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Creep of lumber beams under constant bending load Fouquet, Robert J. M. 1979

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CREEP OF LUMBER BEAMS UNDER CONSTANT BENDING LOAD by ROBERT J . M. FOUQUET Dipl6me d> • Ingenieur en Mecanique, E c o l e N a t i o n a l e Superieure des A r t s e t M e t i e r s , 1977 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ...APPUED SCIENCE  in THE FACULTY OF GRADUATE STUDIES (Department o f C i v i l  Engineering)  We accept t h i s t h e s i s as conforming to the r e q u i r e d standard  THE UNIVERSITY OF BRITISH COLUMBIA October 1979  ©  Robert J . M. Fouquet, 1979  In p r e s e n t i n g t h i s  thesis  in p a r t i a l  fulfilment  an advanced degree at the U n i v e r s i t y of B r i t i s h the L i b r a r y I  further  for  this  freely  available  for  agree t h a t p e r m i s s i o n for e x t e n s i v e  scholarly  by h i s of  s h a l l make it  the  requirements  Columbia,  I agree  reference and copying o f  this  thesis for  It  i s understood that  f i n a n c i a l gain s h a l l  written permission.  C i v i l Engineering  The U n i v e r s i t y o f B r i t i s h  Columbia  2 0 7 5 Wesbrook P l a c e V a n c o u v e r , Canada V6T 1W5  October 15, 1979.  not  copying or  for  that  study. thesis  purposes may be granted by the Head of my Department  representatives.  Department of  of  or  publication  be allowed without my  ABSTRACT  Two set was  sets o f data are analyzed i n the t h e s i s .  The  first  d e r i v e d from the l o n g term deformations of 2 i n x 6 i n x  12 f t (40 mm  x 140 mm  x 3600 mm)  j o i s t s o f D o u g l a s - F i r loaded  under constant bending s t r e s s t o l e v e l s lower than or equal to 3110  p s i (21.44  MPa).  The second s e t was  d e r i v e d from the l o n g term deforma-  t i o n s of 2 i n x 6 i n x 12 f t j o i s t s of Hemlock loaded under cons t a n t bending s t r e s s to l e v e l s o f 3000 p s i (20.68 MPa)  and  4500 p s i ("31.0 2 MPa) . The a n a l y s i s shows t h a t the creep behaviour of s t r u c t u r a l s i z e beams depends upon the m a t e r i a l c h a r a c t e r i s t i c s ;  specifi  c a l l y , m a t e r i a l w i t h a s t r e n g t h lower than 5000 p s i (34.33 appeared t o creep 1.5  MPa)  times more t h a n , m a t e r i a l with a s t r e n g t h  h i g h e r than t h a t l e v e l , over a t h r e e month p e r i o d . In a d d i t i o n , the t e s t r e s u l t s support the  assumption  of a l i n e a r r e l a t i o n s h i p between the creep deformation of a s t r u c t u r a l - s i z e timber beam and a p p l i e d  stress.  A method i s p r e s e n t e d to p r e d i c t the creep behaviour of a s t r u c t u r a l - s i z e specimen month p e r i o d .  at d i s c r e t e times over a three  The method c o n s i s t s o f e x p r e s s i n g the creep de-  formation, <S , i n terms of the e l a s t i c deformation, <5 , or c e q u i v a l e n t l y , the f r a c t i o n a l creep  e (f = 6* /6 ) i n terms of  8 or i n terms of the modulus o f e l a s t i c i t y , e T h i s work i s l i m i t e d t o the s t r e s s l e v e l s i n v e s t i g a t e d and to s p e c i f i c temperature  (10°C< :Q<30°C) and moisture content  (8%<MC<12%) c o n d i t i o n s . While t h i s method c o u l d be employed i n p r e l i m i n a r y design procedures, i t has been e s p e c i a l l y designed f o r more complex s t u d i e s o f the creep behaviour of s t r u c t u r e s i n c l u d i n g f l o o r systems,  trusses, etc...  The advantage  o f the method  i s t h a t i n t h i s kind, of a n a l y s i s the modulus of e l a s t i c i t y o f the i n d i v i d u a l components can be  used.  T h i s t h e s i s a l s o presents a s e t of creep curves t h a t cover a t h r e e year span.  These creep curves show t h a t the  average t o t a l deformation o f beams loaded t o a s t r e s s l e v e l o f 3110  p s i (21.44 MPa),  a t t h i s time, i s approximately 1.6  the e l a s t i c deformation.  times  TABLE OF CONTENTS Page  ABSTRACT  i i i  TABLE OF CONTENTS  V  L I S T OF TABLES  viii  L I S T OF FIGURES  x  ACKNOWLEDGEMENTS  xv  CHAPTER 1.  INTRODUCTION  1  1.1  Objectives  1  1.2  Rheology  2  1.3  Factors Influencing Deformations  .  1.4 2.  Term 4  1.3.1  Time  5  1.3.2  Stress  5  1.3.3  Temperature  8  1.3.4  Moisture Content  8  1.3.5  Volume  9  L o n g Term D e f o r m a t i o n ; F u n c t i o n a l  Form  10  EXPERIMENT IN SURREY  12  2.1  Preliminaries  12  2.2  Material  12  2.2.1  2.2.2 \  Long  2.3  Used  Selection of Material  at Mill  Site  12  Grouping the M a t e r i a l  13  P h y s i c a l Arrangement  16  vi 2.4  17  Deformation Measurement 2.4.1  Expected I n i t i a l D e f l e c t i o n s  2.4.2  Actual  2.4.3  D e f l e c t i o n as a F u n c t i o n o f  I n i t i a l Deflections  .... 17  19  Time 2.5 3.  Interpretations  22  3.1-  22  Materialised  3.1.2  ........ .  Selection of Material at M i l l Site  22  Grouping  22  3.2  Loading C o n f i g u r a t i o n  24  3.3  T e s t i n g Procedure  25  3.4  Deformation Measurements  26  3.4.1  Expected I n i t i a l D e f l e c t i o n s . . . , 26  3.4.2  Actual  3.4.3  Deformation as a F u n c t i o n o f  I n i t i a l Deflections  3.5  Interpretations  .... 27  28  Time o f Results  29  3.5.1  E l a s t i c Deformations  3.5.2  General Comments on Creep Curves 35 Dependence o f Creep Deformation on the S t r e s s L e v e l , the S t r e s s R a t i o and the Modulus o f E l a s t i city 37  3.5.3  5.  20  of Results  EXPERIMENT IN RICHMOND  3.1.1  4.  .. 17  LINEAR VISCOELASTIC MODEL  30  42  4.1  Introduction  42  4.2  Three Parameter S o l i d  43  4.3  S t r e s s Parameter  45  4.4  Comments  53  ANALYTICAL MODEL FITTED TO THE DATA  55  vii  5.1  Model  55  5.2  Comments  57  5.3  Quantitative Analysis  58  5.3.1  Method  58  5.3.2  A n a l y s i s o f Creep a t T h r e e Months  5.3.3  5.3.4 5.4  Final  Analysis  Deformation  of Fractional  59 Creep  a t T h r e e Months  67  Summary  70  Results  6. CONCLUSIONS BIBLIOGRAPHY  72 75 79  APPENDICES A-l  A-2  A-3  B-l  B-2  MATHEMATICAL TREATMENT OF CREEP DATA CREEP DEFORMATION  17 3  MATHEMATICAL TREATMENT OF CREEP DATA CREEP DEFORMATION  178  MATHEMATICAL TREATMENT OF CREEP DATA CREEP DEFORMATION  184  MATHEMATICAL TREATMENT OF CREEP DATA FRACTIONAL CREEP  189  MATHEMATICAL TREATMENT OF CREEP DATA FRACTIONAL CREEP  193  viii LIST OF TABLES Page Table I  S t a r t i n g time a f t e r l o a d i n g f o r measuring creep  81  Grouping o f the m a t e r i a l f o r Surrey experiment a c c o r d i n g t o E-values (Batch 1)  82  Grouping o f the m a t e r i a l f o r Surrey experiment a c c o r d i n g t o E-values (Batch 2)  82  Table IV-a  Data f o r Surrey t e s t  (10 70 p s i )  83  Table IV-b  Data f o r Surrey t e s t  (1410 p s i )  84  Table IV-c  Data f o r Surrey t e s t  (.2.110 p s i )  85  Table IV-d  Date f o r Surrey t e s t  (3110 p s i )  86  Table V  S t a t i s t i c a l i n f o r m a t i o n s on the modulus of e l a s t i c i t y o f boards used i n Surrey experiment  Table I I  Table I I I  Table V l - a Creep data from Surrey (Group 1070 p s i )  experiment  Table V l - b Creep data from Surrey (Group 1410 p s i )  experiment  Table VI-c  Creep data from Surrey (Group 2110 p s i )  experiment  Table V l - d Creep data from Surrey (Group 3110 p s i )  experiment  Table V I I  87  88 89 90  91  Distance between a beam support and an e m p i r i c a l r e f e r e n c e f o r the measurements 92  Table VIII-1 Creep data f o r boards loaded t o 3110 p s i (Surrey experiment) (Board number 1115)  \ . 