n r e l i a b l e because of Connection F a b r i c a t i o n E r r o r  43.8  131 c o n s i d e r e d i n the u l t i m a t e load c a l c u l a t i o n s ;  t h a t i s , the  c a p a c i t y r e d u c t i o n f a c t o r , c|>, was taken t o have the value of 1.0 throughout.  In a d d i t i o n , each c a l c u l a t e d u l t i m a t e  s t r e n g t h i s the maximum p o s s i b l e , because no r e d u c t i o n due to  the presence  of p o s s i b l e t e n s i l e f o r c e s i n i n d i v i d u a l  studs i s c o n s i d e r e d .  I t appears 5000 p . s . i . ,  t h a t f o r v a l u e s of  between 4000 and  shear f r i c t i o n e q u a t i o n 2.3 g i v e s a reasonable  p r e d i c t i o n of the maximum loads c a r r i e d by the c o n n e c t i o n s , while f o r v a l u e s o f f* above 5000 p . s . i . , c v a l u e s become c o n s e r v a t i v e .  I t i s also l i k e l y  v a l u e s o f f ^ below 4000 p . s . i . , prove to  the shear  shear f r i c t i o n  friction  that, f o r theory  will  t o be u n c o n s e r v a t i v e because e q u a t i o n 2.3 i s not l i n k e d  the c o n c r e t e s t r e n g t h f ^ .  In i n s t a n c e s where the end of  the f a c e p l a t e does n o t bear on c o n c r e t e , shear f r i c t i o n t i o n 2.3 may be dangerously  unconservative.  End-bearing  c a p a c i t i e s c a l c u l a t e d from e q u a t i o n 7.7 proved than 4*0% o f the t o t a l  equa-  t o be more  u l t i m a t e l o a d f o r each of the f o u r t e e n  c o n n e c t i o n s under c o n s i d e r a t i o n . not take f a c e p l a t e end-bearing  Shear f r i c t i o n  i n t o account  theory does  and c o u l d thus  o v e r - p r e d i c t u l t i m a t e loads by up t o 40%, w h i l e the loads c u l a t e d by the theory proposed  above would be s u i t a b l y  C o n v e r s e l y , i n cases where f a c e p l a t e end-bearing  cal-  reduced.  areas are  much i n c r e a s e d over the areas o f f a c e p l a t e angles of the conn e c t i o n s under c o n s i d e r a t i o n here, the shear f r i c t i o n may be o v e r l y c o n s e r v a t i v e .  theory  T h i s has been shown by Spencer *  who has measured maximum loads of up t o 50 k i p . f o r connec-  1  5  132 t i o n s w i t h two  1/2-in. diameter  studs and a f a c e p l a t e end-  b e a r i n g area of 2 i n . x 2 i n . i n concrete w i t h a c y l i n d e r s t r e n g t h of 5850 p . s . i . end-bearing will rectify  7.4  E q u a t i o n 7.7  may  not be v a l i d  areas of such p r o p o r t i o n s , but c u r r e n t r e s e a r c h this  deficiency  shortly.  Friction  between F a c e p l a t e and  Friction  f o r c e s between f a c e p l a t e and  been omitted i n the equations proposed the l a b o r a t o r y experiments  Concrete concrete have  above because both  and the computer c a l c u l a t i o n s i n -  d i c a t e d t h a t these were s m a l l and disappeared r a p i d l y cyclic  loading.  the v a l i d i t y  of shear f r i c t i o n e q u a t i o n  Monotonic t e s t s , or c y c l i c t e s t s with a few c y c l e s of  l a r g e displacements may h i g h e r than average  7. 5  reflect  some f a c e p l a t e f r i c t i o n w i t h  maximum l o a d s .  L o a d - d e f l e c t i o n Model f o r a C o n t i n u i n g along the l i n e s  Fig.  7.1,  Connection of the approximation  a complete connection can be approximated  model of F i g . 7.2 of  with  T h i s i s confirmed by the v a l u e s i n Table  VII and a l s o negates 2.3.  for  the type used  which i s comprised  by  of the  of f o u r c o n c r e t e s p r i n g s  i n p r e v i o u s chapters and a c a n t i l e v e r beam  w i t h t r i l i n e a r l o a d - d e f l e c t i o n p r o p e r t i e s as i n F i g . 5.7.  Two  s p r i n g s model the c o n c r e t e around the studs and two model the c o n c r e t e on which the f a c e p l a t e bears. 7.2,  With r e f e r e n c e t o F i g .  the u l t i m a t e shear f o r c e on a c o n n e c t i o n i s g i v e n by S = V + R  2  (7.9)  where V =  U l t i m a t e shear  R =  f o r c e from a l l studs.  U l t i m a t e end-bearing equation (7.7) .  2  From e q u a t i o n  c a p a c i t y from  (7.3) 2M V = n(| +  where n =  (7.10)  t o t a l number of studs i n connection'. i.e. V = R  where R^ =  1  +  R  3  = 2.2f^dLn from e q u a t i o n  = c o n t r i b u t i o n from c o n c r e t e  (7.1) reaction.  L  = d i s t a n c e between p l a s t i c e q u a t i o n (7.4).  hinges,  R_  = ^ p l = c o n t r i b u t i o n from stud L n  S u b s t i t u t i n g i n t o equation S = R  ±  + R  2  bending.  (7.9) + R  A t o t a l l o a d - d e f l e c t i o n curve  (7.11)  3  (R^ versus A) over  d i s t a n c e between p l a s t i c hinges, L, can be c a l c u l a t e d  the  f o r the  studs u s i n g a moment-curvature diagram and c o n v e n t i o n a l Moment Area Theorems.  T h i s curve may  l i n e a r a l g o r i t h m of F i g . 5.7. ducing the model of F i g . 7.2 the v a l u e s of R^  and  R  2  curve, and i s reproduced  be approximated  by the t r i -  A computer s u b - r o u t i n e r e p r o has been w r i t t e n , which r e q u i r e s  and the c o o r d i n a t e s of the R^ versus A i n Appendix B.  The p o s i t i o n of the  134 c o n c r e t e r e a c t i o n f o r c e , \ C , s h i f t s from j u s t behind f a c e - p l a t e at s m a l l loads t o L/2 at maximum l o a d and beyond.  the  away from the f a c e p l a t e  The  s u b r o u t i n e i n c l u d e s de-  g r a d a t i o n of the l o a d - c a r r y i n g c a p a c i t y of the stud by making use of e q u a t i o n 4.1,  and a r e c o r d of  cumulative  p l a s t i c hinge r o t a t i o n i s s t o r e d f o r p r e d i c t i o n of stud fracture..  S e v e r a l l o a d - d e f l e c t i o n curves were c a l c u l a t e d u s i n g t h i s s u b r o u t i n e and are reproduced t i o n curve f o r a s i n g l e 5/8 Fig.  7.3  below. The l o a d - d e f l e c -  i n . diameter  compares f a v o u r a b l y w i t h those of F i g . 6.20,  the curve f o r c o n n e c t i o n E2, F i g . 7.4, to t h a t of F i g . 6.23.  The  t i o n loops of stud bending and may The  i s reasonably  and similar  c a l c u l a t e d degrading l o a d - d e f l e c specimen SB5  appear i n F i g .  be compared w i t h the measured loops i n F i g .  7.5  4.26.  studs i n the a c t u a l specimen f r a c t u r e d towards the end  the t e n t h c y c l e , whereas a cumulative 3.5  stud i n c o n c r e t e of  r o t a t i o n of more than  i n d i c a t e d stud f r a c t u r e f o r the c a l c u l a t e d  b e g i n n i n g of the e l e v e n t h c y c l e . loops of c o n n e c t i o n A3  ( F i g . 7.6)  of  loops at the  Calculated load-deflection have the same.maximum d i s -  placements i n each c y c l e as do the measured loops of F i g . 2.7. The  studs i n t h i s c o n n e c t i o n must have had d i f f e r e n t  t i e s from those used  i n the stud bending  studs w i t h s t o o d a maximum cumulative 11.0  before f r a c t u r e .  proper-  t e s t s , because  these  r o t a t i o n of more than  The measured and c a l c u l a t e d  of c o n n e c t i o n E l are shown i n F i g s . 7.7  and  7.8  loops  respectively,  and both s e t s have the same maximum displacements  i n each  cycle.  Although  the measured and c a l c u l a t e d  loops have  very s i m i l a r maximum l o a d s , the e n c l o s e d area of each culated  loop i s l a r g e r than i t s measured c o u n t e r p a r t , givinc  a h i g h e r energy d i s s i p a t i o n per c y c l e . routine  cal-  indicated  whereas f r a c t u r e fifteenth.  The  computer sub-  stud f r a c t u r e i n the s i x t e e n t h  cycle,  a c t u a l l y o c c u r r e d a t the beginning of the  Deflection  Fig.7.3  Calculated  load-deflection  (in.)  curve  for studs S4, S5 & S6.  O  0.2  0.15  0.1  0.05 Deflection  Fig.7.5  0  0.05  0-15  0.2  (in.)  Calculated load-deflection loops  for s t u d bending s p e c i m e n  SB5. i—  1  CO CO  Fig.7.6  Calculated  load-deflection loops - connection  A3.  CHAPTER 8. CONCLUSION  8.1  Confirmation of I n i t i a l  Assumptions  T h i s i n v e s t i g a t i o n was i n i t i a t e d on the s u p p o s i t i o n t h a t a c o n n e c t i o n t r a n s f e r s a p p l i e d shear rounding  loads t o the s u r -  c o n c r e t e by t h r e e d i s t i n c t mechanisms, namely:  f r i c t i o n between f a c e p l a t e and c o n c r e t e ; f a c e p l a t e on c o n c r e t e ; concrete.  b e a r i n g o f end o f  and i n t e r a c t i o n between studs and  T e s t s on l a b o r a t o r y models designed  i n d i v i d u a l aspects o f these mechanisms confirmed  to i s o l a t e that a l l  three d i d e x i s t .  From the l a b o r a t o r y models i t was found normal t o the f a c e p l a t e , necessary  for friction  that forces f o r c e s be-  tween f a c e p l a t e and c o n c r e t e t o e x i s t , were caused a x i a l s h o r t e n i n g of the studs d u r i n g bending.  mainly by  Measurements o f  s t r a i n s o f the studs o f a r e g u l a r c o n n e c t i o n i n d i c a t e d t h a t some other mechanism,possibly expansion  of the concrete  around  each s t u d i n a d i r e c t i o n p a r a l l e l t o the stud a x i s , enhanced the development of f o r c e s normal t o the f a c e p l a t e . From l a b o r a  t o r y measurements i t was decreased  concluded  that f r i c t i o n forces  r a p i d l y under c y c l e d l o a d i n g , and were n e g l i -  g i b l e compared to the remaining  forces i n a connection.  