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Reliability of slender reinforced concrete columns Bhola, Rajendra Kumar 1985

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RELIABILITY OF SLENDER REINFORCED CONCRETE COLUMNS by RAJENDRA KUMAR BHOLA B. Tech., Indian I n s t i t u t e of Technology, New Delhi,1983 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in FACULTY OF GRADUATE STUDIES Department of C i v i l E n g i n e e r i n g We accept t h i s t h e s i s as conforming to the r e q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA January, 1985 © Rajendra Kumar Bhola, 1985 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the r e q u i r e m e n t s f o r an advanced degree a t the THE UNIVERSITY OF BRITISH COLUMBIA, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r 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 c o p y i n g of t h i s t h e s i s f o r s c h o l a r l y purposes may be g r a n t e d by the Head of my Department or by h i s or her r e p r e s e n t a t i v e s . I t i s un d e r s t o o d t h a t c o p y i n g or p u b l i c a t i o r T - o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department of C i v i l E n g i n e e r i n g THE UNIVERSITY OF BRITISH COLUMBIA 2075 Wesbrook P l a c e Vancouver, Canada V6T 1W5 Date: J a n u a r y , 1985 ABSTRACT The e f f e c t s of the v a r i a b i l i t y i n s t r e n g t h and l o a d i n g on the r e l i a b i l i t y of sle n d e r , r e i n f o r c e d concrete columns are i n v e s t i g a t e d using the Monte C a r l o s i m u l a t i o n technique. The columns are c o n s i d e r e d to be a x i a l l y loaded 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 l o a d . V a r i a b i l i t i e s i n s t r e n g t h , a x i a l l o a d and e c c e n t r i c i t y of a x i a l loads are c o n s i d e r e d . A new procedure c a l l e d the I m p l i c i t U n c o r r e l a t i o n Procedure has been developed to f i n d the values of the f a i l u r e f u n c t i o n from the values of the b a s i c v a r i a b l e s named above. The a l l o w a b l e a x i a l load at v a r i o u s e c c e n t r i c i t y l e v e l s c o rresponding to a p r o b a b i l i t y of f a i l u r e of one i n one hundred thousand has been found f o r three d i f f e r e n t c r o s s s e c t i o n s . Seven d i f f e r e n t s l e n d e r n e s s r a t i o s are c o n s i d e r e d fo r each c r o s s s e c t i o n . The r e s u l t s are compared with those obtained by f o l l o w i n g the code procedures o u t l i n e d i n CAN3-A23.3-M77 and CSA-A23.3 (1984). A change i n the performance f a c t o r f o r moment m a g n i f i c a t i o n , # m , (as given i n CSA-A23.3 (1984)) i s recommended i n order to o b t a i n a more a c c u r a t e and c o n s i s t e n t l e v e l of r e l i a b i l i t y i n the design of slender r e i n f o r c e d c o n c r e t e columns. i i T A B L E OF CONTENTS ABSTRACT i i LIST OF FIGURES v LIST OF TABLES vi ACKNOWLEDGEMENT v i i 1. INTRODUCTION 1 1 . 1 CODE METHOD . 1 1.1.1 Method of CAN3-A23.3-M77 2 1.1.2 Method of CSA A23.3( 1984) 3 2. 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 Variability of Dead Loads 12 2.4.2 Variability of Live Loads .13 2.4.3 Load Combination ;.... 1 3 3. STRENGTH MODEL 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 i i i 3.4 SLENDER COLUMN BEHAVIOUR 22 3.4.1 Failur e Modes 23 3.4.2 Calculation Procedure 25 3. 5 PROGRAM INPUT 28 3.5.1 Cross Section Properties 28 3 .5 .2 Slenderness E f f e c t 30 3.6 PROGRAM OUTPUT 30 4. RELIABILITY EVALUATION 33 4.1 INTRODUCTION 33 4.2 RELIABILITY INDEX CONCEPT 33 4.3 RELIABILITY CRITERION 38 4.3.1 Implicit Uncorrelation Procedure 39 4 . 3 .2 Standard Deviation for E c c e n t r i c i t y 41 4.4 SAMPLE SIZE 43 5. RESULTS • 45 5.1 INTRODUCTION 4 5 5.2 RELIABILITY AND AXIAL LOAD 4 5 5.3 RELIABILITY AND LOAD RATIO . . . 4 7 5.4 RESULT FORMAT 47 5.5 CODE EVALUATION 50 6. CONCLUSIONS AND RECOMMENDATIONS 58 6.1 CONCLUSIONS 58 6.2 RECOMMENDATIONS 59 REFERENCES • 60 i v L I S T OF FIGURES 1 M a t e r i a l R e s i s t a n c e F a c t o r s 4 2 The M o n t e C a r l o S i m u l a t i o n T e c h n i q u e 5 3 C r o s s s e c t i o n b e h a v i o u r 18 4 U l t i m a t e i n t e r a c t i o n d i a g r a m 20 5 C u r v a t u r e c o n t o u r s 20 6 M o m e n t - c u r v a t u r e - a x i a l l o a d r e l a t i o n s h i p s 21 7 C o l u m n w i t h a x i a l l o a d a n d e q u a l e n d e c c e n t r i c i t i e s . . . . 2 2 8 B e h a v i o u r o f an e c c e n t r i c a l l y l o a d e d c o l u m n 24 9 I n t e r a c t i o n d i a g r a m f o r an e c c e n t r i c a l l y l o a d e d c o l u m n . 2 4 10 C o l u m n d e f l e c t i o n c u r v e s •. 27 11 S t r e n g t h i n t e r a c t i o n c u r v e f o r s l e n d e r c o l u m n s 30 12 D e f i n i t i o n o f t h e r e l i a b i l i t y i n d e x 34 13 R e l i a b i l i t y i n d e x a n d t h e p r o b a b i l i t y o f f a i l u r e 36 14 The I m p l i c i t U n c o r r e l a t i o n P r o c e d u r e 38 15 S t a n d a r d d e v i a t i o n o f e c c e n t r i c i t y 41 16 R e l i a b i l i t y i n d e x 0 a n d maximum a l l o w a b l e n o m i n a l a x i a l l o a d 45 17 C o m p u t e r 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 f a c t o r a . . . . . . . 4 5 18 V a r i a t i o n o f e w i t h <t> 53 m 19 R e s u l t s f o r t h e 250mm X 250mm c o l u m n w i t h 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 w i t h 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 w i t h 4% s t e e l 54 v L I S T OF T A B L E S 1 $ Q f o r the 250mm X 250mm column with 2% s t e e l 47 2 $ o f o r the 500mm X 500mm column with 2% s t e e l 47 3 $ f o r the 500mm X 500mm column with 4% s t e e l 47 o 4 $ n f o r the 250mm X 250mm column with 2% s t e e l 48 5 $ f o r the 500mm X 500mm column with 2% s t e e l 48 n 6 $ n f o r the 500mm X 500mm column with 4% s t e e l 48 7 £ for. the 250mm X 250mm column with 2% s t e e l 56 8 f o r the 500mm X 500mm column with 2% s t e e l 56 9 $ f o r the 500mm X 500mm column with 4% s t e e l 56 10 $ 0 f o r the 250mm X 250mm column with 2% s t e e l 57 11 $ ~ f o r the 500mm X 500mm column with 2% s t e e l 57 p2 12 f ° r t n e 500mm X 500mm column with 4% s t e e l 57 13 Comparision of r e s u l t s i n terms of e 58 v i ACKNOWLEDGEMENT I w i s h t o ex p r e s s my deepest g r a t i t u d e t o my s u p e r v i s o r s , P r o f e s s o r N.D.Nathan, and P r o f e s s o r R.O.Foschi f o r t h e i r i n v a l u a b l e a d v i c e and guidance throughout the r e s e a r c h and i n the p r e p a r a t i o n of t h i s t h e s i s . Thanks a r e a l s o due t o a l l those who have re n d e r e d h e l p i n t h i s p r o j e c t , d i r e c t l y or i n d i r e c t l y . The f i n a n c i a l s u p port of the N a t i o n a l R e s e a r c h C o u n c i l of Canada i n the form of a Rese a r c h A s s i s t a n t s h i p i s g r a t e f u l l y acknowledged. ) v i i 1. INTRODUCTION The a c t u a l s t r e n g t h of a r e i n f o r c e d concrete member d i f f e r s from the nominal s t r e n g t h c a l c u l a t e d by the design engineer due to v a r i a t i o n s in the m a t e r i a l s t r e n g t h s and the geometry of the member, as w e l l as the v a r i a b i l i t i e s inherent i n the equations used to compute the member s t r e n g t h . S i m i l a r l y , d e s i g n e r s use constant nominal values of loads i n t h e i r c a l c u l a t i o n t o f f o r c e s , but the a c t u a l loads are v a r i a b l e . T h i s v a r i a b i l i t y i n s t r e n g t h and l o a d i n g i s accounted f o r i n one form or another i n s a f e t y p r o v i s i o n s of a l l e x i s t i n g b u i l d i n g codes. In t h i s study the e f f e c t of these v a r i a b l e s was i n v e s t i g a t e d u sing the Monte C a r l o technique f o r random s i m u l a t i o n . S i m i l a r a n a l y s e s have been a p p l i e d to beams and columns by A l l e n ( l 9 7 0 ) , Ellingwood(1977), Grant et al(1978) and Mirza and MacGregor(1982). 1.1 CODE METHOD Corn e l l ( 1 9 6 9 ) and L i n d ( l 9 7 l ) have shown that i n order to achieve a c o n s i s t e n t l e v e l of r e l i a b i l i t y i n design, the code design c r i t e r i o n should be of the form: <j>R > XU where: R = design s t r e n g t h U = nominal s p e c i f i e d l o a d <p = s t r e n g t h r e d u c t i o n f a c t o r X = loa d f a c t o r 1 2 A summary of the a c t u a l procedure f o l l o w e d by the Code for the Design of Concrete S t r u c t u r e s f o r B u i l d i n g s (CSA-A23.3) i s presented below. 1 . 1 . 1 METHOD OF CAN3-A23.3-M77 The f a c t o r e d design load ( f o r only dead and l i v e loads) i s given by U = X D D + X L L where D and L are the nominal values of the dead and the l i v e loads r e s p e c t i v e l y and X D and X L are the corresp o n d i n g l o a d f a c t o r s . The value f o r X D i s taken as. 1.4 while that f o r XT i s taken as 1.7. The short column s t r e n g t h of the column i s reduced by a c a p a c i t y r e d u c t i o n f a c t o r tf>. The value of 4> f o r pure bending i s 0.90 and f o r a x i a l compression or a x i a l compression combined with bending i t i s 0.70. </> i s l i n e a r l y i n c r e a s e d to 0.90 as the a x i a l design load decreases from 0.1 O f A to zero. c g To account f o r the slenderness e f f e c t of the columns, the members are designed u s i n g a magnified moment M d e f i n e d by: c J M = 8M, c 2 where M 2 i s the l a r g e r d e sign end moment on the member, based on U as d e s c r i b e d above, and i s c a l c u l a t e d from a c o n v e n t i o n a l e l a s t i c frame a n a l y s i s and 5 i s a moment m a g n i f i c a t i o n f a c t o r . M c must be l e s s than the reduced short column s t r e n g t h . 5 i s given by the r e l a t i o n : 5 = 2! > 1.0 1-P /<t>P u' r c where 4> v a r i e s as s t a t e d above, and rr 2EI where k l ^ i s the e f f e c t i v e l e n g t h of the column. EI i s c a l c u l a t e d as: 0.2E I + E I EI c g se s 1 + ^d f o l l o w i n g the n o t a t i o n of CAN3-A23.3-M77. C m i s given by: M, C = 0.6 + 0.4— M 2 where M, and M2 are the sm a l l e r and the l a r g e r d esign end moments r e s p e c t i v e l y at the two ends of the member. 