93  Table VIII-2 Creep data f o r boards loaded t o 3110 p s i (Surrey experiment) (Board number 1609)  94  Table VIII-3 Creep data f o r boards loaded t o 3110 p s i (Surrey experiment) (Board number 1589)  95  ix  Table  VIII-4  Table VII1-5  Table VIII-6  Table VIII-7  Table VIII-8  Table IX Table X Table XI  Table XII  Table X I I I  Table XIV  Creep data f o r boards loaded t o 3110 p s i (Surrey experiment) (Board number 2672)  96  Creep data f o r boards loaded t o 3110 p s i (Surrey experiment) (Board number 2395)  97  Creep data f o r boards loaded t o 3110 p s i (Surrey experiment) (Board number 1896)  98  Creep data f o r boards loaded t o 3110 p s i (Surrey experiment) (Board number 1829)  99  Creep data f o r boards loaded t o 3110 p s i (Surrey experiment) (Board number 1419)  100  Data f o r Richmond experiment group 1)  (Sub101  Data f o r Richmond experiment group 2)  (Sub10 3  F r a c t i o n a l creep expressed a t t h r e e months f o r f i v e ranges o f s t r e s s ratios. (Average values)  107  C o e f f i c i e n t s A and B o f the c o r r e l a t i o n equation between creep deformat i o n and e l a s t i c deformation f o r two stress levels  108  C o e f f i c i e n t s A and B o f the c o r r e l a t i o n e q u a t i o n between creep deformat i o n and e l a s t i c deformation, d i s r e gard o f the s t r e s s l e v e l  109  C o e f f i c i e n t s C and D o f the c o r r e l a t i o n equation between f r a c t i o n a l creep and the modulus o f e l a s t i c i t y , d i s r e g a r d o f the s t r e s s l e v e l  110  X  LIST OF FIGURES Page Figure 1  S t r e s s and s t r a i n versus time d i a grams f o r a specimen o f a h y p o t h e t i cal material I l l  Figure 2  F r a c t i o n a l creep i n Hoop Pine i n bending  112  Devices used t o measure creep o f wood i n bending  113  The i n c r e a s e i n f r a c t i o n a l creep o f the s m a l l and the l a r g e specimens between 1/256 day and 1 day as f u n c tion of stress intensity  113  Creep compliance v e r s u s time f o r 4 s t r e s s l e v e l s expressed as percentrages o f p r o p o r t i o n a l l i m i t  114  T h e o r e t i c a l development o f creep under i n t e r m i t t e n t l o a d i n g  115  Creep curves f o r v a r i o u s types o f temperature  116  F i g u r e 3-a F i g u r e 3-b  Figure 4  Figure 5 Figure 6 Figure 7  Figure 8  Creep versus time f o r hardboard and plywood _3 Creep from 10 hours t o 1 hour o f 1/8" f i b e r b o a r d i n t e n s i o n versus s t r e s s and f r a c t i o n a l s t r e s s  Figure 9  Sample I .  Grouping o f the m a t e r i a l .  F i g u r e 10  S t r e n g t h d i s t r i b u t i o n o f the c o n t r o l sample from Sample 1  F i g u r e 11  T e s t set-up f o r Surrey experiment  F i g u r e 12-1  T e s t set-up f o r Surrey experiment. P i c t u r e as an end-view T e s t set-up f o r Surrey experiment. P i c t u r e as an end-view  F i g u r e 12-2 F i g u r e 13 F i g u r e 14  117  117 118  119 ... 120  121 122  D e f l e c t i o n versus time f o r four stress levels.  123  D e f l e c t i o n versus time f o r four stress levels  124  XI  Figure  15  Sample I I .  Grouping o f m a t e r i a l . . . .  125  Figure  16  Strength d i s t r i b u t i o n of the c o n t r o l sample from p o p u l a t i o n I I  126  Strength d i s t r i b u t i o n o f the c o n t r o l sample from p o p u l a t i o n I I  127  Figure  17  Figure  18  P i c t u r e o f Richmond set-up  128  Figure  19  P i c t u r e of Richmond set-up  12 9  Figure  20  Charts of l o a d and d e f l e c t i o n v a r i a t i o n s with r e s p e c t t o time d u r i n g t r a n s f e r of the l o a d  130  Example o f creep curves p l o t t e d on a b i l i n e a r graph  131  Examples of creep curves p l o t t e d on a f u l l l o g a r i t h m i c graph and a semi l o g a r i t h m i c graph  132  Creep curves f o r boards o f subgroup 1  133  Creep curves f o r boards o f subgroup 2 loaded to 3000 p s i  137  Creep curves f o r boards o f subgroup 2 loaded to 4500 p s i  140  A c t u a l e l a s t i c deformations versus p r e d i c t e d e l a s t i c deformations ...  141  Cumulative p r o b a b i l i t y d i s t r i b u t i o n s of the modulus o f e l a s t i c i t y o b t a i n e d by two d i f f e r e n t methods. Hemlock  142  Cumulative p r o b a b i l i t y d i s t r i b u t i o n s of the modulus o f e l a s t i c i t y o b t a i n e d by two d i f f e r e n t methods. Amabilis f i r  143  Creep d e f l e c t i o n , measured a t 10 d i f f e r e n t times, versus e l a s t i c deformation, f o r two s t r e s s l e v e l s . Raw data  144  Creep curves f o r boards with same v a l u e o f e l a s t i c deformation (36 mm)  145  Creep curves f o r boards w i t h same value o f e l a s t i c deformation (38 mm)  146  Figure Figure  Figure Figure Figure Figure Figure  Figure  Figure  Figure  Figure  21 22  23 24-1 24-2 25 26  27  28  29-1  29-2  xii  F i g u r e 29-3 F i g u r e 29-4 F i g u r e 29-5 F i g u r e 29-6 F i g u r e 30  F i g u r e 31  F i g u r e 32-1  F i g u r e 32-2  F i g u r e 32-3  F i g u r e 32-4  F i g u r e 32-5  F i g u r e 32-6  F i g u r e 32-7  Creep curves f o r boards w i t h same value o f e l a s t i c deformation (41 mm)  147  Creep curves f o r boards w i t h same v a l u e o f e l a s t i c deformation (42 mm)  148  Creep curves f o r boards w i t h same value o f e l a s t i c deformation (4 3 mm)  149  Creep curves f o r boards w i t h same v a l u e o f e l a s t i c deformation (45 mm)  150  F r a c t i o n a l creep expressed a t 10 d i f f e r e n t times versus e l a s t i c deformation, f o r two s t r e s s l e v e l s . Raw data  151  F r a c t i o n a l creep, expressed a t 10 d i f f e r e n t times versus modulus o f e l a s t i c i t y , f o r two s t r e s s l e v e l s . Raw data  152  F r a c t i o n a l creep p l o t t e d a g a i n s t the modulus o f e l a s t i c i t y f o r s p e c i f i c ranges o f s t r e s s r a t i o s , a t time t = 5 minutes  153  F r a c t i o n a l creep p l o t t e d a g a i n s t the modulus o f e l a s t i c i t y f o r s p e c i f i c ranges o f s t r e s s r a t i o s , at time t = 1 hour  154  F r a c t i o n a l creep p l o t t e d a g a i n s t the modulus o f e l a s t i c i t y f o r s p e c i f i c ranges o f s t r e s s r a t i o s , a t time t = 10 hours  155  F r a c t i o n a l Creep p l o t t e d a g a i n s t the modulus o f e l a s t i c i t y f o r s p e c i f i c ranges o f s t r e s s r a t i o s , a t time t = 2 days  156  F r a c t i o n a l creep p l o t t e d a g a i n s t the modulus o f e l a s t i c i t y f o r s p e c i f i c ranges o f s t r e s s r a t i o s , a t time t = 4 days  157  F r a c t i o n a l Creep p l o t t e d a g a i n s t the modulus o f e l a s t i c i t y f o r s p e c i f i c ranges o f s t r e s s r a t i o s , a t time t = 3 weeks  158  F r a c t i o n a l creep p l o t t e d a g a i n s t the modulus o f e l a s t i c i t y f o r s p e c i f i c ranges o f s t r e s s r a t i o s , a t time  xiii  F i g u r e 32-8  F i g u r e 32-9  F i g u r e 32-10  F i g u r e 33  F i g u r e 34  F i g u r e 35  F i g u r e 36  F i g u r e 37  F i g u r e 38  F i g u r e 39  t = 6 weeks  159  F r a c t i o n a l creep p l o t t e d a g a i n s t the modulus o f e l a s t i c i t y f o r s p e c i f i c ranges o f s t r e s s r a t i o s , a t time t = 9 weeks.  160  F r a c t i o n a l creep p l o t t e d a g a i n s t the modulus o f e l a s t i c i t y f o r s p e c i f i c ranges o f s t r e s s r a t i o s , a t time t = 3 months  161  F r a c t i o n a l creep p l o t t e d a g a i n s t the modulus o f e l a s t i c i t y f o r s p e c i f i c ranges o f s t r e s s r a t i o s , a t time t = 13 weeks  162  Creep deformation versus e l a s t i c def o r m a t i o n . A n a l y t i c a l model f i t t e d to data f o r 3000 p s i  163  Creep deformation versus e l a s t i c deformation. A n a l y t i c a l model f i t t e d to data f o r 4500 p s i  164  F r a c t i o n a l creep versus e l a s t i c def o r m a t i o n . A n a l y t i c a l model f i t t e d to data f o r 3000 p s i  165  F r a c t i o n a l creep versus e l a s t i c deformation. A n a l y t i c a l model f i t t e d t o data f o r 4500 p s i  166  F r a c t i o n a l creep versus modulus o f elasticity. A n a l y t i c a l model f i t t e d t o data f o r 3000 p s i  167  F r a c t i o n a l creep versus modulus o f elasticity. A n a l y t i c a l model f i t t e d to data f o r 4500 p s i  168  F r a c t i o n a l creep, expressed a t 10 d i f f e r e n t times, versus s t r e s s r a t i o for 1 stress l e v e l : 3000 p s i . Raw data  169  F i g u r e 40  Creep deformation versus e l a s t i c def o r m a t i o n . A n a l y t i c a l model f i t t e d to data p o i n t s f o r 3000 and 4500 p s i . 170  F i g u r e 41  F r a c t i o n a l creep versus e l a s t i c def o r m a t i o n . A n a l y t i c a l model f i t t e d to data p o i n t s f o r 3000 and 4500 p s i . 171  xiv  Figure  42  F r a c t i o n a l creep v e r s u s modulus o f elasticity. A n a l y t i c a l model f i t t e d t o d a t a p o i n t s f o r 3000 and 4500 p s i . 172  XV  ACKNOWLEDGEMENTS  The  author wishes t o thank P r o f . B. Madsen and Dr.  M. Olson of the U n i v e r s i t y o f B r i t i s h Columbia, Department of  C i v i l E n g i n e e r i n g , whose a d v i c e and c r i t i c a l  review of  the work are g r e a t l y a p p r e c i a t e d . S p e c i a l acknowledgement i s hereby given t o F o r i n t e k Canada C o r p o r a t i o n , Department of Wood E n g i n e e r i n g , f o r the use o f i t s l a b o r a t o r y f a c i l i t i e s .  