On the other hand, d i f f e r e n c e s between c a l c u l a t e d and  ob-  served loads from a few monotonic t e s t s i n d i c a t e d t h a t friction  f o r c e s c o u l d i n c r e a s e the t o t a l u l t i m a t e l o a d of  a c o n n e c t i o n by a s m a l l amount.  Bearing of the end of the f a c e p l a t e on a d j a c e n t c r e t e was  con-  shown to c o n t r i b u t e s i g n i f i c a n t l y t o the t o t a l u l -  timate l o a d of a c o n n e c t i o n and an e q u a t i o n f o r c a l c u l a t i n g t h i s c o n t r i b u t i o n was  derived empirically.  t e r a c t i o n between studs and  Similarly, i n -  surrounding c o n c r e t e , by b e a r i n g  of the studs on concrete and bending  of the studs, was  shown  to make a s i g n i f i c a n t c o n t r i b u t i o n t o the t o t a l l o a d c a p a c i t y of a c o n n e c t i o n . f u l l y modelled  Stud and  concrete i n t e r a c t i o n was  on a computer by means of a beam on  s p r i n g s , l e a d i n g to the development of a simple  success-  Winkler  analytical  model f o r the p r e d i c t i o n of the u l t i m a t e shear l o a d c a p a c i t y of a stud i n c o n c r e t e .  The model was  extended  to a simple  computer model which p r e d i c t e d the l o a d - d e f l e c t i o n  behaviour  of i n d i v i d u a l studs i n c o n c r e t e or complete connections reasonable  with  success under both monotonic and c y c l i c c o n d i t i o n s .  E x i s t e n c e of the three mechanisms whereby a connect i o n t r a n s f e r s shear f o r c e s to the surrounding concrete t r a d i c t s the shear f r i c t i o n concept,  con-  c u r r e n t l y employed i n  144 the d e s i g n of c o n n e c t i o n s , which assumes t h a t a l l of the a p p l i e d shear f o r c e s are t r a n s m i t t e d to the concrete a c r o s s some s o r t of s h e a r i n g plane, the normal f o r c e s being supp l i e d by r e i n f o r c e m e n t normal t o t h i s p l a n e . equations developed  Analytical  from the i n v e s t i g a t i o n i n d i c a t e d t h a t  the s t r e n g t h of a connection i s d i r e c t l y dependent upon the s t r e n g t h of the concrete i n which i t i s c a s t , as opposed t o the e x p r e s s i o n f o r shear f r i c t i o n , e q u a t i o n 2.3,  which does  not c o n t a i n c o n c r e t e s t r e n g t h as a v a r i a b l e .  8. 2  Future The  Research  research described i n t h i s d i s s e r t a t i o n  forms  but a s m a l l p a r t of the o v e r a l l o b j e c t i v e of p r e d i c t i n g behaviour  the  of p r e c a s t c o n c r e t e p a n e l b u i l d i n g s under e a r t h -  quake l o a d s , and r e s e a r c h i n t h i s f i e l d  i s only beginning.  With the a i d of the shear l o a d versus d e f l e c t i o n a l g o r i t h m developed  above, i t may  now  be p o s s i b l e t o analyse a s e t of  panels assembled i n one plane w i t h headed stud c o n n e c t i o n s , under earthquake  l o a d i n g i n the same plane.  Before  analy-  ses of a t h r e e - d i m e n s i o n a l assembly of p a n e l s can be undertaken, the i n t e r a c t i o n between the shear l o a d c a p a c i t y of a c o n n e c t i o n and d i r e c t loads t r a n s m i t t e d by the c o n n e c t i o n ' must be i n v e s t i g a t e d .  Connections  s i m i l a r to ones w i t h  headed studs, and i n p a r t i c u l a r , connections which have deformed bars at 45° to the f a c e p l a t e i n s t e a d of studs,  may  p o s s i b l y be i n v e s t i g a t e d i n a s i m i l a r manner to the r e s e a r c h c a r r i e d out here.  The  d i f f e r e n c e s between the measured  and  calculated  axial tensile strains  E2, d i s c u s s e d i n Chapter  6,  of the development of a x i a l nection.  of the studs i n c o n n e c t i o n  warrant f u r t h e r  investigation  s t r a i n s i n the studs of a con-  146  BIBLIOGRAPHY  1.  PCI Design Handbook. P r e c a s t and P r e s t r e s s e d Concrete, P r e s t r e s s e d Concrete I n s t i t u t e , Chicago, I l l i n o i s , 1971.  2.  Zeck, U.I., " J o i n t s i n Large Panel P r e c a s t Concrete S t r u c t u r e s " , P u b l i c a t i o n No. R76-16, Department of C i v i l E n g i n e e r i n g , Massachusetts I n s t i t u t e of Technology, Camb r i d g e , Massachusetts, Jan. 1976.  3.  Embedment P r o p e r t i e s of Headed Studs, Nelson Stud Welding Company, Design Data 10, 1975.  4.  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Magazine of Concrete Research, V o l . 22, No. 71, June 1970, pp.87-98.  APPENDIX A TRIANGULAR AND QUADRILATERAL PLANE STRESS/STRAIN FINITE ELEMENTS  The obtained  d i s p l a c e m e n t f u n c t i o n s f o r these  by a p p l y i n g  constraints to existing  elements a r e displacement  f u n c t i o n s o f a complete c u b i c t r i a n g l e and a complete c u b i c r e c t a n g l e i n much the same way as proposed by  Tinawi  4 0  for a t r i a n g l e .  A.1  Plane S t r e s s / S t r a i n T r i a n g l e With r e f e r e n c e  t o the complete c u b i c  triangle  3 7  shown i n F i g . A . l , the d i s p l a c e m e n t f u n c t i o n i n terms o f area c o o r d i n a t e s i s : ~(3£i-l) ( 3 ? - 2 ) ? / 2 " 1  r  3  (35 -l) ( 3 £ - 2 K / 2  r  5  9£i?  r  7  r  9  9 C 2 C 3 (352-D/2  r  11  9? C  3  (3? -l)/2  r  13  95i?  3  (3C3-D/2  r  15  (3SX-U/2  r  17  r  19  2  3  3  =  r!  (3£ -l) ( 3 £ - 2 K / 2 2  u  1  9?iC  (3? -l)/2  2  2  (3?i-l)/2  2  2  3  95^3  27?iC C 2  T i . e . u = [Nj [ r  Q d d  ]  3  2  3  (A.l)  T S i m i l a r l y v = T N I [r ~\ [_evenj J  L  (A.2)  J  I f the c e n t r o i d a l nodal displacements are f o r c e d t o be a q u a d r a t i c f u n c t i o n of the corner node d i s p l a c e m e n t s and displacements at the midsides of the t r i a n g l e ,  then  L J ri8  (A.3)  Applying equation  (A.3) t o equations  T " ( 3 S i - 1) (3? i ~ 2 ) 5 i / 2 - 95 i 5 5 / 2 ~ 2  (3£ 2  3  1) (35 - 2 ) 5 / 2 - 95 i 5 5 / 2 2  2  2  3  ( 3 ? - 1) (3? - 2 ) ? / 2 - 95 i 5 5 / 2 3  3  3  2  r9  (35 - l)/2+275i  3  3  (35 - l ) / 2 + 2 7 5 i 5 5 /4  r u  (35 - l ) / 2 + 2 7 5 i 5 5 / 4  r 13  5 /4  r 15  5 /4  r 17  2  3  7  5 /4  2  95l5  5  52  95i5  3  r r  (3? i - l ) / 2 + 2 7 5 i 5  2  r3  5 /4  2  95 5  r1  2  95i5  2  3  (A.l) and (A.2)  2  3  2  (35 - l ) / 2 + 2 7 5 i 3  52  95i5s (3?i- l)/2+275i 5  i.e. u  r1 r  3  r 17  2  3  3  3  3  (A.4)  y.v  4&a» x , U  Fig. A.1  Complete  cubic  triangle.  {j«ju*> X | U  Fig.A.2  A  typical  s i d e of an  element  153 Similarly v =  [M]  r  2  |  ( A . 5)  r is and the dependence upon degrees  of freedom r  1 9  and r o i s 2  removed from the e q u a t i o n s .  C o n s t r a i n t s are a p p l i e d to s i d e node displacements so t h a t they may  be expressed  of the c o r n e r nodes.  i n terms of the  displacements  With r e f e r e n c e t o F i g . A.2,  displacements t o be l i n e a r along any t y p i c a l edge i.e. d  e n  +  7/  enforce AD. (A.6)  2  s o l v i n g f o r 0i and  8  2  d y - d j  whence d d  2d /3 +  3  x  d /3  (A.7)  7  d i / 3 +2d /3 7  5  A l l o w displacements at r i g h t angles t o edge AD remain  cubic. 2 i.e.  d  J^=  0  3  +  Solving for 0  3  9  l  * l r  to  +  0  0  G  5H  +  3 9 n 6  to  154 d  4  =  yyd  2  +  yyd  8  +  -5-731- y y 3  2  ' _  7 ,  . 20,  Transforming and y  (A.8) , 2L  4L„  D  r e s u l t s to d e f l e c t i o n s p a r a l l e l to x  axes:  d  2  d  7  d  8  •-•  cosa  sina  -sina  cosa 0  u a  0  V  cosa  sina  -sina  cosa  V  d  4>  cosa  -sina  d  3  \  sina  cosa  d  4  u c  V  c  0  cosa  -sina  d  5  sina  cosa  d  6  S u b s t i t u t i n g equations (A.10).  (A.9)  a  (A.l),  (A.10)  (A.8)  and  (A.9)  into  2 2 •j cos a 20  .2  +2ysm  —^sinacosa  -^•sinacosa  ^2 0 27 +  2 C O S  27 •sinacosa  . 2  4L 27< :osa  1 . 2 •j s i n a 7 2 -H^cos a  27 s i n a c o s a  ,7.2 -f-jySin a  27 s i n a c o s a  1 . 2 j sin a +.~ cos a 7  2  2L. 27 s i n a  2L -27< :osa  2  2L_ *27 s i n a  a  27 sinacosa  1 2 y cos a  a  1 2 -j cos a 7  —2ySina  a  2 . 2 j sin a  +2ySin  4L .  2L 27^:osa  2 2 j cos a ,20 . 2 +—sin a  —jysmacosa  27 s i n a c o s a  2 . 2 j sin a , 20 2 + ^ycos a  4L^ 27 •sina  4L -27*:osa  (A.11)  156 Using equations  (A.11) f o r a l l t h r e e s i d e s o f the  triangle  r  1  1  •  •  •  •  •  •  •  •  "s  2  •  1  •  •  •  •  •  •  •  s  2  r3  •  •  •  1  •  •  •  •  •  s  3  r  4  •  •  •  •  1  •  •  •  •  S4  r  5  •  •  •  •  •  •  1  •  •  s  5  r  6  •  •  •  •  •  •  • •  1  •  s  6  r  7  a 31  a 32  2a 33  a 34  - a 32  r  8  a2  a 35  2a  32  r  9  a 34  -a 32  a 33  - a 32  a 37  r  r  rio  3  -  a 33  •  •  •  s  7  a 37  -  a 36  •  •  •  s  8  a 31  a2  -  2 a 33  •  •  •  Sg  a 36  a 32  a 35  -2 a 36  •  •  •  2ai3  36  3  a 14  - a 12  -  a 13  2a is - a 12  a 17  -  a i6  an  a 12  -2a  a i6  a 12  a  r ii  •  •  •  an  a 12  r  i 2  •  •  •  a 12  a  r  i 3  •  •  •  a 14  - a 12  a 13  r  i 4  •  •  - a 12  - a 17  r  is  a 24  — a 22  ris  - a 22  is  is  i 3  -2ai  6  -  S 23  •  •  •  a 21  a 22  2a23  a7  -  3 26  •  •  •  a 22  a 25  2a 26  3  •  •  •  a 24  - a 22  a 23  2 6  •  •  •  -a22  a 27  a 26  2  r 17  a 21  a 22  -2a2  r is  a 22  a 25  -2a  (A.12)  2c'  where a. i  i  3L  +  2  157  20b 2 7 L  1  2b.c. i i 27L 2b. I  a. x.  2~  I  ~2~T  7b' 31/ I  +  271/  2bT2 ^ 20c  >  ..f.:.