1.1.2 METHOD O F C S A A 2 3 . 3 ( 1 9 8 4 ) The t o t a l design l o a d f o r the case when only the dead lo a d and the l i v e loads are a c t i n g i s given by a r e l a t i o n s i m i l a r to the one i n CAN3-A23.3-M77 except that i n t h i s case X Q i s taken as 1.25 while X L has the value 1.50. M a t e r i a l r e s i s t a n c e f a c t o r s are a p p l i e d to the nominal s t r e n g t h s of co n c r e t e and r e i n f o r c i n g bars as shown i n f i g u r e 1. The value f o r <j>c i s 0.60 while t h a t f o r <6 i s 0.85. s The slenderness e f f e c t i s taken i n t o account u s i n g a procedure i d e n t i c a l to the one i n CAN3-A23.3-M77 but c Concrete Steel F i g u r e 1. M a t e r i a l r e s i s t a n c e f a c t o r s w i t h the moment m a g n i f i c a t i o n f a c t o r d e f i n e d a s : 6 = -55 ;> , < 0 ' - p / V c where <f> = 0.65. m 2. MONTE CARLO SIMULATION 2.1 INTRODUCTION I f a r e l a t i o n s h i p can be d e r i v e d between the performance of a system and each v a r i a b l e a f f e c t i n g the per f o r m a n c e , and i f s t a t i s t i c a l p r o p e r t i e s of the d i s t r i b u t i o n s of 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 t o use randomly s e l e c t e d v a l u e s of the v a r i a b l e s t o c a l c u l a t e the v a r i a b i l i t y of t h e system p e r f o r m a n c e . T h i s t e c h n i q u e , shown s c h e m a t i c a l l y i n f i g u r e 2 and c a l l e d Monte C a r l o s i m u l a t i o n , was used i n t h i s s tudy t o de t e r m i n e the v a r i a b i l i t y of the s t r e n g t h of r e i n f o r c e d c o n c r e t e columns, because of the c o m p l e x i t y of the s t r e n g t h 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 REPEAT MANY TIMES CALCULATE VALUE OF SYSTEM PERFORMANCE OUTPUT: SUMMARIZE RESULTING VALUES OF SYSTEM PERFORMANCE WITH STATISTICAL ANALYSIS F i g u r e 2. The Monte C a r l o s i m u l a t i o n t e c h n i q u e RELATIONSHIP BETWEEN VARIABLES AND SYSTEM PERFORMANCE 5 6 2.2 DESCRIPTION OF THE METHOD The Monte C a r l o method may be d e s c r i b e d as a means of s o l v i n g problems n u m e r i c a l l y i n mathematics, p h y s i c s , e n g i n e e r i n g , and other s c i e n c e s through sampling experiments. The problem may be posed i n e i t h e r p r o b a b a l i s t i c or d e t e r m i n i s t i c form. In the p r o b a b a l i s t i c case the a c t u a l random v a r i a b l e or f u n c t i o n appearing i n the problem i s simulated, whereas i n the d e t e r m i n i s t i c case an a r t i f i c i a l random v a r i a b l e or f u n c t i o n i s f i r s t c o n s t r u c t e d and then simulated. The s i m u l a t i o n process i s computerised to f o l l o w the d i s t r i b u t i o n p r o p e r t i e s of the v a r i a b l e . The method normally c o n s i s t s of the f o l l o w i n g s t e p s : 1. s i m u l a t i o n of the random v a r i a b l e f u n c t i o n , 2. s o l u t i o n of the d e t e r m i n i s t i c problem f o r a l a r g e number of r e a l i z a t i o n s of the l a t t e r , and 3. s t a t i s t i c a l a n a l y s i s of the r e s u l t s . The present chapter and the two f o l l o w i n g i t d i s c u s s each of these steps i n d e t a i l . 2.3 VARIABILITY OF STRENGTH In order to e v a l u a t e the v a r i a b i l i t y of the s t r e n g t h of slender r e i n f o r c e d c o n c r e t e members a knowledge of the v a r i a b i l i t y of the parameters that a f f e c t the s t r e n g t h i s necessary. These parameters are the concrete s t r e n g t h i n compression and t e n s i o n , the y i e l d s t r e n g t h and the p o s i t i o n of the reinforcement, and the dimensions of the c r o s s s e c t i o n of the member. The v a r i a b i l i t i e s of these parameters 7 used i n t h i s study were based p r i m a r i l y on the data summarized by Mirza et al(1979) and Mirza and MacGregor(1979a,b). Three major assumptions were made in determining the m a t e r i a l s t r e n g t h s to be used i n the d e r i v a t i o n of the s t r e n g t h of the r e i n f o r c e d c o n c r e t e columns. 1. The v a r i a b i l i t y of the concrete p r o p e r t i e s and dimensions corresponds to average q u a l i t y c o n s t r u c t i o n . T h i s assumption i s made so that the r e s u l t s may represent the average of Canadian c o n s t r u c t i o n p r a c t i c e . S i m i l a r l y , the reinforcement was assumed to be drawn from a p o p u l a t i o n r e p r e s e n t i n g a l l sources of reinforcement i n Canada and the Un i t e d S t a t e s . 2. The m a t e r i a l s t r e n g t h s were assumed to correspond to lower l o a d i n g r a t e s than those g e n e r a l l y used i n m a t e r i a l t e s t s or l a b o r a t o r i e s . The c r u s h i n g s t r e n g t h of concrete was based on a 1-hour l o a d i n g to f a i l u r e , and the y i e l d s t r e n g t h of s t e e l was based on a s o - c a l l e d s t a t i c l o a d i n g r a t e . T h i s i s a c o n s e r v a t i v e assumption s i n c e the s t r e n g t h s of co n c r e t e and s t e e l tend to i n c r e a s e at high r a t e s of l o a d i n g . 3. Increase i n the long-time s t r e n g t h of the concre t e due to i n c r e a s e d m a t u r i t y of the c o n c r e t e , as w e l l as p o s s i b l e f u t u r e c o r r o s i o n of the reinforcement were ignored. Gardiner and Hatcher(1970) and Washa 8 and Wendt(l975) report that the mean s t r e n g t h of 20-25 year o l d concrete i s expected to be 150-250% of the mean s t r e n g t h at 28 days. For t h i s study however, t h i s e f f e c t was d i s r e g a r d e d and the concrete s t r e n g t h was r e l a t e d to the 28-day t e s t c y l e n d e r s t r e n g t h . T h i s leads to a c o n s e r v a t i v e estimate of member s t r e n g t h . 2.3.1 PROPERTIES OF CONCRETE Under c u r r e n t d e s i g n , p r o d u c t i o n , t e s t i n g , and q u a l i t y - c o n t r o l procedures, the s t r e n g t h of concre t e i n a s t r u c t u r e may d i f f e r from i t s s p e c i f i e d design s t r e n g t h and may not be uniform throughout the s t r u c t u r e . The major sources of v a r i a t i o n i n concre t e s t r e n g t h are the v a r i a t i o n s i n m a t e r i a l p r o p e r t i e s and p r o p o r t i o n s of the concre t e mix, the v a r i a t i o n s i n mixing, t r a n s p o r t i n g , p l a c i n g and c u r i n g methods, the v a r i a t i o n s i n t e s t i n g procedures, and v a r i a t i o n s due to con c r e t e being i n a s t r u c t u r e r a t h e r than i n c o n t r o l specimens. The f o l l o w i n g data, as suggested by Mirza et al(1979), accounts f o r most of these e f f e c t s . Assuming a r a t e of l o a d i n g corresponding to f a i l u r e i n a t e s t l a s t i n g 1 hour, the mean compressive s t r e n g t h of c o n c r e t e i n a s t r u c t u r e was taken as 23.36 MPa (3388 p s i ) f o r 27.6 MPa (4000 p s i ) c o n c r e t e . The c o e f f i c i e n t of v a r i a t i o n f o r the c a s t - i n - p l a c e c o n c r e t e was taken as 0.175. 9 The mean value of the modulus of e l a s t i c i t y f o r the 27.6 MPa concrete was taken as 22476.91 MPa (3260 k s i ) with a c o e f f i c i e n t of v a r i a t i o n of 0.12. These p r o p e r t i e s were assumed to f o l l o w normal d i s t r i b u t i o n s . 2.3.2 PROPERTIES OF REINFORCEMENT The sources of v a r i a t i o n i n the s t e e l y i e l d s t r e n g t h are the f o l l o w i n g : 1. V a r i a t i o n i n the s t r e n g t h of the m a t e r i a l i t s e l f . 2. V a r i a t i o n i n the area of c r o s s s e c t i o n of the bar. 3. E f f e c t of the r a t e of l o a d i n g . 4. E f f e c t of bar diameter on the p r o p e r t i e s of the bars. 5. E f f e c t of s t r a i n at which y i e l d i s d e f i n e d . The f o l l o w i n g data, as suggested by Mirza and MacGregor (1979b), accounts f o r the e f f e c t of most of these sources of v a r i a t i o n . The mean and the c o e f f i c i e n t of v a r i a t i o n f o r the s t a t i c y i e l d s t r e n g t h f o r the s t e e l reinforcement were taken as 460.6 MPa (66.8 k s i ) and 0.09, r e s p e c t i v e l y , f o r grade 60 hot r o l l e d b a r s . The y i e l d s t r e n g t h was assumed to f o l l o w a Beta d i s t r i b u t i o n . These v a l u e s were assumed to be independent of bar s i z e . 1 0 The mean value f o r the modulus of e l a s t i c i t y was taken as 201000 MPa (29200 k s i ) with a c o e f f i c i e n t of v a r i a t i o n , 0.033. The p r o b a b i l i t y d i s t r i b u t i o n of the modulus of e l a s t i c i t y was c o n s i d e r e d to be normal. The r a t i o of a c t u a l to nominal v a l u e s of the area of c r o s s s e c t i o n of the bars was assumed to f o l l o w a normal d i s t r i b u t i o n t r u n c a t e d at 0.94 with a mean value of 0.99 and a c o e f f i c i e n t of v a r i a t i o n 0.024. Since the s t e e l i n a c o n c r e t e member must be some combination of whole bars, the area of s t e e l a c t u a l l y p r o v i d e d may d i f f e r from that c a l c u l a t e d . As suggested by M i r z a and MacGregor(1979a), t h i s e f f e c t has been c o n s i d e r e d by a m o d i f i e d lognormal d i s t r i b u t i o n having a mean of 1.01, a c o e f f i c i e n t of v a r i a t i o n of 0.04, and a m o d i f i c a t i o n constant of 0.91 below which the m o d i f i e d lognormal d i s t r i b u t i o n equals z e r o . The v a r i a b i l i t y of s t r e n g t h w i t h i n a s i n g l e bar i s r e l a t i v e l y small and i s n e g l e c t e d i n t h i s study. 