In p a r t i c u l a r , thanks t o  Dr. J.D. B a r r e t t , Dr. R.O. F o s c h i , and Dr. P. G l ^ s , whose enthusiasm and suggestions were of g r e a t v a l u e , as was the t e c h n i c a l a s s i s t a n c e o f Mr. B. McKinney and Mr. L. Olson. In a d d i t i o n , I am g r a t e f u l t o Mr. R. G r i g g f o r h i s help with the computer a t the Department o f C i v i l and t o Ms. S. Pennant and J . F r y x e l l f o r t h e i r comments and t y p i n g the manuscript.  Engineering,  editorial  1  CHAPTER 1  INTRODUCTION  1.1  Objectives The design  o f s t r u c t u r e s r e l i e s on the understanding  of t h e i r b a s i c behaviour.  Such a b a s i s i n c l u d e s the m a t e r i a l  p r o p e r t i e s , the behaviour o f the component p a r t s and the behaviour  o f the complete s t r u c t u r e as a u n i t . One requirement o f the design procedure i s t h a t expected  loads be c a r r i e d s a f e l y .  T h i s c o n d i t i o n can be f u l f i l l e d  i f the  s t r e n g t h p r o p e r t i e s o f the m a t e r i a l are known. Another requirement o f the design procedure i s t o provide a s t r u c t u r e t h a t meets f u n c t i o n a l requirements.  In t h i s  context,  i t may be e s s e n t i a l t h a t the appearance o f the s t r u c t u r e , as w e l l as i t s component p a r t s , be c a r e f u l l y c o n s i d e r e d .  This  latter  c o n s i d e r a t i o n w i l l place deformation l i m i t s on s t r u c t u r a l components, and these a r e o f t e n the governing f a c t o r s o f design f o r wood.  F o r new m a t e r i a l s  such as polymers, the design  pro-  cedure i s based upon an expected l i f e t i m e - t h a t i s upon a c e r t a i n deformation by a c e r t a i n time.  F o r wood, some codes  f o r long term deformations by u s i n g c o n s e r v a t i v e  allow  and r a t h e r  2 crude d e s i g n formulae, but f a i l t o i n c o r p o r a t e the t r u e funct i o n a l r e l a t i o n s h i p of these long term deformations.  In order  to a c c u r a t e l y p r e d i c t the behaviour of wooden s t r u c t u r e s , i t i s necessary to know how  the deformations of i n d i v i d u a l com-  ponents vary i n time. T h i s study i s aimed a t c o l l e c t i n g b a s i c i n f o r m a t i o n on creep f o r lumber and a t d e v e l o p i n g simple models which would, i n an approximate  way,  d e s c r i b e the time behaviour under s p e c i f -  i c environmental c o n d i t i o n s (constant humidity and  1.2  temperature).  Rheology Rheology  i s the branch of s c i e n c e t h a t s t u d i e s the l o n g  term d e f o r m a t i o n a l behaviour of m a t e r i a l s . have t r a d i t i o n a l l y emphasized  Rheological studies  d e s c r i p t i v e o b s e r v a t i o n s more than  the a c t u a l mechanisms a t a molecular l e v e l .  T h i s i s not  i n g c o n s i d e r i n g the s t r u c t u r a l complexity of wood.  A  surpris-  complete  molecular theory would be very d i f f i c u l t to develop indeed. R h e o l o g i c a l models are o f t e n used i n such d e s c r i p t i v e studies.  T h i s approach assumes t h a t the deformations o f mate-  r i a l s may  be c l a s s i f i e d as e i t h e r e l a s t i c , v i s c o u s or p l a s t i c .  From t h i s assumption, d e f o r m a t i o n a l behaviour may  mathematical e x p r e s s i o n s f o r the  be d e r i v e d .  These e x p r e s s i o n s c o r -  respond to the deformation o f mechanical systems c o n s i s t i n g o f springs  ( e l a s t i c i t y ) , dashpots  ( v i s c o s i t y ) and f r i c t i o n  elements  (plasticity). There i s g r e a t d i s c r e p a n c y i n the terminology used to d e s c r i b e these phenomena.  For example N i e l s e n  (1972) has  used  terms shown i n F i g u r e 1 where deformation f o r a h y p o t h e t i c a l  3 m a t e r i a l i n a u n i a x i a l t e s t i s p l o t t e d versus time. The instantaneous deformation has two components - the instantaneous e l a s t i c and instantaneous p l a s t i c The  l a t t e r i s i n most cases n e g l e c t e d .  deformations.  As w i l l be shown i n  f u r t h e r chapters where experimental r e s u l t s are presented, i t is  extremely d i f f i c u l t t o q u a n t i f y the instantaneous  deformation.  The main reason i s t h a t the time of a p p l i c a t i o n of the l o a d  has  a f i n i t e v a l u e , and the time-dependent deformation takes p l a c e a l r e a d y before f u l l  l o a d has been t r a n s f e r r e d .  s t u d i e s by Spencer (1978) have shown how  Furthermore,  wood s t r e n g t h i s i n -  f l u e n c e d by the r a t e of a p p l i c a t i o n of the l o a d ; i t i s not known,, however, t o what extent the instantaneous deformation is  affected. In  common r e s e a r c h p r a c t i c e , the i n i t i a l  deformation i s  d e f i n e d as the t o t a l deformation a t a s p e c i f i e d time a f t e r ing  the l o a d i n g .  In Table I, N i e l s e n l i s t s  times f o r v a r i o u s materials..  start-  a number of such  For wood, t h i s time goes up t o  one minute. The deformations  shown i n F i g u r e 1 are s t r a i n s ; i n t h i s  present work, deformations w i l l r e f e r to d e f l e c t i o n s .  The i n -  stantaneous deformation w i l l be c a l l e d e l a s t i c deformation w i l l be the t o t a l deformation, one minute a f t e r s t a r t i n g  and  the  loading. A f t e r the i n i t i a l  time, the deformation continues to i n -  crease and i s r e f e r r e d to as creep deformation.  I t c o n s i s t s of  a r e c o v e r a b l e p a r t , the delayed e l a s t i c deformation, and i r r e c o v e r a b l e p a r t , the v i s c o u s deformation.  an  These components  can o n l y be d i f f e r e n t i a t e d by unloading the s t r u c t u r e ; but i t  i s d i f f i c u l t to unload while s i m u l t a n e o u s l y t a k i n g deformation measurements.  Thus, I d i d not d i f f e r e n t i a t e between the  components of creep deformation i n t h i s study.  two  The creep  de-  formation a t a given time, w i l l be d e f i n e d as the d i f f e r e n c e between the t o t a l deformation a t t h i s p a r t i c u l a r time and elastic  deformation. Another  development:  q u a n t i t y w i l l be used to c h a r a c t e r i z e the creep  f r a c t i o n a l creep, d e f i n e d as the r a t i o of creep  deformation to e l a s t i c  1.3  the  deformation.  F a c t o r s I n f l u e n c i n g Long Term Creep deformation, &  t i o n of v a r i o u s parameters, &  c  = fit,a  Deformations  i s commonly expressed as a funci n the f o l l o w i n g  (t,x),  where t i s the time parameter,  form:  9 ( t , x ) , M.C.,  V}  a the s t r e s s parameter,  x the  space parameter ( d e f i n i n g the l o c a t i o n where the measurements are r e c o r d e d ) , 9 the temperature, and V the volume of the specimen.  M.C.  the moisture  T h i s may  content  not be a t o t a l  p i c t u r e of the dependency, but i s a u s u a l f i r s t  approximation.  I t i s important to know to what e x t e n t and i n what manner creep deformation i s i n f l u e n c e d by these T h i s knowledge may experiments  parameters.  a f f e c t the design of any long term l o a d i n g  as w e l l as the i n t e r p r e t a t i o n of experimental r e -  sults . I w i l l discuss, the p r e c e d i n g parameters assuming the o t h e r s remain  constant.  one a t a time,  1.3.1  Time For a given m a t e r i a l , the magnitude o f creep  increases  with time, as i l l u s t r a t e d i n the f o l l o w i n g f i g u r e :  1.3.2 a)  Stress F o r a given m a t e r i a l , the magnitude o f creep  increases  w i t h i n c r e a s i n g s t r e s s , as i l l u s t r a t e d i n the f o l l o w i n g f i g u r e :  There are s e v e r a l examples i n the l i t e r a t u r e t h a t l u s t r a t e t h i s tendency f o r wood and wood-based Kingston and C l a r k e  il-  materials.  (1961) s t u d i e d s e v e r a l  species.  6 Specimens 3/4  i n wide (19 mm),  span of 12 i n (305 mm) Figure ond  by 7/16  were subjected  i n deep (11 mm)  with a  to four p o i n t bending.  2 shows some of t h e i r r e s u l t s f o r Hoop Pine.  The  sec-  of t h e i r graphs shows f r a c t i o n a l creep p l o t t e d a g a i n s t  s t r e s s expressed as a percentage of u l t i m a t e another way  of i l l u s t r a t i n g the  Cederberg and r e s u l t s on Pine and (Swedish Standard) . ment f o r two 400 mm) Figure  and 3-b  tween 1/256  sizes:  s t r e s s which i s  trend.  