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Reliability of slender reinforced concrete columns Bhola, Rajendra Kumar 1985

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R E L I A B I L I T Y OF SLENDER REINFORCED CONCRETE COLUMNS  by RAJENDRA B. T e c h . ,  KUMAR  Indian I n s t i t u t e  BHOLA  of Technology,  New  Delhi,1983  A THESIS SUBMITTED IN PARTIAL FULFILMENT THE REQUIREMENTS MASTER  FOR THE DEGREE OF  OF APPLIED  SCIENCE  in FACULTY OF GRADUATE Department  of C i v i l E n g i n e e r i n g  We a c c e p t t h i s to  thesis  the r e q u i r e d  THE UNIVERSITY  as c o n f o r m i n g standard  OF BRITISH COLUMBIA  January,  ©  STUDIES  1985  R a j e n d r a Kumar B h o l a ,  1985  OF  In  presenting  requirements BRITISH freely that  this  thesis  in  partial  f o r an a d v a n c e d d e g r e e a t t h e THE UNIVERSITY OF  COLUMBIA,  I  agree  t h a t t h e L i b r a r y s h a l l make i t  a v a i l a b l e f o r r e f e r e n c e and s t u d y . permission  scholarly Department  f o r extensive  purposes or  may  by  be  copying  granted  h i s or  her  by  that copying  or publicatiorT-of  financial  gain  not  shall  be  allowed  permission.  Department of C i v i l  Engineering  THE UNIVERSITY OF B R I T I S H COLUMBIA 2075 W e s b r o o k P l a c e V a n c o u v e r , Canada  1W5  Date: January,  1985  I  further  of t h i s the  Head  this without  agree  thesis for  representatives.  understood  V6T  f u l f i l m e n t of the  of  my  It is  thesis for my w r i t t e n  ABSTRACT The on  effects  of the v a r i a b i l i t y  the r e l i a b i l i t y  of s l e n d e r ,  are  investigated using  The  columns a r e c o n s i d e r e d  end  eccentricities  of  axial  Implicit the  loads  the f a i l u r e  hundred sections.  has  following  CAN3-A23.3-M77 A  magnification, recommended  h a s been d e v e l o p e d  of  found  failure  consistent reinforced  the  #  ,  (as  order  to  level concrete  of  to  of  one  find  in  f o r three d i f f e r e n t  slenderness  the  in  c a l l e d the  levels  in m  eccentricity  load at various e c c e n t r i c i t y  and CSA-A23.3  change  equal  of the  been  Seven d i f f e r e n t  by  l o a d and  A new p r o c e d u r e  e a c h c r o s s s e c t i o n . The r e s u l t s  obtained  with  f u n c t i o n from t h e v a l u e s  to a p r o b a b i l i t y  thousand  technique.  above.  allowable axial  corresponding  loaded  columns  load.  are considered.  of  concrete  simulation  t o be a x i a l l y  in strength, a x i a l  v a r i a b l e s named The  for  t h e Monte C a r l o  U n c o r r e l a t i o n Procedure  values  basic  reinforced  and no l a t e r a l  Variabilities  i n s t r e n g t h and l o a d i n g  code  ratios  are  cross  considered  a r e compared w i t h procedures  one  those  outlined  in  (1984).  performance given obtain  reliability  columns.  i i  factor  in a  for  CSA-A23.3 more  (1984)) i s  accurate  i n the design  moment  and  of slender  TABLE  OF CONTENTS  ABSTRACT  i i  LIST OF FIGURES  v  LIST OF TABLES  vi  ACKNOWLEDGEMENT 1.  vii  INTRODUCTION  1  1 . 1 CODE METHOD  2.  .  1  1.1.1 Method of CAN3-A23.3-M77  2  1.1.2 Method of CSA A23.3( 1984)  3  MONTE CARLO SIMULATION  5  2.1 INTRODUCTION  5  2.2 DESCRIPTION OF THE METHOD  6  2.3 VARIABILITY OF STRENGTH  6  2.3.1 Properties of concrete  8  2.3.2 Properties of reinforcement  9  2.3.3 Geometric Properties  10  2.4 VARIABILITY OF LOADS  11  2.4.1 V a r i a b i l i t y of Dead Loads  12  2.4.2 V a r i a b i l i t y of Live Loads  .13  2.4.3 Load Combination 3.  STRENGTH MODEL  .... 1 3  ;  16  3.1 INTRODUCTION  .16  3.2 ASSUMPTIONS  16  3.3 CROSS SECTION BEHAVIOUR  17  3.3.1 Ultimate Interaction Diagram  17  3.3.2 Curvature Contours  19  3.3.3 Moment-Curvature Curves  21  iii  3.4 SLENDER COLUMN BEHAVIOUR  22  3.4.1 F a i l u r e Modes  23  3.4.2 C a l c u l a t i o n Procedure  25  3 . 5 PROGRAM INPUT  4.  28  3.5.1 Cross S e c t i o n P r o p e r t i e s  28  3.5.2  30  Slenderness E f f e c t  3 . 6 PROGRAM OUTPUT  30  RELIABILITY EVALUATION  33  4.1 INTRODUCTION  33  4.2 RELIABILITY INDEX CONCEPT  33  4.3 RELIABILITY CRITERION  38  4.3.1  39  I m p l i c i t U n c o r r e l a t i o n Procedure  4 . 3 . 2 Standard D e v i a t i o n  for Eccentricity  41 43  4.4 SAMPLE SIZE 5.  6.  •  RESULTS  45  5.1 INTRODUCTION  45  5.2 RELIABILITY AND AXIAL LOAD  45  5 . 3 RELIABILITY AND LOAD RATIO  ...47  5.4 RESULT FORMAT  47  5 . 5 CODE EVALUATION  50  CONCLUSIONS AND RECOMMENDATIONS  58  6.1 CONCLUSIONS  58  6 . 2 RECOMMENDATIONS  59 •  REFERENCES  iv  60  LIST  OF F I G U R E S  1  Material  Resistance Factors  2  The Monte  3  Cross  4  Ultimate  5  Curvature  6  Moment-curvature-axial  load  7  Column w i t h  and equal  8  Behaviour  9  Interaction  Carlo Simulation  section  Technique  5  behaviour  interaction  18  diagram  20  contours  axial  of  20  load  diagram  of  13 R e l i a b i l i t y 14 T h e I m p l i c i t  15 S t a n d a r d d e v i a t i o n 16 R e l i a b i l i t y  for  index  slender  loaded  of  column.24 27  columns  30  index  and the p r o b a b i l i t y  Uncorrelation  24  •.  the r e l i a b i l i t y  index  loaded column  curves curve  21  end e c c e n t r i c i t i e s . . . . 2 2  f o r an e c c e n t r i c a l l y  interaction  12 D e f i n i t i o n  relationships  an e c c e n t r i c a l l y  10 C o l u m n d e f l e c t i o n 11 S t r e n g t h  4  34 of  failure  36  Procedure  38  eccentricity  41  0 a n d maximum a l l o w a b l e  nominal  load  axial 45  17 C o m p u t e r  result  18 V a r i a t i o n  of  and the dead load  e with  ratio  factor  a.......45  <t> m  53  19 R e s u l t s  f o r t h e 250mm X 250mm c o l u m n  with  2% s t e e l  53  20 R e s u l t s  f o r t h e 500mm X 500mm c o l u m n  with  2% s t e e l  54  21 R e s u l t s  f o r t h e 500mm X 500mm c o l u m n  with  4% s t e e l  54  v  LIST  OF  TABLES  1  $  Q  f o r t h e 250mm X 250mm column  with  2% s t e e l  47  2  $  o  f o r t h e 500mm X 500mm column w i t h  2% s t e e l  47  3  $ o $  f o r t h e 500mm X 500mm column  with  4% s t e e l  47  f o r t h e 250mm X 250mm column  with  2% s t e e l  48  5  $ n  f o r t h e 500mm X 500mm column  with  2% s t e e l  48  6  $  f o r t h e 500mm X 500mm column  w i t h 4% s t e e l  48  7  £  4  n  n  for. t h e 250mm X 250mm column  8 9  $  10  $  0  with  2% s t e e l  56  f o r t h e 500mm X 500mm column  with  2% s t e e l  56  f o r t h e 500mm X 500mm column  with  4% s t e e l  56  f o r t h e 250mm X 250mm column  11  with  2% s t e e l  $ ~ f o r t h e 500mm X 500mm column w i t h 2% s t e e l p2 12 f° 500mm X 500mm column w i t h 4% s t e e l 13 C o m p a r i s i o n o f r e s u l t s i n t e r m s o f e rt  n  e  vi  57 57 57 58  ACKNOWLEDGEMENT  I  wish  supervisors, for  their  research also  to  express  Professor  my  project,  gratitude  N.D.Nathan, a n d P r o f e s s o r  invaluable  advice  and i n the p r e p a r a t i o n  due  deepest  t o a l l those  who  and guidance of t h i s have  R.O.Foschi  throughout the  thesis.  rendered  t o my  Thanks a r e help  in this  Research  Council  d i r e c t l y or i n d i r e c t l y .  The  financial  of C a n a d a gratefully  )  support of the N a t i o n a l  i n t h e form  of a  acknowledged.  vii  Research  Assistantship i s  1. INTRODUCTION The  a c t u a l s t r e n g t h of  differs  from  the nominal  engineer  due t o v a r i a t i o n s  a  the  member,  inherent  in  the  equations  of  loads  in their  is  accounted  of  a l l existing In  designers  building  Similar  the  t  of  Mirza  1.1  CODE METHOD  effect Monte  analyses  by A l l e n ( l 9 7 0 ) ,  to  the  variabilities  compute  use c o n s t a n t forces,  the  member  nominal  values  but  the  actual  i n s t r e n g t h and l o a d i n g in safety provisions  of Carlo  these  variables  technique  f o r random  have been a p p l i e d t o beams  Ellingwood(1977),  Grant  was  et  and  al(1978)  and MacGregor(1982).  Cornell(1969) achieve  code d e s i g n  a n d L i n d ( l 9 7 l ) have  a consistent level criterion  should  be o f t h e f o r m :  where: R = design = nominal  strength specified  <p = s t r e n g t h r e d u c t i o n X = load  shown t h a t  of r e l i a b i l i t y  <j>R > XU  U  by t h e d e s i g n  codes.  the  using  and  to  used  as  This v a r i a b i l i t y  study  investigated  columns  well  f o r i n one form o r a n o t h e r  this  simulation.  as  calculation  loads are v a r i a b l e .  member  i n t h e m a t e r i a l s t r e n g t h s and t h e  of  Similarly,  concrete  strength calculated  geometry  strength.  reinforced  load factor  factor  1  in  order  i n design, the  2  A  summary  of the a c t u a l  the Design is  of Concrete  presented  1.1.1  procedure  f o l l o w e d by t h e Code f o r  Structures for Buildings  below.  METHOD OF CAN3-A23.3-M77 The  loads)  factored  design  load  ( f o r o n l y dead and l i v e  i s g i v e n by U = X  D  D  X  +  L  L  where D a n d L a r e t h e n o m i n a l  values  live  and  loads  respectively  corresponding 1.4 w h i l e The by  (CSA-A23.3)  a  load factors.  that  for X  s h o r t column  capacity  reduction  combined  moment M  with  is  as  the  i t  o f 4> f o r  0.1 O f A to zero. c g account f o r the slenderness  or  is  axial  reduced  members  are  0.70. < / > is  design  load  designed  effect  of  the  using a magnified  d e f i n e d by: J  M i s the larger  c  = 8M, 2  where M  2  based  on U a s d e s c r i b e d a b o v e , a n d i s c a l c u l a t e d  conventional magnification short  axial  from  the c  i s t a k e n as.  