2.3.3 GEOMETRIC PROPERTIES Geometric i m p e r f e c t i o n s i n r e i n f o r c e d c o n c r e t e members are mainly caused by d e v i a t i o n from the s p e c i f i e d values of the c r o s s s e c t i o n a l shape and dimensions, the p o s i t i o n i n g of r e i n f o r c i n g bars, the h o r i z o n t a l i t y and v e r t i c a l i t y of c o n c r e t e l i n e s , the alignments of columns and beams, and the grades and s u r f a c e s of the c o n s t r u c t e d s t r u c t u r e s . The data used to 1 1 account f o r these e f f e c t s i n t h i s study were as suggested by Mirza and MacGregor(1979a). I t has been assumed that a normal d i s t r i b u t i o n can be used to represent the d i s t r i b u t i o n of the geometric i m p e r f e c t i o n s of r e i n f o r c e d c o n c r e t e members. The mean d e v i a t i o n of the a c t u a l c r o s s s e c t i o n a l dimensions from the s p e c i f i e d dimension was taken as +1.52 mm (+0.06 i n ) 1 . The standard d e v i a t i o n of the cr o s s s e c t i o n a l dimensions was taken as 6.35mm (0.25 in) . The l o c a t i o n of v e r t i c a l reinforcement i n columns i s a f f e c t e d by t o l e r a n c e s i n the t i e s , forms, column alignment from f l o o r to f l o o r and care taken to center the reinforcement cage w i t h i n the form. The mean d e v i a t i o n of concret e cover f o r s t e e l bars was taken as +8.13 mm (+0.32 i n ) 2 . The standard d e v i a t i o n of the conc r e t e cover f o r s t e e l bars was taken as 4.32mm (0. 1 7 i n ) . 2.4 VARIABILITY OF LOADS In the a n a l y s i s of s a f e t y i t i s necessary to d e a l with l o a d e f f e c t s such as moments, e t c . r a t h e r than the loads themselves. I t i s t h e r e f o r e necessary to have a d i s t r i b u t i o n of l o a d e f f e c t s . These are found by combining the 1 i . e . s p e c i f i e d dimension + 1.52 mm. 2 i . e . s p e c i f i e d cover + 8.13 mm. 1 2 v a r i a b i l i t y of the loads themselves with the v a r i a b i l i t y i n t r o d u c e d by the s t r u c t u r a l a n a l y s i s . The l a t t e r component i s small and can be ignored except i n the case of dead l o a d . The v a r i a b i l i t y of the loads themselves i s in turn the combined v a r i a b i l i t y of the magnitude and the d i s t r i b u t i o n of the lo a d , and t h i s i n f l u e n c e s the load e f f e c t s on the columns. In the f o l l o w i n g , v a r i a b i l i t y from both the components i s c o n s i d e r e d . 2.4.1 VARIABILITY OF DEAD LOADS Except i n cases where the lower p a r t s of the b u i l d i n g have to be designed before the upper p a r t i s w e l l d e f i n e d , dead loads are known a c c u r a t e l y i n comparision with other l o a d s . The r a t i o , R Q of a c t u a l to nominal dead l o a d was represented by a normal d i s t r i b u t i o n with the mean equal to 1.05 and the c o e f f i c i e n t of v a r i a t i o n equal to 0.07. T h i s was based on s t u d i e s of the v a r i a b i l i t y of dead l o a d e f f e c t s i n con c r e t e s t r u c t u r e s r e s u l t i n g from v a r i a t i o n s i n dimensions, d e n s i t i e s , superimposed l o a d s , and a n a l y s i s (Ellingwood et al 1980). A l l e n ( l 9 7 5 ) assumed v a l u e s of 1.0 and 0.07 whereas L i n d et al (1978) used values of 1.0 and 0.05. Nowak and L i n d ( l 9 7 9 ) assumed the v a l u e s as 1.05 and 0.08 f o r s i t e - c a s t c o n c r e t e b r i d g e s t r u c t u r e s . 1 3 2.4.2 VARIABILITY OF LIVE LOADS The r a t i o , R L of the maximum 30 year l i v e load to the nominal l o a d was assumed to have a mean of 0.70, and as suggested by Allen(1975) i s independent of the t r i b u t a r y a r e a. On the b a s i s of load survey r e s u l t s of M i t c h e l l and Woodgate(1971), the c o e f f i c i e n t of v a r i a t i o n of maximum l i v e load i s taken to be 0.30, and i s independent of the t r i b u t a r y a r e a . Nowak and C u r t i s ( l 9 8 0 ) suggest a gamma d i s t r i b u t i o n f o r the l i v e l o a d s , while A l l e n ( l 9 7 5 ) does not s p e c i f y the d i s t r i b u t i o n . In the present study an extreme type I d i s t r i b u t i o n was assumed f o r the maximum l i v e load i n 30 ye a r s . 2.4.3 LOAD COMBINATION It i s important to combine the d i f f e r e n t load e f f e c t s p r o p e r l y so as to achieve a more r e a l i s t i c assessment of r e l i a b i l i t y . Load e f f e c t s are u s u a l l y random f u n c t i o n s of time. When the design i s r e s i s t i n g g r a v i t y l o a d s , one p o s s i b l e l o a d combination i s the dead loa d (which would be constant i n time) and the maximum l i v e l o a d (or the maximum occupancy load) i n the l i f e t i m e of the s t r u c t u r e . T h i s combination of the loads has been c o n s i d e r e d i n t h i s study with the nominal va l u e s of the l i v e and the dead loads equal to each o t h e r . 1 4 Define a dead load r a t i o f a c t o r , a as: Nominal dead load a = T o t a l nominal lo a d where the t o t a l nominal load i s the sum of the nominal v a l u e s of the dead and the l i v e l o a d s . Hence f o r our case a=l/2. Two more cases with a=1/3 and a=2/3 ( i . e . with the nominal value of the l i v e load equal to twice the nominal value of the dead l o a d , and v i c e - v e r s a ) were a l s o c o n s i d e r e d f o r one c r o s s s e c t i o n i n order to i n v e s t i g a t e the e f f e c t of t h i s f a c t o r on the r e l i a b i l i t y . The l o a d e f f e c t s of wind, snow and earthquake have not been c o n s i d e r e d . No load f a c t o r s have been a p p l i e d . The l o a d combination procedure may be summarized as f o l l o w s : D + L = R D D N + R L L N where D = a c t u a l dead l o a d L = a c t u a l l i v e l o a d DXT = nominal dead load N L.T = nominal l i v e load and R D and R^ are as d e s c r i b e d p r e v i o u s l y . Then D D + L = L N {R D + R L } LN 1 5 Now we have: DN a = D N + L N Rearranging D N L N 1-a Hence, the load e f f e c t can f i n a l l y be w r i t t e n as: D + L = LN { RD ( "FS> + R L } T h i s f o r m u l a t i o n was used to simulate the load e f f e c t f o r t h i s study. 3. STRENGTH MODEL 3.1 INTRODUCTION T h i s chapter d e s c r i b e s a t h e o r e t i c a l model f o r p r e d i c t i n g the s t r e n g t h of slender r e i n f o r c e d columns. The model uses the s t r e s s s t r a i n behaviour f o r concret e as given by Desai and Krishnan (1964). 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 f o r s t e e l i s assumed to be e l a s t i c - p e r f e c t l y p l a s t i c . The theory and assumptions i n the model are d e s c r i b e d , along with a b a s i s f o r the computer program used to o b t a i n the s t r e n g t h of s l e n d e r , r e i n f o r c e d concrete columns. The o r g a n i z a t i o n of the program i s s i m i l a r to one developed by Nathan (1972) f o r r e i n f o r c e d and p r e s t r e s s e d c o n c r e t e , and v e r i f i e d f o r those m a t e r i a l s by Alcock and Nathan(1977). 3.2 ASSUMPTIONS The f o l l o w i n g assumptions are made : 1. Plane s e c t i o n s remain pl a n e . 2. M a t e r i a l p r o p e r t i e s are constant along the len g t h of the column. 3. Dimensions and e r r o r i n p l a c i n g of s t e e l bars are constant along the l e n g t h of the column. 4. I f moment v a r i e s along a member, f a i l u r e occurs at the c r o s s s e c t i o n s u b j e c t e d to maximum moment, or by i n s t a b i l i t y of the member. 5. Bending i n only one plane i s c o n s i d e r e d . 16 1 7 6. E f f e c t s of shrinkage are n e g l e c t e d . 7 . No t o r s i o n a l or out - o f - p l a n e deformations are co n s i d e r e d . Duration of load e f f e c t s are not co n s i d e r e d . Shear f a i l u r e s are not c o n s i d e r e d . 8. There i s no s l i p between the concre t e and the r e i n f o r c i n g s t e e l . 3.3 CROSS SECTION BEHAVIOUR The u l t i m a t e i n t e r a c t i o n diagram and the moment-c u r v a t u r e - a x i a l load r e l a t i o n s h i p s f o r a c r o s s - s e c t i o n are d e r i v e d by using a simple step-by-step procedure to o b t a i n a x i a l l o a d and moment c a p a c i t i e s f o r a range of n e u t r a l a x i s depths and c u r v a t u r e s . R e c a l l t h a t the u l t i m a t e i n t e r a c t i o n diagram shows the l i m i t i n g combinations of a x i a l load and bending moment that a s e c t i o n can r e s i s t . 3.3.1 ULTIMATE INTERACTION DIAGRAM For the r e i n f o r c e d c o n c r e t e c r o s s s e c t i o n such as the one shown i n f i g u r e 3 ( a ) , the c a l c u l a t i o n begins by c o n s i d e r i n g the top f i b r e at u l t i m a t e s t r a i n ( f a i l u r e s t r a i n f o r c o n c r e t e ) . The n e u t r a l a x i s i s then marched ac r o s s the s e c t i o n . A t y p i c a l l o c a t i o n of the n e u t r a l a x i s produces a s t r a i n d i s t r i b u t i o n a c r o s s the s e c t i o n as shown i n f i g u r e 3 (b). The f o l l o w i n g procedure i s then used to determine what combination of a x i a l load and bending moment would produce t h i s c o n d i t i o n . 18 • • / t m TP (•Ml IT 7 p n.o_ f 1 (train f 1 (train / J v , • • section strain to material properties stress segment forces net actions t o Figure 3. Cross section behaviour The depth of the section is divided into a number of segments. The strain at the mid-height of each segment is evaluated. The appropriate stress-strain law is then used to find the stress for each segment as shown in figure 3(d). This stress is multiplied by the area of the segment to give the force acting on each segment, as shown in figure 3(e). The forces in the steel bars, C and T are found in a similar manner, s s The forces for a l l the segments are added to give the required axial force, P. The forces in 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 this stage, the following information is stored, before repeating the above calculation with increased neutral axis depth. The stored information i s : 1. Curvature (input) 2. Net axial load (output) 1 9 3. Bending moment (output) For a number of n e u t r a l a x i s l o c a t i o n s , the procedure d e s c r i b e d above i s repeated, t i l l the n e u t r a l a x i s has marched acr o s s the s e c t i o n . Hence we o b t a i n the a x i a l load-bending moment i n t e r a c t i o n diagram. The u l t i m a t e i n t e r a c t i o n diagram f o r a s e c t i o n of dimensions 500mm X 500mm with 2% s t e e l i s shown i n f i g u r e 4. Any po i n t i n s i d e the curve r e p r e s e n t s a combination of a x i a l f o r c e and bending moment that the c r o s s s e c t i o n can r e s i s t . Any p o i n t o u t s i d e the curve r e p r e s e n t s f a i l u r e . 