Danielsson  .(1970) presented experimental  Spruce, c o n s t r u c t i o n grade T200 and Figure 0.8  3-a  i n x 15.8  i n (50 mm  in  x 100  (20 mm mm  shows the i n c r e a s e of f r a c t i o n a l creep day  and  one  day  T300  shows the experimental arrange-  i n x 0.8  2 i n x 4 i n x 118  x 20 mm  x 3000  be-  as a f u n c t i o n o f s t r e s s i n t e n s i t y differ-  I t can be seen t h a t the f r a c t i o n a l creep o f  small specimens i n c r e a s e s  x  mm).  (e^/e^)  expressed as the r a t i o to the u l t i m a t e s t r e s s , f o r two ent s i z e s .  the  s i g n i f i c a n t l y more with the  i n t e n s i t y than t h a t of l a r g e specimens.  Nielsen  the  stress  (1972) argues  t h a t the i n f l u e n c e of s i z e , v i a i r r e g u l a r i t i e s . : e x i s t i n g i n the m a t e r i a l such as knots and  g r a i n d i s t o r t i o n , i s not very  signif-  i c a n t on the f r a c t i o n a l creep, probably because the creep  and  the i n i t i a l deformations are i n f l u e n c e d to the same e x t e n t . c o n t r a s t t h i s may  In  have a marked.influence on the f r a c t u r e s t r e s s  a_., l a r g e beams being up to 50% weeker than small beams. Nakai  (1978) s t u d i e d e i g h t wood-based m a t e r i a l s .  mens 2 i n (50 mm) ranging  from .32  wide by 18 i n (450 mm) i n (8 mm)  four p o i n t bending. and  2/4  long, w i t h a  to 1 i n (.25 mm),  Speci-  thickness  were s u b j e c t e d  to  S t r e s s l e v e l s were s e l e c t e d at 5/4,4/4,3/4,  o f p r o p o r t i o n a l l i m i t , d e f i n e d as t h a t p o i n t a t which  7  the r e l a t i o n between s t r e s s and deformation ceases to be F i g u r e 4 shows some of h i s r e s u l t s : the creep  linear.  compliance.,  J , d e f i n e d as the s t r a i n per u n i t s t r e s s , i s p l o t t e d versus f o r s e l e c t e d s t r e s s l e v e l s , f o r s t r u c t u r a l plywood mark 12  time Ko.  From these graphs, i t can be seen t h a t the magnitude of creep i n c r e a s e s w i t h i n c r e a s i n g a p p l i e d  stress.  b) S t r e s s i n f l u e n c e s the creep development i n a second  way:  the creep behaviour of a m a t e r i a l a t a< c e r t a i n moment depends upon the p r e v i o u s s t r e s s h i s t o r y . Some examples of s t r e s s h i s t o r i e s and t h e i r i n f l u e n c e on creep deformation can be  found  i n the l i t e r a t u r e . U n f o r t u n a t e l y , these s t u d i e s do not apply to wood d i r e c t l y . N i e l s e n (1972) shows the t h e o r e t i c a l d e v e l opment of creep under i n t e r m i t t e n t l o a d i n g (Figure 5); the creep tendency  i s decreased upon each new l o a d i n g .  Work by Lundgren  (1968) on hardboard  seems t o c o n f i r m t h i s  t r e n d f o r wood-based m a t e r i a l s . A creep experiment  i s based on a constant magnitude  of a p p l i e d l o a d . I d i d not c o n s i d e r l o a d . v a r i a t i o n s such as i n t e r m i t t e n t l o a d i n g or stepwise v a r i a t i o n s . However,a;-structure  i n r e a l l i f e i s subjected t o  a load h i s t o r y which i s very seldom constant; hence the long-term d e f o r m a t i o n a l behaviour would be b e t t e r d e s c r i b e d under these a c t u a l c o n d i t i o n s . On the other hand, constant s t r e s s l o a d i n g w i l l y i e l d an upper bound of deformation  likely  ( possibly fatigue loading  excluded ), and i t would be v e r y d i f f i c u l t t o d e f i n e l o a d h i s t o r i e s which would r e p r e s e n t the g r e a t v a r i e t y of loads on a s t r u c t u r e .  8 1.3.3  Temperature Temperature i n f l u e n c e s creep development i n two ways:  a)  For a given m a t e r i a l , the magnitude of creep i n c r e a s e s  w i t h i n c r e a s i n g temperature.  An i l l u s t r a t i o n of t h i s p o i n t i s  seen i n F i g u r e 2 (Kingston and C l a r k e 1961) . Sauer and Haygreen (196 8) t e s t e d hardboard i n f l e x u r e and found t h a t at a s t r e s s i n t e n s i t y o f 30%, an i n c r e a s e i n temp e r a t u r e from 2 2.2 t i o n of about b)  C t o 33.3  C gave an i n c r e a s e i n creep d e f l e c -  50%.  Changes i n temperature  d u r i n g a creep experiment w i l l give  r i s e to i n t e r n a l s t r e s s e s i n a specimen  (the i n t e r n a l  stresses  depend on the g r a d i e n t of temperature w i t h r e s p e c t to time as w e l l as to specimen  dimensions)  and thereby i n f l u e n c e the  creep. An i l l u s t r a t i o n of t h i s p o i n t i s g i v e n by K i t a h a r a and Yukawa (1964) who 0.4  i n x 0.4  studied. Chamaecyparis obtusa i n s m a l l s i z e :  i n x 16.8  i n (10 mm  x 10 mm  x 420 mm).  Specimens  were t e s t e d u s i n g f o u r p o i n t bending at a s t r e s s l e v e l of 1370  p s i (9.45 MPa).  temperature  1.3.4  F i g u r e 6 shows the d i f f e r e n t types of  used as w e l l as p l o t s of d e f l e c t i o n versus time.  Moisture Content A g r e a t d e a l of r e s e a r c h has been c a r r i e d out to study  the i n f l u e n c e o f moisture content upon creep deformation. Schniewind  (1968) p r e s e n t s a good survey o f the r e s u l t s  ed up t o 1968.  obtain-  More s p e c i f i c r e s u l t s c o n c e r n i n g wood i n s t r u c -  t u r a l s i z e s can be found i n the work by N i e l s e n  (1972),  appendix  B. As a summary, one can say t h a t moisture i n f l u e n c e s the  9 creep development i n two ways: a)  A specimen i n e q u i l i b r i u m a t a h i g h r e l a t i v e humidity  i n the surrounding a i r creeps more than a specimen i n e q u i l i b rium a t a lower b)  RH.  I f a change i n the moisture content occurs d u r i n g the  creep experiment,  i t w i l l a l s o cause a change, most o f t e n an  i n c r e a s e , i n the magnitude o f creep. These two phenomena are i l l u s t r a t e d i n the f o l l o w i n g figures:  Creep deformation  Creep deformation <5  6  R e l a t i v e humidity  1.3.5  Time t  Volume The  specimen volume i n f l u e n c e s creep development i n  two ways: a)  Volume e x e r t s an i n f l u e n c e on v a r i o u s other f a c t o r s  as g r a d i e n t o f temperature b)  such  or moisture content.  The g r e a t e r the volume, the more l i k e l y the m a t e r i a l w i l l  c o n t a i n flaws and, t h e r e f o r e , deform more e a s i l y . w i t h f u l l s i z e members, t h i s phenomenum i s d i r e c t l y f o r ; however, f o r a given volume there i s s t i l l  In t e s t s accounted  a discrepancy  10 in  the types  size, kind  location...  1.4  previous  practice,  In  did  this  n o t appear  these  the space parameter, be c o n s t a n t . factor,  i n the functional  on a I n common  some v a r i a b l e s defor-  temperature H o w e v e r , we  and moiswill  t h e grade e f f e c t ,  expression of 5 , given  An e a s y way t o do t h i s  t h e modulus o f e l a s t i c i t y  In  i t s f u n c t i o n a l , form,  as  follows: &  = f{t,  creep  a(t,x),  MOE,  <5 = f { t , a ( t , x ) ,  MOE,  i n the following  i s to consider  that i n the two  and t h e s t r e n g t h .  Both  w h e r e SR i s t h e s t r e s s r a t i o , t o short-term  deformation  c a n now b e e x -  strength}  way:  c  stress  and i t s study  v a r i a b l e s depend upon t h e grade o f t h e material:. .  pressed  or  deformation  variables.  f o r another  section.  new v a r i a b l e s ,  for i n  F u n c t i o n a l Form  parameters w i l l  t o account  previous  accounted  and s t u d y i n g t h e dependence o f creep  study,  to the  effect,  basis i s therefore rather difficult.  values  content  attempt  that this  s e c t i o n h a s shown t h a t t h e c r e e p  m a t i o n upon a few s e l e c t e d  ture  according  one n a r r o w s t h e p r o b l e m down b y s e t t i n g  constant  number,  deformation.  h i g h l y dependent upon s e v e r a l f a c t o r s  phenomenological  to  Note  i s not directly  expression o f creep  Long Term D e f o r m a t i o n ; The  is  of flaws allowed.  t o as grade e f f e c t ,  functional  i n the material, their  Commercial m a t e r i a l i s graded  and q u a n t i t i e s  referred the  of defects present  SR}  d e f i n e d as t h e r a t i o  s t r e n g t h ; MOE  of applied  i s t h e modulus o f  elasticity.  I f the s t r e s s r a t i o s are computed at one l e v e l of a p p l i e d  stress,  as w i l l be the case i n t h i s t h e s i s , then the two e x p r e s s i o n s are  e q u i v a l e n t i n t h a t s t r e n g t h and s t r e s s r a t i o are i n v e r s e l y  proportional. The f u n c t i o n f w i l l be determined by f i t t i n g e x p r e s s i o n s to creep curves.  