D  compression  bending  i n c r e a s e d t o 0.90  columns,  a r e the  L  f a c t o r tf>. The v a l u e  compression  To  for X  s t r e n g t h of t h e column  i s 0.90 and f o r a x i a l  decreases  X  a s 1.7.  pure bending  linearly  and  D  The v a l u e  i s taken  T  X  o f t h e dead and t h e  column  elastic  design  end moment on t h e  frame a n a l y s i s a n d 5  factor.  M  c  must be l e s s  is  than  member,  a  from a moment  the reduced  s t r e n g t h . 5 i s g i v e n by t h e r e l a t i o n :  5 =  2! > 1-P /<t>P u' c  1.0  r  where 4> v a r i e s  as s t a t e d  a b o v e , and rr EI 2  where k l ^ i s t h e e f f e c t i v e l e n g t h EI  i s calculated as: 0.2E EI  c  I 1  following C  m  of the column.  the notation  i s given  + E  g  I se s  ^d  +  o f CAN3-A23.3-M77.  by: M, = 0.6 + 0 . 4 —  C  M where end  M,  and  are the smaller  2  and t h e l a r g e r  design  moments r e s p e c t i v e l y a t t h e two e n d s o f t h e member.  1.1.2  METHOD  The dead  OF  total  load  relation  value  1.50.  design  load  the l i v e  t h i s case X  Material  shown  A23.3(1984)  s i m i l a r to the  in  nominal  CSA  and  that  for  M  2  figure  loads one  when  are acting  in  only  the  i s given  by a  CAN3-A23.3-M77  i s t a k e n a s 1.25 w h i l e  resistance  strengths in  Q  f o r the case  factors  o f c o n c r e t e and 1. The v a l u e  are  L  has t h e  applied  to the  reinforcing  X  except  bars  f o r <j> i s 0.60 w h i l e c  as that  <6 i s 0.85. s The  slenderness  a procedure  effect  i s taken  i d e n t i c a l t o t h e one i n  i n t o account CAN3-A23.3-M77  using but  c  Steel  Concrete Figure with  1. M a t e r i a l  resistance  t h e moment m a g n i f i c a t i o n  6 = ' where <f> = 0.65. m  factor defined  -55 p  factors  / V c  ;> ,  < 0  as:  2. MONTE CARLO SIMULATION  2.1 INTRODUCTION If  a  relationship  performance  of  performance,  a  and  can  system i f  be  and  derived  between  the  each v a r i a b l e a f f e c t i n g t h e  statistical  properties  of  the  d i s t r i b u t i o n s o f a l l t h e v a r i a b l e s a r e known, i t i s p o s s i b l e to  use  randomly  selected  values  the variables  to  o f t h e system performance.  This  calculate  the v a r i a b i l i t y  technique,  shown s c h e m a t i c a l l y  Carlo  simulation,  variability  was  i n f i g u r e 2 and c a l l e d  of r e i n f o r c e d concrete  columns,  of the strength r e l a t i o n s h i p s .  INPUT: STATISTICAL PROPERTIES OF VARIABLES  SELECT A RANDOM VALUE OF EACH VARIABLE  RELATIONSHIP BETWEEN VARIABLES AND SYSTEM PERFORMANCE  REPEAT MANY TIMES CALCULATE VALUE OF SYSTEM PERFORMANCE  OUTPUT: SUMMARIZE RESULTING VALUES OF SYSTEM PERFORMANCE WITH STATISTICAL ANALYSIS  Figure  Monte  used i n t h i s study t o determine t h e  of the strength  because of the complexity  of  2. The M o n t e C a r l o  5  simulation  technique  6 2.2 DESCRIPTION  OF THE METHOD  The Monte C a r l o solving  method may be d e s c r i b e d  problems  numerically  engineering,  and  other  experiments.  The  problem  probabalistic case  problem  and  then  to  follow  the  or f u n c t i o n  in  or function  The s i m u l a t i o n  appearing  is first  process  the d i s t r i b u t i o n p r o p e r t i e s  method n o r m a l l y c o n s i s t s  is  simulation  2.  s o l u t i o n of the d e t e r m i n i s t i c  i n the  case  an  constructed computerised  of t h e v a r i a b l e . The  of t h e f o l l o w i n g  1.  either  probabalistic  whereas i n t h e d e t e r m i n i s t i c  random v a r i a b l e  simulated.  sampling  posed  In  of  physics,  through be  form.  random v a r i a b l e  i s simulated,  artificial  may  means  mathematics,  sciences  or d e t e r m i n i s t i c  the a c t u a l  in  as a  steps:  o f t h e random v a r i a b l e  function,  problem  for  a  large  number o f r e a l i z a t i o n s o f t h e l a t t e r , a n d 3.  s t a t i s t i c a l analysis  The p r e s e n t each of t h e s e  chapter  steps  of the r e s u l t s .  and t h e two  following  i t  discuss  in detail.  2.3 V A R I A B I L I T Y OF STRENGTH In slender  order  to evaluate  reinforced  variability  of  the v a r i a b i l i t y  concrete  the  members  parameters that  n e c e s s a r y . These parameters a r e compression of  the  section  and t e n s i o n ,  reinforcement,  the  the y i e l d and  the  a  o f t h e s t r e n g t h of knowledge  of  the  a f f e c t the strength i s concrete  strength dimensions  o f t h e member. The v a r i a b i l i t i e s  strength  in  and t h e p o s i t i o n of  of these  the  cross  parameters  7 used  in  this  summarized  study  by  were  based  et  Mirza  primarily  al(1979)  on  and  the  data  Mirza  and  MacGregor(1979a,b). T h r e e major material strength 1.  assumptions  strengths  to  of the r e i n f o r c e d The  variability  used  concrete of  the  T h i s assumption  construction was  assumed  representing and  practice. to  to  average  of the  drawn  the from  and  quality  so  average  Similarly,  be  the  properties  i s made  the  a l l sources  the United  the d e r i v a t i o n  concrete  construction.  represent  determining  columns.  corresponds  may  in  in  dimensions  results  2.  be  were made  that  of  the  Canadian  reinforcement a  population  of reinforcement  in  Canada  States.  The m a t e r i a l s t r e n g t h s were assumed t o c o r r e s p o n d  to  lower  in  l o a d i n g r a t e s than  material strength to  tests  of c o n c r e t e  failure,  based  or  and  conservative concrete  was b a s e d yield  static  assumption  and s t e e l  generally  laboratories.  the  on a s o - c a l l e d  those  tend  on a  used  The 1-hour  crushing loading  s t r e n g t h of s t e e l  loading rate. since  the  This  is a  strengths  to increase at high  was  of  r a t e s of  loading. 3.  Increase  in  the long-time  due  to increased maturity  as  possible  were  future  ignored. Gardiner  s t r e n g t h of the concrete  of the c o n c r e t e ,  as  well  c o r r o s i o n of the r e i n f o r c e m e n t and H a t c h e r ( 1 9 7 0 )  and  Washa  8  and  Wendt(l975) r e p o r t  20-25 of  year  o l d concrete  t h e mean s t r e n g t h  however,  this  concrete  strength  cylender estimate  2.3.1  at  the  i s expected 28  effect  days.  was  was r e l a t e d  strength. o f member  mean  This  strength t o be  For  to  leads  the to  current  quality-control  study  and  the  28-day  a  test  conservative  strength.  structure  design,  production,  procedures,  the strength  may  differ  strength  and  structure.  The m a j o r  strength  are  proportions mixing,  of  may  concrete  of  concrete  in  accounts  throughout in  a test  of  concrete  psi)  for  lasting  variations  than  27.6 MPa  in  as suggested  of these  to  in control  by M i r z a  et  effects. to failure  t h e mean c o m p r e s s i v e  strength  i n a s t r u c t u r e was t a k e n a s 23.36 MPa  variation  0.175.  1 hour,  concrete  p r o p e r t i e s and  Assuming a r a t e of l o a d i n g c o r r e s p o n d i n g in  the  and v a r i a t i o n s due  structure rather  f o r most  design  and c u r i n g methods, t h e  procedures, a  the  and  of concrete i n  variation  mix,  placing  s p e c i m e n s . The f o l l o w i n g d a t a , al(1979),  uniform  the v a r i a t i o n s i n m a t e r i a l the  testing,  i t s specified  be  sources  in testing being  from  not  transporting,  variations  of  150-250%  this  disregarded  of  PROPERTIES OF CONCRETE Under  a  that  (4000 p s i ) c o n c r e t e .  (3388  The c o e f f i c i e n t  f o r the c a s t - i n - p l a c e concrete  was t a k e n a s  9  The mean v a l u e 27.6 MPa c o n c r e t e  o f t h e modulus o f e l a s t i c i t y  was t a k e n a s 22476.91  with a c o e f f i c i e n t  of v a r i a t i o n  These p r o p e r t i e s  were  MPa  f o r the  (3260  ksi)  of 0.12.  assumed  to  follow  normal  distributions.  2.3.2 PROPERTIES OF REINFORCEMENT The  sources  strength are the 1.  of  variation  in  the  steel  yield  following:  Variation  in  the  strength  in  the  area  of  the  material  itself. 2.  Variation  of c r o s s  section  of the  bar. 3.  Effect  of t h e r a t e  of l o a d i n g .  4.  Effect  of b a r d i a m e t e r  on t h e p r o p e r t i e s  of  the  bars. 5. The  Effect  following  (1979b),  data,  accounts  sources of The  mean  yield  taken  as  grade  a t which y i e l d  i s defined.  a s s u g g e s t e d by M i r z a  and  for  most  the  effect  of  MacGregor of these  variation.  static  for  of s t r a i n  and t h e c o e f f i c i e n t  strength  460.6  MPa  f o r the s t e e l  of v a r i a t i o n  reinforcement  (66.8 k s i ) and 0.09,  60 h o t r o l l e d  bars.  assumed  to follow  assumed  t o be i n d e p e n d e n t  The  f o r the  yield  a Beta d i s t r i b u t i o n . of bar s i z e .  were  respectively, strength  was  These v a l u e s  were  10  The taken  mean v a l u e  as  201000  variation,  f o r t h e modulus MPa  (29200 k s i ) w i t h  0.033. The  probability  modulus of e l a s t i c i t y The of  ratio  cross  0.99  section  and  Since  of  the  provided  differ  may  and  considered mean  of  by  lognormal The  2.3.3  t o be  whole  the  of  constant  of  area  of  variability s m a l l and  value  be  steel As  suggested  distribution of  some  actually  effect  below w h i c h  equals  of  this  of v a r i a t i o n  0.91  area  0.024.  