3.3.2 CURVATURE CONTOURS A number of values f o r cu r v a t u r e (30 i n t h i s case) are s e l e c t e d between zero and <2> , where <b i s the max max maximum cu r v a t u r e on the u l t i m a t e i n t e r a c t i o n diagram. The c u r v a t u r e contours are found f o r each of these c u r v a t u r e v a l u e s . The procedure followed f o r f i n d i n g the cu r v a t u r e contours i s d e s c r i b e d below. For a s e l e c t i o n of c u r v a t u r e , the n e u t r a l a x i s i s marched acr o s s the c r o s s s e c t i o n . The net a x i a l l o a d and bending moment f o r each of these n e u t r a l a x i s l o c a t i o n s i s found by a procedure s i m i l a r to the one d e s c r i b e d i n the p r e v i o u s s e c t i o n . F i g u r e 5 shows the r e l a t i o n s h i p between a x i a l f o r c e and bending moment f o r 30 cu r v a t u r e c o n t o u r s . Each l i n e r e p r e s e n t s a s i n g l e value of s e c t i o n c u r v a t u r e . D i f f e r e n t p o i n t s on a l i n e of constant 20 9000 8000-_ j 4000-< X < 3000-2000-1000-0 100 200 300 400 500 600 700 800 900 B E N D I N G M O M E N T k N - m F i g u r e 4 . U l t i m a t e i n t e r a c t i o n d iagram 9000-T 8000-100 200 300 400 500 600 700 B E N D I N G M O M E N T k N - m ' i 1 1 800 900 F i g u r e 5 . C u r v a t u r e c o n t o u r s 21 c u r v a t u r e r e p r e s e n t the c o m b i n a t i o n s of a x i a l l o a d and moment r e q u i r e d t o produce t h a t c u r v a t u r e f o r v a r i o u s n e u t r a l a x i s l o c a t i o n s . Note t h a t f i g u r e 4 i s the envelope of a l l the p o i n t s shown i n f i g u r e 5. 3.3.3 MOMENT-CURVATURE CURVES F i g u r e 6 shows the moment-curvature r e l a t i o n s h i p s f o r s e v e r a l l e v e l s of a x i a l l o a d s , P . A l l the c u r v e s s t a r t a t the o r i g i n but some have been s h i f t e d f o r c l a r i t y of p r e s e n t a t i o n . P 0 i s the a x i a l c o m p r e s s i o n s t r e n g t h of the column. These c u r v e s a r e computed from a l r e a d y s t o r e d d a t a . A h o r i z o n t a l l i n e on f i g u r e 5 900 800- P/Po=0.3 0.2 0.1 0.02 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 C U R V A T U R E 1/m F i g u r e 6 . M o m e n t - c u r v a t u r e - a x i a l l o a d r e l a t i o n s h i p s 22 represents a level of axial load. Each intersection of this horizontal line with a curvature contour provides a value of bending moment and curvature, which is plotted on figure 6 . The final point for each moment-curvature curve represents the point on the ultimate interaction curve for that level of axial load. For high values of axial load, a falling branch of the moment-curvature curve exists, beyond the maximum moment, but this is not plotted as this information is 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 eccentricities and no lateral load as shown in figure 7 . P P Figure 7 . Column with axial load and equal end eccentricities 23 3.4.1 FAILURE MODES In order to f a c i l i t a t e the understanding of the computer model, the behaviour of e c c e n t r i c a l l y loaded compression members w i l l be reviewed i n t h i s s e c t i o n . Consider the p o s s i b l e behaviour of the member shown i n f i g u r e 7, as the a x i a l l o a d P i s i n c r e a s e d to f a i l u r e . If the end e c c e n t r i c i t y f o r the a x i a l l o a d i s e, then the bending moment at the ends i s Pe, while the bending moment at the mid-span i s P(e+A), where A i s i s the d e f l e c t i o n at mid-span, as shown. Fi g u r e 8 i s an i n t e r a c t i o n diagram of a x i a l l o a d vs bending moment. The outer curved l i n e i s the u l t i m a t e i n t e r a c t i o n curve f o r the c r o s s - s e c t i o n , and i t rep r e s e n t s m a t e r i a l f a i l u r e . Consider a loa d path f o r a x i a l load and end moment, as a x i a l l o a d P i s i n c r e a s e d , represented by the l i n e 0-A, to P,. The corresponding load path f o r mid-span moment i s shown by the curved l i n e 0-B. The h o r i z o n t a l d i s t a n c e between l i n e s 0-A and 0-B rep r e s e n t s the amount by which the i n i t i a l mid-span moment, Pe has magnified to P(e+A). In t h i s case the member f a i l s at an a x i a l l o a d P 1 # when the mid-span l o a d path 0-B i n t e r s e c t s the m a t e r i a l s t r e n g t h i n t e r a c t i o n diagram at the p o i n t B. T h i s i s d e s c r i b e d as m a t e r i a l f a i l u r e . If the same member i s loaded with a smal l e r e c c e n t r i c i t y , the loa d path f o r end moments c o u l d be Figure 9. Interaction diagram for an eccentrically loaded column 2 5 shown by l i n e O-C, and the load path f o r mid-span moments by l i n e 0-D. In t h i s case an i n s t a b i l i t y f a i l u r e occurs when the a x i a l load reaches a maximum value P 2. The mid-span moment at f a i l u r e i s shown by po i n t D, which i s w e l l i n s i d e the m a t e r i a l s t r e n g t h curve. I f the member were loaded with a system under lo a d c o n t r o l ( f o r example, g r a v i t y loads) to loa d P 2, deformations would i n c r e a s e r a p i d l y and a m a t e r i a l f a i l u r e would f o l l o w immediately. I f the member were loaded under c o n d i t i o n s of c o n t r o l l e d displacement the load path shown by the ext e n s i o n of l i n e 0-D would be followed to eventual m a t e r i a l f a i l u r e at p o i n t E. If t h i s process i s repeated many times f o r the same member using a range of e c c e n t r i c i t i e s , f i g u r e 9 can be produced. The s o l i d l i n e i s the u l t i m a t e i n t e r a c t i o n diagram. The c h a i n - d o t t e d l i n e P u~C-A-M u i s the locu s of p o i n t s such as A and C i n f i g u r e 8, r e p r e s e n t i n g combinations of a x i a l l o a d and end moment (unmagnified moment) j u s t c a u s i n g f a i l u r e . 3.4.2 CALCULATION PROCEDURE A computer program f o r c a l c u l a t i n g p o i n t s on the curves shown i n f i g u r e 8 i s d e s c r i b e d below. The program can c o n s i d e r a column of any l e n g t h made up of a number of segments of equal l e n g t h . A method d e s c r i b e d by Galambos(1968) i s used to develop column d e f l e c t i o n curves f o r a given a x i a l l o a d , to 26 determine the maximum end e c c e n t r i c i t y , e, at which that l o a d can be a p p l i e d to that column. As the column i s symmetrically loaded, the slope i s co n s i d e r e d to be zero at mid-span. For the a x i a l load under c o n s i d e r a t i o n , the moment at mid-span i s i n i t i a l l y set to the m a t e r i a l f a i l u r e moment f o r that load (a po i n t on the u l t i m a t e i n t e r a c t i o n diagram). The corresp o n d i n g mid-span d e f l e c t i o n , e+A, (from the l i n e of a x i a l load) i s the f a i l u r e moment d i v i d e d by the a x i a l l o a d . To f i n d the a c t u a l values of e and A i t i s necessary to c a l c u l a t e the d e f l e c t e d shape of the member. A column d e f l e c t i o n curve i s obt a i n e d by proceeding along the column, segment-by-segment from mid-span, c a l c u l a t i n g the d e f l e c t i o n at each node. Consider the c a l c u l a t i o n s f o r a t y p i c a l segment of l e n g t h Ax, such as that shown i n f i g u r e 10. I f the d e f l e c t i o n v 0 and slope V Q are known at the s t a r t i n g node x 0 , then the moment M,, at the mid-point of the segment (point x,) i s approximately M, = P(v 0+v 0-Ax / 2 ) The c u r v a t u r e , 0 , , at p o i n t x, can be ob t a i n e d from the moment-curvature-axial l o a d r e l a t i o n s h i p ( f i g u r e 5 ) . The c u r v a t u r e i s assumed to be constant along the segment. The d e f l e c t i o n s are assumed to be small such that <j>=v". The displacement, v 2 , and slope, v 2 , at the next node, x 2 are c a l c u l a t e d from v 2 = v 0 + v 0 ( A x ) - 0 1 ( A x ) 2 / 2 27 Moment IM, (b). Figure 1 0 . Column deflection curves and: V j = vj - ^,(Ax) The moment M2 at node x 2 is the product of P and v 2. For the f i r s t calculation of the column deflection curve, (with the moment at mid-span equal to the failure moment at that load) the end eccentricity is calculated to be e 0. The calculation is then repeated for a lower mid-span moment. If the new end eccentricity, e 1 # is less than e 0 (figure 10(b)), then this also represents material failure and this calculation is ignored. However, i f e, is greater than e 0 (figure 10(c)) then this represents instability failure, and the calculation 28 i s repeated with smaller r e d u c t i o n s i n the mid-span moment t i l l a peak value f o r the end e c c e n t r i c i t y i s observed. T h i s peak value i s then the end e c c e n t r i c i t y at which i n s t a b i 1 i t y ' f a i l u r e o c c u r s , and when m u l t i p l i e d by the a x i a l l o a d , i t g i v e s the end moment f o r i n s t a b i l i t y f a i l u r e . 3.5 PROGRAM INPUT The f o l l o w i n g i n f o r m a t i o n i s r e q u i r e d as input to t h i s computer model. Each of these items w i l l be d i s c u s s e d i n d e t a i l below. 