}: :?  >  i , j,k are c y c l i c permutations of 1,2,3  2  < .i3) A  27L  3L  2c. I  ~2T b  2  Z  a.  ,  I  +  7c  2  i  27L  3L:  b. + c. x  x  and b. = y. - y, x  j  J  k  c. = x. - x. i k 2  ^  Degrees of freedom s^ i n e q u a t i o n s (A.12) r e f e r t o the degrees of freedom of the c o n s t r a i n e d t r i a n g l e shown i n F i g . A.3.  S t r a i n s i n the element are g i v e n by  e  °/3x  X  e y Y  0  9 /  xy  3y  0  u  ^Sy  v  (A.15)  ^3x  The r e l a t i o n s h i p between d e r i v a t i v e s w i t h r e s p e c t t o x and y and the d e r i v a t i v e s w i t h r e s p e c t t o area  coordi-  nates i s g i v e n by 3/ 3x 2A  3/ 3y  hi  b  2  3/  Ci  c  2  3/  2A = b i c 2  (A.4) and  (A.5)  (A.16)  3?:  b c  A p p l y i n g equations equations  3?:  (A.17)  2  (A.15),  (A.16) and  (A.17) t o  159  Values  from  reference  A c c u r a t e up to q u i n t i c  Fig. A.4  Numerical  41  terms  d=0.1012865073234563 in i n t e g r a n d .  integration  points for t r i a n g l e .  160  T 0  3l - 4 > 3 2  <f>  0 <f>l i-<t> i  4>  4>2 l <f>2 -  3 5 - 4> 63  <f> 15-4>  0  1 6  ~  2  r  3  r  4  r  5  r  6  3 7 + 3 4>e 3/ 2  r  7  2/2 <P 3 3 + 3(J)  r  8  8 + 34> 3 / 2  r  9  3 4 + 34>23/ 2  r  10  4> 17+34> 1 e / 2  r  11  4> 13+34> 1 2 / 2  r  12  18 + 34) 1 e / 2  r  13  4) 1 1 + 3 4 ) 1 2 / 2  r  14  7 + 34)2 / 2  r  15  1 6  12  4> 11~4>  4>2 5 4>2 6 -  4>2 1~4>2  4>2 5~4>2 6  r i r  32  4> 1 5-4>  0  2  3 5-$ 3 6  4> 31-4>  0  2  0  4>  2  3  0  33 + 3 4 ) 3 2 / 2  4>  0  3 + 3(J)3 6 / 2  (J)  7  3 + 3c{) 3 2 / 2  <J>  0  lt  y  0 4>1  4>1  4  1 7 + 3 i0 e / 2 0  + 3<j) 1 2 / 2 0  0  (J)  2  4  4>2  + 3 cj> 2 2 / 2 0  4>2  7 + 34>  2  S  / 2  0 (J)2  <j>  <f>  4> 18+34> i e / 2  4>2 3 + 34)2 2 / 2 0  3 6/2  0 4>  3  4> 3  4> 3 8+34>  3 + 3 4 )12 / 2 0  4>  2  6  6  4>2 3 + 34>22 / 2 4>2  8 + 34) / 2  S  8 + 34)2 / 2 6  4>2 i* + 34)2 2 / 2  r le r  i  7  r is  (A.18)  161 where  i i  b. ( ^ d - 9 5 , + D / 2 A J ^ J J  12  2 ¥j k (  ?  9  It  {  k iS Vk i ?  j^ S-i  9 { b  1 3  + b  ( 3  +  ) + b  j ? k l ? k - i  b  {  ?  ) / 2 A  k j IS"^ ?  ,  +  (  V j  (  ^ - i  3  )  ) } / 2 A  /  }  2  A  >  c. (^ -? -95 +D/2A 7  2  (A.19)  j  15  2  1 6  (  c  i 5 j ^  9{c.  1?  9 { c  18  C k  +  C  k ^ i ? j  +  C  j V i ) /  2  A  (3C -|) c 5.(|5 -|)}/2A j  +  ^k4Vl  k  ) + C  j  kV ^ 3  ) } / 2 A  i , j , k are c y c l i c permutations o f 1,2,3  From e q u a t i o n s  (A.12) and (A.18)  T <I»3K  ^ 3 7~  s1  ty  3 2 ^3 5  ^ 3 8  s  2  ty  3 3 ^ 3 6  ^3 9  s  3  ty  11  ^ 14  ty  ty  1 2  4* 1 5  <P  s  5  ^1 3  ty  ty  s  s  tyz 1  ^2 4  ty  s  7  ^2 2  ^2 5  ^ 28  s  8  ^2 3  <J>2 6  ^2 9  s  9  •  e X  e y Y  I  xy  1 6  1 7  18  1 9  2  7  (A.20)  where ty 11  =  (ty -ty ) + l l 12 l  (cj> +h> )+a. l I 3 Z 12 4  iK 12  =  ty. 1  = -^(9<j) +ity +2ty )-*±(9ty ^ 12 13 14 ^  ty. li|  = a . ((j). -<J> )+a (<J>, -<J) ) 12 17 18 K2 K-S K 7  ty.  =  a  a. 1  2  1  (ty -ty. )+a, I3 lit K  r  X  Kt,  16  a. ty. = -^(9ty  iy i .  x  /  -  = lg  16  (ty. -ty.  15  7  (4>-  )+a.  ~ty.  15  16  r  9  K  15  (<j) +|<J) K3 Z K2  )+a Kj  (ty +h>, K t, Z K2  )  (<J>.  )  (*,  )  3  17 2 1  +2<j>  )+a.  ((J).  l e  11  18  K2  a, )-JSs-(9<J> z  +- r4>. T  )+a.  3  17 2  i  k7  T  Ke  +2ty, + 4<j> K7  (c|).  J<8  +^4 .. Y  i i , i s 2  6  )+a.  +lty.  18 2 1 6  17  6  + 2ty + 4ty ) K3 K!i|  (4>-  )+a.  T  +4<j)  17  +4<)|-  i  a  i  k7 2 k6  k s 2 k6  ks  )  (cL  )fa.  kii k  6  +  t 7  3  ^ Y.  )+a.  2 k  6  (Y<J>.  k i k  (ty. -ty. )+a. (4). +|4>. )+a (<j>. + | 4 > . ) + a (ty +fcj> )+a l i I2 15 13 z I 2 17 1t ^ ! K7 K 3 Z K-2 K5 k  2  a. ty. i  K  a  (ty-  5  )+a t  1  (<J>, -<J>. ) 2  a. 3  (<b. +|<J> 4 ^ 12  a.  = -^(9ty. 2  Y  a  i,j,k  r  are c y c l i c  T  2  permutations  o f 1,2,3  3  7  4  ,  Y  )  1  2 k  5  .  12  +~ty )+ty Ki* Z K 2 li)  a  + 4ty. +2ty. )+-^-(9ty. +44>. +2ty. )-J£3-(9<|>. i 17 i s 2 i 13 i t 2 k6 6  + 8  +  2  4>^  k7  + 4*.  ks  ) - - 4 M 9 T< J > .  2  k2  + 2$. k  3  + 4*.  T  ki»  )  (A.21)  c  ro  163 From and  the  (A.20)  equation  stiffness  matrix  is  (e) given  [f](s)  = by  T  [K] = t | [ Y ]  [E] [ f ] d A  I  = t  F(5i  ,£ ,S )dA 2  3  A  Integrating  numerically  n  [K]  Cw  =  F(?t,?2,?t)At  (A.22)  i=l  where  The fourth point  A. 2  weighting  A  =  area  t  =  thickness  of  n  =  number  integrating  and  integration at  a  for  Plane  an  exact  erratic  of  shown  in  element  terms  numerical in  in  points  the  £ s  up  1  integration  Fig.  A.4  is  to  the  necessary.  with  less  element  were  abandoned.  integration  Quadrilateral  function Fig.  and  the seven  integration  Stress/Strain  3 7  function  element  shown  order  displacement  quadrilateral  of  contains  scheme  lower  yielded  The  is:  =  integrand  power  Attempts points  w_^  A.5,  for in  the local  complete s,t  cubic  coordinates  164  Fig. A.6  Constrained  cubic  quadrilateral.  (1-s) (1-t){ -10+9(s +t ) }/32  ri  (1+s) (1-t){ - 1 0 + 9 ( s + t ) >/32  r  (1+s) (1+t){ - 1 0 + 9 ( s + t ) 1/32  r  (1-s) d + t ) { - 1 0 + 9 ( s + t ) }/32  r  2  2  2  2  , -> 2.  2  2  2  3  5  7  9(1-t) (1-s ) (l-3s)/32  r  9(1-t) ( 1 - s ) (l+3s)/32  r ii  9(1+s) ( 1 - t ) ( l - 3 t ) / 3 2  r i  9(1+s) ( 1 - t ) (l+3t)/32  r 15  9(1+t)  (l+3s)/32  r i  9(1+t) ( 1 - s ) (l-3s)/32  r 19  9(1-s) ( 1 - t ) (l+3t)/32  r i  9(1-s) ( 1 - t ) ( l - 3 t ) / 3 2  r  2  u  2  2  2  (1-S  )  2  2  2  i.e. u = [N] ( r  Q d d  )  T  S i m i l a r l y v = TNI (r ) even  9  3  7  2  23  (A.23)  (A.24)  J  A p p l y i n g equations (A.11) t o a l l four s i d e s of the q u a d r i l a t e r a l of F i g . A.5, a r e l a t i o n s h i p between the r degrees of freedom and the s degrees of freedom f o r the c o n s t r a i n e d c u b i c q u a d r i l a t e r a l of F i g . A.6 i s o b t a i n e d .  166 ri ~  1  •  •  •  •  Sl  r  2  •  1  •  •  -  s  2  r  3  •  •  1  •  S  3  r<4  •  •  •  1  S4  r  5  •  •  •  •  r  6  •  •  •  •  r  7  •  •  •  •  r  8  •  •  •  •  r  9  a n -a 22 -2a  rw  a 14  a^  a is  2ai  a 12  a 17  -a 13 an  —a^  rn  a 14  a i2  ri2  ai2  a 17  ri  13  6  a i6 —a 12  1  1 1 a 13  —a.22  rn  •  •  •  —a.22  a5  ris  •  •  •  a24  a22  ris  •  •  •  a.22  a7  r  X7  •  •  •  •  rie  •  •  •  •  —a32  r  •  •  •  •  •  •  • a43  -2a  a4  a22  a3  2a26  5.22  a7  -a^  -a 3  ai  —az2  2a  a5  -2a  2  23  2  2  a26 - a.22  2  2  23  26  a 31 -a 32 -2a 33  a34  a^  a 33  a7  -a se  a35  2a36 a32  a34  a  -a 33 a 31 -a 32  •  a32  a 37  •  •  •  •  •  32  3  2a  33  a36 -a32  a35  -2aa6  •  ai  -a42  -2a43  •  •  -a42  a 45  2a46  a 1*  r22  a^ . a 7  -a45  r  a 41  2a4  •  •  •  •  a44  a42  -a 43  a 45 -2a46  •  •  •  •  a4  a 47  a46  2 3  r4 2  —a 42  -a42  3  8  2  r i  4  s  Sl2  a 21  42  7  Sll  3  •  a  s  2a i  •  2  6  s io  •  i 9  s  -a i6  3  2  5  Sg  a is -2 a is  2  s  4  2  (A.25)  c.  where a,  1  x1  L.  l b.c. _2_ x 1 27  12  2_0 + 27 L. l  2b. l 27  a. 13  f  > z fb.l 7 c. + X L.X 27 L. 1 iJ 1 ij 2 c. 2 X + 20 l 3 L. 27 L i 1 i j I i 2c. l 27  it  is  a. 16  ct • 17  —  2  1 3  X  +  L.  1  The r e l a t i o n s h i p  iJ  i  = 1,2,3,4 (A.26)  c. X  L.  1  i j  between d e r i v a t i v e s  x and y and the d e r i v a t i v e s by the c h a i n  7 27  >  with respect to  w i t h r e s p e c t t o s and t i s g i v e n  rule  9s 9  9x 9s  9y 9s  9x 9t  9y_ 9t m i  9x  (A.27)  Choose a l i n e a r t r a n s f o r m a t i o n between the x,y s y s tem and the s , t s y s t e m  37  160  x = i { ( l - s ) ( l - t ) x i + (1+s) ( l - t ) x + (1 + s) ( l + t ) x + (1-s) (l+t)x„} 2  y = i{(1-s) (l-t)  Y l  3  + ( l + s ) ( l - t ) y + ( l + s ) (1+t)y +(1-s) (l+t)y } 2  3  4  (A.28)  Equations tuted  (A.28) may  be d i f f e r e n t i a t e d  and  substi-  i n (A.27) t o g i v e  9 9s 9  This  may  be i n v e r t e d  9 9x 9  Note a l s o t h a t  J 12  J 21  J 22  9 9x 9  LsyJ  to give  In  L YJ 9  J n  I 21  I 12  9 9s  I 22  3 _ L  (A.30)  9T  dxdy = d e t [ j ] d s d t  (A.31)  169 A p p l y i n g equations  (A.15) t o the above equations  0 I 11  4 >2 1+1  I 2  I  10 1 1 + I 20 1 it 2  0  12^21  0  T  0  I 1 10 1 1+1 1 2 0 1 4  0 I 2 1  0  0  0 1 1+1 1 2 0 1 4  r  2  r  3  0 2 1 + 1 1 2 024  r  4  I  0 31+1 2 20 3 4  r  5  0 3 1+1 1 2 0 3 4  re  0 4 1+12 2 0 4 4  r  II  0 4 1+1 1 20 4 4  r  I  0 1 2 + 12 2 0 1 5  I  2  r1  II 2  2  I21041+I 2044  0 1 1 + 12 2 0 1 4  2 0 2 1 + 1 2 2 02 4  0 3 1 + I 2ty3 4 II 0  0  II I  I 2 102 1 + 1 2 2 0 2 4  I 1 1 0 3 1+1 1 2 0 3 4  2  2  2  8  9  r  I 1 1 <> f 1 2+1 1 2 <J> 1 5  0  I 2 1 0 1 2 + 1 2 2ty1 5  0  I 1 1 0 1 3+1 1 2 0 1 6  0  II xy  I  0  0  I 2 1022 +122025  I l 10 2 3 + 1 1 2 0 2 6  0  0  I 2 1  Il 10 32+ 1 120 3 5  0  II  1042+Il2045  0  12  10 3 2 + 1 2 2 0 3 5  0 1 2  r  10  I  0 1 3+12 2 0 1 6  r  11  013+112 016  r  12  02 2 + 1 2 2 02 5  r i  2  I  2  • 0 121042+1220^5  •2 1 0 4 3 + 1 2 2 0 4 6  3  II  02 2 + 1 1 2 02 5  r  14  I  02 3 + 1 2 2 0 2 6  r  15  02 3+1 1 2 0 2 6  r is  0 32+ I  r  2  I  2  2  20 3 5  17  II  0 3 2+1 1 2 0 3 5  r is  I  0 3 3 + 12 2 0 3 6  r  19  0 3 3+1 1 2 0 3 6  r  20  042+122045  r i  II  042+Il2045  r  22  I  0 4 3 + 12 2 0 4 6  r  23  0 4 3+1 1 2 0 4 6  r 4  2  1 0 3 3 +12 2 0 