member must  lognormal  a coefficient  the  a mean  from t h a t c a l c u l a t e d .  a modified  the  normal.  with  of v a r i a t i o n  bars,  of  of  assumed t o f o l l o w a  0.94  in a concrete  distribution  relatively  was  MacGregor(1979a),  1.01,  modification  bars  truncated at  steel  of  Mirza  the  was  a coefficient  distribution  considered  a coefficient  combination  by  was  elasticity  of a c t u a l t o n o m i n a l v a l u e s  normal d i s t r i b u t i o n of  of  has  having  0.04,  the  been  and  a a  modified  zero.  strength within a single  i s neglected  in this  bar  is  study.  GEOMETRIC PROPERTIES Geometric  members  are  specified  mainly  values  dimensions,  the  horizontality alignments surfaces  imperfections  of  caused  of  the  positioning and  in by  the c o n s t r u c t e d  deviation  of  beams,  concrete  from  the  sectional  shape  and  reinforcing  bars,  the  of  lines,  the  cross  verticality  of c o l u m n s and  reinforced  concrete and  the  s t r u c t u r e s . The  grades data  used  and to  1 1  account  f o r these  suggested  by M i r z a a n d  assumed  that  represent  a  The  +1.52  mm  cross  of r e i n f o r c e d  from  the  (+0.06 i n ) . 1  sectional  this  study  distribution  distribution  mean d e v i a t i o n  dimensions  in  MacGregor(1979a).  normal  the  imperfections  effects  has  be  the  as been  used t o geometric  c o n c r e t e members.  of the a c t u a l specified The  It can  of  were  dimension  standard  dimensions  cross  was  sectional  was t a k e n a s  deviation  taken  of the  a s 6.35mm (0.25  in) . The is  location  affected  alignment the  of v e r t i c a l  by t o l e r a n c e s i n from  floor  reinforcement  cage  +8.13  (+0.32  in) . 2  the  for steel  The  for steel  ties,  i n columns  forms,  and c a r e taken  within  of concrete cover  concrete cover  the  to floor  deviation mm  reinforcement  form.  to center The  was  mean  b a r s was t a k e n a s  standard deviation  bars  column  taken  as  of  the  4.32mm  (0.17in).  2.4  VARIABILITY OF LOADS In  load  the analysis  effects  themselves. of  load  of s a f e t y  i t i s necessary  s u c h a s moments, e t c . I t i stherefore  effects.  These  1  i.e. specified  dimension  2  i.e. specified  cover  rather  necessary are  found  + 1.52 mm.  + 8.13 mm.  than  to deal with the  loads  t o have a d i s t r i b u t i o n by  combining  the  12 variability  of the l o a d s  introduced  by t h e s t r u c t u r a l  is The  themselves  s m a l l and c a n be i g n o r e d variability  of  the  with  analysis.  except loads  i n the case themselves  variability  o f t h e m a g n i t u d e and  of  load,  this  columns.  In  components  2.4.1  and the  component  o f dead  i s i n turn the  the  distribution  i n f l u e n c e s the load e f f e c t s  following,  variability  load.  from  on t h e  both  the  i s considered.  VARIABILITY OF DEAD LOADS Except  building well  in  defined,  nominal  cases  where  have t o be d e s i g n e d  comparision  with  dead  distribution coefficient on  variability  The l a t t e r  combined the  the  dimensions, (Ellingwood  are  known  l o a d s . The r a t i o , was  the  densities, al  parts  equal  equal  to  of the  upper  R  part  is  accurately  in  Q  represented  mean  variation  structures  of a c t u a l t o by  a  1.05  normal and  the  t o 0.07. T h i s was b a s e d  o f dead  resulting  load  from  superimposed  effects  in  variations  in  l o a d s , and  analysis  1980). A l l e n ( l 9 7 5 ) assumed v a l u e s o f  1.0 a n d 0.07 whereas L i n d et Nowak  the  other  with  et  before  loads  s t u d i e s of the v a r i a b i l i t y  concrete  lower  dead  load  of  the  and  0.05.  and  1.05  and 0.08 f o r s i t e - c a s t  al (1978) u s e d v a l u e s  Lind(l979)  o f 1.0  assumed t h e v a l u e s a s  concrete  bridge structures.  1 3 2.4.2  VARIABILITY OF L I V E The  ratio,  the nominal as  load  suggested  tributary  o f t h e maximum 30 y e a r  was assumed  by  and  variation  Curtis(l980)  while  distribution  live  of  suggest  distribution.  is  independent  the  results  of  the  coefficient  of  i s taken  t o be 0.30,  and  of l o a d  load  the  survey  tributary  area.  a gamma d i s t r i b u t i o n  Allen(l975) In the p r e s e n t  was assumed  load to  of  Woodgate(1971),  o f maximum  live  t o have a mean o f 0.70, and  Allen(1975)  independent  loads,  L  a r e a . On t h e b a s i s  Mitchell  is  R  LOADS  does an  and  the  live  specify  the  for  not  study  Nowak  extreme  f o r t h e maximum l i v e  type  I  load  i n 30  different  load  years.  2.4.3  LOAD COMBINATION It  effects  is  important  properly  assessment  of  so  to  combine  as  to  reliability.  random  functions  gravity  l o a d s , one p o s s i b l e  load  (which  live  load  lifetime has  the  other.  the  effects  live  in  occupancy  study  usually  i s resisting i s t h e dead  a n d t h e maximum load)  This combination  this  and  i n time)  realistic  are  load combination  maximum  of the s t r u c t u r e .  of  a more  o f t i m e . When t h e d e s i g n  been c o n s i d e r e d  values  achieve  Load  would be c o n s t a n t (or  the  with  i n the  of the l o a d s the  nominal  t h e dead l o a d s e q u a l  t o each  14 D e f i n e a dead  load  values case  total  of  nominal  the  factor, a  as:  N o m i n a l dead l o a d = T o t a l nominal load  a where the  ratio  dead  load and  i s the  sum  of  the  live  loads.  more c a s e s w i t h a=1/3  and  a=2/3  the  nominal  Hence  for  our  ( i . e . with  the  to  the  a=l/2. Two  nominal v a l u e of nominal also  value  the of  considered  investigate  live  the  for the  load  dead  one  equal  load,  cross  effect  and  vice-versa)  section  of  twice  this  in factor  were  order  to  on  the  earthquake  have  reliability. The not  load  been No  wind,  snow and  considered. load  The  e f f e c t s of  f a c t o r s have been  load  combination  applied.  p r o c e d u r e may  be  follows: D  L = R  +  D  D  N  R  +  L  L  N  where D  = actual  dead  load  L  = actual  live  load  D  = nominal dead  load  L.  = nominal  load  R  and  XT  N T  and  D  R^  live  are  as  described  previously.  Then D D  +  L = L  N  {R  + R  D L  N  L  }  summarized  as  15  Now  we  have: D  a  N  = D  N  + L  N  Rearranging D  N  L Hence, t h e  load D  This for  formulation this  study.  effect +  L  = N  was  L  can { R  D  used  1-a  N  finally (  to  "FS>  +  be R  L  simulate  written  as:  }  the  load  effect  3.  STRENGTH MODEL  3.1 INTRODUCTION This  chapter  predicting  the  describes  strength  for  Desai  and K r i s h n a n  steel  i s assumed  The along the  theory  with  strength The  concrete,  f o r concrete  t o be e l a s t i c - p e r f e c t l y  as given  of  Nathan  and v e r i f i e d  program u s e d  r e i n f o r c e d concrete the  plastic.  i n t h e model a r e d e s c r i b e d ,  f o r the computer  of s l e n d e r ,  by  for  ( 1 9 6 4 ) . The s t r e s s s t r a i n r e l a t i o n s h i p  organization  developed  model  r e i n f o r c e d c o l u m n s . The  behaviour  and a s s u m p t i o n s  a basis  theoretical  of slender  model u s e s t h e s t r e s s s t r a i n by  a  program  is  to  obtain  columns. similar  to  one  (1972) f o r r e i n f o r c e d a n d p r e s t r e s s e d f o r those  materials  by  Alcock  and  Nathan(1977).  3.2 ASSUMPTIONS The  f o l l o w i n g a s s u m p t i o n s a r e made :  1.  Plane  2.  M a t e r i a l p r o p e r t i e s are constant the  3.  sections  Dimensions and e r r o r along  If  moment  along  the  cross  Bending  i n p l a c i n g of  the length of  steel  a member, f a i l u r e  section subjected  are  one p l a n e  16  occurs a t  t o maximum moment, o r by  o f t h e member.  i n only  bars  the l e n g t h of the column.  v a r i e s along  instability 5.  plane.  column.  constant 4.  remain  i s considered.  1 7  6.  E f f e c t s of s h r i n k a g e a r e  7.  No  torsional  considered.  There  is  out-of-plane  Duration  considered. 8.  or  Shear no  reinforcing  neglected.  of  load  failures  slip  deformations effects  a r e not  between  the  are  are not  considered. concrete  and t h e  steel.  3.3 CROSS SECTION BEHAVIOUR The  ultimate  interaction  curvature-axial  load  derived  a simple  axial  by u s i n g  load  the  b e n d i n g moment t h a t  the  step-by-step procedure  a n d moment c a p a c i t i e s  shows  and  moment-  relationships for a cross-section are  depths and c u r v a t u r e s . diagram  diagram  Recall  limiting a section  to  obtain  f o r a range of n e u t r a l  that  the ultimate  combinations  axis  interaction  of a x i a l l o a d and  can r e s i s t .  3.3.1 ULTIMATE INTERACTION DIAGRAM For the  the r e i n f o r c e d  one shown  considering strain across axis as  for  in figure  the top f i b r e concrete).  produces  at  to  determine  ultimate  The n e u t r a l  a strain  in figure  section  such  as  3 ( a ) , t h e c a l c u l a t i o n b e g i n s by  the section. A t y p i c a l  shown  used  concrete cross  axis  location  i s then of  d i s t r i b u t i o n across  3 ( b ) . The f o l l o w i n g what  strain  combination  b e n d i n g moment would p r o d u c e t h i s  the  (failure marched neutral  the s e c t i o n  procedure  i s then  of a x i a l l o a d a n d  condition.  