1. Cross s e c t i o n p r o p e r t i e s : a. Cross s e c t i o n dimensions b. P r o p e r t i e s of co n c r e t e c. P r o p e r t i e s of reinforcement 2. Slenderness e f f e c t : a. Column l e n g t h b. Segment le n g t h f o r column d e f l e c t i o n curves c. Reduction r a t i o f o r mid-span moment 3.5.1 CROSS SECTION PROPERTIES The program can be used to o b t a i n column p r o p e r t i e s f o r any polygonal shape of c r o s s s e c t i o n up t o twenty c o r n e r s . However, f o r the present study i t has been used only f o r r e c t a n g u l a r s e c t i o n s . The program can c o n s i d e r v a r i o u s s t r e s s s t r a i n r e l a t i o n s h i p s f o r concret e as w e l l as f o r s t e e l . 29 However, f o r t h i s study 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 f o r concrete i n compression i s assumed to f o l l o w the m o d i f i e d Hognestad r e l a t i o n s h i p , as presented by Desai and Krishnan (1964), by the f o l l o w i n g r e l a t i o n : f 2(e/e0) fo 1 + ( e / e 0 ) 2 Here f 0 and e 0 are the peak s t r e s s and peak s t r a i n r e s p e c t i v e l y . The compressive s t r e n g t h of c o n c r e t e , £' i s i nput, along with the modulus of e l a s t i c i t y , E c . The peak s t r a i n i s c a l c u l a t e d by the f o l l o w i n g r e l a t i o n suggested by Desai and Krishnan(1964): A f t e r an u l t i m a t e s t r a i n of 0.0038 the c o n c r e t e was assumed to have no s t r e n g t h even though i t may s t i l l be c o n f i n e d by the column t i e s and capable of s u p p o r t i n g some l o a d . The s t r e n g t h of concrete i n t e n s i o n i s n e g l e c t e d . The Youngs Modulus and the y i e l d s t r e s s f o r the s t e e l reinforcement are i n p u t . The reinforcement i s assumed to f o l l o w the e l a s t i c - p e r f e c t l y p l a s t i c s t r e s s - s t r a i n r e l a t i o n s h i p . The e l a s t i c - p e r f e c t l y p l a s t i c s t r e s s - s t r a i n r e l a t i o n s h i p i s somewhat c o n s e r v a t i v e because the e f f e c t of s t r a i n hardening i s n e g l e c t e d . 30 3.5.2 SLENDERNESS EFFECT The program can handle a column of any length and slenderness r a t i o . The e f f e c t of the slen d e r n e s s on the s t r e n g t h of the column i s accounted f o r as d e s c r i b e d e a r l i e r . Chen and A s t u t a ( l 9 7 6 ) have shown that a segment l e n g t h of four times the r a d i u s of g y r a t i o n g i v e s s u f f i c i e n t l y a c c u r a t e r e s u l t s . For a r e c t a n g u l a r s e c t i o n t h i s corresponds to 1.16 times the s e c t i o n depth. A segment le n g t h equal to the s e c t i o n depth has been c o n s i d e r e d throughout t h i s study. The program a l s o r e q u i r e s as input the ra t e at which the mid-span moment i s to be reduced i n the step-by-step procedure f o r the c o n s t r u c t i o n of the column d e f l e c t i o n curves. A value of 0.05 times the maximum moment has been used. 3.6 PROGRAM OUTPUT The output from t h i s computer model t y p i c a l l y c o n s i s t s of a x i a l load-end moment i n t e r a c t i o n curves f o r s e v e r a l column l e n g t h s . The c a l c u l a t i o n s d e s c r i b e d i n the pr e v i o u s s e c t i o n s have been c a r r i e d out f o r columns of s e v e r a l s l e n d e r n e s s r a t i o s , each at e i g h t d i f f e r e n t l e v e l s of a x i a l l o a d between 0.02 to 0.60 times the maximum l o a d . F i g u r e 11 shows an i n t e r a c t i o n diagram showing the combination of a x i a l l o a d P and end moment Pe j u s t producing f a i l u r e f o r a t y p i c a l 500mm X 500mm c r o s s s e c t i o n column with 2% s t e e l . 31 9000-8000-7000-2 6000-Ul ^ 5000-O u. _j 4000-< X < 3000-2000-1000-0-0 Figure 11. Strength interaction curve for slender columns The outer line is the ultimate interaction diagram representing material strength (slenderness effects for a slenderness ratio of 10 are negligible). The inner curves correspond to the curve Pu~C-A-Mu in figure 9. For low slenderness ratios the inner curves are close to the ultimate interaction curve. As the slenderness ratio increases, the curves move inside the ultimate interaction diagram, indicating that behaviour under these loads is governed by instability failures. These curves have been plotted directly from the computer output. Only eight levels of axial load were considered and linear interpolation was used to find the 32 l i m i t i n g bending moment f o r any intermediate a x i a l l o a d . T h i s i s c o n s i d e r e d to give s u f f i c i e n t l y a c c urate r e s u l t s f o r the purpose of t h i s study. 4. RELIABILITY EVALUATION 4.1 INTRODUCTION T h i s chapter d e s c r i b e s the theory behind the method used to e v a l u a t e the r e l i a b i l i t y of the r e i n f o r c e d concrete columns, as w e l l as the b a s i s f o r the computer program used. The e v a l u a t i o n of the r e l i a b i l i t y i s done by f i n d i n g the value of the r e l i a b i l i t y index f o r a number of values f o r the nominal a x i a l l o a d . L i n e a r i n t e r p o l a t i o n i s then used to f i n d the value of the a l l o w a b l e nominal a x i a l l o a d c o r r e s p o n d i n g to the t a r g e t r e l i a b i l i t y . 4.2 RELIABILITY INDEX CONCEPT To f a c i l i t a t e a b e t t e r understanding of the t h e o r e t i c a l model f o r the r e l i a b i l i t y program, the s a f e t y index concept i s b r i e f l y reviewed. The d e s c r i p t i o n presented below i s s i m i l a r to the one by F o s c h i ( 1 9 7 9 ) . A design problem i n s t r u c t u r a l a n a l y s i s t y p i c a l l y has a set of r e l e v a n t v a r i a b l e s (X 1 rX 2,...,X ), some of which are r e l a t e d to the r e s i s t a n c e of the m a t e r i a l , while others are r e l a t e d to the e f f e c t of the design l o a d s . These v a r i a b l e s must s a t i s f y a design equation of the form G(X,,X 2,...,X n ) < 0 In g e n e r a l , s e v e r a l such c o n d i t i o n s c o u l d be c o n s i d e r e d , each of them d e f i n i n g the a p p r o p r i a t e l i m i t st at es. 33 34 In g e n e r a l , the v a r i a b l e s X are random v a r i a b l e s obeying some d i s t r i b u t i o n f u n c t i o n . Hence the design process i s not d e t e r m i n i s t i c , and the design c o n d i t i o n s w i l l not be s a t i s f i e d f o r a l l values of the v a r i a b l e s X. The design s t r a t e g y u s u a l l y f o l l o w e d i s to s p e c i f y a ' t o l e r a t e d ' range f o r which the v a r i a b l e s should s a t i s f y the design c o n d i t i o n s . T h i s g i v e s a ' r e l i a b i l i t y ' range, which can be expressed i n terms of mathematical p r o b a b i l i t i e s , p r o v i d e d the i n d i v i d u a l d i s t r i b u t i o n s of the design v a r i a b l e s X are known. Veneziano (1974) d e s c r i b e s some methods of doing t h i s i n d e t a i l . Consider the simple case of only two design v a r i a b l e s , R and U. R i s the r e s i s t a n c e f o r the problem, while U re p r e s e n t s the l o a d e f f e c t . R and U are c a l l e d 'basic v a r i a b l e s ' of the design problem (Hasofer and L i n d (1974)). The f a i l u r e c r i t e r i o n can be w r i t t e n as U > R where the e q u a l i t y would represent the boundary between s u r v i v a l and f a i l u r e . I f 0 and are the mean and standard d e v i a t i o n of U, and 1 and o R are the mean and standard d e v i a t i o n of R, we can d e f i n e non-dimensional random v a r i a b l e s x and y such that (R - 5) x = aR (U - 0) y = 35 Hence the f a i l u r e c r i t e r i o n can be w r i t t e n as (R-0) Note that t h i s i s the equation of a s t r a i g h t l i n e with a slope O p / c y and an i n t e r c e p t (R-\2)/o^ on the y - a x i s as shown i n f i g u r e 12. If the v a r i a b l e s R and U are not c o r r e l a t e d , the p o i n t s (x,y) r e p r e s e n t i n g a combination (R,U) w i l l be d i s t r i b u t e d over the e n t i r e x-y pla n e . The f a i l u r e c r i t e r i o n d i v i d e s the plane i n t o two p a r t s . The upper p a r t corresponds to combinations of R and U producing f a i l u r e and the lower p a r t corresponds t o combinations r e p r e s e n t i n g s u r v i v a l . The minimum d i s t a n c e between the o r i g i n 0, corresponding to the mean value s of R and U, and the boundary I - I , the ' f a i l u r e s u r f a c e ' i s given by the segment F i g u r 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 index 36 OA. T h i s minimum d i s t a n c e , d e f i n e d as the r e l i a b i l i t y index /3, can be c a l c u l a t e d from the f i g u r e as (R - 0 ) The p o i n t A i s c a l l e d the most l i k e l y f a i l u r e p o i n t . Any combination of R and U below the f a i l u r e s u r f a c e (which i s at a minimum d i s t a n c e of (3 from the mean p o i n t ) w i l l not produce f a i l u r e . T h i s i s a r e l i a b i l i t y statement on the design problem and t h e r e f o r e , /3 may be used as a r e l i a b i l i t y measure, which can be r e l a t e d to the p r o b a b i l i t y of f a i l u r e , . Even f o r the case when there are more than two b a s i c v a r i a b l e s and when the f a i l u r e f u n c t i o n , G, i s n o n - l i n e a r , (3 i s d e f i n e d as the minimum d i s t a n c e from the mean p o i n t to the f a i l u r e s u r f a c e . If we d e f i n e : Y = R-U the f a i l u r e c r i t e r i o n becomes the event Y < 0. Then Y = P>0 and Hence, /3 can be w r i t t e n as (3 = ? / a y Hence, as shown i n f i g u r e 13, the measure 0 becomes the number of standard d e v i a t i o n s , a v , that the mean Y" i s from the f a i l u r e event Y=0. The p r o b a b i l i t y of f a i l u r e P^ i s then given by P(Y<0), and corresponds to the area of the shaded p o r t i o n of f i g u r e 13. If R and U are normally d i s t r i b u t e d , Y 37 P R O B A B I L I T Y D E N S I T Y Figure 13. Reliability Index and the probability of failure w i l l be normally distributed and can easily 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 distributions which minimizes the error in the calculation of 0 due to non-normal distributions. This transformation is the basis of the Rackwitz-Fiessler algorithm. This algorithm uses an iterative procedure on the variables to locate the most likely failure point, and hence to evaluate the r e l i a b i l i t y index. 