3 6 II  I l 104 3 + I l 2 0 4 6  0  0 1 2+1 1 2 0 1 5  0 2 3 + 1 2 2 0 2 6 II .0  I 1 1 0 3 3 + 1 1 2$ 3 6  0  II  2 10 1 3+1 2 2 4> 1 6 II  1 0 2 2+1 1 2 0 2 5  7  I  2  2  II  2  2  (A.32)  170 where  - (1-t) {-10+9(s +t )}/32+9s(1-s)(l-t)/16 2  2  0 11  =  0 12  =  -9s (1-t) (l-3s)/16-27(1-t) ( l - s ) / 3 2  0 13  =  -9s (1-t) ( l + 3 s ) / 1 6 + 2 7 ( 1 - t ) ( l - s ) / 3 2  4> it  =  2  2  - (1-s) [-10+9(s +t )}/32+9t(1-s)(1-t)/16 2  2  0 15  =  -9 (1-s  0 16  =  -9  021  =  022  =  9 (1-t ) (l-3t)/32  0 23  =  9 ( 1 - t ) (l+3t)/32  024  =  - (1+s)^[-10+9(s +t )}/32+9t(1+s)(1-t)/16  025  =  -9t (1+s) (l-3t)/16-27(1+s) ( l - t ) / 3 2  0 26  =  -9t (1+s) ( l + 3 t ) / 1 6 + 2 7 ( 1 + s ) ( l - t ) / 3 2  0 31  =  0 32  =  -9s (1+t) (l+3s)/16+27(1+t) ( l - s ) / 3 2  0  33  =  -9s (1+t) (l-3s)/16-27(1+t) ( l - s ) / 3 2  0 34  =  0 35  =  9 (1-s ) (l+3s)/32  0 36  =  9 (1-s ) (l-3s)/32  0 41  =  0  =  -9 d - t ) (l+3t)/32  043  ~  -9 [ 1 - t ) ( l - 3 t ) / 3 2  044  =  0 45  =  -9t :i-s) (l+3t)/16+27(1-s) ( l - t ) / 3 2  0 46  =  -9t '1-s) (l-3t)/16-27(1-s) ( l - t ) / 3 2  42  (l-3s)/32  M  (l+3s)/32 (1-s , (1-t) 1[-10+9 ( s + t ) } / 3 2 + 9s(1+s) ( l - t ) / 1 6 2  2  2  2  2  2  2  2  2  (1+t)\[-10+9(s +t )}/32+9s(1+s)(l+t)/16 2  2  2  2  (1+s) [-10 + 9 ( s + t ) }/32 + 9t(1+s) (l+t)/16 J  2  2  - (1+t)\[-10+9 ( s + t ) } / 3 2 + 9s(1-s) (l+t)/16 2  2  2  2  (1-s)<'-10+9(s +t )}/32 + 9t(1-s) (l+t)/16 2  2  2  2  (A.33)  1 7 1  From equations  (A.25) and (A.32)  4)  14  4) 1 7  4)  15  4)18  s  2  16  4)19  s  3  ^21  4)24  4)27  S  4  ^ 22  4)25  4)28  s  5  ^ 23  4)26  4)29  s  6  4)  34  4)  37  s  7  ^ 32  4)  35  4)  38  s  8  IP  4)36  4)39  Sg  4^1  4)44  4)47  S 10  4*42  4>45  4)48  S 11  4)43  4)46  4)49  S 12  4> 4  1  12  13  4> xy  T  31  33  _  ~Sl  (A.34)  where  \p. -  I  l l  <j>. +1 <J>. 11 l l 12 11  +a iK !2  =  a  (I  1  11  <j>  +1  rt. = -a 1  =  a  ll*  (I  £  a. (2(1  6  rt . 17  I  +1  cj).  11 l l 0  +a  0.  X,  3  +1  (I  +1  Cj)  11 *2 cj)  21  )+a  12 16  (I  0  * 4  <J> 2 0  11  +1  cf> ) ^5 0  12  X  cj) 2 1 12 * 5  Cj)  15  c(>. 15  cj)  -I  11 *> 3  +1  cj)  11 *• 3  . +1  cj>  21 13  17  )-a  cj). 1 6  cj) . 1 5  22  X.  $ 12  c j > )+I 12  22  ) ^6  cj) +1  11 * 2  6  <j) ) p  12  ^5  cj> . ) 16  3  (I 11  +1  cj).  >i  cj).  +1 3  cj)  Cj)  12  X,  cj) . 16  22  lz  22  *  ?  )  X, 5  )  . +1  < J > . ) 12  16  )+rt 6  1 •*  )+a. (2(1 < J > . +1  22 16 16 +1 cj) )+i cj) +1 cj) ^3 2 2 X-6 21 &2 22 * 5  l  12  11 13  17  +1  <f> » e) +12 1 <J>. +1 22  )+rt  cj) 22 * 6  )+a. (I Cj)  cj) . 1 3  2 1  14  12 15 £ 5  2 1 Jt 3  )+a. (I  21 cj).  (2(1 < j > +1  X, 6  (I cj) +1  +1  21 1  (2(1  * 3  )+a. (I  22  . +1  )+a  )+I  -I  cj>  12 * 5  cj) +i cj) ) 2 1 * 3 2 2 X. 6  *. 1  cj).  +1  cj)  11 X- 2  )+a  d>  +l  cj).  11 12  (I  (I  <}>„ )+a  <j). 22 cj,  22  21 1 3  12 * 5  . +1  4>.  . +1  <f>. )+a. (I d> +1 ch - i cj> -I cj) ) 1 5 2 21 * 2 2 2 *• 5 21 X, 3 2 2 X< 6  *5  )+i  22  15  +l a  * 5  +a. (I  (2(1  0  l  X-2  22  )+.  cj)  15  l  c|>  11 13  l  *2  12 16  -I 2  +a. (I  0  21 12  13  cj). 22  +1  cj>  cj) .  )+a  (J)  11 13  22  12 14  X/7  rt. = -a. (2(1  +i  X<2  l  12 15  21 12  15  cj) . li|  (I  p  +a  lg  -I  (J>  +a. (I  +1  21  18  cj)  0  cj).  I cj). +1 21 l l 22 +a  rt. =  -I  )+I  2 1 12  16  =  <j>  +1  21  )+a. (I  p  12 *6  12 15  (I 7  cj). 12 15  4> )  +1 3  cj) +1 <j>. - I 0. 21 1 3 22 16 21 1  +a„  l  . +1  cj)  11 12  11 12  cj>. +1 cj) . 21 l l 22 14  rt. =  l  12 16  I  15  r  d>  p  X,  11 12  12  =  Ui.  (2(1  13  3  <J>i  11 13  12  l  4>  (I  p  JO  +a. (I  11 12  3  )-a„ (2(1  i,j,k,x\ are c y c l i c permutations of 1,2,3,4  * 6  cj> .  )+1  12 15 cj)  11 * 3  +1  0.  11 13  +l  0. )  12  16 cj) +1 cj> 11 * 2 12 X .  )+l  cj)  12 *< 6  ) 5  (A.35) —J  From e q u a t i o n and  the  (A.34)  (e) =  s t i f f n e s s matrix i s given  [K] = t  [V]  T  n  (A.31)  n  [K] = t /__, > j=l  /__, H H F ( s , t ) d e t [ j ] i=i 3 i i  (A.36)  1  where i  H.,H. 3  n  4x4  from equation  n u m e r i c a l l y 37  Integrating  integrand  s i x t h power, and  by:  [E] [y]dxdy  t / F(s,t)det [j]dsdt  The  [y](s)  weighting  functions  thickness  of element  number of i n t e g r a t i n g i n each d i r e c t i o n  c o n t a i n s terms i n s and  t up  to  the  f o r exact n u m e r i c a l i n t e g r a t i o n a set  Gauss quadrature p o i n t s would be  required.  37  order i n t e g r a t i o n on a 3 x 3 g r i d of i n t e g r a t i o n yielded  a successful  A. 3  Patch Tests ' 1  The  points  element and  2  on  displacement  was  therefore  of  Lower points  retained.  Elements f u n c t i o n s f o r the  triangular  and  174 quadrilateral value  tests  three  zero  elements  on  stiffness  eigenvalues  dent  of element  ence  of three  the  rotation  shape  rigid  restriction  a  mesh  under  angle.  least  one  strain  a pure  with  of triangles.  ently  o f mesh  ments  consistently  In  value.  refinement.  strain  using  f o r the s t r a i n  The  strain  that  are too s t i f f  dent  o f mesh  would  tests  refinement.  This  always  of the element  that  independ-  i t would  never  to the  correct  functions.  elements  because It i s  indicate are  i t possible strain  ele-  stress.  t o o low  which  the  t o o low, i n -  direct  be  loads  t o o low f o r  4.9%  elements  at  t o o low f o r  t o converge  makes  o f any  the triangular  pure  amounts,  i s zero  under  constant  energies  on t h e s e  by c o n s t a n t  r e l e v a n t columns  remained  these  placed  with  found  9.1%  of the displacement  patch  they  i t was  a n d 9.3%  under  energy  energy  the incompleteness  elements,  In addition,  refinement,  any a n a l y s i s  significant  values  yielded  stress  low i n a c o r n e r  consistently  of quadrilaterals  that  I n any element i n  the shear  condition,  was  These  found  and u n r e s t r a i n e d node,  shear  exist-  o f an e l e m e n t  o f assembled  4 2  unloaded  o f mesh  possible  condition,  indepen-  the  i t was  there.  and markedly  of a patch  those  dependently  corner  Eigen-  yielded  t o be  indicating  a t each strain  elements  proved  However,  shear  a pure  consisting  which  modes.  For a patch  energy  patches  the  body  but incomplete.  of both  or orientation,  on t h e shear  internal  consistent  of  f o r each,  a right-angled corner  other  be  matrices  constraints  a  in  are compatible  matrix  that  indepen-  to  multiply  by a c o n -  s t a n t i n order t h a t the r e s u l t a n t s t i f f n e s s matrix may c o r r e s p o n d i n g l y more f l e x i b l e i n the pure s t r e s s The  c o n s t a n t s t h a t were a p p l i e d were determined  a n d - e r r o r b a s i s and are c l e a r l y v i s i b l e presented  i n the next s e c t i o n .  i n the  175  be  states. on a  trial-  subroutines  In a d d i t i o n , the  shear  s t r a i n s i n the t r i a n g u l a r element were made c o n s t a n t  through-  out by u s i n g the c e n t r o i d a l value of shear s t r a i n f o r a l l i n t e g r a t i o n p o i n t s i n each element.  S i m i l a r l y , the s t r a i n c a l c u l a t i o n s u b r o u t i n e s f o r these elements have c o n s t a n t s a p p l i e d t o the r e l e v a n t columns of the s t r a i n m a t r i x t o g i v e much-improved s t r a i n s .  The  shear s t r a i n s of the q u a d r i l a t e r a l element are l i n e a r l y i n t e r p o l a t e d from the s t r a i n s c a l c u l a t e d a t the f o u r p o i n t s w i t h c o o r d i n a t e s s=t=±0.55735.  These p o i n t s were found  to  have shear s t r a i n s very c l o s e to the c o r r e c t v a l u e s under a l l t h r e e d i f f e r e n t constant s t r e s s  states.  Patch t e s t s on the improved elements i n d i c a t e d ,  with  some e x c e p t i o n s , t h a t s t r e s s e s and d e f l e c t i o n s were c o n s i s t e n t w i t h boundary c o n d i t i o n s and a p p l i e d l o a d s .  The  excep-  t i o n s were d i r e c t s t r e s s e s a t the t r i a n g l e nodes which were too low and the P o i s s o n e f f e c t of the t r i a n g l e s , which consistently  85% of what i t should have been.  The  was  strain  e n e r g i e s f o r a t y p i c a l s e t of p a t c h t e s t s are presented i n Table V I I I .  Coarser t r i a n g u l a r meshes y i e l d  low s t r a i n  ener-  g i e s i n a l l three c o n s t a n t s t r a i n nodes, but appear t o improve r a p i d l y w i t h f u r t h e r mesh refinement.  TABLE V I I I STRAIN ENERGIES FROM PATCH TESTS ON ELEMENTS  STRAIN ENERGY  (Kip.  in.)  QUADRILATERAL FINITE ELEMENT Finite Element Mesh  Pure Shear x=8k.s.i. Pure Compression o=-8k.s.i. Pure Bending ~ P ^bot , a =-o =10k.s.i. to  c  CALC.  EXACT  CALC.  EXACT  CALC.  EXACT  1.33333  1.33333  1.33463  1.33333  2.66666  2.66667  0.53333  0.53333  0.53409  0.53333  1.06666  1.06667  0.27327  0.27778  0.27337  0.27778  0.54146  0.55556  0 1  X  X  TRIANGULAR FINITE ELEMENT Finite Element Mesh  Pure Shear x=8k.s.i. Pure Compression o =-8k.s.i.  CALC.  EXACT  CALC.  EXACT  CALC.  