18  •  •  /  t  (•Ml  TP IT f  n.o_  •  m  1  (train  1  (train  J  •  section  f  p  /  v ,  material properties  strain  7 stress  segment  net  forces  actions  to  to  Figure 3. Cross section behaviour The  depth  of segments. segment  of the section i s divided into a number  The  strain  at  the mid-height  each  i s evaluated. The appropriate s t r e s s - s t r a i n law  i s then used to find the stress shown  of  for each  segment  as  in figure 3(d). This stress i s multiplied by the  area of the segment to give the force segment,  as  shown  in figure  acting  on  each  3(e). The forces in the  steel bars, C and T are found i n a similar manner, s s The  forces  for a l l the segments are added to give  the required a x i a l force, P. The forces i n the segments are  multiplied  by  the distance from the mid-height of  the segment to the centroidal axis, and  then  added to  give the bending moment, M, acting on the cross section. At t h i s stage, the following information i s stored, before  repeating  the above c a l c u l a t i o n with increased  neutral axis depth. The stored information i s : 1.  Curvature  (input)  2.  Net a x i a l load  (output)  1 9  3.  B e n d i n g moment  For  a  procedure  number  (output)  of  described  neutral  above  a x i s h a s marched a c r o s s Hence  we  interaction for is  obtain  figure  a  moment t h a t  the c r o s s  outside  A  the curve  values.  curvature  contours  bending is the  curvature.  Each  the  steel curve  f o r c e and b e n d i n g resist.  f o r curvature and < 2 > max  are  Any  point  i s described  the cross  section.  interaction  diagram.  for  followed  Figure  represents  Different  each  of  points  these  f o r f i n d i n g the  below. the n e u t r a l a x i s i s  s e c t i o n . The n e t a x i a l  similar  case)  i s the  l o a d and  neutral axis locations  t o t h e one d e s c r i b e d  in  5 shows t h e r e l a t i o n s h i p  f o r c e a n d b e n d i n g moment line  (30 i n t h i s  , where <b max  found  The p r o c e d u r e  f o u n d by a p r o c e d u r e  contours.  inside  2%  failure.  moment f o r e a c h o f t h e s e  between a x i a l  diagram  can  s e l e c t i o n of curvature,  previous  interaction  axial  on t h e u l t i m a t e  contours  marched a c r o s s  moment  CONTOURS  curvature  a  of  represents  curvature  For  load-bending  point  section  s e l e c t e d between z e r o  curvature  the neutral  500mm X 500mm w i t h  Any  number o f v a l u e s  maximum The  axial  ultimate  combination  3.3.2 CURVATURE  are  the  4.  represents  till  the  the s e c t i o n .  d i a g r a m . The  in  locations,  i s repeated,  a s e c t i o n of dimensions shown  axis  f o r 30  a single value on  a  line  curvature of s e c t i o n  of  constant  20 9000 8000-  _j < X <  4000300020001000-  0  100  200  300  400  500  BENDING MOMENT  Figure  4. Ultimate  600  700  800  900  kN-m  interaction  diagram  9000-T 8000-  100  200  Figure  300  400  500  BENDING MOMENT  5.  Curvature  600 kN-m  contours  700  'i  11  800  900  21  curvature  represent  the combinations of a x i a l  moment r e q u i r e d t o p r o d u c e t h a t  curvature  neutral  that  axis  locations.  Note  load and  f o r various 4  figure  i s the  e n v e l o p e o f a l l t h e p o i n t s shown i n f i g u r e 5 .  3.3.3 MOMENT-CURVATURE CURVES Figure for  several  6 shows t h e levels  start a t the o r i g i n clarity  of  moment-curvature  of  axial  b u t some  presentation.  P  loads,P. have  stored  data.  A  A l l the curves  been  shifted for  i s the a x i a l  0  s t r e n g t h o f t h e column. These c u r v e s a r e already  relationships  horizontal  compression  computed  from  on f i g u r e 5  line  900  800-  P/P =0.3 o  0.2  0.1  0.02  0.00 Figure  0.01  0.02  0.03  0.04  CURVATURE  1/m  0.05  0.06  0.07  6. Moment-curvature-axial load r e l a t i o n s h i p s  22  represents a level of a x i a l load. Each  intersection  of  t h i s horizontal line with a curvature contour provides a value of bending moment and curvature, which i s plotted on  figure  6 . The f i n a l point for each moment-curvature  curve represents the point on the ultimate curve  interaction  for that l e v e l of a x i a l load. For high values of  a x i a l load, a f a l l i n g  branch  of  the moment-curvature  curve exists, beyond the maximum moment, but this i s not plotted as t h i s information i s not used  in subsequent  calculations.  3.4 SLENDER COLUMN BEHAVIOUR This  section describes the procedure used to establish  the behaviour of columns of any length under the action of eccentric  axial  loads with equal end e c c e n t r i c i t i e s and no  l a t e r a l load as shown in figure 7 . P  P  Figure 7 . Column with a x i a l load and equal end eccentricities  23  3.4.1  FAILURE MODES In  computer  order  to  model,  the behaviour  compression Consider figure If  facilitate  members  will  the p o s s i b l e 7,  of  as t h e a x i a l  load  P i s increased  f o r the a x i a l  moment a t t h e m i d - s p a n deflection  moment.  path  line  0-B.  outer  for  material  where  diagram  curved  the  by  the  Consider  the  amount  is  between  by  which  is  e,  then  the bending  A  is  i s the  of a x i a l  load vs  member  load  a t an a x i a l  p a t h 0-B i n t e r s e c t s at  the  i s the ultimate and  a load  i t  path f o r  P i s increased,  the  point  P  1 #  material B. T h i s  by  lines  the  0-A  In  curved  and  the i n i t i a l  t o P(e+A).  fails  load  shown  moment, Pe h a s m a g n i f i e d  If  in  l i n e 0-A, t o P,. The c o r r e s p o n d i n g  f o r m i d - s p a n moment  represents  failure  shown  cross-section,  failure.  The h o r i z o n t a l d i s t a n c e  diagram  line  and e n d moment, a s a x i a l  represented load  The  curve  represents load  i s P(e+A),  section.  to f a i l u r e .  i s Pe, w h i l e  8 i s an i n t e r a c t i o n  interaction  axial  load  loaded  a t m i d - s p a n , a s shown.  Figure bending  in this  o f t h e member  b e n d i n g moment a t t h e ends  of t h e  eccentrically  be r e v i e w e d  behaviour  t h e end e c c e n t r i c i t y  the  the understanding  this  0-B  mid-span case  the  when t h e m i d - s p a n  load  strength  i s described  interaction as m a t e r i a l  . the  eccentricity,  same  member  the load  path  is for  loaded  with  end  moments  a  smaller could  be  Figure 9. Interaction diagram for an e c c e n t r i c a l l y loaded column  25  shown by l i n e  O-C,  moments by l i n e occurs The  load  moment  i s well  the  0-D. In t h i s  when t h e a x i a l  mid-span  which  and  load  r e a c h e s a maximum  at  with  failure  a system  example, g r a v i t y l o a d s )  to load  increase  a  rapidly  immediately. of  material If  failure this  member  using  0-D  would  process  combinations moment) j u s t  as  line  A  of  and axial  causing  The made  is  of  method d e s c r i b e d column  P ,  curve.  D,  I f the  deformations  2  failure  would  under  would follow  conditions shown by t h e  be  to  followed  many t i m e s  the  eventual  C  P ~C-A-M u  in  figure  f o r t h e same  f i g u r e 9 c a n be  ultimate  interaction  i s the locus of  u  8,  representing  l o a d a n d end moment  (unmagnified  PROCEDURE for calculating  i n figure 8 i s described  a  2  the load path  line  program can c o n s i d e r  up  P .  failure.  A computer p r o g r a m shown  value  load c o n t r o l( f o r  a range of e c c e n t r i c i t i e s ,  3.4.2 CALCULATION  curves  under  i s repeated  d i a g r a m . The c h a i n - d o t t e d such  failure  a t point E.  p r o d u c e d . The s o l i d  points  strength  material  displacement  of l i n e  mid-span  i s shown by p o i n t  I f t h e member were l o a d e d  controlled  extension  and  for  c a s e an i n s t a b i l i t y  i n s i d e the m a t e r i a l  member were l o a d e d  path  number  of  a  curves  of  any  segments o f e q u a l  for a  is  on  the  below.  column  by G a l a m b o s ( 1 9 6 8 )  deflection  points  used  given  length. A  to  axial  length  develop load, to  26  determine load  t h e maximum end e c c e n t r i c i t y , e, a t w h i c h  c a n be a p p l i e d As  t h e column  considered under set  t o be z e r o  t o the material on  the  corresponding axial  axial  necessary member.  proceeding  t h e moment a t m i d - s p a n failure  is  To f i n d  moment  for  the  calculate  along  failure  the  (from  shape  is  length  deflection  v  node x , t h e n 0  segment  and  0  shown  line  by t h e i t is of  the  segment-by-segment  from  in  node.  figure  segment o f  10.  I f the  V Q a r e known a t t h e s t a r t i n g  slope  t h e moment M,, a t  (point  The  by  the c a l c u l a t i o n s f o r a t y p i c a l  Ax, s u c h a s t h a t  (a  obtained  mid-span, c a l c u l a t i n g the d e f l e c t i o n a t each Consider  the  of e and A  curve  column,  load  diagram).  deflected  deflection  load  is initially  moment d i v i d e d  values  the  axial  that  interaction  the actual  column  the slope i s  the  m i d - s p a n d e f l e c t i o n , e+A,  to A  loaded,  a t mid-span. For  ultimate  load)  load.  column.  i s symmetrically  consideration,  point  of  to that  that  the  mid-point  of  the  x,) i s a p p r o x i m a t e l y M, = P ( v + v - A x / 2 ) 0  The  0,,  curvature,  at point  moment-curvature-axial curvature The  is  load  0  x, c a n be o b t a i n e d  r e l a t i o n s h i p ( f i g u r e 5 ) . The  assumed t o be c o n s t a n t  along  d e f l e c t i o n s a r e assumed t o be s m a l l The  node, x  2  displacement,  v ,  are calculated  from  v  2  = v  2  and s l o p e ,  v , a t the next 2  + v (Ax) - 0 (Ax) /2 0  1  t h e segment.  