mathematical  A l a r g e number of mathematical  p r e s s i o n s have been a p p l i e d s u c c e s s f u l l y to creep  ex-  (more than  30 e x p r e s s i o n s f o r metals, 10 f o r c o n c r e t e ) , however these exp r e s s i o n s are e s s e n t i a l l y time f u n c t i o n s . few have been a p p l i e d t o wood. of  Nielsen  Furthermore,  only a  (1972) showed an example  time f u n c t i o n f o r hardboard and plywood s u b j e c t to t e n s i o n .  The power f u n c t i o n e = At c was  b  used, f i t t i n g the data shown i n F i g u r e 7. K i n g s t o n and C l a r k e (1961) assumed t h a t r e c o v e r a b l e  time-dependent  deformation of wood w i l l , as to f i r s t a p p r o x i -  mation, f o l l o w the h y p e r b o l i c s i n e law, as developed by E y r i n g ' s (1935) a p p l i c a t i o n of s t a t i s t i c a l mechanics to r a t e p r o c e s s e s . T h i s law i s expressed i n terms o f thermodynamic c o n s t a n t s . T h i s law i s w r i t t e n : dt where K^ and ^  = K, s i n h 1  K~a 2  are experimental c o n s t a n t s .  an a p p l i c a t i o n o f t h i s theory to f i b e r b o a r d (Lundgren  1968).  F i g u r e 8 shows (masonite i n tension)  The agreement between theory and experimental  r e s u l t s i s very good. Due  to time l i m i t a t i o n s I was  r e f i n e d mathematical  treatment o f t h i s  not a b l e to perform a sort.  12  CHAPTER 2  EXPERIMENT IN SURREY  2.1  Preliminaries The  p r e v i o u s c h a p t e r h a s shown t h a t  t i o n s depend upon s e v e r a l v a r i a b l e s temperature,  such  long term  as m o i s t u r e  content The  and t e m p e r a t u r e  original  purpose  stress  and t i m e  thesis,  were a p p r o x i m a t e l y  o f the experiment,  S u r r e y , B.C., was t o i n v e s t i g a t e plied  content,  stress, etc.  I n t h e e x p e r i m e n t a l work p r e s e n t e d i n t h i s moisture  deforma-  constant.  carried  outi n  t h e r e l a t i o n s h i p between a p -  f o r commercial  lumber.  The p h y s i c a l  a r r a n g e m e n t was d e s i g n e d a c c o r d i n g l y . Some d e f o r m a t i o n measurements were t a k e n . o v e r in  order to get a coarse idea of creep.  Although  ments a r e n o t v e r y a c c u r a t e , t h e y c o n s t i t u t e data s e t that  c o v e r s a l o n g time  2.2  Used  2.2.1  Material  Selection  of Material  span.  at Mill  Site  three  these  a rather  years  measure-  unique  13 J o i s t s o f Douglas F i r 2 i n x 6 i n x 12 f t (40 mm x 140 mm x 3600 mm) were s e l e c t e d i n 1973 from a s i n g l e m i l l i n the  i n t e r i o r o f B r i t i s h Columbia.  M a t e r i a l was s e l e c t e d as i t  came from the p l a n e r m i l l a f t e r the m a t e r i a l had been k i l n d r i e d t o the standard maximum 19%. "no. 2 and b e t t e r " included.  T h i s m a t e r i a l was e s s e n t i a l l y  grade w i t h o n l y a few boards o f no. 3 grade  The lumber was grade marked a t the m i l l , stacked and  wrapped b e f o r e shipment t o the l a b o r a t o r y i n Vancouver.  2.2.2- G r o u p i n g 'tne M a t e r i a 1 The sub-sample o f lumber used i n the a c t u a l experiments was  s e l e c t e d from the i n i t i a l shipment u s i n g the f o l l o w i n g p r o -  cedure  (Figure 9 ) .  Step 1:  The Shipment o f t h i s m a t e r i a l was made i n two batches; the  first  1973.  load a r r i v e d i n the l a b o r a t o r y on J u l y 17,  The load c o n s i s t e d o f e i g h t packages, each con-  t a i n i n g 128 boards, f o r a t o t a l o f 1024 boards.  The  second load a r r i v e d i n the l a b o r a t o r y on August 16, 1973. Step 2:  I t c o n s i s t e d of 12 packages  400 boards, out o f the f i r s t  (1536 b o a r d s ) .  1024, were randomly  s e l e c t e d , and the r e s t sent to storage. out  600 boards,  o f the next 1536, were randomly s e l e c t e d and the  r e s t sent to s t o r a g e . Step 3:  a)  Boards were v i s u a l l y i n s p e c t e d t o determine the weaker o f the two edges.  b)  M o i s t u r e content readings were taken w i t h a r e s i s t a n c e type moisture meter.  Readings were  14 taken a t 3 f e e t , 6 f e e t and 9 f e e t from the end of the boards. c)  The boards were s u b j e c t e d to t h i r d p o i n t and t h e i r E-value c a l c u l a t e d .  load  They were con-  sequently a s s i g n e d t o E groups depending upon the c a l c u l a t e d E - v a l u e .  Table IT and III' show  the grouping f o r the f i r s t and second batches, respectively. The formula used f o r the c a l c u l a t i o n o f the modulus of elasticity i s : . 0  23 P I  3  =  648 EI where  P = 50.3 l b s (226.4 N) 1 = 138 i n (3500 mm) I = 20.8 i n (8.66 x 10 mm g  4  4  ) f o r average 2 i n x 6 i n section.  <5 = d e f l e c t i o n measured a t the c e n t r e o f the span. Step 4:  The boards of each E-category were ranked a c c o r d i n g to t h e i r apparent s t r e n g t h .  (The boards were v i s u -  a l l y graded f o r s t r e n g t h u s i n g slope o f g r a i n , knot s i z e and l o c a t i o n e t c . as s t r e n g t h  indicators  ( N a t i o n a l Lumber Grades A u t h o r i t i e s , 1970)):. A f t e r b e i n g ranked from s t r o n g e s t to weakest, the boards from each category were d i v i d e d i n t o  four  sub-groups s t a r t i n g from the lowest r a n k i n g s t r e n g t h . Step 5:  The boards from each sub-group  ( f o r a t o t a l o f 44  sub-groups) were randomly a l l o c a t e d to f o u r groups. The end r e s u l t i s f o u r groups o f 250 boards each. Step 6:  For each group, 10 boards out o f 250 were d i s c a r d e d randomly.  Step 7:  One o f the 4 groups was  s e t a s i d e and c o n s t i t u t e d  15 a control  sample.  The 3 remaining p i l e s were s u b d i v i d e d by a s s i g n i n g the f i r s t board t o sub-group to sub-group  "a", the second board  "b", the t h i r d board t o sub-group " c " ,  the f o u r t h board t o sub-group  "a", e t c . The end  r e s u l t i s 9 groups o f 80 boards each. The boards from the c o n t r o l group were re-weighed, and a moisture content reading taken a t the c e n t r e of the board.  The dimensions were a l s o r e c o r d e d :  the width and depth a t the c e n t r e t o the n e a r e s t 0.01 i n and the l e n g t h t o the n e a r e s t h a l f - i n c h . The 240 specimens were then t e s t e d t o f a i l u r e , u s i n g a t h i r d - p o i n t l o a d i n g c o n f i g u r a t i o n and a r a t e of l o a d i n g o f 5000 psi/mn (34.5 MPa/mn), which produced an average time t o f a i l u r e of approxi m a t e l y one minute. F i g u r e 10 shows the s t r e n g t h d i s t r i b u t i o n o f the c o n t r o l sample, p l o t t e d u s i n g the normalized r a n k i n g method. Step 8:  Four groups out o f the nine f i n a l groups were r a n domly s e l e c t e d f o r the purpose o f the experiment. The e x p e r i m e n t a l groups were decreased from 80 boards to 33, 44, 72 and 72 boards, due to space l i m i t a t i o n s . T h i s was done by e x c l u d i n g the boards t h a t had a h i g h modulus o f e l a s t i c i t y as w e l l as small knots and were t h e r e f o r e u n l i k e l y t o f a i l d u r i n g the l i f e - t i m e o f the experiment.  2.3  P h y s i c a l Arrangement The experiment  was  s t a r t e d on January  4, 19 75.  Two  hundred and nineteen boards were loaded a t v a r i o u s s t r e s s u s i n g the set-up i n F i g u r e 11.  'Photographs of the  arrangement are?seen i n F i g u r e s 12-1  and 12-2.  p s i (7.38  MPa)  times the f i f t h p e r c e n t i l e v a l u e on the  s t r e n g t h d i s t r i b u t i o n of the c o n t r o l sample. 0.71  experimental  Thirty-three  boards were s u b j e c t e d to a s t r e s s l e v e l of 1070 corresponding t o 0.71  levels,  The  f a c t o r of  i s the s t r e n g t h r e d u c t i o n p r e d i c t e d a c c o r d i n g t o the  Madison hypothesis f o r a constant l o a d of 7 day d u r a t i o n . f a c t o r was  i n t r o d u c e d i n p r e v i o u s experiments  designed to study  the t i m e - s t r e n g t h r e l a t i o n s h i p over a 7 day p e r i o d , and kept f o r c o n s i s t a n c y i n these t r i a l s  This  was  (Madsen, 1976).  