EXACT  1.35422  1.33333  1.34652  1.33333  2.80364  2.66667  0.53284  0.53333  0.53064  0.53333  1.04648  1.06667  0.26986  0.27778 0.277037 0.27778  0.55191  0.55556  X  Pure Bending top hot , o ^--o =10k.s.x. X  X  A. 4  Element S t i f f n e s s and S t r a i n C a l c u l a t i o n  A.4.1  L i s t of V a r i a b l e s  PLSTRI  i n Subroutine  Subroutines  Input/Output  PLSQUA  X,Y  X,Y  C o o r d i n a t e s of element nodes  EXY  EXY  Lower t r i a n g l e of e l a s t i c i t y s t o r e d as column v e c t o r  T  TH  Element t h i c k n e s s  S  SM  Lower t r i a n g l e of s t i f f n e s s matrix s t o r e d as column v e c t o r  TRISTN  Lists  matrix  QUASTN  X,Y  X,Y  C o o r d i n a t e s of element nodes  DEF  DEF  V e c t o r of nodal  EPS  EPS  M a t r i x of s t r a i n s w i t h c e n t r o i d a l s t r a i n s i n f i r s t row and nodal s t r a i n s i n succeedi n g rows  II  IJ  I f v a l u e = 1, c e n t r o i d a l s t r a i n s o n l y are c a l c u l a t e d , otherwise c e n t r o i d a l and nodal s t r a i n s are c a l c u l a t e d  displacements  PLSTRI - PLANE STRESS/STRAIN TRIANGLE STIFFNESS MATRIX SUBROUTINE PLSTRI(X,Y,EXY,T,S) IMPLICIT REAL*8(A-H,0-Z) DIMENSION X(3),Y(3),S(45),A(3,7),B(3),C(3),RL(3),P(3,8),Q(3,9) ,SPA CD (3,9) ,SPAD1(45),Z(7,3),W(7) ,EXY (6) DATA Z/.3 333333333333333D0,.0597158717897698D0,.4701420641051151D0 C,.4701420641051151D0,.7974269853530 872D0,.1012865073234563D0,.1012 C8 65 07 3 23 45 6 3D0,.3333333333333333D0,.470142 0641051151D0,.05 9715 8 717 C8 976 98D0,.4701420641051151D0,.1012 865 07 32 34 563D0,.797 42 6985353 0872 CDO,.1012865073234563D0,.3333333333333333D0,.47 0142 06 41051151D0,.4 7 C0142 06 41051151D0,.0597158717897698D0,. 1012865073234563D0,.1012 865 0 C7 32 345 6 3D0,.7 97 42 6 985 353 0872D0/ DATA W/.225D0,3*.1323941527885062D0,3*.1259391805448271D0/ DO 1 1=1,45 S (I)=0.D0 DO 2 J=l,3 I=J+2-(J/2)*3 K=J+1-(J/3)*3 B(I)=Y(J)-Y(K) C(I)=X(K)-X(J) RL(I)=DSQRT(B (I)**2+C(I)**2) A(I,1)=2.D0*C(I)**2/(3.D0*RL(I)**2)+20.D0*B(I)**2/(2 7.D0*RL(I)**2) A(I,2)=2.D0*B(I)*C(I)/(27.D0*RL(I)**2) A(I,3)=2.D0*B(I)/27.D0 A(I,4)=C(I)**2/(3.D0*RL(I)**2)+7.DO*B(I)**2/(27.DO*RL(I)**2) A(I,5)=2.D0*B(I)**2/(3.D0*RL(I)**2)+2 0.D0*C(I)**2/(27.D0*RL(I)**2) A(I 6)=2.D0*C(I)/2,7.D0 A(I,7)=B(I)**2/(3.D0*RL(I)**2)+7.D0*C(I)**2/(27.DO*RL(I)**2) AREA=(B(1)*C(2)-B(2)*C(1))/2.D0 DO 3 L=l,7 DO 4 J=l,3 I=J+2-(J/2)*3 K=J+1-(J/3) *3 P(I,1)=B(J)*(13.5D0*Z(L,J)**2-9.D0*Z(L,J)+1.DO)/(2.D0*AREA) P(I,2)=4.5D0*(B(I)*Z(L,J)*Z(L,K)+B(K)*Z(L,I)*Z(L,J)+B(J)*Z(L,K)*Z ( CL,I))/(2.D0*AREA) /  P(I,3)=9.D0*(B(J)*Z(L,K)*(3.D0*Z(L J)-0.5D0)+B(K)*Z(L,J)*(1.5D0*Z ( CL,J)-0.5D0))/(2.DO*AREA) P (I,4)=9.D0*(B(J)+Z(L,K)*(1.5D0*Z(L,K)-0.5D0)+B(K)*Z(L,J)*(3.D0*Z( CL,K)-0.5D0))/(2.DO *AREA) P (I,5)=C (J)*(13.5D0*Z(L, J)**2-9.D0*Z(L,J)+1.DO)/(2.DO*AREA) P(I,6)=4.5D0*(C(I)*Z(L,J)*Z(L,K)+C(K)*Z(L,I)*Z(L,J)+C(J)*Z(L,K)*Z ( CL,I))/(2.DO*AREA) P(I,7)=9.DO*(C(J)*Z(L,K)*(3.D0*Z(L,J)-0.5D0)+C(K)*Z(L,J)*(1.5D0*Z ( CL,J)-0.5D0))/(2.D0*AREA) P (I,8)=9.D0*(C(J)*Z(L,K)*(1.5D0*Z(L,K) -0.5D0)+C(K)*Z(L,J)*(3.DO*Z( CL,K)-0.5D0))/(2.D0*AREA) CONTINUE DO 5 J=l,3 I=J+2-(J/2)*3 f  K=J+1-(J/3)*3  M1=1+(J-1)*3 M2=M1+1 M3^=Ml + 2 Q(1,M1) = (P (I,1)-P(I,2) )+A(I,l) * (P(I,3)+1.5D0*P(I,2) )+A (1, 4 ) * (P (1, 4 C)+1.5D0*P(1,2))+A(K,4)*(P(K,3)+1.5D0*P(K,2))+A(K,1)*(P(K,4)+1.5D0* CP(K,2) ) Q(1,M2)=A(I,2) * (P(I,3)-P (1,4) )+A(K,2) * (P (K, 4 )-P (K, 3 ) ) Q(1,M3)=A(1,3)*(4.5D0*P(I,2)+2.DO*P(I,3)+P(I,4))-A(K,3)*(4.5D0*P(K C,2)+P(K,3)+2.D0*P(K,4) ) Q(2,M1)=A(I,2) *(P(I,7)-P(I,8) )+A (K, 2 ) * (P (K, 8 )-P (K, 7 ) ) Q(2,M2) = (P(I,5)-P(1,6))+A(I,5)*(P(1,7)+1.5D0 *P(1,6) )*A(1,7)*(P(1,8 C)+1.5D0*P(1,6))+A(K,7)*(P(K,7)+1.5D0 *P(K,6))+A(K,5)*(P(K,8)+1.5D0* CP(K,6) ) Q(2,M3)=A(I,6)*(4.5D0*P(1,6)*2.D0*P(1,7)+P(1,8))-A(k,6)*(4.5D0*P (K C,6)+P(K,7)+2.D0*P(K,8)) Q(1,M1)=Q(1,M1)/1.025D0 Q(1,M2)=Q(1,M2)/1.025D0 Q(1,M3)=Q(1,M3)/1.025D0 Q(2,M1)=Q(2,M1)/1.025D0 Q(2 M2)=Q(2,M2)/l.025D0 Q(2,M3)=Q(2,M3)/1.025D0 /  IF(l-L)5,7,7 Q(3,Ml) = (P(I,5)-P (1,6) )+A(I,l) * (P (I,7)+1.5D0*P (1, 6 ) )+A (1, 4 ) * (P (1, 8 C)+1.5D0*P(I,6))+A(K,4)*(P(K,7)+1.5D0 *P(K,6))+A(K,1)*(P(K,8)+l.5D0* CP(K,6))+Q(l,M2) Q(3 M2)=(P(I,l)-P(1,2))+A(I,5)*(P(1,3)+1.5D0*P(I,2))+A(I,7)*(P(I,4 C)+1.5D0*P(1,2))+A(K,7)*(P(K,3)+l.5D0*P(K,2))+A(K,5)*(P(K,4)+1.5D0* CP (K,2) )+Q(2,Ml) Q(3,M3)=A(I,3) * (4.5D0*P(I,6)+2.D0*P(I,7)+P (I,8))+A(I,6)* (4.5D0*P (I C,2)+2.D0*P(I,3)+P(1,4))-A(K,3)*(4.5D0*P(K,6)+P(K,7)+2.DO*P (K,8))-A C (K,6)*(4.5D0*P(K,2)+P(K,3)+2.DO*P(K,4)) Q(3,M1)=Q(3,M1)/l.165D0 Q(3,M2)=Q(3,M2)/1.165D0 Q(3,M3)=Q(3,M3)/1.165D0 CONTINUE CALL DIMULT(EXY,Q,SPAD,3,9,3,3,0) CALL REMULT(Q,SPAD,SPAD1,9,3,3,3,1,0,NC) DO 6 1=1,45 S (I)=S (D+SPAD1 (I) *W(L) *AREA*T CONTINUE RETURN END f  C  TRISTN - CALCULATION OF STRAINS IN TRIANGLE SUBROUTINE TRISTN(X,Y,DEF,EPS,II) IMPLICIT REAL*8(A-H,0-Z) DIMENSION X(3) ,Y(3) ,DEF(9) ,EPS (5,3) ,B(3),C(3),P(3,8),Q(3,9),Z(4,3) CSPAD (3) ,RL(3) ,A(3,7) DATA Z/.3 33 3 3 33 33 333 3 333D0,1.D0,0.D0,0.D0,.3333333333333333D0,0.DO C,1.D0,0.D0,.33 333 333 3333 3333D0,0.D0,0.D0 1.D0/ DO 2 J=l,3 I=J+2-(J/2)*3 K=J+1-(J/3)*3 B(I)=Y(J)-Y(K) C (I)=X(K)-X(J) RL(I)=DSQRT(B(I)**2+C(I)**2) A (1,1)=2.DO*C(I)**2/(3.D0*RL(I)**2)+2 0.D0*B(I)**2/(27.DO*RL(I)**2) A(I,2)=2.D0*B(I)*C(I)/(27.D0*RL(I)**2) A(I,3)=2.D0*B(I)/27.D0 A(I,4)=C(I)**2/(3.D0*RL(I)**2)+7.D0*B(I)**2/(27.D0*RL(I)**2) A(I,5)=2.D0*B(I)**2/(3.D0*RL(I)**2)+2 0.D0*C(I)**2/(27.D0*RL(I)**2) A(I,6)=2.D0*C(I)/27.D0 A(I 7)=B(I)**2/(3.D0*RL(I)**2)+7.DO*C(I)**2/(27.D0*RL(I)**2) AREA= (B (1) *C(2)-B(2) *C(1) )/2.D0 IF(II-1)7,8,7 NN=4 GO TO 9 NN=1 DO 3 L=1,NN DO 4 J=l,3 I=J+2-(J/2)*3 K=J+1-(J/3)*3 P(I,1)=B(J)*(13.5D0*Z(L,J)**2-9.D0*Z(L,J)+1.DO)/(2.D0*AREA) P (I,2)=4.5D0*(B(I)*Z(L,J) *Z(L,K)+B(K)*Z(L,I)*Z(L J)+B(J)*Z(L,K)*Z ( CL,I))/(2.D0*AREA) P (I,3)=9.DO*(B(J)*Z(L,K)*(3.D0*Z(L,J)-0.5D0)+B(K)*Z(L,J)*(1.5D0*Z( CL,J)-0.5D0))/(2.D0*AREA) P (I,4)=9.D0*(B (J)*Z(L,K)*(1.5D0*Z(L,K)-0.5D0)+B(K)*Z(L,J)*(3.D0*Z( CL,K)-0.5D0))/(2.DO *AREA) f  2 7 8 9  /  #  4  P(I,5)=C(J)*(13.5D0*Z(L,J)**2-9.D0*Z(L,J)+1.DO)/(2.DO*AREA) P (I,6)=4.5D0*(C ( I ) * Z ( L , J ) * Z ( L , K ) + C ( K ) * Z ( L , I ) * Z ( L , J ) + C ( J ) * Z ( L , K ) * Z ( CL,I) )/(2.D0*AREA) P(I,7)=9.DO*(C(J)*Z(L,K)*(3.D0*Z(L,J)-0.5D0)+C(K)+Z(L,J)*(1.5D0*Z ( CL,J)-0.5D0))/(2.D0*AREA) P (I,8)=9.D0*(C(J)*Z(L,K)*(1. 5D0*Z(L,K)-0.5D0)+C(K)*Z(L,J)* (3.DO*Z ( CL,K)-0.5D0))/(2.DO *AREA) CONTINUE DO 5 J=l,3 I=J+2-(J/2)*3 K=J+1- (J/3)*3 Ml=l+ ( J - D + 3 M2=M1+1 M3=Ml+2 Q(1,M1) = (P(I,1)-P(I,2) )+A(I,l) *(P(I,3)+1.5D0*P (1, 2 ) )+A (1, 4 ) * (P (I 4 C)+1.5D0*P(1,2) )+A(K,4) *(P(K,3)+1.5D0*P(K,2))+A(K,1)*(P(K,4)+1.5D0* CP (K,2)) Q(1,M2)=A(I,2) *(P(I,3)-P(I,4) )+A(K,2) * (P (K, 4)-P (K, 3) ) Q(1,M3)=A(I,3) *(4.5D0*P(1,2)+2.DO*P(1,3)+P(1,4))-A(K,3)*(4.5D0*P (K C,2)+P(K,3)+2.D0*P(K,4)) Q(2,M1)=A(I,2)*(P(I,7)-P(I,8) )+A(K,2)* (P (K, 8)-P (K , 7) ) Q(2,M2)=(P(I,5)-P(I,6) )+A(I,5)*(P(I,7)+1.5D0*P(I,6) )+A(I,7) *(P(I,8 C)+1.5D0*P (1,6) )+A(K,7) * (P (K, 7 )+1. 5D0* P (K, 6 ) )+A(K,5) * (P (K, 8 )+1. 5D0* CP (K,6)) Q(2 M3)=A(I 6)*(4.5D0*P(I,6)+2.D0*P(1,7)+P(1,8))-A(K,6)*(4.5D0*P (K C,6)+P(K,7)+2.D0*P(K,8)) Q(1,M1)=Q(1,M1)*.9316D0 Q(1,M2)=Q(1,M2)*.9316D0 Q(1,M3)=Q(1,M3) *.9316D0 Q(2,M1)=Q(2,M1) *.9316D0 Q(2,M2)=Q(2,M2)*.9316D0 Q(2,M3)=Q(2,M3) *. 9316D0 IF(1-L)5,10,10 Q(3 Ml) = (P(I,5)-P(I,6) )+A(I,l) * (P (I,7)+1.5D0*P (I,6))+A(I,4)*(P(I,8 C)+1.5D0*P(I,6))+A(K,4)*(P(K,7)+1.5D0*P(K,6))+A(K,1)*(P(K,8)+1.5D0* CP (K,6) )+Q(l,M2) f  f  10  #  f  M  CO  to  Q(3,M2) = (P (I,1)-P(I,2) )+A(I, 5) * (P (1, 3)+1. 5D0*P (1,2) )+A(I,7) * (P (1,4 C)+1.5D0*P(1,2))+A(K,7)*(P(K,3)+1. 5D0*P(K,2))+A(K,5)*(P(K,4)+1.5D0* CP(K,2) )+Q(2,Ml) Q(3,M3)=A(I,3) * (4.5D0*P (1, 6)+2 . DO *P (1, 7 )+P (1,8) ) +A (I,6)*(4.5D0*P(I C,2)+2.D0*P(I,3)+P(1,4))-A(K,3)*(4.5D0*P(K,6)+P(K,7)+2.D0*P(K,8))-A C(K,6)*(4.5D0*P(K,2)+P(K,3)+2.D0*P(K,4)) Q(3,Ml)=Q(3,Ml)/1.131D0 Q(3,M2)=Q(3,M2)/1.131D0 Q(3,M3)=Q(3,M3)/1.131D0 CONTINUE CALL MAMULT(Q,DEF,SPAD,3,9,1,3,9,3,0,0) DO 6 1=1,3 EPS(L,I)=SPAD(I) CONTINUE RETURN END  PLSQUA - PLANE STRESS/STRAIN QUADRILATERAL STIFFNESS MATRIX SUBROUTINE PLSQUA(X,Y,EXY,TH,SM) IMPLICIT REAL*8(A-H,0-Z) DIMENSION X(4),Y(4),SM(78) ,EXY(6) ,SS(3),W(3),RL(4),P(4,6),A(4,7),Q C(3,12) ,SPAD(3,12) , SPADl (78),B(4),C(4) DATA SS/-.774596669241483D0,0.DO,.774596669241483D0/ DATA W/.555555555555556D0,.888888888888889D0,.555555555555556D0/ DO 1 1=1,7 8 SM(I)=0.D0 DO 2 1=1,4 J=I+1-(1/4)*4 B(I)=Y(J)-Y(I) C (I)=X(J)-X(I) RL(I)=DSQRT(B(I) *B (I)+C (I) *C (I) ) A (1,1) = ( (C(I) /RL(i') ) **2) *2.D0/3.D0+( (B (I)/RL(I) ) **2)*20.DO/27.DO A(I,2) = (B(I) * C ( I ) / ( R L ( I ) **2) ) *2. DO/27. DO A(I,3)=B(I)*2.DO/27.DO A (I, 4)= ( (C(I)/RL(I) ) **2)/3.D0+( (B(I)/RL(I) ) **2)*7.DO/27.DO A (1,5)= ( (B(I)/RL(I))**2)*2.D0/3.D0+((C(I)/RL(I) )**2)*20.DO/27.DO A(I,6)=C(I) *2.D0/27.D0 A (I,7) = ( (B(I)/RL(I))**2)/3.D0+((C(I)/RL(I)) **2)*7.DO/27.DO DO 3 K=l,3 DO 4 J=l,3 T=SS(K) S=SS (J) RJ11=( (T-1.D0) * (X(l)-X(2) ) + (T+l.D0) * (X (3)-X ( 4 ) ) ) / 4 . DO RJ12=( (T-1.D0)*(Y(1)-Y(2) ) + (T+l. DO) * (Y (3)-Y (4 ) ) )/4.D0 RJ21=((S-l.DO)*(X(1)-X(4))+(S+1.D0)*(X(3)-X(2)))/4.DO RJ22=((S-l.DO)*(Y(l)-Y(4))+(S+1.D0)*(Y(3)-Y(2)))/4.D0 DET=RJ11*RJ2 2-RJ12*RJ21 RI11=RJ22/DET RI12=-RJ12/DET RI21=-RJ21/DET RI22=RJ11/DET CON=-l0.DO+9.DO*(S*S+T*T) P (1,1) = ((T-1.D0)*CON+18.D0*S*(1.D0-S)*(1.D0-T))/32.D0  P (1,2) = (T-1. DO) * (18.D.0*S* (l.D0-3.D0*S)+2 7.D0* (l.DO-S*S) J/32.D0 P (1,3)= (T-1.DO)*(18.D0*S*(l.D0+3.D0*S)-2 7.D0*(l.DO-S*S) J/32.D0 P (1,4)=((S-l.DO)*CON+18.D0*T*(l.DO-S)*(l.DO-T)J/32.D0 P (1,5)=-9.DO*(l.DO-S*S)*(l.D0-3.D0*S)/32.D0 P (1,6)=-9.DO* (l.DO-S*S)*(l.D0+3.D0*S)/32.D0 P (2,1)= ( (l.DO-T)*CON+18.D0*S*(l.DO+S)*(l.DO-T)J/32.D0 P (2,2) = 9.DO*(l.DO-T*T)*(1.DO-3.DO*T)/32.DO P(2,3)=9.DO*(l.DO-T*T)*(1.DO+3.DO*T)/32.DO P (2,4)= (- (l.DO+S)*CON+18.D0*T*(l.DO+S)*(l.DO-T)J/32.D0 P (2,5)=- (l.DO+S)*(18.D0*T*(1.DO-3.DO*T)+27.DO*(l.DO-T*T)J/32.D0 P (2,6)=-(l.DO+S)*(18.D0*T*(1.DO+3.DO*T)-27.DO*(l.DO-T*T)J/32.D0 P (3,1) = ( (l.DO+T)*CON+18.D0*S*(l.DO+S)*(l.DO+T)J/32.D0 P(3,2)=(l.DO+T)*(-18.D0*S*(1.DO+3.DO*S)+27.DO*(l.DO-S*S))/32.D0 P (3,3)= (l.DO+T)*(-18.D0*S*(1.DO-3.DO*S)-27.DO*(l.DO-S*S))/32.D0 P (3,4)= ( (l.DO+S)*CON+18.D0*T*(l.DO+S)*(l.DO+T))/32.D0 P (3,5)=9.DO* (l.DO-S*S)*(1.DO+3.DO*S)/32.DO P(3,6)=9.D0* (l.DO-S*S)*(1.DO-3.DO*S)/32.DO P(4,1)=(-(l.DO+T)*CON+18.D0*S*(l.DO-S)*(l.DO+T))/32.D0 P(4,2)=-9.DO*(l.DO-T*T)*(1.DO+3.DO*T)/32.DO P(4,3)=-9.DO*(l.DO-T*T)*(1.DO-3.DO*T)/32.DO P (4,4)=((l.DO-S)*CON+18.D0*T*(l.DO-S) *(l.DO+T))/32.D0 P (4,5)=- (l.DO-S)*(18.D0*T*(1.DO+3.DO*T)-27.DO*(l.DO-T*T))/3 2.D0 P(4,6)=-(l.DO-S)*(18.D0*T*(1.DO-3.DO*T)+27.DO*(l.DO-T*T))/32.D0 DO 5 1=1,4 Ml=l+ (1-1)*3 M2=M1+1 M3=Ml+2 L=I-1+(1/1)*4 Q (1,M1)=RI11*P(I,1)+RI12*P(1,4)+A(1,1)*(RI11*P(I,2)+RI12*P(1,5))+A C(I,4)*(RI11*P(1,3)+RI12 *P(1,6))+A(L,4)*(RIll*P(L,2)+RI12*P(L,5))+A C(L,l)*(RI11*P(L,3)+RI12 *P(L,6)) Q(1,M2)=A(I,2)*(RI11*(P(I,3)-P(1,2))+RI12*(P(I,6)-P(I,5)))+A(L,2)* C(Rill*(P(L,2)-P(L,3))+RI12*(P(L,5)-P(L,6))) Q(1,M3)=-A(I,3)*(Rill*(2.D0*P(I,2)+P(1,3))+RI12*(2.DO*P(1,5)+P (1,6 C)))+A(L,3)*(Rill*(2.D0*P(L,3)+P(L,2))+RI12*(2.D0*P(L,6)+P(L,5))) Q(2,M1)=A(I,2)*(RI21*(P(1,3)-P(1,2))+RI22 *(P(1,6)-P(1,5)))+A(L,2)*  C(RI21*(P(L,2)-P(L,3))+RI2 2*(P(L,5)-P(L,6))) Q(2,M2)=RI21*P(I,1)+RI22*P(1,4)+A(1,5)*(RI21*P(I,2)+RI22*P (1,5))+A 0(1,7)*(RI21*P(I,3)+RI22*P(I,6))+A(L,7)*(RI21*P(L,2)+RI2 2*P(L,5))+A C(L,5)*(RI21*P(L,3)+RI2 2*P(L,6)) Q(2,M3)=A(I,6)*(RI21*(2.D0*P(I,2)+P(I,3))+Rl22*(2.D0*P(I,5)+P(1,6) C))-A(L,6)*(RI21*(2.D0*P(L,3)+P(L,2))+RI22*(2.DO*P(L,6)+P(L,5))) Q(3,M1)=RI21*P(I,1)+RI22*P(I,4)+A(I,1)*(RI21*P(I,2)+RI2 2*P(1,5))+A C(1,4)*(RI21*P(I,3)+RI22*P(I,6))+A(L,4)*(RI21*P(L,2)+RI22*P(L,5))+A C(L,1)*(RI21*P(L,3)+RI22*P(L,6))+Q(1,M2) Q(3,M2)=RI11*P(I,1)+RI12*P(1,4)+A(1,5)*(RI11*P(I,2)+RI12*P (1,5))+A C(I,7)*(RI11*P(I,3)+RI12*P(1,6))+A(L,7)*(RIll*P(L,2)+RI12*P(L,5))+A C(L,5)*(RI11*P(L,3)+RI12*P(L,6))+Q(2,M1) Q(3,M3)=-A(I,3)*(RI21*(2.D0*P(I,2)+P(1,3))+RI22*(2.DO*P(1,5)+P(1,6 C)))+A(I,6)*(Rill*(2.D0*P(I,2)+P(1,3))+RI12*(2.DO*P(1,5)+P(1,6)))+A C(L,3)*(RI21*(2.D0*P(L,3)+P(L,2))+Rl22*(2.DO*P(L,6)+P(L,5)))-A(L,6) C*(Rill*(2.D0*P(L,3)+P(L,2))+RI12*(2.D0*P(L,6).+P(L,5))) Q(3,M1)=Q(3,M1)/0.1048 808D+01 Q(3,M2)=Q(3,M2)/0.10488 08D+01 Q(3,M3)=Q(3,M3)/0.104 88 0 8D+01 CONTINUE CALL DIMULT(EXY,Q,SPAD,3,12,3,3,0) CALL REMULT(Q,SPAD,SPADl,12,3,3,3,1,0,NC) DO 6 1=1,78 SM(I)=SM(I)+SPAD1 (I) *W(K) *W(J) *TH*DET CONTINUE CONTINUE RETURN END  C  120 130 140  2  7  12 13  QUASTN - CALCULATION OF STRAINS IN QUADRILATERAL SUBROUTINE QUASTN(X,Y,DEF,EPS,IJ) IMPLICIT REAL*8(A-H,0-Z) DIMENSION X(4) ,Y(4) ,DEF (12) ,EPS(5,3) ,B(4) ,C (4) ,SS (5) ,TT(5) ,P (4,6) , 00(3,12),SPAD(3),A(4,7),RL(4),R(4,12) DATA SS/0.D0,-1.D0,1.D0,1.D0,-1.D0/,TT/0.D0,-1,D0,-1.D0,1.D0,1.D0/ DATA CT/.57735D0/ IF(IJ-1)130,120,130 NN=1 GO TO 140 NN=5 DO 2 1=1,4 J=I+1- (1/4)*4 B(I)=Y(J)-Y(I) C(I)=X(J)-X(I) RL(I)=DSQRT(B(I) *B(I)+C(I) *C(I) ) A(I,1) = ( (C(I)/RL(I) ) **2) *2.D0/3.D0+( (B(I)/RL(I) )* *2 ) *2 0. DO/27 . DO A (I, 2) = (B.(I) * C ( I ) / ( R L ( I ) **2) ) *2. DO/27. DO A(I,3)=B(I)*2.DO/27.DO A(I,4)=((C(I)/RL(I))**2)/3.D0+((B(I)/RL(I))**2)*7.DO/27.DO A (I,5)=((B(I)/RL(I))**2)*2.D0/3.D0+((C(I)/RL(I))**2)*20.DO/27.DO A(I,6)=C(I)*2.D0/27.D0 A(I,7)=((B(I)/RL(I))**2)/3.D0+( (C(I)/RL(I))**2)*7.DO/27.DO M=0 DO 15 0 K=1,NN IF (M)7,7,8 S=-CT T=S N=l GO TO 9 S=-T N=2 GO TO 9 T=S N=3 GO TO 9 h-  1  oo  N=4 GO TO 9 M=l N=5 T=TT(K) S=SS(K) RJ11=((T-1.D0)*(X(l)-X(2))+(T+1.D0)*(X(3)-X(4)))/4.DO RJ12= ( (T-1.D0) *(Y(1)-Y(2) ) + (T+l.D0) * (Y ( 3)- Y (4 ) ) )/4 . DO RJ21=((S-l.DO)*(X(l)-X(4))+(S+l.DO)*(X(3)-X(2)))/4.D0 RJ22=((S-l.DO)*(Y(l)-Y(4))+(S+l.DO)*(Y(3)-Y(2)))/4.DO DET=RJ11*RJ22-RJ12*RJ21 RIll=RJ22/DET RI12=-RJ12/DET RI21=-RJ21/DET RI22=RJ11/DET CON=-10.D0+9.D0*(S*S+T*T) P (1,1)= ( (T-l.DO)*CON+18.D0*S*(l.DO-S)*(l.DO-T))/32.D0 P(1,2)=(T-l.DO)*(18.D0*S*(l.D0-3.D0*S)+27.D0*(l.DO-S*S))/32.D0 P(1,3)=(T-l.DO)* (18.D0*S*(l.D0+3.D0*S)-27.DO*(l.DO-S*S))/32.D0 P (1,4)=((S-l.DO)*CON+18.D0*T*(l.DO-S)*(l.DO-T)J/32.D0 P (1,5)=-9.DO*(l.DO-S*S)*(l.D0-3.D0*S)/32.D0 P (1,6)=-9.DO*(l.DO-S*S)*(l.D0+3.D0*S)/32.D0 P(2,1)=((l.DO-T)*CON+18.D0*S*(l.DO+S)*(l.DO-T))/32.D0 P(2,2)=9.D0*(l.DO-T*T)*(1.DO-3.DO*T)/32.DO P(2,3)=9.DO*(l.DO-T*T)*(1.DO+3.DO*T)/32.DO P(2,4)=(-(l.DO+S)*CON+18.D0*T*(l.DO+S)*(l.DO-T))/32.D0 P(2,5)=-(l.DO+S)*(18.D0*T*(1.DO-3.DO*T) +27. ODO*(l.DO-T*T))/32.DO P (2 ,6)=- (l.DO+S)*(18.DO*T*(1.DO+3.DO*T)-27. DO*(l.DO-T*T) )/32.D0 P (3,1)=( (l.DO+T)*CON+18.D0*S*(l.DO+S)*(l.DO+T)J/32.D0 P(3,2)=(l.DO+T)*(-18.D0*S*(1.DO+3.DO*S)+27.DO*(l.DO-S*S)J/32.D0 P(3,3)=(l.DO+T)*(-18.D0*S*(1.D0-3.DO*S)-27.DO*(l.DO-S*S))/32.D0 P(3,4)=((l.DO+S)*CON+18.D0*T*(l.DO+S)*(l.DO+T)J/32.D0 P (3, 5) = 9. DO* (l.DO-S*S) * (1. DO+3 . DO*S)/32 . DO' P (3,6) = 9.DO*(l.DO-S*S)*(1.DO-3.DO*S)/32.DO P(4,1)=(-(1.DO+T)*CON+18.D0*S*(l.DO-S)*(l.DO+T))/32.D0  P (4,2)=-9.DO*(l.DO-T*T)*(1.DO+3.DO*T)/32.DO P (4,3)=-9.DO* (l.DO-T*T)*(1.DO-3.DO*T)/32.DO P (4,4)=((l.DO-S)*CON+18.D0*T*(l.DO-S)*(l.DO+T))/32.D0 P (4,5)=- (l.DO-S)*(18.D0*T*(1.DO+3.DO*T)-27.DO*(l.DO-T*T))/32.D0 P (4,6)=- (l.DO-S)*(18.D0*T*(1.DO-3.DO*T)+27.DO*(l.DO-T*T))/32.D0 DO 5 1=1,4 Ml=l+(1-1)*3 M2=M1+1 M3=Ml+2 L=I-1+(1/1)*4 IF (M)10,10,11 11 Q(1,M1)=RI11*P(I,1)+RI12*P(1,4)+A(1,1)*(RI11*P(I,2)+RI12*P (1,5))+A C (1,4)*(RI11*P(I,3)+RI12*P(1,6))+A(L,4)*(RIll*P(L,2)+RI12*P(L,5))+A C(L,1)*(RI11*P(L,3)+RI12*P(L,6)) Q(1,M2)=A(I,2) * ( R i l l * (P(I,3)-P(I,2) )+RI12* (P (1, 6 )-P (1, 5 ) ) )+A (L, 2 ) * C(Rill*(P(L,2)-P(L,3))+RI12*(P(L,5)-P(L,6))) Q(1,M3)=-A(I,3)*(Rill*(2.DO*P(1,2)+P(1,3))+RI12*(2.DO*P(I,5)+P (1,6 C)))+A(L,3)*(Rill*(2.D0*P(L,3)+P(L,2))+RI12*(2.DO*P(L,6)+P(L,5))) Q(2,M1)=A(I,2) * (RI21* (P (1, 3 ) -P (1, 2 ) )+RI22* (P (1, 6 ) -P (1, 5 ) ) )+A (L, 2) * C(RI21*(P(L,2)-P(L,3))+Rl22*(P(L,5)-P(L,6))) Q (2,M2)=RI21*P(1,1)+RI22*P(1,4)+A(I,5)*(RI21*P(1,2)+RI22*P(1,5))+A C (1,7)*(RI21*P(I,3)+RI22*P(1,6))+A(L,7)*(RI21*P(L,2)+RI22*P(L,5))+A C(L,5)*(RI21*P(L,3)+RI22*P(L,6)) Q(2,M3)=A(I,6)*(RI21*(2.D0*P(1,2)+P(I,3) )+RI22 *(2.DO*P(1,5)+P (1,6) C))-A(L,6)*(RI21*(2.D0*P(L,3)+P(L,2))+RI2 2*(2.D0*P(L,6)+P(L,5))) DO 17 II=M1,M3 Q(3,II)=R(1,II)*(l.DO-(S+T-S*T/CT)/CT)/4.D0+R(2,II)*(1.D0+(S-T-S*T C/CT)/CT)/4.D0+R(3,II)*(1.D0+(S+T+S*T/CT)/CT)/4.DO+R(4,II)* (l.DO-(S C-T+S*T/CT)/CT)/4.DO 17 CONTINUE GO TO 5 10 Q(1,M2)=A(I,2)*(RIll*(P(1,3)-P(1,2))+RI12*(P(I,6)-P(I,5)))+A(L,2)* C(RI11*(P(L,2)-P(L,3))+RI12*(P(L,5)-P(L,6))) Q (2,M1)=A(I,2) * (RI21* (P (1, 3 )-P (1, 2 ) ) +RI22* (P (1, 6 )-P (1, 5) ) )+A (L, 2 ) * C(RI21*(P(L,2)-P(L,3))+Rl2 2*(P(L,5)-P(L,6))) R(N,M1)=RI21*P(I,1)+RI22*P(I,4)+A(I,1)*(RI21*P(I,2)+RI2 2*P(1,5))+A 00  5 16 110 15 0  C(1,4)*(RI21*P(I,3)+RI22*P(1,6))+A(L,4)*(RI21*P(L,2)+RI22*P(L,5))+A C(L,1)*(RI21*P(L,3)+RI22*P(L,6))+Q(1,M2) R(N,M2)=RI11*P(I,1)+RI12*P(1,4)+A(1,5)*(RI11*P(I,2)+RI12*P(1,5))+A C (1,7)*(RI11*P(1,3)+RI12*P(1,6))+A(L,7)*(RI11*P(L,2)+RI12*P(L,5))+A C(L,5)*(RI11*P(L,3)+RI12*P(L,6))+Q(2,Ml) R(N,M3)=-A(I,3)*(RI21*(2.DO*P(I,2)+P(I,3))+RI22*(2.DO*P(I,5)+P(I,6 C)))+A(I,6)*(Rill*(2.D0*P(I,2)+P(1,3))+RI12*(2.D0*P(1,5)+P(1,6)))+A C(L,3)*(RI21*(2.D0*P(L,3)+P(L,2))+RI22*(2.DO*P(L,6)+P(L,5)))-A(L,6) C* (Rill*(2.D0*P(L,3)+P(L,2))+RI12*(2.DO*P(L,6)+P(L,5))) CONTINUE GO TO(12,13,14,15,16) ,N CALL MAMULT(Q,DEF SPAD,3,12,1,3,12,3,0,0) DO 110 1=1,3 EPS(K,I)=SPAD(I) CONTINUE RETURN END /  O  MAMULT - MATRIX MULTIPLICATION SUBROUTINE MAMULT(A,B,C,L,M,N,NDIMA,NDIMB,NDIMC,NAT,NBT) IMPLICIT REAL*8(A-H,0-Z) DIMENSION A(NDIMA,1),B(NDIMB),C(NDIMC,1) DO 1 1=1,L DO 2 J=1,N C (I,J)=0.DO DO 3 K=1,M IF(NAT)4,5,4 IF (NBT)6,7,6 C (I, J) =C (I, J) +A (I,K) *B (K, J) GO TO 3 C(I,J)=C(I,J)+A(I,K) *B(J,K) GO TO 3 IF (NBT)8,9,8 C (I, J)=C (I, J)+A(K,I) *B (K, J) GO TO 3 C (I, J)=C(I,J)+A(K,I) *B(J,K) CONTINUE CONTINUE CONTINUE RETURN END  DIMULT - MULTIPLICATION OF SQUARE SYMMETRIC MATRIX BY A RECTANGULAR MATRIX WHERE UPPER TRIANGLE OF SYMMETRIC MATRIX IS STORED AS VECTOR SUBROUTINE DIMULT(A,B,C,N,M,NDIMB,NDIMC,NBT) IMPLICIT REAL*8(A-H,0-Z) DIMENSION A(l),B(NDIMB,1),C(NDIMC,1) DO 1 1=1,N DO 2 J=1,M C (I, J) =0. DO 11=1 DO 3 K=1,N IF (NBT)7,6,7 C(I,J)=C(I,J)+A(II) *B(K,J) GO TO 8 C(I,J)=C(I,J) + A (II) *B(J,K) IF (I-K)4,4,5 II=II+N-K GO TO 3 11=11+1 CONTINUE CONTINUE CONTINUE RETURN END  C C  5 7 6 4 9 8 3 2 1  REMULT - MULTIPLICATION OF TWO RECTANGULAR MATRICES WHERE THE PRODUCT IS KNOWN TO BE A SYMMETRIC SQUARE MATRIX STORED AS A COLUMN VECTOR SUBROUTINE REMULT(A,B,C,M,N,NDIMA,NDIMB,NAT,NBT,NC) IMPLICIT REAL*8(A-H,0-Z) DIMENSION A(NDIMA,1),B(NDIMB,1),C(1) 11=0 DO 1 1=1,M DO 2 J=I,M 11=11+1 C (II) = 0.DODO 3 K=1,N IF (NAT)4,5,4 IF (NBT)6,7,6 C(II)=C(II)+A(I,K)*B(K,J) GO TO 3 C(II)=C(II)+A(I,K)*B(J,K) GO TO 3 IF (NBT)8,9,8 C(II)=C(II)+A(K,I)*B(K,J) GO TO 3 C(II)=C(II)+A(K,I)*B(J,K) CONTINUE CONTINUE CONTINUE NC=II RETURN END  A.5  Performance of Elements i n S e l e c t e d  Problems  Both the t r i a n g u l a r and q u a d r i l a t e r a l elements are used i n t h r e e d i f f e r e n t  A.5.1  problems.  C a n t i l e v e r w i t h P a r a b o l i c End Load: The dimensions  of the c a n t i l e v e r and the d i f f e r e n t g r i d s used are shown i n F i g . A.7.  In F i g . A.8 the s t r a i n energy convergence  toward  the e x a c t v a l u e w i t h p r o g r e s s i v e g r i d refinement i s compared w i t h the well-known  c o n s t a n t and l i n e a r s t r a i n  Note the non-monotonic  triangles.  