s u c h t h a t <j>=v".  2  0  from t h e  27  Moment IM,  (b).  Figure 1 0 . Column deflection curves and: V j = v j - ^,(Ax) The moment M  2  For  at node x  2  i s the product of P and v . 2  the f i r s t c a l c u l a t i o n of the column deflection  curve, (with the moment at mid-span equal to the f a i l u r e moment  at that load) the end e c c e n t r i c i t y i s calculated  to be e . The c a l c u l a t i o n i s then repeated for a 0  mid-span  moment.  less than e material  0  If the new end e c c e n t r i c i t y , e  (figure 10(b)), then t h i s  failure  and  this  also  calculation  However, i f e, i s greater than e  0  (figure  lower 1 #  is  represents i s ignored. 10(c))  then  this represents i n s t a b i l i t y f a i l u r e , and the c a l c u l a t i o n  28  is  repeated  moment  till  observed. at  with  smaller  a peak v a l u e This  peak  reductions  for  value  the  axial  instability  load,  end  i s then  which i n s t a b i 1 i t y ' f a i l u r e  by  the  it  i n t h e mid-span is  t h e end e c c e n t r i c i t y  occurs, gives  eccentricity  and when m u l t i p l i e d  the  end  moment  for  failure.  3.5 PROGRAM INPUT The  following information  computer detail  model.  Each  of  i s r e q u i r e d as input  these  items w i l l  to  this  be d i s c u s s e d i n  below. 1.  Cross  2.  3.5.1  section properties:  a.  Cross  s e c t i o n dimensions  b.  P r o p e r t i e s of concrete  c.  P r o p e r t i e s of  Slenderness  reinforcement  effect:  a.  Column  length  b.  Segment  c.  Reduction  length ratio  f o r column  d e f l e c t i o n curves  f o r mid-span  CROSS SECTION PROPERTIES The p r o g r a m c a n be u s e d t o o b t a i n  for  any  corners. only  moment  polygonal However,  f o r the present  f o r rectangular The p r o g r a m  relationships  shape o f c r o s s  column  properties  s e c t i o n up t o t w e n t y  study  i t h a s been u s e d  sections.  can  consider  f o r concrete  various as  well  stress as  for  strain steel.  29  However, for  for  concrete  modified and  this  study  the  in compression  Hognestad  Krishnan  stress  is  assumed  relationship,  (1964),  by  the  0  and  e  0  respectively. is  peak  strain  suggested  After  an  by  compressive  confined  by  The  n e g l e c t e d . The the  steel  assumed  stress-strain plastic  neglected.  strength  the  follow  E  .  c  £' The  the  are  the  concrete i t may  still  was be  and  capable  of  supporting  of  concrete  in  tension i s  the y i e l d  i n p u t . The  stress  reinforcement  elastic-perfectly  relationship.  because  strain  following relation  0.0038  Youngs M o d u l u s and  stress-strain  conservative  peak  s t r e n g t h of c o n c r e t e ,  by  of  ties  reinforcement  to  and  s t r e n g t h even t h o u g h  the column  load.  stress  Krishnan(1964):  ultimate strain t o have no  Desai  2  t h e modulus of e l a s t i c i t y ,  and  assumed  some  peak  calculated  Desai  by  the  0  The  is  as p r e s e n t e d  1+(e/e )  the  with  follow  0  are  input, along  to  2(e/e )  fo f  relationship  following relation:  f  Here  strain  The  of  is  plastic  elastic-perfectly  relationship the e f f e c t  for  is  strain  somewhat  hardening  is  30 3.5.2  SLENDERNESS The  EFFECT  program  slenderness strength  c a n h a n d l e a column  ratio.  of  the  o f any l e n g t h and  The e f f e c t o f t h e s l e n d e r n e s s column  i s accounted  on  the  f o r as d e s c r i b e d  earlier. Chen length  and  of  four  sufficiently this  Astuta(l976) times  accurate  corresponds  segment  length  considered The which  column  to this  a segment  gyration  the  section  is  for  d e f l e c t i o n curves.  section  depth.  A  d e p t h h a s been  study.  moment  procedure  section  gives  A  as  to  the  input  be  the rate at  reduced  i n the  construction  value  of  0.05  of  times  the the  used.  PROGRAM OUTPUT The  of  of  the  requires  maximum moment h a s been  3.6  radius  times  also  the mid-span  shown t h a t  r e s u l t s . For a rectangular  equal  program  step-by-step  the  t o 1.16  throughout  have  output  axial  column  from t h i s  load-end lengths.  sections  have  moment  computer  model  interaction  typically  curves  The c a l c u l a t i o n s d e s c r i b e d  been  carried  out  for  f o r columns  ratios,  load  0.02 t o 0.60 t i m e s t h e maximum l o a d .  shows  an  axial  load  typical  interaction  diagram  X  500mm c r o s s  of  several  d i f f e r e n t l e v e l s of a x i a l  showing  Figure  11  the combination of  P a n d end moment Pe j u s t p r o d u c i n g  500mm  several  i n the previous  slenderness between  each a t e i g h t  consists  failure  s e c t i o n column w i t h  for a  2% s t e e l .  31  9000800070002  6000-  Ul ^  5000-  O u. _j < X <  400030002000100000  Figure 11. Strength interaction curve for slender columns The  outer  line  representing  i s the  material  ultimate  strength  interaction  (slenderness e f f e c t s for a  slenderness r a t i o of 10 are n e g l i g i b l e ) . correspond  to  slenderness ultimate  the curve  ratios  the  interaction  P ~C-A-M u  inner curve.  The  indicating  that  curves  in figure 9. For low  u  curves As  inner  are close  the  behaviour  to the  slenderness  increases, the curves move inside the ultimate diagram,  diagram  under  ratio  interaction  these loads i s  governed by i n s t a b i l i t y f a i l u r e s . These  curves  computer output. considered  and  have Only  linear  been eight  plotted levels  interpolation  directly of  axial  from load  the were  was used to f i n d the  32  limiting This  bending  i s considered  t h e p u r p o s e of  this  moment  for  to give  sufficiently  study.  any  intermediate accurate  axial  load.  results  for  4.  4.1  used  d e s c r i b e s the  to evaluate  the  c o l u m n s , as w e l l as The  of  the  for  the  nominal  used  to  find  To  model f o r t h e  similar  set  axial  index  of  target  one  program, the  The  relevant variables  at  interpolation  i s then  axial  1 r  of  the  load  a design  of  presented  analysis  equation  of  general, e a c h of  several them  n  such  defining  es.  33  loads.  the  concept below i s  typically  has  ), some of w h i c h  2  the d e s i g n  theoretical  s a f e t y index  (X X ,...,X  2  st  values  r e s i s t a n c e of t h e m a t e r i a l , w h i l e  to the e f f e c t  considered,  finding  of  description  G(X,,X ,...,X In  number  a better understanding  in structural  satisfy  i s done by  used.  reliability.  A design problem  related  concrete  INDEX CONCEPT  reviewed.  to the  method  program  the a l l o w a b l e n o m i n a l  Foschi(1979).  of  the  reinforced  for a  load. Linear  reliability  to the  the  reliability  by  related  must  the  reliability  facilitate  briefly  of  t o the  RELIABILITY  of  behind  t h e b a s i s f o r t h e computer  the v a l u e  corresponding  theory  reliability  evaluation  the v a l u e  is  EVALUATION  INTRODUCTION This chapter  4.2  RELIABILITY  These  others  a  are are  variables  form  ) < 0 conditions the  could  appropriate  be  limit  34  In  general,  obeying is  not d e t e r m i n i s t i c ,  strategy for  for  which  expressed  are  of  the v a r i a b l e s  i s to specify  variables  individual  should  t e r m s of m a t h e m a t i c a l distributions  known. V e n e z i a n o  variables process  conditions will  This gives a ' r e l i a b i l i t y ' in  random  Hence t h e d e s i g n  and t h e d e s i g n  followed  the  X  function.  a l l values  usually  conditions.  in  variables  some d i s t r i b u t i o n  satisfied  the  the  not  X. The d e s i g n  a 'tolerated' satisfy range,  the  range design  which  can  probabilities,  of t h e d e s i g n  be  be  provided  variables  X  are  (1974) d e s c r i b e s some methods o f d o i n g  this  detail. Consider  R and U. R i s represents  the  the  variables' The  the simple  o f o n l y two d e s i g n  resistance  load  for  effect.  of the design  failure  case  R  problem  criterion  the  and  U  (Hasofer  variables,  problem,  while  are c a l l e d and L i n d  U  'basic  (1974)).  c a n be w r i t t e n a s U > R  where  the  survival  equality  would  and f a i l u r e .  deviation  of  U,  and  deviation  of  R,  we  variables  x and y such  represent  I f 0 and 1  a r e t h e mean a n d  and  can  o  R  define  non-dimensional  (R - 5) a  R  (U - 0 ) y =  between standard  a r e t h e mean and s t a n d a r d  that x =  the boundary  random  35 Hence  the f a i l u r e  criterion  c a n be w r i t t e n a s  (R-0)  Note a  slope  that  Op/cy  this  i s the equation  and an i n t e r c e p t  of a s t r a i g h t l i n e  (R-\2)/o^  on  the  y-axis  with as  shown i n f i g u r e 1 2 . If (x,y) over  t h e v a r i a b l e s R and U a r e n o t c o r r e l a t e d , t h e p o i n t s representing  a combination  t h e e n t i r e x-y p l a n e .  plane  into  combinations  two  parts.  criterion  The  part  corresponding  to  the  failure  representing  distance mean  boundary I - I , t h e ' f a i l u r e  Figure  upper  o f R and U p r o d u c i n g  minimum  be d i s t r i b u t e d  The f a i l u r e  corresponds t o combinations The  (R,U) w i l l  between values  surface'  of  d i v i d e s the  corresponds  and the lower  to part  survival. the R  i s given  origin  and  U, and t h e  by t h e  1 2 . D e f i n i t i o n of the r e l i a b i l i t y  0,  index  segment  36  OA.  