Forty-two boards were s u b j e c t e d to a s t r e s s l e v e l of 1410  p s i (9.72 MPa)  corresponding to 0.71  times the t e n t h p e r -  c e n t i l e value. Seventy-two boards were s u b j e c t e d to a s t r e s s l e v e l of 2110  p s i (14.55 MPa)  corresponding t o 0.71  times the t w e n t y - f i f t h  percentile value. Seventy-two boards were s u b j e c t e d to a s t r e s s l e v e l of 3110  p s i (21.44 MPa)  corresponding to 0.71  times the  fiftieth  percentile value. I analyzed the measurements obtained from a random sample of the boards t h a t were s t i l l T h i s sub-sample c o n s i s t e d o f : 16 boards s u b j e c t e d t o 1410 and  unbroken i n February  sub1979.  15 boards s u b j e c t e d to 1070 p s i ,  p s i , 25 boards s u b j e c t e d t o 2110 p s i ,  8 boards subjected t o 3110  psi.  Tables IV a, b, c and d  show" a l i s t of these boards with t h e i r E-value c a l c u l a t e d as  mentioned i n Step 3 o f s e c t i o n 2.2.2. Table V shows the mean and standard the modulus o f e l a s t i c i t y  d e v i a t i o n values o f  f o r the f o u r groups.  The c o e f f i c i e n t  of v a r i a t i o n , C.V., i s a l s o shown.  2.4  Deformation Measurement  2.4.1  Expected I n i t i a l The  ing  Deflections  l o a d i n g c o n f i g u r a t i o n i s represented  i n the f o l l o w -  figure:  Where deformations are measured.  .x -(—5 i n 46 i n  5 in—f46 i n  46 .in 138 i n 144 i n  I used the simple homogeneous beam theory the i n i t i a l  to p r e d i c t  d e f l e c t i o n a t x = 5 i n from- the p o i n t o f a p p l i c a -  t i o n of the l o a d .  I used E-values as w e l l as c r o s s - s e c t i o n a l  dimensions shown i n Tables IV a, b, c and d i n the c a l c u l a t i o n s . D e f l e c t i o n due t o shear was n e g l e c t e d . d show these values  2.4.2  Tables IV a, b, c and  of i n i t i a l deflection.  Actual I n i t i a l  Deflections  A scheme o f measurements t h a t were done i s represented  18  as  follows: 138  in  SOUTH  NORTH  s i  Empirical 6  n  and  various  c5 were measured w i t h i n h a l f an hour of each other, at s times d u r i n g  the t e s t i n g p e r i o d , w i t h a h i g h e r frequency  in  the b e g i n n i n q . ^ =  6 . and ni  6  and  Y  n  references  <5 . s  Y and n  the t e s t set-up. *  s  6 . are the si  first  values a v a i l a b l e  of  were measured once as c h a r a c t e r i s t i c s o f  C a l l i n g f . and ^ nx  t i o n s of a board a t i t s n o r t h and  f . the a c t u a l i n i t i a l si south ends, one  obtains  deflecthe  following:  f nx. = Yn • - 5 n i. 'si  The  ' S  r e s u l t s f o r f . and nx  S I  f ., as w e l l as t h e i r average hif . + f .) si nx si y  are presented i n T a b l e s IV a, b, c and levels.  E r r o r on i n i t i a l  d e f l e c t i o n , defined  the d i f f e r e n c e between measured and predicted  d f o r the four  d e f l e c t i o n i s a l s o shown.  predicted  stress  as the r a t i o of d e f l e c t i o n s to  the  2.4.3  D e f l e c t i o n as a F u n c t i o n o f Time I f we  d e f i n e the d e f l e c t i o n of one group  (defined by i t s  s t r e s s l e v e l ) as the average d e f l e c t i o n of the boards of the group, we can see i n Tables IV a, b, c and d the d e f l e c t i o n o f each group a t v a r i o u s times throughout the experiment.  Standard  d e v i a t i o n s as w e l l as c o e f f i c i e n t s of v a r i a t i o n s are a l s o shown. As an example o f the c a l c u l a t i o n s t h a t have y i e l d e d these r e s u l t s , c o n s i d e r the group o f e i g h t beams loaded a t 3110 p s i . Table VII shows Y and Y f o r each board, as c h a r a c t e r n 's i s t i c s o f the t e s t set-up a t the s p e c i f i c l o c a t i o n o f the board. Table V I I I - i the  calculated f  ( i = 1,8)  and f  shows the measured 6  and the average %(f  not p o s s i b l e to measure y  three groups.  n  and y  s  and 6 ,  + f ) f o r a given  board i over the course of the experiment. was  n  Unfortunately, i t  f o r every board i n the  In such a case I used the average v a l u e s o f  a v a i l a b l e Y ' S and v s to estimate Y„ and Y . n 's 'n s 1  1  1  The steps y i e l d i n g the r e s u l t s p r e s e n t e d i n Tables VI a, b, and c (groups 1070  p s i , 1410  p s i , 2110  from the U n i v e r s i t y o f B r i t i s h Columbia,  p s i ) are a v a i l a b l e  Department of  Civil  Engineering. The raw data presented i n Tables VI a, b, c and d are p l o t t e d i n F i g u r e 13. Points  ( i = 1,4)  r e p r e s e n t the expected i n i t i a l  f l e c t i o n s f o r the f o u r groups: M*  (0.1 hr, 0.52  in)  (13.2  mm)  M^  (0.1 hr, 0.68  in)  (17.3  mm)  (0.1 hr, 0.90  in)  (22.9  mm)  M  3  de-  20 (0.1 h r , 1.30 Time t = 0.1 ( i = 1,4)  hr was  in)  (33.0  mm)  chosen as the o r i g i n of times.  represent actual i n i t i a l  Points  d e f l e c t i o n s f o r the f o u r  groups:  (1 hr, 0.29 in)  ( .7.4  mm)  (1 hr, 0.60 in)  (15.2  mm)  MQ  (1 h r , 1.02 in)  (25.9  mm)  MQ  (1 hr, 1.20 in)  (30.5  mm)  MJ 2 M  Q  Time t = 1 hr was  chosen a r b i t r a r i l y s i n c e I d i d not know when  the f i r s t measurements were taken.  2.5  I n t e r p r e t a t i o n s of R e s u l t s As mentioned i n the p r e v i o u s s e c t i o n , the p h y s i c a l  arrangement of t h i s experiment  i n t r o d u c e d l i m i t a t i o n s on the  type of deformations, recorded, as w e l l as t h e i r accuracy.  For  these reasons, the r e s u l t s presented, r e f e r to a deformation d e f i n e d as the average  of the d e f l e c t i o n s of i n d i v i d u a l  measured a t a convenient l o c a t i o n of  a p p l i c a t i o n of the l o a d ) .  of  e l a s t i c deformations. ( i = 1,4),  (5 inches away from the p o i n t  I d i d not measure a c t u a l v a l u e s  T h e r e f o r e , I assumed that p o i n t s  shown i n F i g u r e 13,  e l a s t i c deformation.  r e f e r to the a c t u a l v a l u e s of  I t i s then p o s s i b l e t o redraw the f o u r  creep curves, shown i n F i g u r e 13,  from these p o i n t s M ^ ( i =  The redrawn curves are shown i n F i g u r e 14. experimental e r r o r remained have been drawn as w e l l . of  boards,  I assumed t h a t  constant over time.  Trend  1,4). any  lines  They i n d i c a t e t h a t the r a t e of change  creep deformation seems to i n c r e a s e w i t h i n c r e a s i n g a p p l i e d  stress. I t should be s t r e s s e d t h a t the p r e s e n t a t i o n of the r e s u l t s on a s e m i - l o g a r i t h m i c graph accentuates the slope o f the curve i n the upper r e g i o n o f the X-axis  (time).  Had  the curves  been p l o t t e d on a b i l i n e a r graph, they would be very f l a t  indeed,  i n d i c a t i n g t h a t any s u b s t a n t i a l i n c r e a s e i n deformation takes p l a c e over a long p e r i o d of time.  22  CHAPTER 3  EXPERIMENT IN RICHMOND  3.1  3.1.1  M a t e r i a l Used  Selection of Material at M i l l Hem-fir  joists,  Site  2 i n x 6 i n x 12 f t (40 mm x 140 mm x  3600 ram), were s e l e c t e d i n August 1977 from a m i l l Vancouver area. Hemlock.) mill  (30% of the boards were A m a b i l i s - F i r ;  70% were  2947 p i e c e s were s e l e c t e d as they came from the p l a n e r  a f t e r they had been k i l n d r i e d t o the standard maximum 19%.  T h i s m a t e r i a l was "no. 2 and b e t t e r "  3.1.2  i n the Greater  grade.  Grouping A f t e r the boards had a r r i v e d i n the Laboratory, they  were numbered, re-graded by a COFI lumber i n s p e c t o r ; s p e c i e s i d e n t i f i c a t i o n c h i p s were taken.  A l l p i e c e s were d r i e d a t the  l a b o r a t o r y to a moisture content r a n g i n g between 8% and 12%. F i g u r e 15 shows the three steps y i e l d i n g the f i n a l sub-  23 samples. Step 1:  The two s p e c i e s were separated,  and badly warped  p i e c e s were removed. Step 2:  The boards were s u b j e c t e d to t h i r d p o i n t l o a d and t h e i r E-value  calculated.  ranked a c c o r d i n g t o these The  They were  subsequently  values.  f i r s t 18 boards were then randomly a l l o c a t e d t o  18 d i f f e r e n t groups.  The next  18 boards were s i m i -  l a r l y a l l o c a t e d t o the 18 groups.  This r e s u l t e d i n  18 Hemlock groups and 5 A m a b i l i s - F i r groups s t a t i s t i c a l l y s i m i l a r w i t h i n each s p e c i e s with regard t o the value of the modulus o f e l a s t i c i t y . Step 3:  Random s e l e c t i o n o f s i x groups.  Groups no. 2 f o r  A m a b i l i s - F i r and no. 15 f o r Hemlock were used t o i n v e s t i g a t e E-values  (Chapter  3.5.1).  