convergence of the q u a d r i l a t e r a l s ,  which i s a t t r i b u t a b l e t o the lower o r d e r i n t e g r a t i o n p r e v i o u s l y d e s c r i b e d . A h i g h e r o r d e r i n t e g r a t i o n scheme e n s u r i n g exact i n t e g r a t i o n o f a l l terms i n the i n t e g r a n d m a t r i x r e s u l ted i n monotonic  convergence f o r t h i s problem.  Some of the  c a l c u l a t e d s t r e s s e s i n the c a n t i l e v e r are compared w i t h e n g i n e e r i n g beam t h e o r y i n F i g . A.9.  A.5.2  P a r a b o l i c a l l y Loaded Square P l a t e : T h i s problem  has  become a p o p u l a r t e s t of the c a p a b i l i t i e s o f a p l a n e s t r e s s element. The dimensions of the p l a t e and the d i f f e r e n t used are shown i n F i g . A.10, w h i l e s t r a i n energy  convergence  i s again compared w i t h the c o n s t a n t and l i n e a r s t r a i n angles i n F i g . A.11.  grids  tri-  A comparison o f s t r e s s e s and d e f l e c -  t i o n s w i t h e x a c t v a l u e s i s t a b u l a t e d i n ^ T a b l e IX.  A.5.3  Short Deep Beam: T h i s problem was i n c l u d e d  u n l i k e the p r e v i o u s two, a r e l a t i v e l y  because,  l a r g e p r o p o r t i o n of the  t o t a l s t r a i n energy comes from shear. B e a r i n g i n mind the p r o -  195 blems t h a t were encountered elements i n pure shear,  w i t h the c o n s t r a i n e d c u b i c  i t was c o n s i d e r e d t h a t the s h o r t  deep beam would p r o v i d e a r i g o r o u s t e s t of these elements. The  beam dimensions and f i n i t e element g r i d s are shown i n  F i g . A.12.  U n f o r t u n a t e l y an exact s o l u t i o n t o t h i s problem  does not e x i s t , and although v a r i a t i o n of s t r a i n energy w i t h g r i d s i z e i s compared w i t h the constant and l i n e a r  strain  t r i a n g l e s i n F i g . A.13, t h e r e i s no i n d i c a t i o n o f how c l o s e l y the s t r a i n energy approaches the exact v a l u e .  Calculated  s t r e s s e s compare f a v o u r a b l y with an approximate s e r i e s t i o n by von Kantian * * i n F i g . A. 14. 1  1  solu-  196  T h i c k n e s s = 1 in. c CM  Y  E = 30x10  k.s.i.  V=0.25  • 1 -ag  48in. 40 k i p . Support x = y=fixed  nodes  r o t a t ion = f r e e  1 x 4 grid  2 x 8 grid  \/ / \  I / /  \ 4 x 16 g r i d  Fig.A.7  C a n t i l e v e r with p a r a b o l i c  end  load.  110 +  80  70 + A Constrained 60  0 Linear  cubic  strain  ©Constant  triangle  strain  • Constrained  50  triangle  triangle  cubic  quadrilateral  j  1x4  2x8  4x16 Grid  Fig.A.8  Strain  energy  8x32  size.  convergence  for  cantilever.  A Interpolated  from  •  quadrilateral  Averaged  at  Interpolated  6  triangle  centroids nodes  from quadrilateral  centroids  -A  1  CD C I  CL>  4  i_  c  3  +  Q) U r  -  c CD CD  > o  < (cju nd  cd  LO  Q  Beam  Beam  theory  theory  0  -As 0  20 X- stress  60  40 (k.s.i.)  Fig. A.9 section  -10  1-  0  10  40  from  1  in c a n t i l e v e r at support  for  + 2  Shear  Y - s t r e s s (k.s.i.)  Stresses 12in.  H  4x16  crossgrid.  +3 stress  l  4  5  — h  (k.s.i.)  6  F i g . A.10 parabolically  One  quadrant  of  l o a d e d square plate.  100  j3  A Constrained 0  90-  Linear  triangle.  strain triangle.  0 Constant •  cubic  strain triangle.  Constrained  cubic  quadrilateral  88 + 1x1  2x2  4x4 Grid  Strain  energy  convergence  8x8  size.  for p a r a b o l i c a l l y l o a d e d s q u a r e  plat  TABLE IX COMPARISON OF STRESSES AND DEFLECTIONS FOR PARABOLICALLY LOADED SQUARE PLATE Refer t o F i g . A.10 f o r Key  TRIANGLE  RECTANGLE  Element  EXACT  Grid  u  V  d in.  V  0  0  c in. .005077  xa k.s.i. .015849 -1.79  k.s.i. 47.10  xb k.s.i.  yb k.s.i.  -10.41  12.63  a xc k.s.i. 8.45  U  lxl  -.004521  c in. .001373  2x2  -.004382  .001154  .004452  .016273  -4.81  47.36  - 5.00 18.57  1.58  7.2055  4x4  -.004339  .001152  .004233  .016381 -6.81  46.23  - 1.79  20.89  -1.01  7.2239  8x8  -.004328  .001162  .004162  .016407  45.92  - 0.60  21.61  -1.31  7.2292  Lxl  -.003272  .000932  .006353  .014668  2x2  -.004087  .001103  .005052  .016211 -6.41  43.19  - 1.41 28.19  0.84  7.2655  4x4  -.004087  .001386  .004574  .016463 -8.21  44.88  -0.92  24.83  0.61  7.2859  8x8  -.004086  .001527  .004367  .016522  -8.73  45.45  -0.63  23.35  0.20  7.2982  -.005066  .000595  .004258  .016912  -7.518  45.816  21.902  0  7.4495  in.  -7.34  ya  kip. i n . 7.1088  6.9910  0  202  2 00 k i p .  F i g . A.12  One  half of s h o r t  deep  beam.  6.0  5.0  ^  ^ ^ ^ ^ ^  \^^^  c c  i A Constrained  2.0  0 Linear  quadrilateral j.  1  2x2  UxU Grid  S t r a i n energy  cubic  triangle  l  I  1x1  triangle  triangle  strain  • Constrained  1.0  Fig. A.13  strain  ©Constant  (  c: u b i c  versus  8x8  size.  grid  size  for  short  deep  beam. o  A Interpolated from triangle c e n t r o i d s . •  A v e r a g e d at q u a d r i l a t e r a l  nodes.  ® Interpolated from quadrilateral centroids.  f-H  1  1  1  1  1  1  -30 -20  -10  0  10  20  30  X - s t r e s s (k.s.i.)  Fig.A.K  Stresses  for 8 x 8 grid.  1—  40  —!  1  -10 - 5  1—  0  Y - s t r e s s (k.s.i.)  —I  0 Shear  1  1  {-  5  10  15  stress  (k.s.i.)  in short deep beam at cross-section 3 in. f r o m midspan  R e s u l t s are c o m p a r e d w i t h an  a p p r o x i m a t e series s o l u t i o n .  APPENDIX B LOAD-DEFLECTION CURVE GENERATOR FOR HEADED STUD CONNECTIONS  B.l  DELTA  L i s t o f V a r i a b l e s i n Subroutine Input/Output  List  Shear d e f l e c t i o n increment a p p l i e d t o connection I  PR(I,J)  M a t r i x which s t o r e s l o a d - d e f l e c t i o n and materi a l p r o p e r t y h i s t o r y of c o n n e c t i o n I. I dimension = number o f c o n n e c t i o n s i n problem or  greater.  J dimension = 18 NDIMPR  The f i r s t dimension of PR(I,J)  I  Connection number.  RL  Stud bending l e n g t h as c a l c u l a t e d  i n Chapter 7  T o t a l u l t i m a t e f o r c e i n c o n c r e t e under studs a R1MAX  c a l c u l a t e d i n Chapter 7. U l t i m a t e end-bearing r e s i s t a n c e o f f a c e p l a t e a  R2MAX  c a l c u l a t e d i n Chapter 7.  206 XC(J),YC(J) Three X and Y c o o r d i n a t e s o f t r i l i n e a r  total  l o a d - d e f l e c t i o n curve, f o r a l l studs i n a conn e c t i o n , over bending  l e n g t h RL, d e f i n e d i n  F i g . 5.7. P  Shear l o a d on c o n n e c t i o n a f t e r a p p l i c a t i o n o f d e f l e c t i o n increment DELTA.  D  Shear d e f l e c t i o n of c o n n e c t i o n a f t e r  applica-  t i o n o f d e f l e c t i o n increment DELTA. IFNEW  =-ve,0, elements  i n PR(I,J) are not r e s e t ,  a l l o w i n g f o r another t r i a l  on the next  itera-  tion. =+ve, elements i n PR(I,J) are r e s e t . Rl  That p o r t i o n o f P r e s i s t e d by c o n c r e t e under studs.  R2  That p o r t i o n of P r e s i s t e d by end-bearing o f f a c e p l a t e on c o n c r e t e .  R3  That p o r t i o n o f P r e s i s t e d by studs i n bending.  B. 2  Notes  B.2.1  The shear l o a d Po on a c o n n e c t i o n I, b e f o r e a p p l i c a t i o n of d e f l e c t i o n increment DELTA, i s s t o r e d i n PR(I,D • The shear d e f l e c t i o n D  0  o f a c o n n e c t i o n I, b e f o r e  a p p l i c a t i o n o f d e f l e c t i o n increment DELTA, i s s t o r e d i n PR(I, 2) .  B. 2. 2  The cumulative r o t a t i o n o f the studs i n a connec-  207 tion,  equation  (4.2), i s s t o r e d i n PR(I,16) and  may be used as an i n d i c a t o r o f stud  B.2.3  A l l elements of PR(I,J) must be s e t t o zero before s u b r o u t i n e STUDCO i s f i r s t  B.2.4  fracture.  called.  The s u b r o u t i n e makes use of the c o n c r e t e degradat i o n model of F i g . 5.3 and the t r i l i n e a r  hystere-  s i s model of F i g . 5.7.  B.2.5  A l l i n p u t loads may be i n any u n i t o f measurement d e s i r e d , p r o v i d e d t h a t the same u n i t i s used cons i s t e n t l y throughout.  The u n i t f o r d e f l e c t i o n s i s  the i n c h , but may be changed t o any o t h e r u n i t by changing  the DATA l i n e i n STUDCO.  The v a r i a b l e s  D10, D20 and D2MAX i n t h i s l i n e a r e s p e c i f i e d i n i n c h u n i t s and should be m u l t i p l i e d by the approp r i a t e f a c t o r f o r c o n s i s t e n c y w i t h the new u n i t o f l e n g t h measurement.  B. 3  Subroutine  STUDCO  A l i s t i n g o f the s u b r o u t i n e appears on the pages that follow.  