This  minimum d i s t a n c e ,  /3, c a n be c a l c u l a t e d from  defined  Any is  point  A i s called  combination  produce design  - 0)  t h e most  of R and U below  a t a minimum d i s t a n c e failure.  This  likely  failure  the f a i l u r e  is  a  reliability  variables is the  and when t h e f a i l u r e  defined failure If  we  than  not  on t h e  reliability  m e a s u r e , w h i c h c a n be r e l a t e d t o t h e p r o b a b i l i t y o f a r e more  (which  will  statement  /3 may be u s e d a s a  f o r t h e c a s e when t h e r e  point.  surface  o f (3 f r o m t h e mean p o i n t )  p r o b l e m and t h e r e f o r e ,  . Even  index  the f i g u r e as (R  The  as the r e l i a b i l i t y  failure,  two  basic  f u n c t i o n , G, i s n o n - l i n e a r , (3  a s t h e minimum d i s t a n c e  from t h e  mean  point  to  surface. define: Y = R-U  the  failure  criterion  becomes t h e e v e n t  Y < 0. Then  Y = P>0 and  Hence,  /3 c a n be w r i t t e n a s (3 = ? / a  Hence, a s shown i n f i g u r e number the  of standard  failure  given portion  by  event  deviations,  y  13, t h e measure 0 becomes t h e a  v  , that  t h e mean Y" i s from  Y=0. The p r o b a b i l i t y o f f a i l u r e  P(Y<0), a n d c o r r e s p o n d s t o t h e a r e a  of f i g u r e  13. I f R and U a r e n o r m a l l y  P^ i s t h e n  of the shaded distributed, Y  37  P R O B A B I L I T Y D E N S I T Y  Figure 13. R e l i a b i l i t y Index and the probability of f a i l u r e will  be  normally d i s t r i b u t e d and  can e a s i l y be obtained  from the normal tables, knowing the value of 0. The  relationship between 0 and P^ becomes very complex  for the following cases: 1.  G being a non-linear function  2.  R and U being non-normal  3.  R and U being correlated  Rackwitz and Fiessler(1978) suggest a transformation to the R and U d i s t r i b u t i o n s which minimizes the error calculation  of  transformation  0  due  i s the  to  non-normal  basis  of  the  in the  d i s t r i b u t i o n s . This Rackwitz-Fiessler  algorithm. This algorithm uses an i t e r a t i v e procedure on the variables to locate the most l i k e l y f a i l u r e point, and hence to evaluate the r e l i a b i l i t y  index.  38  If to  R and  be  U are  dependent  uncorrelated  before  on  each o t h e r ,  using  the  then  they  have  Rackwitz-Fiessler  algorithm. For  this  study,  a  computer  Rackwitz-Fiessler  algorithm  of  index,  the  reliability  finction,  4.3  RELIABILITY  described  used  concrete  eccentricity  at  of  columns  the  which  column p r o p e r t i e s . The represent  the  parameters P written  0  it  the  of  the  value  failure  reliability  of  the  i s analogous to the  column is  depends  loaded,  interaction  and  M.  Hence, t h e  0  as  curves  column p r o p e r t i e s , and  one  upon  well  as  the  upon  the  shown i n f i g u r e  may  be  capacity  of  defined the  by  14 the  column  may  as P is  the  b e n d i n g moment M The  values  for evaluating  slender  capacity  c  the  the  the  above.  The  P  /3 from  to evaluate  on  CRITERION  method  reinforced  where  used  based  G.  The  be  was  program  failure  =  c  axial  f(M,M ,P ) 0  0  force capacity  i s a p p l i e d at f u n c t i o n may G  the be  = P  c  of  column when a  ends.  written -  the  as:  P,  or: G where P-^  i s the  axial  function  represents  =  f(M,M ,P )-P 0  0  load. A negative  failure.  But  1  value  for a given  f o r the  failure  eccentricity,  a  39  higher a x i a l load corresponds t o while  a  lower  axial  moment. T h u s , M a n d  load  a  higher  bending  moment  corresponds t o a lower  bending  are correlated, or, P  and  c  P^ a r e  correlated. T h i s p r o b l e m may be  solved  by  t h e method  described  below.  4.3.1 IMPLICIT UNCORRELATION PROCEDURE In  this  explicitly for  procedure  uncorrelated.  solving  t h e v a r i a b l e s R and U a r e n o t The p r o c e d u r e  t h e problem  at  i s presented here  h a n d . However, w i t h some  modifications the Implicit Uncorrelation be  generalized  f o r any  s e t of  Procedure  random  correlated  variables.  MOMENT Figure  14. The I m p l i c i t U n c o r r e l a t i o n  can  Procedure  40  Recall  that  i n our c a s e ,  moment M a r e c o r r e l a t e d corresponds of  to a high value  P-^ c o r r e s p o n d s In  the  load  and  and  high  bending  value  o f M. S i m i l a r l y ,  of  P^  a low v a l u e  o f M.  j3 f o r g i v e n mean v a l u e s  eccentricity.  deviation  be  that a  t o a low v a l u e  index  i s known.  section.  load  t h e f o l l o w i n g i t i s assumed t h a t we n e e d t o f i n d  reliability  load  such  axial  The  for  The  method  for  eccentricity  The d i s t r i b u t i o n  distribution  is  finding  axial  of the a x i a l the  described  of e c c e n t r i c i t y  for  standard  i n t h e next  i s assumed  to  normal. The I m p l i c i t  U n c o r r e l a t i o n Procedure  is  described  below. 1.  Simulate  random  eccentricity  values  for  the  for axial given  l o a d and f o r  mean  values  and  di stribut ions. 2.  Select (say  3.  one  P^ 1  1  value  a s shown  Multiply  P^  each  values  to  range o f b e n d i n g 4.  get  interpolation,  for  each  bending  Subtract axial  load  curve. a s shown  from  the  simulated  the corresponding  values.  linear  capacities 5.  moment  of  Using  interaction  load to investigated  i n f i g u r e 14).  by  1  eccentricity  of a x i a l  moment  find  the  value  capacity for  each  This gives a d i s t r i b u t i o n f o r in figure  14.  each of the c a p a c i t y v a l u e s , the  value  under  investigation  to get  41  6.  values  f o r the  stored  i n an  Repeat axial  The values  the  algorithm  as  be  be  due 1.  measure  0,  using  u n c e r t a i n i t y i n the  Due  are  simulated  lead  to  a set  Note t h a t t h e  These v a l u e s  outlined earlier  t o two  which  considered.  distributed  STANDARD DEVIATION FOR The  G,  a l l the  will  f u n c t i o n G.  distribution.  evaluate  4.3.2  not  5 till  have been  described  failure  will  standard  2 through  load values  f o r the  function  array.  steps  procedure  function  failure  according are  then  of  failure t o some used  to  the R a c k w i t z - F i e s s l e r  in t h i s  chapter.  ECCENTRICITY bending  moment  can  mainly  reasons. to  the  exact  m a g n i t u d e of  the  loads  being  the  loads  being  unknown. 2.  Due  t o the  exact  distribution  of  unknown. The has has is the  been  variability discussed  of  the  f o r the  i n s e c t i o n 2.4.  been assumed t h a t t h e e x a c t l y known, and e v a l u a t i o n of  loads  that  the  In t h e  distribution i t entails  bending  first  of no  moment  reason  following i t  the  dead  load  uncertainity in acting  on  the  column. A survey distributions  of d i f f e r e n t on  kinds  of  a beam r e v e a l t h a t  assume a v a r i a b i l i t y  of  about  5%  possible  loading  i t i s reasonable in  evaluating  to the  42  mean eccentricity  MOMENT  Figure 15. Standard deviation  of eccentricity  bending moment when the exact force i s known. The  method used for finding the standard  deviation  for the e c c e n t r i c i t y i s as follows. 1.  Select  the mean  eccentricity  for which the  v a r i a b i l i t y i s to be investigated. 2.  Simulate an a x i a l average mean  axial  load  load  eccentricity  bending  moment  (Figure 15).  distribution  value.  Hence, knowing the  the average  corresponding  for some  value  of the  to a p a r t i c u l a r  value of a x i a l load i s fixed. 3.  Simulate values for the bending moment a v a r i a b i l i t y caused only by the  assuming  d i s t r i b u t i o n  of  43 the  load.  Hence,  corresponding 4.  Repeat axial  It  steps  Uncorrelation  of  level  simulation  the  Hence, due  this  to the  which the  4.4  SAMPLE The  value the  of  the  considered.  the  of  nominal a x i a l  s t e p 3 of values  load.  the  of  Implicit  the  directly,  the  bending  without  using the  moments would be  different  for  leads to  l a r g e number of  the  axial  load,  values  remains the  saving  i n the  nominal a x i a l  whereas  the  same.  computer  time  load levels  at  i s t o be i n v e s t i g a t e d .  d e s c r i b e d above  reliability axial  times  corresponding  in order to  is  sample  sizes.  the  sample  size  of  200  to  find  target  et  to al  column  of  finding  /3 f o r a p a r t i c u l a r  the  i s t o be nominal  the  value  of  repeated axial  in order  determine (1978)  the  have  to  save  a  load  j3 f o r e a c h c r o s s s e c t i o n ,  r a t i o . Hence,  necessary Grant  index,  i s capable  load. This procedure  each slenderness it  simulated  However,  nominal  reliability  the  eccentricity.  eccentricity  method  of  SIZE  nominal  time  bending  the  procedure  number o f  for  of  the  set  variability  simulated  of  the  of  of  Procedure,  variability  each  the  using  been  all  have been  that  the  values.  until  i n s t e a d of  moments c o u l d had  simulation  4  i s independent  Note t h a t  the  3 and  found  eccentricity  eccentricity  load values  was  calculate  and  computer  smallest possible suggested  s e c t i o n s i s adequate t o  that  a  represent  44  the  variability  cross  s e c t i o n s were The  sample load for of  i n s t r e n g t h . In t h e p r e s e n t  axial  size  study  250 column  simulated.  