Groups no.  2, 6, 8 and 9 f o r Hemlock were used i n the creep experiment. F i g u r e s 16 and 17 show the s t r e n g t h d i s t r i b u t i o n o f the c o n t r o l sample  (Group no. 17). The boards o f t h i s group  were broken a t a constant r a t e o f l o a d i n g o f 5000 psi/mn (34.5 MPa/mn) t h a t produced an average time to f a i l u r e o f approximately one minute. raw  In F i g u r e 16, two curves were f i t t e d t o the  data; the s o l i d l i n e r e p r e s e n t s a three parameter W e i b u l l  d i s t r i b u t i o n ; the dashed l i n e r e p r e s e n t s a two parameter W e i b u l l distribution.  In F i g u r e 17, a lognormal  to the data p o i n t s .  d i s t r i b u t i o n was  fitted  These three d i s t r i b u t i o n s have been widely  used t o approximate data on v a r i o u s p r o p e r t i e s o f wood.  In  p a r t i c u l a r , the three parameter W e i b u l l d i s t r i b u t i o n has been  put forward data  as being a good r e p r e s e n t a t i o n f o r timber  ( P i e r c e 1976).  strength  This i s v e r i f i e d i n the p r e s e n t work;  F i g u r e s 16 and 17 show t h a t a b e t t e r f i t i s obtained with the three parameter W e i b u l l d i s t r i b u t i o n . cumulative  p r o b a b i l i t y o f f a i l u r e , F, and the a p p l i e d s t r e s s , a ,  i s then w r i t t e n as f o l l o w s :  1.987  r  /a - 1428\  F = 1 - exp  \6156 where a i s expressed  3.2  The r e l a t i o n between the  /  in psi.  Loading C o n f i g u r a t i o n  46 i n  46 i n  46 i n  i t " 'h  A f o u r p o i n t s l o a d i n g c o n f i g u r a t i o n i s represented i n the above f i g u r e .  The span has a constant value of 138 i n and  the loads are a p p l i e d a t the two t h i r d p o i n t s . I computed the value of F f o r each loaded beam such t h a t the beam was subjected t o a maximum s t r e s s o f e i t h e r 3000 p s i (20.68 MPa) o r 4500 p s i (31.02 MPa).  The maximum s t r e s s t h a t  occurs between the t h i r d p o i n t i s constant i s due t o bending o n l y .  i n t h i s r e g i o n and  25 These two s t r e s s l e v e l s were d e r i v e d from the f i f t h and t w e n t i e t h p e r c e n t i l e s o f the s t r e n g t h d i s t r i b u t i o n o f t h e c o n t r o l sample  (Group no. 17) seen i n F i g u r e s 16 and 17.  From one o f these v a l u e s o f maximum s t r e s s , the f o r c e Fi  f o r the i n d i v i d u a l board i i s computed from the f o l l o w i n g  equation: 2  a  F• = b . h . ° ^ i i 1 i x 2L where: a max i s the maximum s t r e s s  3.3  b. l h^  i s the width o f the board  L  i s the span  i s the depth o f t h e board (138 in)  (3505  mm)  T e s t i n g Procedure The experiment was i n i t i a t e d by the Western  Forest  Products L a b o r a t o r y , S t r u c t u r e D i v i s i o n , i n Richmond, B.C., i n e a r l y 1978. Group numbers 2, 6, 8 and 9 (Hemlock) t a i n i n g 100 beams each were used. of  a randomly  s e l e c t e d sub-sample  (Figure 15) con-  The deformation measurements o f boards were taken over a  three month p e r i o d w i t h a h i g h e r frequency o f measurements i n the  beginning. T h i s experiment was c a r r i e d out i n two s t a g e s .  group o f 97 boards  The f i r s t  (79 o f which belonged to group number 8,  18 to group number 2) were loaded i n August 1978.  The f i r s t de-  formation measurements were o b t a i n e d two t o three hours a f t e r the full  l o a d had been t r a n s f e r r e d t o the beams.  t h i s group o f 97 boards as sub-group 1.  I w i l l refer to  Sub-group 2 c o n s i s t e d of 105 boards to  (76 o f which  belonged  group number 2; 11 t o group number 6 and 18 to group number 9),  loaded i n November 1978.  Deformation i n f o r m a t i o n s were o b t a i n e d ,  t h i s time, from the b e g i n n i n g o f l o a d  transfer.  The beams were s t o r e d i n the Western F o r e s t Products Laboratory f o r about one year p r i o r to t r a n s f e r to a l a b o r a t o r y in  Richmond where they were loaded.  They had by then reached  a s t a b l e l e v e l o f moisture content (8% to 12%) temperature.  ambient  These c o n d i t i o n s were kept approximately c o n s t a n t  d u r i n g the Richmond Laboratory experiments. i s i n d i c a t e d by the photographs experimental data f o r sub-groups IX and X.  and  The t e s t set-up  i n F i g u r e s 18 and 19.  The  1 and 2 are i n d i c a t e d i n Tables  The board number, group number, c r o s s s e c t i o n a l dimen-  s i o n s , l e v e l o f a p p l i e d s t r e s s and modulus o f e l a s t i c i t y are shown f o r each i n d i v i d u a l board.  3.4  Deformation Measurements  3.4.1  Expected I n i t i a l  Deflections  Using simple homogeneous beam theory, one can p r e d i c t i n i t i a l d e f l e c t i o n of a beam a t the c e n t r e o f i t s span. the  l o a d i n g c o n f i g u r a t i o n p r e s e n t e d i n s e c t i o n 3.2, 23 F . L  one  For obtains:  3  s  ' i =  648 E . I . l l  where E^ i s the modulus o f e l a s t i c i t y o f a s p e c i f i c board i . Tables IX and X show the p r e d i c t e d i n i t i a l for  sub-groups  1 and 2, r e s p e c t i v e l y .  deflection  3.4; 2  Actual I n i t i a l a)  Deflections  Sub-group 1 For  the f i r s t  sub-group 1 the a c t u a l i n i t i a l  d e f l e c t i o n r e f e r s to  deformation measurement o b t a i n e d .  I t i s not the e l a s t i c  deformation, d e f i n e d as the deformation o f a beam immediately after f u l l  l o a d has been t r a n s f e r r e d to i t (the load  takes about one minute). to  The deformation was  transfer  recorded by  ruler,  the n e a r e s t m i l l i m e t r e . Table IX shows these v a l u e s of i n i t i a l b)  deflections.  Sub-group 2 For  the boards of t h i s sub-group g r e a t e r care was  taken  i n m o n i t o r i n g the d e f l e c t i o n at the centre o f the span, from time t = 0, d e f i n e d as the time when l o a d t r a n s f e r was The r a t e of l o a d i n g was red  i n approximately one minute.  such t h a t f u l l  started.  l o a d was  Charts of load and  transfer-  deflection  v a r i a t i o n s w i t h r e s p e c t t o time were recorded w h i l e the l o a d  was  t r a n s f e r r e d and up t o the moment when the d e f l e c t i o n reached a stable value.  A measurement o f the d e f l e c t i o n was  a r u l e r , t o the n e a r e s t m i l l i m e t r e .  C l o s e agreement was  between t h i s value and the v a l u e read, on the c h a r t . was  the d e f l e c t i o n a t time t = 4 minutes.  t i o n s a t t = 2 minutes  then taken w i t h reached  This value  I estimated the d e f l e c -  and t = 1 minute by i n t e r p o l a t i n g from t h i s  r e f e r e n c e p o i n t on the c h a r t .  The d e f l e c t i o n at t = 1 minute i s  the e l a s t i c deformation of the beam a t the c e n t r e of i t s span, shown i n Table X. F i g u r e 20 shows a c h a r t o f l o a d and d e f l e c t i o n v a r i a t i o n s w i t h r e s p e c t to time f o r board number 2673 from group number 2. P o i n t s M^, minutes  M 2 and  r e f e r to the deformations a t times t = 4  (0.067 h r ) , t = 2 minutes  (0.033 h r ) , and t = 1 minute  28 (0.017 h r ) .  3.4.3  Deformation  as a F u n c t i o n o f Time  The d e f l e c t i o n of the beams a t t h e i r h a l f - s p a n was r e corded r e g u l a r l y over a p e r i o d of three months.  A numerical  d e s c r i p t i o n of these data p o i n t s i s not presented i n t h i s  thesis,  but i s a v a i l a b l e from the U n i v e r s i t y of B r i t i s h Columbia,  Depart-  ment o f C i v i l E n g i n e e r i n g . Curves can be presented i n three ways:  deflection  time p l o t t e d on a b i l i n e a r graph, d e f l e c t i o n versus time on a s e m i - l o g a r i t h m i c graph, and d e f l e c t i o n versus time on a f u l l - l o g a r i t h m i c graph. given i n F i g u r e 21.  versus  plotted plotted  An example o f a b i l i n e a r graph i s  Seven creep curves are shown.  i n m i l l i m e t r e s i s p l o t t e d a g a i n s t time i n hours  Deflection  f o r boards  number  1513, 2498, 2238, 2207, 2497, 2687 and 2756, s e l e c t e d a t random. Values o f a p p l i e d s t r e s s , e l a s t i c deformation and f i n a l measured deformation are shown. T h i s method of p r e s e n t i n g creep curves i s r e p r e s e n t a t i v e of  the t r u e e v o l u t i o n o f the deformation i n time, but i s not very  p r a c t i c a l i n t h a t any s u b s t a n t i a l i n c r e a s e i n deformation  takes  p l a c e over a long p e r i o d of time. Examples o f semi- and f u l l - l o g a r i t h m i c p l o t s are shown i n F i g u r e 22.  