38 If R and U are dependent on each other, then they have to be u n c o r r e l a t e d before using the R a c k w i t z - F i e s s l e r a l g o r i t h m . For t h i s study, a computer program based on the R a c k w i t z - F i e s s l e r a l g o r i t h m was used to evaluate the value of the r e l i a b i l i t y index, /3 from the v a l u e s of the f a i l u r e f i n c t i o n , G. 4.3 RELIABILITY CRITERION The method used f o r e v a l u a t i n g the r e l i a b i l i t y of the r e i n f o r c e d c o n c r e t e s l e n d e r columns i s analogous to the one d e s c r i b e d above. The c a p a c i t y of the column depends upon the e c c e n t r i c i t y at which i t i s loaded, as w e l l as upon the column p r o p e r t i e s . The i n t e r a c t i o n curves shown in f i g u r e 14 represent the column p r o p e r t i e s , and may be d e f i n e d by the parameters P 0 and M 0. Hence, the c a p a c i t y of the column may be w r i t t e n as P c = f(M,M 0,P 0) where P c i s the a x i a l f o r c e c a p a c i t y of the column when a bending moment M i s a p p l i e d at the ends. The f a i l u r e f u n c t i o n may be w r i t t e n as: G = P c - P, or: G = f(M,M 0,P 0)-P 1 where P-^  i s the a x i a l l o a d . A negative value f o r the f a i l u r e f u n c t i o n r e p r e s e n t s f a i l u r e . But f o r a given e c c e n t r i c i t y , a 39 h i g h e r a x i a l l o a d c o r r e s p o n d s t o a h i g h e r bending moment w h i l e a lower a x i a l l o a d c o r r e s p o n d s t o a lower bending moment. Thus, M and a r e c o r r e l a t e d , o r , P c and P^ a r e c o r r e l a t e d . T h i s problem may be s o l v e d by the method d e s c r i b e d below. 4.3.1 IMPLICIT UNCORRELATION PROCEDURE In t h i s p r o c e d u r e the v a r i a b l e s R and U a r e not e x p l i c i t l y u n c o r r e l a t e d . The proced u r e i s p r e s e n t e d here f o r s o l v i n g t he problem a t hand. However, w i t h some m o d i f i c a t i o n s the I m p l i c i t U n c o r r e l a t i o n P r o c e d u r e can be g e n e r a l i z e d f o r any s e t of random c o r r e l a t e d v a r i a b l e s . MOMENT F i g u r e 14. The I m p l i c i t U n c o r r e l a t i o n P r o c e d u r e 40 R e c a l l that i n our case, a x i a l l o a d and bending moment M are c o r r e l a t e d such that a high value of P^ corresponds to a high value of M. S i m i l a r l y , a low value of P-^  corresponds to a low value of M. In the f o l l o w i n g i t i s assumed that we need to f i n d the r e l i a b i l i t y index j3 f o r given mean va l u e s f o r a x i a l l o a d and e c c e n t r i c i t y . The d i s t r i b u t i o n of the a x i a l l o a d i s known. The method f o r f i n d i n g the standard d e v i a t i o n for e c c e n t r i c i t y i s d e s c r i b e d i n the next s e c t i o n . The d i s t r i b u t i o n of e c c e n t r i c i t y i s assumed to be normal. The I m p l i c i t U n c o r r e l a t i o n Procedure i s d e s c r i b e d below. 1. Simulate random valu e s f o r a x i a l l o a d and f o r e c c e n t r i c i t y f o r the given mean value s and d i s t r i b u t i o n s . 2. S e l e c t one value of a x i a l l o a d to i n v e s t i g a t e d (say P1^1 as shown i n f i g u r e 14). 3. M u l t i p l y P^1 by each of the simulated e c c e n t r i c i t y v a l u e s to get the corres p o n d i n g range of bending moment v a l u e s . 4. Using l i n e a r i n t e r p o l a t i o n , f i n d the c a p a c i t y fo r each bending moment va l u e f o r each i n t e r a c t i o n curve. T h i s g i v e s a d i s t r i b u t i o n f o r c a p a c i t i e s as shown i n f i g u r e 14. 5. Subtract from each of the c a p a c i t y v a l u e s , the a x i a l load value under i n v e s t i g a t i o n to get 41 values f o r the f a i l u r e f u n c t i o n G, which are s t o r e d i n an a r r a y . 6. Repeat steps 2 through 5 t i l l a l l the simulated a x i a l load values have been c o n s i d e r e d . The procedure d e s c r i b e d w i l l l e a d to a set of v a l u e s f o r the f a i l u r e f u n c t i o n G. Note that the f a i l u r e f u n c t i o n w i l l not be d i s t r i b u t e d a c c o r d i n g to some standard d i s t r i b u t i o n . These values are then used to e v a l u a t e the measure 0, using the R a c k w i t z - F i e s s l e r a l g o r i t h m as o u t l i n e d e a r l i e r i n t h i s c h a p ter. 4.3.2 STANDARD DEVIATION FOR ECCENTRICITY The u n c e r t a i n i t y i n the bending moment can mainly be due to two reasons. 1. Due to the exact magnitude of the loads being unknown. 2. Due to the exact d i s t r i b u t i o n of the loads being unknown. The v a r i a b i l i t y of the loads f o r the f i r s t reason has been d i s c u s s e d i n s e c t i o n 2.4. In the f o l l o w i n g i t has been assumed that the d i s t r i b u t i o n of the dead l o a d i s e x a c t l y known, and that i t e n t a i l s no u n c e r t a i n i t y i n the e v a l u a t i o n of the bending moment a c t i n g on the column. A survey of d i f f e r e n t kinds of p o s s i b l e l o a d i n g d i s t r i b u t i o n s on a beam r e v e a l that i t i s reasonable to assume a v a r i a b i l i t y of about 5% i n e v a l u a t i n g the 42 mean eccentricity MOMENT Figure 15. Standard deviation of eccentricity bending moment when the exact force is known. The method used for finding the standard deviation for the eccentricity is as follows. 1. Select the mean eccentricity for which the variability is to be investigated. (Figure 15). 2. Simulate an axial load distribution for some average axial load value. Hence, knowing the mean eccentricity the average value of the bending moment corresponding to a particular value of axial load is fixed. 3. Simulate values for the bending moment assuming a variability caused only by the d i s t r i b u t i o n of 43 the l o a d . Hence, c a l c u l a t e the set of corresponding e c c e n t r i c i t y v a l u e s . 4. Repeat steps 3 and 4 u n t i l a l l the simulated a x i a l load values have been c o n s i d e r e d . It was found that the v a r i a b i l i t y of the e c c e n t r i c i t y i s independent of the nominal a x i a l l o a d . Note that i n s t e a d of u s i n g step 3 of the I m p l i c i t U n c o r r e l a t i o n Procedure, the values of the bending moments c o u l d had been s i m u l a t e d d i r e c t l y , without u s i n g the v a r i a b i l i t y of the e c c e n t r i c i t y . However, the s i m u l a t i o n of the bending moments would be d i f f e r e n t f o r each l e v e l of the nominal a x i a l l o a d , whereas the s i m u l a t i o n of the e c c e n t r i c i t y values remains the same. Hence, t h i s method leads to s a v i n g i n the computer time due to the l a r g e number of nominal a x i a l l o a d l e v e l s at which the r e l i a b i l i t y i s to be i n v e s t i g a t e d . 4.4 SAMPLE SIZE The procedure d e s c r i b e d above i s capable of f i n d i n g the value of the r e l i a b i l i t y index, /3 f o r a p a r t i c u l a r value of the nominal a x i a l l o a d . T h i s procedure i s to be repeated a number of times i n order to f i n d the nominal a x i a l l o a d corresponding to the t a r g e t j3 f o r each c r o s s s e c t i o n , and f o r each slenderness r a t i o . Hence, i n order to save computer time i t i s necessary to determine the s m a l l e s t p o s s i b l e sample s i z e s . Grant et al (1978) have suggested that a sample s i z e of 200 column s e c t i o n s i s adequate to represent 44 the v a r i a b i l i t y i n s t r e n g t h . In the present study 250 column c r o s s s e c t i o n s were simulated. The a x i a l l o a d was simulated by 50 valu e s while the sample s i z e f o r the e c c e n t r i c i t i e s was 20. T h i s g i v e s 1000 loa d p o i n t s . The Code f o r the Design of Concrete S t r u c t u r e s f o r B u i l d i n g s (CSA A23.3) aims at a p o s s i b i l i t y of overloads of 1 i n 1000, so a sample s i z e of 1000 simulated l o a d p o i n t s seems to be adequate. The above sample s i z e s w i l l l e a d to 250,000 values f o r the f a i l u r e f u n c t i o n , G, which are to be p l a c e d i n ascending order, and then used to eva l u a t e the r e l i a b i l i t y index. In order to save computer time i n t h i s c a l c u l a t i o n only a l t e r n a t e ranked v a l u e s of the f a i l u r e f u n c t i o n were c o n s i d e r e d a f t e r p l a c i n g them i n ascending order (a t o t a l of 125,000 values) and the r e s u l t was compared with that obtained by using a l l the 250,000 v a l u e s . A v a r i a t i o n of about 5% was observed. Hence, the c a l c u l a t i o n s f o r the r e l i a b i l i t y index were performed using a l l the 250,000 va l u e s of the f a i l u r e f u n c t i o n . 5. RESULTS 5.1 INTRODUCTION The r e s u l t s o b tained by the method d e s c r i b e d i n the prev i o u s c h a p t e r s are i n the form of the value of the r e l i a b i l i t y index, /3 v/s the corresp o n d i n g nominal a x i a l l o a d . The code aims at a p r o b a b i l i t y of overloads f o r columns of one i n a thousand, and a p r o b a b i l i t y of understrength of one i n a hundred. Now P r o b ( f a i l u r e ) = Prob(overloads) X Prob(understrength) Hence the the code aims at a c h i e v i n g a p r o b a b i l i t y of f a i l u r e of 1 i n 100,000 f o r columns. The method f o r doing t h i s i s by s p e c i f y i n g s t r e n g t h r e d u c t i o n f a c t o r s and some load f a c t o r s . The r e l i a b i l i t y index 0 corresponding to a p r o b a b i l i t y of f a i l u r e of 1 i n 100,000 i s 4.265. 