load  was  simulated  f o r the e c c e n t r i c i t i e s  by 50 v a l u e s  was  p o i n t s . The Code f o r t h e D e s i g n  20. T h i s  gives  of Concrete  B u i l d i n g s (CSA A23.3) a i m s a t a p o s s i b i l i t y 1 i n 1000, so a sample s i z e  while the  Structures  of  o f 1000 s i m u l a t e d  1000  overloads  load points  seems t o be a d e q u a t e . The the  above sample s i z e s w i l l  failure  order, order  and to  alternate  f u n c t i o n , G, w h i c h a r e t o be p l a c e d then save  used  to evaluate  computer  ranked  considered  after  time  values  of  the  obtained  by u s i n g a l l t h e 250,000  5%  reliability values  was  this  and  the  observed.  index  were  of the f a i l u r e  result  failure  Hence, performed  function.  was  function  order  compared  values. the  i n d e x . In  calculation  p l a c i n g them i n a s c e n d i n g  values)  i n ascending  the r e l i a b i l i t y  in  125,000  about  l e a d t o 250,000 v a l u e s f o r  A  were  (a t o t a l o f with  variation  calculations  using  only  a l l the  that of  f o r the 250,000  5. RESULTS  5.1 INTRODUCTION The  results  previous  obtained  chapters  reliability  index,  load.  code  The  columns  of  in  the  failure this load  a  the  the  at  method form  described of  probability  thousand,  and  the  by  = Prob(overloads) code  aims  specifying  factors.  probability  of the  nominal  axial  of o v e r l o a d s f o r  a  at  X  probability  of  Prob(understrength)  achieving  o f 1 i n 100,000 f o r c o l u m n s . The  is  the value  corresponding  a  i n the  o f one i n a h u n d r e d . Now  Prob(failure) Hence  in  /3 v / s aims  one  understrength  are  by t h e  method  for  strength reduction factors  The r e l i a b i l i t y  of f a i l u r e  a p r o b a b i l i t y of  index  0  doing  and some  corresponding  to  a  o f 1 i n 100,000 i s 4.265.  5.2 RELIABILITY AND AXIAL LOAD The obtain that  first  the value would  step  correspond  250mm  eccentricity used of  between  cross  of a x i a l  This  corresponding result  of t h e r e s u l t s  i s to  axial  load  t o t h e t a r g e t 0 o f 4.265. F i g u r e 16 j3 a n d  section  with  the  axial  2%  l o a d o f 50mm. L i n e a r  to obtain the a x i a l 4.265.  the a n a l y s i s  o f t h e maximum a l l o w a b l e n o m i n a l  shows t h e r e l a t i o n 250mm X  in  maximum  steel,  nominal  t o a /3 o f 4.265 i s r e f e r r e d  i n the following.  45  for a  f o r a mean  i n t e r p o l a t i o n was  load corresponding allowable  load  to the target 0 axial  load  t o a s t h e computer  46  5.5  x' o C  5  250mmX250mm(2%steel) •cc=50mm 5^  4.5  H  <-! l/r=BO  I 60  I 50  I 40  30  2010  3.5-  I ' ' ' I ' • ' I ' ' ' I ' •  100  200  300  400  1  I  1  500  1  i  600  700  NOMINAL AXIAL LOAD k N  Figure  16. R e l i a b i l i t y i n d e x /3 a n d maximum a l l o w a b l e a x i a l load  nominal  800 0C =2/3 0C =1/2  700-  0C =1/3 600-  50mm  80mm  150mm  250mm  250mmX250mm(2%steel) 1 1 1  I  1  1  20  1  1  I  1  1  •  25  1  I  11  ''I '  30  BENDING MOMENT  Figure  35  45  50  kN-m  17. Computer r e s u l t a n d t h e d e a d l o a d r a t i o  factor,a  47 5.3  R E L I A B I L I T Y AND LOAD RATIO As d e s c r i b e d e a r l i e r ,  load  and  the  dead  load r a t i o  this  study  live  the load e f f e c t s  l o a d were c o n s i d e r e d  factor  a was d e f i n e d  a was p r i m a r i l y  considered  However, t o c h e c k t h e v a r i a t i o n values one  of  a  equal  5.4 RESULT In  effect  the r e s u l t s  X  as  study  The computer with  evaluation  17. Note  by more t h a n  modified  i s allowed The  cross  of the l o a d  1/2. a,  of a  a b o u t 7%.  the  by  than  the  load  express the  by t h e p r e s e n t  the  allowable  Hognestad  that  effects.  obtained  found  code  axial  stress-strain  study  method.  were  For the  load c a p a c i t i e s  by t h e  was assumed  follow  to  relationship.  the rectangular s t r e s s  block  This i s  relationship  by t h e c o d e .  results  are  sections  presented  and  eccentricities  of  CAN3-A23.3-M77  and  results  the value  i n t h e code a d e q u a t e l y  results  those  of  more a c c u r a t e and  with  In  f o r slenderness  that  assumed  c o d e method, t h e s t r e n g t h o f c o n c r e t e the  the value  of the r e l i a b i l i t y  i t was  presented  of v a r i a b i l i t y  compared  t o have  2.4.3.  FORMAT  the present  factors  s t u d y . The  section  are presented  o f 10 a n d 60 i n f i g u r e  does not a f f e c t  in this  t h e dead  t o 1/3 a n d 2/3 were a l s o c o n s i d e r e d f o r  c r o s s s e c t i o n . The r e s u l t s  ratio's  in  of only  slenderness  loading in  for  in  the  various  ratios  tables  for 1  to  specimen different 3  for  t a b l e s 4 t o 6 f o r t h e new c o d e . The  a r e i n terms o f t h e f a c t o r  £ where  48  Mean E c c  Slenderness Ratio  (mm)  10  20  30  40  25  0 .87  0.91  0.99  1.12  1 .28  1.15  1 .38  50  0 .91  0.96  1.01  1 .06  1 .07  1 .06  1 .09  80  0 .95  0.94  0.93  0.95  0.95  1.01  0.94  1 50  0 .93  0.94  0.93  0.92  0.92  0.94  0.92  250  1 .10  1 .06  0.99  0.92  0.91  0.90  0.89  Table  1.  for  250mm X 250mm column  Mean E c c  60  50  w i t h 2% s t e e l ,  80  7=0.52  Slenderness Ratio  (mm)  10  20  30  50  1 . 14  1.15  1.18  1 .24  1 .33  1 .45  1 .69  100  1 .16  1.19  1 .22  1 .27  1.31  1 .33  1 .47  160  1 .17  1.19  1 .21  1 .22  1 .24  1 .28  1 .30  300  1 .14  1.17  1.18  1.19  1 .20  1 .21  1 .20  500  1 .24  1 .25  1 .26  1 .24  1.17  1.10  1 .03  for  T a b l e 2.  40  500mm X 500mm column  60  50  w i t h 2% s t e e l ,  80  7=0.76  Slenderness Ratio  Mean E c c  40  50  60  80  10  50  1 .06  1 .05  1 .03  1.01  1 .00  1 .04  1.15  100  1 .05  1 .05  1 .04  1 .04  1 .06  1 .09  1 .20  160  1 .06  1 .07  1 .06  1 .09  1 .09  1.12  1.19  300  1 .03  '' 1 .03  1 .03  1 .04  1 .06  1.12  1.21  500  1 .03  1 .04  1 .05  1 .06  1 .09  1.10  1.12  Table  3. $  20  30  (mm)  f o r 500mm X 500mm column  with  4% s t e e l ,  7=0.73  49  Mean E c c  Slenderness  Ratio  (mm)  10  20  30  40  50  60  80  25  0.86  0.87  0.94  1.06  1.21  1.12  1.32  50  0.86  0.91  0.95  1.01  1.04  1.03  1.05  80  0.89  0.91  0.91  0.94  0.93  0.97  0.91  150  0.92  0.92  0.91  0.90  0.89  0.91  0.92  250  1.07  1.05  1.00  0.92  0.90  0.87  0.91  Table  4. $ f o r 250mm X 250mm column n  Mean E c c  Slenderness  with  2% s t e e l ,  7=0.52  Ratio  (mm)  10  20  30  40  50  60  80  50  1.11  1.10  1.11  1.16  1.25  1.36  1.59  100  1.08  1.11  1.13  1.18  1.22  1.24  1.35  160  1.07  1.10  1.11  1.12  1.13  1.15  1.17  300  0.97  0.98  0.99  1.00  1.01  1.02  1.03  500  0.96  0.97  0.97  0.98  0.95  0.91  0.90  Table  5. $  n  f o r 500mm X 500mm column  Slenderness  Mean E c c  with  2% s t e e l ,  7=0.76  Ratio  (mm)  10  20  30  40  50  60  80  50  1.02  0.99  0.94  0.91  0.91  0.96  1.07  100  0.93  0.94  0.93  0.93  0.95  0.99  1.10  160  0.93  0.94  0.94  0.95  0.97  1.00  1.08  300  0.89  0.90  0.90  0.90  0.92  0.96  1.04  500  0.81  0.82  0.83  0.85  0.88  0.89  0.92  Table  6. $  f o r 500mm X 500mm column  with  4% s t e e l ,  7=0.73  50  /R  actual where study  R  are  actual  f o r a 0 of 4.265. R  obtained  by  results  the  as  compared  code  compared  CAN3-A23.3-M77,  5.5  the a x i a l  with  the a x i a l  method. with  while  those  load c a p a c i t i e s are  code  $  code  $  those  obtained  to  by  this  capacities  t o t h e computer  obtained  refers  n  load  refers  Q  found  by  following  t h e computer  results  by f o l l o w i n g C S A - A 2 3 . 3 ( 1 9 8 4 ) .  CODE EVALUATION MacGregor  late  1960s  al  and  slenderness code  et  found  ratio  less  calibration  l e s s must  (1970) s u r v e y e d that than  be  realized  the  ones t h a t u s u a l l y  with  t h a t columns w i t h create  into  categories  separately. slenderness consists  The  range percent  of  30  of  50  16in.(406mm)  of t h e c o l u m n s have  0.015. Hence,  study  each  or  them  any  ratios i t  as  had  consists  less.  The  percent to  at  should  also  r a t i o s are such  was  their divided  considered  of c o l u m n s second  with  category  r a t i o s g r e a t e r than  a use s t u d y  o f column  of  the  sizes  column  ratios  between  o u t of t h e s e c t i o n s c o n s i d e r e d  and  widths  0.005  for this  30.  results  24in.(610mm) and more t h a n  steel  a  of 30 or  t h e c o l u m n s were  category  the  attempt  slenderness and  in  v a r i o u s b u i l d i n g s i n A l b e r t a . The  t h a t more t h a n  from  higher  slenderness  conducted  in  However,  category  of columns w i t h  ratios  indicate  first  ratios  Grant(l976) steel  and  a l l of  problems  i n c r e a s e s . In t h i s  columns  slenderness  important.  importance two  nearly  30. Hence,  columns  be c o n s i d e r e d  22000  50 and  study,  51  the of  500mm  X 500mm c r o s s s e c t i o n w i t h  t h e more w i d e l y  which  can  lead  this  particular  the  results  practically For the  used to  cross  section  acceptable  Tables  i s performed over  close  to  increasing  Hence,  present  4>  0  c  a different  consistent For  would  be  a  t h e p r o p o s a l and  2  a l l the values  falling  0  computer  result  f o r low  increasing between  i t was  felt  slenderness, the that  code  there  and t h e  retaining  the  ( i . e . 0.60 a n d 0.85 r e s p e c t i v e l y ) , b u t  g  w o u l d be a r e a l i s t i c  m  (1984) l e a d s t o  proposal  f o r a more  code.  this  purpose, values  considered  presented  the  and  the  difference  results.  