STUDCO - LOAD-DEFLECTION CURVE GENERATOR FOR HEADED STUD CONNECTIONS SUBROUTINE STUDCO(DELTA,PR,NDIMPR,I,RL,RlMAX,R2MAX,XC,YC,P,D,IFNEW C,R1,R2,R3) IMPLICIT REAL*8(A-H,0-Z) DIMENSION PR(NDIMPR,1),XC(3),YC(3) DATA DlO/0.5D-01/,D20/0.25D-01/,D2MAX/0.9D-01/ DTHETA=DELTA/RL DD2=RL*DTHETA DD3=DD2 THETA=PR(I,2)/RL D2=RL*THETA D3 = D2 D=PR(I,2)+DELTA PR(I,16)=PR(1,16)+DABS(DELTA/RL) IF(.1D0-DABS(D))60,60,61 FACT=2.D0 GO TO 62 FACT=.1D+01+.3D+03*D**2-.2D+04*DABS(D**3)) D1=PR(I,15) DDl=D/FACT-Dl IF(R1MAX-.1D-05)90,90,91 R1=0.D0 GO TO 5 IF(D1+DD1)1,2,2 IF(PR(I,3) )3,4,4 F0=0.DO GO TO 7 0 F0=PR(1,3)/RlMAX S0=D1/D10 IF(DABS(S0)-.lD-05)80,8 0,81 S0=.1D-05 DELS=DD1/D10 SE=PR(I,4) SEHAT=PR(I,5) J=l CALL STRESN(F0,S0,DELS,0.DO,SE,SEHAT,J,F,S,DFDS)  9  8 1 7 6 71 82 83  11  10 5 92 93 22 23 24 72 84  IF(IFNEW)8,8,9 PR(I,3)=F*R1MAX PR(I,4)=SE PR(I,5)=SEHAT PR(I,15)=S*D10 R1=F*R1MAX GO TO 5 IF(PR(I,3))6,6,7 F0=0.D0 GO TO 71 FO-PR (1, 3) /RlMAX S0=-D1/D10 IF(DABS(SO)-.1D-05)82,82,83 S0=.lD-05 DELS=-DD1/D10 SE=PR(I,6) SEHAT=PR(1,7) J=l CALL STRESN(FO,S0,DELS,0.DO,SE,SEHAT,J,F,S,DFDS) IF(IFNEW)10,10,11 PR(I,3)=-F*R1MAX PR(I,6)=SE PR (1,7)=SEHAT PR(I,15)=-S*D10 Rl=-F*RlMAX IF(R2MAX-.1D-05)92,92,93 R2=0.D0 GO TO 25 IF(D2+DD2)21,22,22 IF(PR(I,8))23,24,24 F0=0.D0 GO TO 72 F0=PR(I,8)/R2MAX S0=D2/D20 IF(DABS(S0)-.lD-05)84,84,85 S0=.lD-05  o >^3  85  DELS=DD2/D2 0 SE=PR(I 9) SEHAT=PR(I,10) J=l CALL STRESN(FO,SO,DELS,O.DO, SE,SEHAT,J,F,S,DFDS) IF(S-D2MAX/D20)51,50,50 F=0.D0 SE=10.D0 IF (IFNEW)28,28,29 PR(I,8)=F*R2MAX PR (I, 9)=SE PR(I,10)=SEHAT R2=F*R2MAX GO TO 25 IF(PR(I,8))26,26,27 F0=0.D0 GO TO 7 3 F0=-PR(I 8)/R2MAX S0=-D2/D20 IF(DABS(SO)-.1D-05)86,86,87 S0=.lD-05 DELS=-DD2/D20 SE=PR(I,11) SEHAT=PR(I,12) J=l CALL STRESN(F0,S0,DELS,0.D0, SE,SEHAT,J,F,S,DFDS) IF (S-D2MAX/D20) 53,52,52 F=0.D0 SE=10.DO IF(IFNEW)30,30,31 PR(I,8)=-F*R2MAX PR(I,11)=SE PR(I,12)=SEHAT R2=-F*R2MAX Y0=PR(I,13) XP=PR(I,14) f  50 51 29 28 21 27 26 73 86 87  52 53 31 30 25  /  42 41  44 43  CALL TRILIN(D3,Y0,DD3,XP,XC,YC,X,Y,DYDX) IF(IFNEW)41,41,42 PR(I,13)=Y PR(I,14)=XP R3=Y DO=XC(2) CALL DEGRAD(PR,NDIMPR,R3,D,DO,I,IFNEW,FACT) P=R1+R2+R3*FACT IF(IFNEW)43,43,44 PR(I,1)=P PR(I,2)=D RETURN END  r—  1  STRESN - STRESS-STRAIN DEGRADATION MODEL FOR CONCRETE SUBROUTINE STRESN(FO,SO,DELS,FTENS,SE,SEHAT,IFCRAK, F , S ,DFDS) IMPLICIT REAL*8(A-H,0-Z) S=S0+DELS I F ( 4 . D 0 - S ) l 1,2 SE=5.D0 F=0.D0 DFDS=0.D0 GO TO 3 IF(4.D0-SE)4,4,5 IF(DELS)6,7, 7 ** LOADING CURVE **** IF(DABS(FE(SO)-F0)-l.D-6)8,8,9 F=FE(S) DFDS=DFDSl (S) SE=S GO TO 3 IF(SP(SE)-S)10,10,11 IF(F0)12,4,4 IF(SR(SE)-S)8,8,13 IF(SC(SE)-S)14,15,15 F=(SO-SP(SE))*DFDS2(SE) IF (FO-F)16,12,12 IF(SE-.250+00)12,17,17 SI= (FO+SP(SE)*DFDS2(SE)-S0*DFDS3(SE))/(DFDS2(SE)-DFDS3(SE)) IF(S-SI)18,12,12 F=F0+(S-SO)*DFDS3(SE) DFDS=DFDS3(SE) GO TO 3 F=(S-SP(SE))*DFDS2(SE) DFDS=DFDS2(SE) GO TO 3 IF(FO-FC(SC(SE)))35,35,36 F=F0+(S-SO)*DFDS4(SE) IF(F-FE(S)) 37 ,37,8 F=FC(SC(SE)) + (S-SC(SE) ) *DFDS4 (SE) f  37 19 C 6 32 31 20 23 22 24 25 21 26 28 27 29 30 3  DFDS=DFDS4(SE) IF(SE-.25D0)3,3,19 SEHAT=(F-FE(SE)+SE*DFDS1(SE)-S*DFDS3(SE))/(DFDS1(SE)-DFDS3(SE)) GO TO 3 **** UNLOADING CURVE **** IF(SEHAT-SE)31,31,32 SE=SEHAT IF(S-SP(SE))20,21,21 . IF(IFCRAK)23,22,23 F=0.D0 DFDS=0.DO GO TO 3 F=(S-SP(SE))*DFDS2(SE) IF(F-FTENS)2 4,24,25 IFCRAK=1 GO TO 23 DFDS=DFDS2(SE) GO TO 3 F=FT(ST(SE))+(SO-ST(SE))*DFDS2(SE) IF (FO-F)26,26,27 F=FT(ST(SE))+(S-ST(SE))*DFDS2(SE) DFDS=DFDS2(SE) IF(F)28,28,3 F=0.D0 DFDS=0.DO GO TO 3 IF (SE-.2500)26,26,29 SI= (FO-FT (ST (SE) )+ST(SE) *DFDS2 (SE) -S.0*DFDS3 (SE) )/ (DFDS2 (SE) -DFDS3 ( CSE) ) IF (S-SI) 26,26, 30 F=F0+(S-SO)*DFDS3(SE) DFDS=DFDS3(SE) RETURN END FUNCTION FE(A) IMPLICIT REAL*8(A-H,0-Z)  1 2 3 4 5  FE=A*(2.7182 8182 85D+00**(l.DO-A)) RETURN END FUNCTION DFDS1(A) IMPLICIT REAL*8(A-H,0-Z) DFDS1=(l.DO-A)*(2.7182 818285D+00**(l.DO-A)) RETURN END FUNCTION SP(A) IMPLICIT REAL*8(A-H,0-Z) SP=(0.145D+00*A*A)+(0.13D+00*A) RETURN END FUNCTION SR(A) IMPLICIT REAL* 8(A-H,0-Z) SR=(-.091D+00+DSQRT(.8281D-02+(.372D+00*SP(A))))/.186D+00 RETURN END FUNCTION SC(A) IMPLICIT REAL*8(A-H,0-Z) SC=(-0.141D+00+DSQRT(.19881D-01+(.68D+00*SP(A))))/.3 4D.00 RETURN END FUNCTION DFDS2(A) IMPLICIT REAL*8(A-H,0-Z) IF (A-.1D-01)1,2,2 DFDS2=2.7182818285D+00 GO TO 5 IF(A-.25D+00)3,4,4 DFDS2=SLOPE(A) GO TO 5 DFDS2=FC(SC(A))/(SC(A)-SP(A)) RETURN END FUNCTION FC(A) IMPLICIT REAL*8(A-H,0-Z) i—'  FC=A*(2. 7182818285D+00**(1.DO-1.17D0*A)) RETURN END FUNCTION SLOPE(A) IMPLICIT REAL*8(A-H,0-Z) SLOPE=FE(A)/(A-SP(A)) RETURN END FUNCTION DFDS3(A) IMPLICIT REAL*8(A-H,0-Z) IF(A-.25D+00)1,2,2 DFDS3=DFDS2(A) GO TO 3 DFDS3=(FE(A)-FC(SC(A)))/(A-SC(A)) RETURN END  FUNCTION DFDS4(A) IMPLICIT REAL*8(A-H,0-Z) IF(A-.05D+00)1,2,2 DFDS4=DFDS2(A) GO TO 3 DFDS4= (FE (SR (A) ) -FC (SC (A) ) ) / (SR (A) -SC (A) ) RETURN END FUNCTION ST(A) IMPLICIT REAL*8(A-H,0-Z) Sl= (-.218D+00+DSQRT(. 47 52 4D-01+(.64D0*SP(A))))/.32D+00 DFDS=.675D0*(1.DO-1.17D0*A)*2.7182818285D0**(1.DO-1.17D0*A) ST=(FT(SI)-FE(A)+A*DFDS3(A)-S1*DFDS)/(DFDS3(A)-DFDS) RETURN END FUNCTION FT(A) IMPLICIT REAL*8(A-H,0-Z) FT=.675D+00*A*(2.7182818285D+00**(1.DO-1.17D0*A)) RETURN END  C  10 11 2  1 3 5 6  7 12  4 13 .8 14  T R I L I N - TRILINEAR HYSTERESIS LOOP GENERATOR SUBROUTINE T R I L I N ( X O , Y O , D X , X P , X C , Y C , X , Y , D Y D X ) IMPLICIT R E A L * 8 ( A - H , 0 - Z ) DIMENSION X C ( 3 ) , Y C ( 3 ) X=X0+DX IF(XC(3)-DABS(XP))9,9,10 IF(XC(3)-DABS(X))9,9,11 IF(XP-X)1,1,2 IFBEL=-1 XP=-XP X=-X DX=-DX X0=-X0 Y0=-Y0 GO TO 3 IFBEL=1 IF(DX)4,5,5 IF ( X Q ( X C , Y C , X P ) - X ) 6 , 6 , 7 Y=Y3(XC,YC,X) DYDX=DFDX3(XC,YC) GO TO 8 IF ( X T ( X C , Y C , X P ) - X ) 1 2 , 1 3 , 1 3 Y=Y2(XC,YC,XP,X) DYDX=DFDX2(XC,YC) GO TO 8 XP=XPNEW(XC,YC,X0,Y0) Y=Y1(XC,YC,XP,X) DYDX=DFDX1(XC,YC) IF(IFBEL)14,15,15 XP=-XP Y=-Y X=-X DX=-DX X0=-X0 Y0=-Y0 GO TO 15 NJ r1  9 15  XP=2*XC (3) Y=O.DO DYDX=O.DO RETURN END FUNCTION DFDX1(X,Y) IMPLICIT REAL*8(A-H,0-Z) DIMENSION X(3),Y(3) DFDX1=Y(1)/X(1) RETURN END FUNCTION DFDX2(X,Y) IMPLICIT REAL*8(A-H,0-Z) DIMENSION X(3),Y(3) DFDX2= (Y(l)-Y(2) ) / ( X ( l ) - X ( 2 ) ) RETURN END FUNCTION DFDX3(X,Y) IMPLICIT REAL*8(A-H,0-Z) DIMENSION X (3) ,Y (3) DFDX3=(Y(2)-Y(3))/(X(2)-X(3)) RETURN END FUNCTION XT(X,Y,XP) IMPLICIT REAL*8(A-H,0-Z) DIMENSION X(3),Y(3) XT=(Y(1)+DFDX1(X,Y)*XP-DFDX3(X,Y)*X(1))/(DFDX1(X,Y)-DFDX3(X,Y)) RETURN END FUNCTION XQ(X,Y,XP) IMPLICIT REAL*8(A-H,0-Z) DIMENSION X(3),Y(3) XQ=(Y(2)-Yl(X,Y,XP,XT(X,Y,XP) )+DFDX2(X,Y)*XT(X,Y,XP)-DFDX3(X,Y)*X ( C2) ) / (DFDX2 (X, Y) -DFDX3 (X, Y) ) RETURN END NJ  FUNCTION Yl(X,Y,XP,XX) IMPLICIT REAL*8(A-H,0-Z) DIMENSION X(3),Y(3) Y1=DFDX1(X,Y)*(XX-XP) RETURN END FUNCTION Y2(X,Y,XP,XX) IMPLICIT REAL*8(A-H,0-Z) DIMENSION X(3),Y(3) X O X T (X,Y,XP) Y2=Y1(X,Y,XP,X0)+DFDX2(X,Y)*(XX-XO) RETURN END FUNCTION Y3(X,Y,XX) IMPLICIT REAL*8(A-H,0-Z) DIMENSION X(3),Y(3) Y3=Y(2)+DFDX3(X,Y)*(XX-X(2) ) RETURN END FUNCTION XPNEW(X,Y,XN,YN) IMPLICIT REAL*8(A-H,0-Z) DIMENSION X(3) ,Y(3) XPNEW=XN-YN/DFDX1(X,Y) RETURN END  OO  C  1 3 2 7 8 9  6 4 5  DEGRAD - CONTROLS DEGRADATION OF STUD LOAD-DEFLECTION LOOPS SUBROUTINE DEGRAD(PR,NDIMPR,R3,D,DO,I,IFNEW,FACT) IMPLICIT REAL*8(A-H,0-Z) DIMENSION PR(NDIMPR,1) IF(R3)1,2,3 IF(PR(I,1) )4,4,2 IF(PR(I,1))2,4,4 IF(DABS(R3-PR(I,1))-.1D-10)7,7,8 CROSS=D GO TO 9 CROSS=D+(R3/DABS(R3-PR(I,1)))*(PR(I,2)-D) DIST=DABS(PR(1,17)-CROSS)/2.DO DEL=PR(I,18)+(.822D+01*(DIST/DO)+0.935D+02*(DIST/DO)**2)*0.5D-03 FACT=1.DO-DEL IF(IFNEW)5,5,6 PR(I,17)=CROSS PR(I,18)=DEL GO TO 5 FACT=1.D0-PR(I,18) RETURN END  APPENDIX C CONTROL OF AXIAL LOAD AND MOMENT INTERACTION FOR STUDS IN ITERATIVE CALCULATIONS  Suppose t h a t a t some c r o s s - s e c t i o n i n a stud the v a l u e o f a x i a l load i s P', and the bending moment M', a t a p a r t i c u l a r stage o f a c a l c u l a t i o n .  The next  increment  i n the c a l c u l a t i o n , u s i n g tangent s t i f f n e s s e s from the p r e v i o u s increment, y i e l d s approximate and AM' i n P' and M  1  respectively.  increments o f AP  1  This r e s u l t s i n a point  on the i n t e r a c t i o n diagram o f F i g . C . l w i t h c o o r d i n a t e s  M +AM M , Pl 1  1  P'+AP  1  V  >  The s l o p e , e, o f the l i n e AB i n F i g . C . l i s P'+AP  1  e --  Any  pl M +AM M Pl 1  (C.l)  1  adjustments t o AP' and AM' w i l l be c a r r i e d out  along l i n e AB. F i g . 5.9 which  F i g . C . l i s the same i n t e r a c t i o n diagram as i s d e s c r i b e d by equations  (5.12) and (5.13).  221  By s u b s t i t u t i n g e -  it  i n equation  J2i M M Pi  (5.12)  can be shown t h a t l+4e 2e  Pi  = A+4e  V  2  2  1  (C2)  - 1  (C.3)  2e  The d i s t a n c e  i n F i g . C . l i s g i v e n by  From equations  (C.2) and (C.3)  r = e  'l+4e - 1 2e  2r  2^ 1+e 2 I e J  Using a s i m i l a r s u b s t i t u t i o n o f e i n e q u a t i o n  pl  1+e  M M  Pl  1+e  (C.4)  (5.13) (C.5)  (C.6)  whence r  i  1+e' (1+e)  (C.7)  222 The d i s t a n c e r i n F i g . C . l i s g i v e n by  P'+AP  M'+AM  1  r =  Let F  p  and 1.0  pi J  and F  m  be s c a l i n g  f o r F , and  Pl purpose of these s c a l i n g  (i)  (C.8)  f a c t o r s with values  between  M  between  and 1.0 f o r F . The pl m f a c t o r s w i l l become c l e a r s h o r t l y , M  For any p o i n t which l i e s below the s t r a i g h t tion line,  interac-  0 £ r £ r ,  F  = F = m  p (ii)  1  +  1.0  For any p o i n t which l i e s o u t s i d e of the curved i n t e r action line, r ^ r , e' F  (iii)  P  =  F Pl  m  =  M M  Pl  For any p o i n t which l i e s between curved i n t e r a c t i o n r-r r -r e i between  1.0  1.0-  and  the s t r a i g h t and  lines, r < r < r , i e which i s a l i n e a r  'Pl P Pl  interpolation  S i m i l a r l y F. m  r-r  1.0-  r -r e  i  M M  Pl  The s c a l i n g f a c t o r s F and F are a p p l i e d t o the ^ p m ^ s t r e s s - s t r a i n and moment-curvature  diagrams r e s p e c t i v e l y ,  f o r the c r o s s s e c t i o n under c o n s i d e r a t i o n .  This  forces  the diagrams t o conform t o the i n t e r a c t i o n curve as y i e l d i n g commences.  C o n s i d e r , f o r example, the  moment-curvature  diagram of F i g . C.2, which i s s e l f - e x p l a n a t o r y .  224  Curvature  Fig. C.2  Moment-curvature  diagram.  

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