5.2 RELIABILITY AND AXIAL LOAD The f i r s t s tep i n the a n a l y s i s of the r e s u l t s i s to ob t a i n the value of the maximum a l l o w a b l e nominal a x i a l l o a d that would correspond to the t a r g e t 0 of 4.265. F i g u r e 16 shows the r e l a t i o n between j3 and the a x i a l l o a d f o r a 250mm X 250mm c r o s s s e c t i o n with 2% s t e e l , f o r a mean e c c e n t r i c i t y of a x i a l load of 50mm. L i n e a r i n t e r p o l a t i o n was used to o b t a i n the a x i a l load c o r r e s p o n d i n g to the t a r g e t 0 of 4.265. T h i s maximum a l l o w a b l e nominal a x i a l l o a d c o r r e s p o n d i n g to a /3 of 4.265 i s r e f e r r e d to as the computer r e s u l t i n the f o l l o w i n g . 45 46 5.5 x' 5^ o C 4.5 H 5 <-! 3.5-250mmX250mm(2%steel) •cc=50mm l/r=BO I 60 I 50 I 40 30 2010 I ' ' ' I ' • ' I ' ' ' I ' • 1 I 1 1 i 100 200 300 400 500 600 700 N O M I N A L A X I A L L O A D k N F i g u r e 16. R e l i a b i l i t y index /3 and maximum a l l o w a b l e nominal a x i a l l o a d 800 700-600- 5 0 m m 8 0 m m 0C =2/3 0C =1/2 0C =1/3 150mm 2 5 0 m m 250mmX250mm(2%steel) 1 1 1 I 1 1 1 1 I 1 1 • 1 I 1 1 ' ' I ' 20 25 30 35 B E N D I N G M O M E N T k N - m 45 50 F i g u r e 17. Computer r e s u l t and the dead l o a d r a t i o f a c t o r , a 47 5.3 RELIABILITY AND LOAD RATIO As d e s c r i b e d e a r l i e r , the load e f f e c t s of only the dead l o a d and the l i v e l o a d were c o n s i d e r e d i n t h i s study. The dead l o a d r a t i o f a c t o r a was d e f i n e d i n s e c t i o n 2.4.3. In t h i s study a was p r i m a r i l y c o n s i d e r e d to have the value 1/2. However, to check the v a r i a t i o n of the r e l i a b i l i t y with a, values of a equal to 1/3 and 2/3 were a l s o c o n s i d e r e d f o r one c r o s s s e c t i o n . The r e s u l t s are presented f o r sl e n d e r n e s s r a t i o ' s of 10 and 60 i n f i g u r e 17. Note that the va l u e of a does not a f f e c t the r e s u l t s by more than about 7%. 5.4 RESULT FORMAT In the present study i t was assumed that the l o a d f a c t o r s X as presented i n the code adequately express the e f f e c t of v a r i a b i l i t y of the l o a d e f f e c t s . The computer r e s u l t s o b t a i n e d by the present study were compared with those found by the code method. For the e v a l u a t i o n of the a l l o w a b l e a x i a l load c a p a c i t i e s by the code method, the s t r e n g t h of co n c r e t e was assumed to f o l l o w the m o d i f i e d Hognestad s t r e s s - s t r a i n r e l a t i o n s h i p . T h i s i s more ac c u r a t e than the r e c t a n g u l a r s t r e s s block r e l a t i o n s h i p and i s allowed by the code. The r e s u l t s are presented f o r the v a r i o u s specimen c r o s s s e c t i o n s and slenderness r a t i o s f o r d i f f e r e n t e c c e n t r i c i t i e s of l o a d i n g i n t a b l e s 1 to 3 f o r CAN3-A23.3-M77 and i n t a b l e s 4 to 6 f o r the new code. The r e s u l t s are i n terms of the f a c t o r £ where 48 Mean Ecc Slenderness R a t i o (mm) 10 20 30 40 50 60 80 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. fo r 250mm X 250mm column with 2% s t e e l , 7=0.52 Mean Ecc Slenderness R a t i o (mm) 10 20 30 40 50 60 80 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 Table 2. fo r 500mm X 500mm column with 2% s t e e l , 7=0.76 Mean Ecc Slenderness R a t i o (mm) 10 20 30 40 50 60 80 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. $ f o r 500mm X 500mm column with 4% s t e e l , 7=0.73 49 Mean Ecc (mm) Slenderness R a t i o 10 20 30 40 50 60 80 25 50 80 150 250 0.86 0.87 0.94 1.06 1.21 1.12 1.32 0.86 0 .91 0.95 1.01 1.04 1.03 1.05 0.89 0.91 0.91 0.94 0.93 0.97 0.91 0.92 0.92 0 .91 0.90 0.89 0.91 0.92 1.07 1.05 1.00 0.92 0.90 0.87 0 .91 Table 4. $ f o r 250mm X 250mm column with 2% s t e e l , 7=0.52 n Mean Ecc (mm) Slenderness R a t i o 10 20 30 40 50 60 80 50 100 160 300 500 1.11 1.10 1.11 1.16 1.25 1.36 1.59 1.08 1.11 1.13 1.18 1.22 1.24 1.35 1.07 1.10 1.11 1.12 1.13 1.15 1.17 0.97 0.98 0.99 1.00 1.01 1.02 1.03 0.96 0.97 0.97 0.98 0.95 0.91 0.90 Table 5. $ n f o r 500mm X 500mm column with 2% s t e e l , 7=0.76 Mean Ecc (mm) Slenderness R a t i o 10 20 30 40 50 60 80 50 100 160 300 500 1.02 0.99 0.94 0.91 0 .91 0.96 1.07 0.93 0.94 0.93 0.93 0.95 0.99 1.10 0.93 0.94 0.94 0.95 0.97 1.00 1.08 0.89 0.90 0.90 0.90 0.92 0.96 1.04 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 a c t u a l /R code where R a c t u a l are the a x i a l load c a p a c i t i e s found by t h i s study f o r a 0 of 4.265. R code are the a x i a l l o a d c a p a c i t i e s o b t ained by the code method. $ Q r e f e r s to the computer r e s u l t s as compared with those obtained by f o l l o w i n g CAN3-A23.3-M77, while $ n r e f e r s to the computer r e s u l t s compared with those o b t a i n e d by f o l l o w i n g CSA-A23.3(1984). 5.5 CODE EVALUATION MacGregor et al (1970) surveyed 22000 columns i n the l a t e 1960s and found that n e a r l y a l l of them had a sl e n d e r n e s s r a t i o l e s s than 30. Hence, i n any attempt at code c a l i b r a t i o n columns with slenderness r a t i o s of 30 or l e s s must be c o n s i d e r e d important. However, i t should a l s o be r e a l i z e d that columns with higher slenderness r a t i o s are the ones that u s u a l l y c r e a t e problems and as such t h e i r importance i n c r e a s e s . In t h i s study the columns were d i v i d e d i n t o two c a t e g o r i e s and each category was c o n s i d e r e d s e p a r a t e l y . The f i r s t c ategory c o n s i s t s of columns with s l e n d e r n e s s r a t i o s of 30 or l e s s . The second category c o n s i s t s of columns with s l e n d e r n e s s r a t i o s g r e a t e r than 30. Grant(l976) conducted a use study of column s i z e s and s t e e l r a t i o s of 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 r e s u l t s i n d i c a t e that more than 50 percent of the column widths range from 16in.(406mm) to 24in.(610mm) and more than 50 percent of the columns have s t e e l r a t i o s between 0.005 and 0.015. Hence, out of the s e c t i o n s c o n s i d e r e d f o r t h i s study, 51 the 500mm X 500mm c r o s s s e c t i o n with 2% s t e e l would be one of the more widely used ones. Hence, a code m o d i f i c a t i o n which can lead to some improvement i n the value of £ f o r t h i s p a r t i c u l a r c r o s s s e c t i o n without a d v e r s e l y a f f e c t i n g the r e s u l t s f o r the other c r o s s s e c t i o n s would be a p r a c t i c a l l y a c c e p t a b l e p r o p o s a l . For an o b j e c t i v e comparison between the pr o p o s a l and the code procedure the f o l l o w i n g method was adopted. The e r r o r e i s d e f i n e d as: c _ , Z($ - 1.0) 2  e " ^ N where the summation i s performed over a l l the v a l u e s f a l l i n g i n a p a r t i c u l a r c a t e g o r y . Tables 4, 5 and 6 r e v e a l that CSA-A23.3 (1984) leads to r e s u l t s that are c l o s e to the computer r e s u l t f o r low sle n d e r n e s s , e s p e c i a l l y f o r the 500mm X 500mm c r o s s s e c t i o n with 2% s t e e l . However, with i n c r e a s i n g s l e n d e r n e s s , there i s an i n c r e a s i n g d i f f e r e n c e between the code and the computer r e s u l t s . Hence, i t was f e l t that r e t a i n i n g the present 4>c and 0 g ( i . e . 0.60 and 0.85 r e s p e c t i v e l y ) , but with a d i f f e r e n t 0 m would be a r e a l i s t i c p r o p o s a l f o r a more c o n s i s t e n t code. For t h i s purpose, v a l u e s of c/>m between 0.55 and 1.05 were c o n s i d e r e d at i n t e r v a l s of 0.05. The r e s u l t i s presented i n f i g u r e 18 i n terms of the f a c t o r e. In f i g u r e 18 note that f o r the 500mm X 500mm (2% s t e e l ) c r o s s s e c t i o n , the cumulative e r r o r e i s minimum f o r d =1.0. However, at m 52 0.35 1.10 Figure 18. Variation of e with 4> high values of <t> some values of K became as low as 0.78. 3 m for this column (an error of up to 22% from the computer result). Such low values of p" are incompatible with the code aim as they would lead to a very low r e l i a b i l i t y . It is proposed that for such common sections, the minimum value of $ should not be less than 0.90, i.e. an error of more than 10% from the computer result is not allowed. With this condition, a value for # of 0.70 was found to be optimum. ' m However, this entails restricting the use of the moment magnification formula to columns with slenderness ratios of equal to or less than 60. 53 Mean Ecc S l e n d e r n e s s R a t i o (mm) 10 20 30 40 50 60 80 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. 250mm X 250mm column w i t h 2% s t e e l , 7=0.52 Mean Ecc S l e n d e r n e s s R a t i o (mm) 10 20 30 40 50 60 80 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 T a b l e 8. 500mm X 500mm column w i t h 2% s t e e l , 7=0.76 Mean Ecc S l e n d e r n e s s R a t i o (mm) 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 column w i t h 4% s t e e l , 7=0.73 54 900 B E N D I N G M O M E N T 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 =0.70. Note that c s m this proposal does not have any significant adverse effect on the two less common cross sections (250mm X 250mm (2% steel) and 500mm X 500mm (4% steel)). In addition to the above, note from Table 5 that CSA-A23.3 (1984) leads to a £ less than 1.00 for high eccentricities while for low eccentricities the $ is greater than 1.00. This is true even for the short column, which suggests a change in the material resistance factors, 4>c and 0 . By increasing <f> and decreasing # this difference can s c s 4000 COMPUTER RESULT £SAJLA?3.3_1984 _ MODIFIED C S A - A 2 3 . 3 ecc=500mm I < i i i — I 1 1 1 1 I—1 i • i | • i i i 0 100 200 300 400 500 B E N D I N G M O M E N T kN-m Figure 20. Results for the 500mm X 500mm column with 2% steel 5000 3500 3000 o or < 2000 X 500 H 0 ecc=50mm COMPUTER RESULT C S A - A 2 3 . 3 1984 l/r = 10 / f V w C S A - A 2 3 . 3 - M 7 7 MODIFIED C S A - A 2 3 . 