note  affecting  f o r t h e 500mm X 500mm c r o s s s e c t i o n  However, w i t h  computer  18  between  _ , Z($ - 1.0) " ^ N  especially  2% s t e e l .  were  sections  4, 5 a n d 6 r e v e a l t h a t CSA-A23.3  slenderness,  with  adversely  of £ f o r  category.  that are  an  i n the value  error e i sdefined as:  a particular  is  modification  code  t h e f o l l o w i n g method was a d o p t e d .  where t h e summation  with  cross  comparison  e  results  a  proposal.  c  in  would be one  without  other  objective  code p r o c e d u r e  Hence,  some improvement  f o r the  an  The  ones.  2% s t e e l  at  in figure  that  o f c/> between  intervals  error  of  0.05.  18 i n t e r m s o f t h e f a c t o r  f o r t h e 500mm X 500mm  cumulative  m  e i s minimum  0.55 The e.  and  1.05  result In  is  figure  (2% s t e e l ) c r o s s s e c t i o n , f o r d =1.0. m  However,  at  52  0.35  1.10  of e with 4>  Figure 18. Variation high 3  values  of  <t> m some values of K became as low as 0.78.  for t h i s column (an error of up to  22%  from  the  computer  r e s u l t ) . Such low values of p" are incompatible with the code aim as they would lead to a  very  low  reliability.  It i s  proposed that for such common sections, the minimum value of $ should not be less than 0.90, i . e . an error of 10%  from  the computer  result  this  entails  restricting  be  optimum.  the use of the moment  magnification formula to columns with slenderness ratios equal to or less than 60.  than  i s not allowed. With this  condition, a value for # of 0.70 was found to ' m However,  more  of  53  Mean E c c  Slenderness Ratio  (mm)  10  20  30  25  0.86  0.86  0.93  1 .03  1.16  1 .06  1 .24  50  0.86  0.90  0.94  0.99  1.01  0.99  0.98  80  0.89  0.90  0.90  0.92  0.90  0.93  0.87  150  0.92  0.91  0.90  0.88  0.87  0.89  0.89  250  1 .07  1 .04  0.99  0.91  0.89  0.85  0.87  T a b l e 7.  40  50  250mm X 250mm c o l u m n w i t h 2%  Mean E c c  60  80  s t e e l , 7=0.52  Slenderness Ratio  (mm)  10  20  30  40  50  50  1.11  1.10  1.10  1.14  1.21  1 .30  1 .49  100  1 .08  1.10  1.12  1.16  1.19  1 .20  1 .28  160  1 .07  1 .09  1.10  1.11  1.11  1.11  1.11  300  0.97  0.98  0.98  0.99  0.99  0.99  1 .00  500  0.96  0.97  0.96  0.97  0.94  0.90  0.89  500mm X 500mm c o l u m n w i t h 2%  T a b l e 8.  Mean E c c (mm)  60  80  s t e e l , 7=0.76  Slenderness Ratio 10  20  30  40  50  60  80  50  1 .02  0.96  0.93  0.90  0.89  0.92  1.01  100  0.93  0.93  0.93  0.92  0.93  0.96  1 .05  160  0.93  0.94  0.93  0.94  0.95  0.97  1 .03  300  0.89  0.89  0.89  0.89  0.90  0.94  1 .00  500  0.81  0.82  0.83  0.84  0.86  0.87  0.90  T a b l e 9.  500mm X 500mm c o l u m n w i t h 4%  s t e e l , 7=0.73  54  900  BENDING MOMENT k N - m  Figure 19. Results for the 250mm X 250mm column with 2% steel The results for a l l the three sections in terms of the factor  $ (denoted  in this case by the $ ^ ) are presented in  Tables 7-9 for <f> =0.60, A =0.80 and 6 c s m this  proposal  =0.70.  Note  does not have any s i g n i f i c a n t adverse  on the two less common cross sections  (250mm  X  that effect  250mm (2%  steel) and 500mm X 500mm (4% s t e e l ) ) . In addition to the above, CSA-A23.3  (1984)  leads  to a  note £  less  from  Table  5  that  than 1.00 for high  e c c e n t r i c i t i e s while for low e c c e n t r i c i t i e s the $ i s greater than  1.00. This  i s true even for the short column, which  suggests a change in the material resistance factors, 4> and 0s . By increasing <f> c and decreasing #s t h i s difference can c  4000 COMPUTER RESULT £SAJLA?3.3_1984  _  MODIFIED C S A - A 2 3 . 3  ecc=500mm  I  0  100  <  i i—I  1  i  1  1  i  I—  1  1  200  •  i  |  300  •  i  i  i  400  500  BENDING MOMENT kN-m  Figure 20. Results for the 500mm X 500mm column with 2% steel 5000  ecc=50mm  COMPUTER RESULT C S A - A 2 3 . 3 1984 CSA-A23.3-M77  l/r = 1 0 f w V  /  3500  o or < X  MODIFIED C S A - A 2 3 . 3  /  3000  -* X\ -  / / /.  *  L '•  *  \  N  N  \_ \ .  \_  IA=60/SN--.. 2000  /  / / // / / / / /  *  *  \  '• * *  *  *  *.•. *  /  / / / / / /  *  *  *  '•  \%  '• ^^^* • • '  *  *  * %  \ *  \  \ X \ %  *» X > X * * \ ^s\\  / 500  «  \  \  \ \  \ % \  I i1  \ .v— «cc=500mm  ^i—•"""*  H  0 100  200  300  400  500  600  700  800  BENDING MOMENT kN-m  F i g u r e 21. R e s u l t s f o r t h e 500mm X 500mm column w i t h 4% steel  56  •  Mean Ecc  Slenderness Ratio  (mm)  10  20  30  40  50  60  80  25  0.86  0.86  0.90  1 .01  1.14  1 .05  1 .24  50  0.83  0.87  0.91  0.97  ' 1 .00  0.98  0.99  80  0.86  0.88  0.88  0.90  0.89  0.94  0.87  1 50  0.92  0.92  0.91  0.89  0.88  0.89  0.88  250  1 .09  1 .05  0.99  0.92  0.92  0.90  0.85  Table 10. $ - for 250mm X 250mm column with 2% s t e e l , 7=0.52  Mean Ecc  Slenderness Ratio  (mm)  10  20  30  40  50  60  80  50  1.11  1.10  1 .07  1.11  1.18  1 .29  1 .49  100  1 .04  1 .07  1 .09  1.13  1.17  1.18  1 .28  1 60  1 .04  1 .06  1 .07  1 .08  1 .09  1.11  1.12  300  0.96  0.98  0.99  1 .00  1 .00  1.01  1 .02  500  1 .00  1.01  1 .00  1 .00  0.97  0.94  0.89  Table 11. $ - for 500mm X 500mm column with 2% s t e e l , 7=0.76  Mean Ecc  Slenderness Ratio  (mm)  10  20  30  40  50  60  80  50  1.00  0.97  0.92  0.89-  0.88  0.92  1.01  100  0.92  0.92  0.91  0.91  0.93  0.95  1.05  160  0.92  0.93  0.93  0.93  0.95  0.97  1.03  300  0.89  0.89  0.89  0.89  0.91  0.94  1.02  500  0.83  0.84  0.85  0.86  0.89  0.90  0.93  Table 12. 5 , for 500mm X 500mm column with 4% s t e e l , 7=0.73  57  be  r e d u c e d . To t h i s e n d t h e v a l u e s o f £, c o r r e s p o n d i n g  = 0.65, ),  4> =0.80 a n d s  a r e presented  10-12. A g a i n , 500mm  X  affecting  m  (2% steel)  19-21  result  cross section without  f o r the other show  sections f o r slenderness  the results  1  r a t i o s o f 10 a n d 60.  i s t h e one  i n Table  with only the  2%  steel  500mmX500mm 2%  steel  500mmX500mm 4%  steel Table  Category  adversly  cross  The  modified  <t>  changed t o  m  f r o m CSA-A23.3  m  i s changed w h i l e  2 a l l the three strength reduction factors  250mmX250mm  p 2  (both  CSA-A23.3  13 i n t e r m s o f e. I n t a b l e 13 P r o p o s a l  i s t h e one i n w h i c h o n l y #  Section  $  f o r the  f o r the three  a n d 1984 v e r s i o n s ) a n d t h e p r o p o s e d m o d i f i e d  i s presented  by  two s e c t i o n s .  0.70. A c o m p a r i s o n b e t w e e n t h e r e s u l t s 1977  i n t h i s case  c  f o r the three cross sections i n Tables  the results  CSA-A23.3  =0.70 ( d e n o t e d  note t h a t t h i s l e a d s t o b e t t e r r e s u l t s  500mm  Figures  <t>  t o <P  f o r Proposal  a r e changed.  CSA-A23.3 CSA-A23.3 P r o p o s a l  Proposal  1  2  M77  (1984)  1  0.0698  0.0920  0.0998  0. 1 105  2  0.1319  0.1151  0.1093  0. 1054  1  0.1936  0.0822  0.0800  0.0594  2  0.3063  0.2159  0.1801  0.1708  1  0.0472  0.1030  0.1066  0.1025  2  0.1045  0.0791  0.0861  0.0809  13. C o m p a r i s o n o f r e s u l t s i n t e r m s o f e  6.  6.1  CONCLUSIONS  AND  RECOMMENDATIONS  CONCLUSIONS  From t a b l e (CSA-A23.3 would for (i.e.  the root  a modification  column  respectively)  would  However, a change  and  s  lead  <A  to  m  to  members o t h e r  than  c o m p r e s s i o n . Hence,  suggested  to  evaluate the f i r s t  the  0.65,  still  i n the m a t e r i a l  with a x i a l  Nevertheless,  by a s  much  as  more  16%  and  more a c c u r a t e  those with  code  modification  0.80  resistance  a  the  0.65 t o 0.70  m  of common u s a g e . A more d r a s t i c c  affect  in  o f tf> from  mean s q u a r e e r r o r  t o change 4> , <j>  also  that  1984) t o change t h e v a l u e  reduce a  13 we n o t e  0.70  results.  factors flexure  detailed  would  combined study  is  consequences of such a change.  modification  should  be  a  practical  suggest i o n . It become and  c a n a l s o be n o t e d increasingly  code  with  now p o s s i b l e columns.  for large  results  Moreover,  the  slender,  more from t h e computer  cause of concern the  are  Hence,  i t  u s e o f t h e moment  eccentricities  is  of loading in  o f computer  predict  because  such  cases.  software,  the behaviour  of  a  i t  is  slender  magnification  columns  ratio  formula  t o or l e s s than  lowest  $  58  to  60. Note t h a t  of, up t o 60, t h e p r o p o s a l a  especially  to restrict  slenderness  to  This  more  i s d e s i r a b l e and p r a c t i c a l  r a t i o s equal  leads  result.  non-conservative  to e f f e c t i v e l y  as the columns  t h e code r e s u l t s d e v i a t e  the development  slenderness  0.70  i n the r e s u l t s that  t o change  of  fora 4>  m  to  o f 0.90 f o r t h e commonly u s e d  59 section  6.2  (Table  RECOMMENDATIONS  It  is  recommended  condition, Code  the  The  1984).  case  followed.  to  0 should m  Concrete  ratio  condition evaluate  A computer  thesis  this end.  of  subject  of  to  the  be t a k e n  Structures  as 0.70 in the for  t o be s a t i s f i e d the  column  following  Buildings  is:  should  not  be  than 60.  this  procedure  that  The c o n d i t i o n  slenderness  greater  this  value  f o r the Design of  (CSA-A23.3  In  8).  would  cannot the  program be  be  satisfied,  slenderness similar  a  effect  rational s h o u l d be  t o the one d e s c r i b e d  an a c c e p t a b l e  rational  procedure  in to  REFERENCES  Alcock,W.J., tests  and  Nathan,N.D.,  1977,  Moment  magnification  o f p r e s t r e s s e d c o n c r e t e c o l u m n s , J o u r n a l P C I , 2 2 , No.  4, p p . 5 0 - 6 1 .  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