3 / - X N / * \ -/ * \ _ /. * \ . N L '• \ \_ \ I A =60 / S N - - . . \ \ / * '• * / * / * *. « \ \ X \ / * •. / * ' • / * ' • % \ \ / * ^ ^ ^ * • • / * ' / * % \ % \ \ / *» X> / X \ s I \ / * ^ \ / * \ / * \ i 1 .v— «cc=500mm / * * / * / % / ^i—•"""* 100 700 800 200 300 400 500 600 B E N D I N G M O M E N T kN-m Figure 21. R e s u l t s f o r the 500mm X 500mm column with 4% steel 56 Mean Ecc (mm) • Slenderness Ratio 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% steel, 7=0.52 Mean Ecc (mm) Slenderness Ratio 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% steel, 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% steel, 7=0.73 57 be reduced. To t h i s end the v a l u e s of £, c o r r e s p o n d i n g t o <Pc = 0.65, 4>s =0.80 and <t>m =0.70 (denoted i n t h i s case by $ p 2 ), a r e p r e s e n t e d f o r the t h r e e c r o s s s e c t i o n s i n T a b l e s 10-12. A g a i n , 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 f o r the 500mm X 500mm ( 2 % s t e e l ) c r o s s s e c t i o n w i t h o u t a d v e r s l y a f f e c t i n g the r e s u l t s f o r the o t h e r two s e c t i o n s . F i g u r e s 19-21 show the r e s u l t s f o r the t h r e e c r o s s s e c t i o n s f o r s l e n d e r n e s s r a t i o s of 10 and 60. The m o d i f i e d CSA-A23.3 r e s u l t i s the one w i t h o n l y the <t>m changed t o 0.70. A comparison between the r e s u l t s from CSA-A23.3 (both 1977 and 1984 v e r s i o n s ) and the proposed m o d i f i e d CSA-A23.3 i s p r e s e n t e d i n T a b l e 13 i n terms of e. In t a b l e 13 P r o p o s a l 1 i s the one i n which o n l y # m i s changed w h i l e f o r P r o p o s a l 2 a l l the t h r e e s t r e n g t h r e d u c t i o n f a c t o r s a r e changed. S e c t i o n C a t e g o r y CSA-A23.3 CSA-A23.3 P r o p o s a l P r o p o s a l M77 (1984) 1 2 250mmX250mm 1 0.0698 0.0920 0.0998 0. 1 105 2% s t e e l 2 0.1319 0.1151 0.1093 0. 1054 500mmX500mm 1 0.1936 0.0822 0.0800 0.0594 2% s t e e l 2 0.3063 0.2159 0.1801 0.1708 500mmX500mm 1 0.0472 0.1030 0.1066 0.1025 4% s t e e l 2 0.1045 0.0791 0.0861 0.0809 T a b l e 13. Comparison of r e s u l t s i n terms of e 6. C O N C L U S I O N S AND RECOMMENDATIONS 6.1 C O N C L U S I O N S From t a b l e 13 we note that a m o d i f i c a t i o n i n the code (CSA-A23.3 1984) to change the value of tf>m from 0.65 to 0.70 would reduce the root mean square e r r o r by as much as 16% f o r a column of common usage. A more d r a s t i c m o d i f i c a t i o n ( i . e . to change 4>c , <j>s and <Am to 0.65, 0.80 and 0.70 r e s p e c t i v e l y ) would l e a d to s t i l l more acc u r a t e r e s u l t s . However, a change i n the m a t e r i a l r e s i s t a n c e f a c t o r s would a l s o a f f e c t members other than those with f l e x u r e combined with a x i a l compression. Hence, a more d e t a i l e d study i s suggested to eva l u a t e the consequences of such a change. N e v e r t h e l e s s , the f i r s t m o d i f i c a t i o n should be a p r a c t i c a l suggest i o n . I t can a l s o be noted i n the r e s u l t s that as the columns become i n c r e a s i n g l y s l e n d e r , the code r e s u l t s d e v i a t e more and more from the computer r e s u l t . T h i s i s e s p e c i a l l y a cause of concern f o r l a r g e e c c e n t r i c i t i e s of l o a d i n g because the code r e s u l t s are non-conservative i n such cases. Moreover, with the development of computer software, i t i s now p o s s i b l e to e f f e c t i v e l y p r e d i c t the behaviour of slender columns. Hence, i t i s d e s i r a b l e and p r a c t i c a l to r e s t r i c t the use of the moment m a g n i f i c a t i o n formula to columns of slend e r n e s s r a t i o s equal to or l e s s than 60. Note that f o r a slend e r n e s s r a t i o of, up to 60, the pro p o s a l to change 4>m to 0.70 leads to a lowest $ of 0.90 f o r the commonly used 58 59 s e c t i o n ( T a b l e 8 ) . 6.2 RECOMMENDATIONS I t i s r e c o m m e n d e d t h a t s u b j e c t t o t h e f o l l o w i n g c o n d i t i o n , t h e v a l u e o f 0 s h o u l d be t a k e n a s 0 . 7 0 i n t h e m C o d e f o r t h e D e s i g n o f C o n c r e t e S t r u c t u r e s f o r B u i l d i n g s ( C S A - A 2 3 . 3 1 9 8 4 ) . The c o n d i t i o n t o be s a t i s f i e d i s : The s l e n d e r n e s s r a t i o o f t h e c o l u m n s h o u l d n o t be g r e a t e r t h a n 6 0 . I n c a s e t h i s c o n d i t i o n c a n n o t be s a t i s f i e d , a r a t i o n a l p r o c e d u r e t o e v a l u a t e t h e s l e n d e r n e s s e f f e c t s h o u l d be f o l l o w e d . A c o m p u t e r p r o g r a m s i m i l a r t o t h e o n e d e s c r i b e d i n t h i s t h e s i s w o u l d be a n a c c e p t a b l e r a t i o n a l p r o c e d u r e t o t h i s e n d . REFERENCES Alc o c k , W . J . , and Nathan,N.D., 1977, Moment m a g n i f i c a t i o n t e s t s of p r e s t r e s s e d c o n c r e t e columns, J o u r n a l P C I, 22, No. 4, pp. 50-61. A l l e n , D . E . , 1970, P r o b a b a l i s t i c study of c o n c r e t e i n b e n d i n g , ACI J o u r n a l , 67, pp. 989-993. A l l e n , D . E . , 1975, L i m i t S t a t e s Design - a p r o b a b a l i s t i c s t u d y , Canadian J o u r n a l of C i v i l E n g i n e e r i n g , 2, pp. 36-49. Chen,W.F., and A s t u t a , T . , 1976, Theory of Beam Columns, V o l  I I , Space Beh a v i o u r and D e s i g n , M c G r a w - H i l l Book Company, New Y o r k , 732p. C o r n e l l , C . A . , 1969, A p r o b a b i l i t y based s t r u c t u r a l code, Proceedings ACI, 66(12), pp 974-985. D e s a i , P . and K r i s h n a n , S . , 1964, E q u a t i o n f o r the s t r e s s - s t r a i n c u r v e of c o n c r e t e , ACI J o u r n a l , 61, No 36, pp 345. E l l i n g w o o d , B r u c e , 1977, S t a t i s t i c a l A n a l y s i s of RC Beam-Column I n t e r a c t i o n , Proceedings, ASCE, 103, ST7, pp. 1377-1388. 60 6 1 Ellingwood,B., Galambos,T.V., MacGregor,J.G. and C o r n e l l , C . A . , 1980, Development of a p r o b a b i l i t y based load c r i t e r i o n f o r American N a t i o n a l Standard A58, NBS S p e c i a l P u b l i c a t i o n 577, N a t i o n a l Bureau of Standards, Washington, D.C., 222p. Foschi,R.O., 1979, A d i s c u s s i o n on the a p p l i c a t i o n of the s a f e t y index concept to wood s t r u c t u r e s , Canadian J o u r n a l of C i v i l E n g i n e e r i n g , V o l 6, No 1, pp 51-58. Gardiner,R.A. and Hatcher,D.S., 1970, M a t e r i a l and dimensional p r o p e r t i e s of an e l e v e n - s t o r e y r e i n f o r c e d c o n c r e t e b u i l d i n g . S t r u c t u r a l D i v i s i o n Research Report No. 52, Washington U n i v e r s i t y , S a i n t L o u i s , MO, 98p. Grant,L.H., 1976, A Monte C a r l o study of the s t r e n g t h v a r i a b i l i t y of r e c t a n g u l a r t i e d r e i n f o r c e d concrete columns, MSc t h e s i s , Department of C i v i l E n g i n e e r i n g , U n i v e r s i t y of A l b e r t a , Edmonton, 208 pp. Grant,L.H., Mirza,S.A., and MacGregor,J.G., 1978, Monte C a r l o Study of srength of concrete columns, ACI J o u r n a l , 7 5 , No. 8, pp. 348-358. L i n d , N . C , 1971, C o n s i s t e n t P a r t i a l S a f e t y f a c t o r s , Proceedi ngs, ASCE, 97(ST6), pp 1651-1670. 6 2 L i n d , N . C , Nowak,A.S. and Moss,B.G., 1978, Risk a n a l y s i s Procedures I I , Research r e p o r t submitted to N a t i o n a l Research C o u n c i l of Canada, D i v i s i o n of B u i l d i n g Research, Department of C i v i l E n g i n e e r i n g , U n i v e r s i t y of Waterloo, Waterloo, Ont., 51p. MacGregor,J.G., Breen,J.E., and Pfrang,E.O., 1970, Design of sl e n d e r c o n c r e t e columns. Proceedings, ACI, 67, pp 6-28. Mirza,S.A., H a t z i n i k o l a s , M . and MacGregor,J.G., 1979, S t a t i s t i c a l d e s c r i p t i o n of the s t r e n g t h of c o n c r e t e , J o u r n a l of the S t r u c t u r a l D i v i s i o n , ASCE, 105, ST6, pp. 1021-1037. Mirza,S.A. and MacGregor,J.G., 1979a, V a r i a t i o n s i n dimensions of r e i n f o r c e d c o n c r e t e members, J o u r n a l of the S t r u c t u r a l D i v i s i o n , ASCE, 105, ST4, pp. 751-766. Mirza,S.A. and MacGregor,J.G., 1979b, V a r i a b i l i t y of mechanical p r o p e r t i e s of r e i n f o r c i n g bars, J o u r n a l of the S t r u c t u r a l D i v i s i o n , ASCE, 105, ST5, pp. 921-937. Mirza,S.A. and MacGregor,J.G., 1979c, S t a t i s t i c a l study of shear s t r e n g t h of r e i n f o r c e d c o n c r e t e s l e n d e r beams, ACI J o u r n a l , 76, pp. 1159-1177. Mirza,S.A. and MacGregor,J.G., 1982, P r o b a b a l i s t i c study of s t r e n g t h of r e i n f o r c e d c o n c r e t e members, Canadian J o u r n a l of 63 C i v i l E n g i n e e r i n g , V o l 9, pp 431-448. Mi t c h e l l , G . R . and Woodgate,R.W., 1971, F l o o r l o a d i n g s in o f f i c e s - the r e s u l t s of a survey, Dept. E n v i r o n . Bldg. Res. S t a t i o n , Garston, England, Current Paper 3/71. Nathan,N.D., 1972, Slenderness of p r e s t r e s s e d concrete beam-columns, J o u r n a l PCI, 1 7 , No. 6, pp. 45-57. Nowak,A.S. and C u r t i s , J . D . , 1980, Risk a n a l y s i s computer program, Research Report, Department of C i v i l E n g i n e e r i n g , U n i v e r s i t y of Michigan, Ann Arbor, MI, 20p. Nowak,A.S. and L i n d , N . C , 1979, P r a c t i c a l b r i d g e code c a l i b r a t i o n , J o u r n a l of the S t r u c t u r a l D i v i s i o n , ASCE, 1 0 5 , ST12, pp. 2497-2510. Rackwitz,R. and F i e s s l e r , B . , 1978, S t r u c t u r a l r e l i a b i l i t y under Combined Random Load Sequences, Computers and S t r u c t u r e s , V o l 9, pp 484-494. Veneziano,D., 1974, C o n t r i b u t i o n s to second moment r e l i a b i l i t y theory, 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, S t r u c t u r a l P u b l i c a t i o n No. 389. Washa,G.W. and Wendt,K.F., 1975, F i f t y year p r o p e r t i e s of 64 c o n c r e t e , ACI J o u r n a l , 72, pp. 20-28. 

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