In the lower graph, d e f l e c t i o n i s s t i l l p l o t t e d on  a l i n e a r s c a l e i n m i l l i m e t r e s w h i l e time i s p l o t t e d on a l o g a r i t h mic s c a l e i n hours.  I have presented curves f o r the same seven  boards as i n the p r e v i o u s F i g u r e .  In the top graph, both d e f l e c -  t i o n and time a r e p l o t t e d on a l o g a r i t h m i c s c a l e . of  such a graph i s t h a t an e x p r e s s i o n f o r the creep  The advantage deformation  such as  ( a l s o c a l l e d power f u n c t i o n ) , i s transformed t o a s t r a i g h t Log in a log-log plot.  e  c  line  = Log A + b l o g t  T h i s e x p r e s s i o n has been widely used t o de-  s c r i b e the time dependence o f creep deformation.  Although we  can f i n d some examples o f i t s a p p l i c a t i o n t o wood-based materials  (Sugiyama 1957; C l o u s e r 1959; N i e l s e n 1968), the power func-  t i o n has been mainly used f o r b u i l d i n g m a t e r i a l s other than wood. An example o f the a p p l i c a t i o n o f the power f u n c t i o n i s shown i n F i g u r e 7.  F i g u r e 22 shows l i n e a r i t y up to time t = 200 hours,  but one cannot e x t r a p o l a t e f u r t h e r , and more i n f o r m a t i o n i s needed on the creep deformation a t times g r e a t e r than 5000 hours. I chose the s e m i - l o g a r i t h m i c c o n f i g u r a t i o n t o p r e s e n t the data because o f i t s c l a r i t y ;  the r a t e o f change o f deformation w i t h  r e s p e c t t o time i s c l e a r l y shown as the s l o p e o f the curve. F i g u r e 2 3 shows creep curves f o r the boards o f sub-group 1.  For each creep curve, three numbers a r e shown.  The number a t  the l e f t end of the curve i s the f i r s t a v a i l a b l e v a l u e o f deformation  Cs ); the number a t the r i g h t end o f the curve i s the l a s t  measured v a l u e o f deformation number  (_BN) .  ( 6 ) and the t h i r d number i s board fc  F i g u r e 2 4 - i ( i = 1,2) shows creep curves f o r the  boards of sub-group 2 t h a t were loaded t o 3000 p s i , and 4500 p s i respectively.  The creep curves have been p o s i t i o n e d a c c o r d i n g t o  increasing value of e l a s t i c  3.5  deformation.  I n t e r p r e t a t i o n s of R e s u l t s  30 3.5.1  E l a s t i c Deformations In the f o l l o w i n g  s e c t i o n , I w i l l address myself to the  r e s u l t s obtained f o r sub-group 2 s i n c e the a c t u a l i n i t i a l deformations f o r boards o f sub-group 1 are not known, as in. s e c t i o n 3.4.2.  Tables^ X - i  discussed  ( i = 1,3) show t h a t t h e r e i s a  d i s c r e p a n c y between expected and a c t u a l e l a s t i c deformations. This i s b e t t e r  seen i n F i g u r e  t i o n s are p l o t t e d a g a i n s t  25, where a c t u a l e l a s t i c deforma-  expected e l a s t i c deformation f o r the  two- s t r e s s l e v e l s .  Regression l i n e s have been f i t t e d t o the two  s e t s of data p o i n t s  a c c o r d i n g t o the l e a s t squares c r i t e r i o n .  As  shown i n F i g u r e  25 t h e i r e q u a t i o n as w e l l as the squares o f  the c o r r e l a t i o n c o e f f i c i e n t s are ^actual Rsq  0*4378 + 1.073 ^ p  rec  j£ t j c  ec  = 0.9127  and 6 . = 2.957 + 1.017 6 . , actual predicted =  n  Rsq  = 0.8950  f o r 3000 p s i and 4500 p s i r e s p e c t i v e l y .  In these equations,  6 , and 6 ^ are expressd i n m i l l i m e t r e s . actual predicted * and  5  p  r e c  jicted  a regression  w  e  r  e  i d e n t i c a l , then the c o e f f i c i e n t s A and B o f  equation such as «  .  ,  actual  = A + B  would be 0 and 1 r e s p e c t i v e l y . ^predicted  w  o  I f 6 . .. actual  u  l  d  {  ,.  .  ,  predicted  A graph o f 6  show a s t r a i g h t l i n e a t 45  s i d e s of the axes.  , plotted  against  between the p o s i t i v e  A c o e f f i c i e n t A w i t h non-zero value  indicates  a v e r t i c a l displacement of the s t r a i g h t l i n e ; i t r e p r e s e n t s a d i s c r e p a n c y t h a t i s constant throughout the range o f e l a s t i c deformations.  A c o e f f i c i e n t B. g r e a t e r  that 1 indicates  31 a r o t a t i o n of the s t r a i g h t l i n e around the o r i g i n of the axes. I t i n d i c a t e s a d i s c r e p a n c y t h a t i n c r e a s e s w i t h i n c r e a s i n g value of e l a s t i c deformation. mentation, point. a)  T h i s d i s c r e p a n c y r e s u l t s from e x p e r i -  e i t h e r i n a p h y s i c a l sense or from a t h e o r e t i c a l view-  There are s e v e r a l p o t e n t i a l sources of t h i s  discrepancy,  F i r s t l y , there are two main types of e r r o r s i n h e r e n t to the  measurements.  The  f i r s t type w i l l be repeated throughout  experiment l i f e t i m e :  the  the e r r o r of the measuring device t h a t  reads to the n e a r e s t m i l l i m e t r e .  T h i s i n c l u d e s the e r r o r due  to the p o s i t i o n i n g o f the measuring d e v i c e a t each r e c o r d i n g time.  The  second type o f e r r o r i s due  supporting s t e e l s t r u c t u r e . be reproduced  to the bending  of the  T h i s e r r o r occurs once only and  a t each c a l c u l a t i o n of the d e f l e c t i o n .  will  Furthermore,  i t depends upon the l e v e l of a p p l i e d s t r e s s , as i l l u s t r a t e d i n the f o l l o w i n g f i g u r e :  m  Vi  T w  2  7777777777777777777/  Before  ////////////////'//  Loading  After loading  The recorded deformation 6  =  %  2  -  i s 8, such t h a t l  1  I t i n c l u d e s a component b due structure. as 2.5  to the bending  of the s u p p o r t i n g  T h i s component has been estimated and  to 3 mm,  can be as high  f o r the h i g h e s t l e v e l of a p p l i e d s t r e s s .  32 b)  Secondly, i t has been e s t a b l i s h e d t h a t the presence o f de-  f e c t s such as knots and zones o f i n c l i n e d g r a i n causes r e d u c t i o n i n modulus o f e l a s t i c i t y  (Curry 1976) .  localized  Consequently,  s i n c e gross deformations a r e used i n the c a l c u l a t i o n s , the c a l c u l a t e d v a l u e s o f the modulus o f e l a s t i c i t y depend of  on the type  l o a d i n g , on the number and s i z e o f d e f e c t s p r e s e n t , and t h e i r  l o c a t i o n w i t h i n the s t r u c t u r a l member.  S p e c i f i c a l l y , I have  h y p o t h e s i z e d t h a t a v a l u e o f the modulus o f e l a s t i c i t y computed from a deformation recorded a t the t h i r d p o i n t s o f a member i n a t h i r d p o i n t l o a d i n g c o n f i g u r a t i o n might  l e a d t o an under e s t i -  mation o f the maximum deformation o f the member. t r a t e d i n the f o l l o w i n g  This i s i l l u s -  figure:  46 i n  46 i n  46 i n  theoretical deflected shape actual deflected shape The valuei^o-f the modulus o f e l a s t i c i t y , is  f o r a s p e c i f i c beam,  computed from the f o l l o w i n g e q u a t i o n : 8  -  5  F  i  L  3  162 E . I . l l where 6^ i s the d e f l e c t i o n expressed as t h e average o f the def l e c t i o n o f the two t h i r d p o i n t s . specific test  The value E^, o b t a i n e d i n a  ( s e c t i o n 3.1.2, step 2 ) , i s then used t o compute  the e l a s t i c deformation o f the beam under f u l l  t r a n s f e r o f the  33 load i n the d u r a t i o n 23 6  F.L  of l o a d experiment.  The  equation used i s :  3  = 648 E . I .  e  1  l  Such a procedure i n v o l v e s the hypothesis t h a t the beam, under load, assumes a t h e o r e t i c a l d e f l e c t e d shape. may  not be  c o r r e c t and  An  easy way  to  deter-  t h i s phenomenon gives r i s e to a component o f  e r r o r between a c t u a l and i n the  ex-  a c t u a l d e f l e c t e d shapes do not match  even though they have common t h i r d p o i n t s .  sent  hypothesis  the p r e v i o u s f i g u r e i l l u s t r a t e s an  ample where t h e o r e t i c a l and  mine whether or not  This  p r e d i c t e d e l a s t i c deformations i s p r e -  following.  Subject a beam to loads a p p l i e d a t i t s t h i r d p o i n t s record  the d e f l e c t i o n a t these t h i r d p o i n t s as w e l l as a t  half point.  The  l a t t e r measurement i s used to compute E  2  and  the in  the f o l l o w i n g  equation: 2.3F..L = — 648 E I . 3  1  8 2  0  2 l The  average of the f i r s t measurements i s used to compute E^ i n  the f o l l o w i n g  equation: 5F L  3  ±  S  l/3  =  I f E^ and  E  162— ~E . I . 1 l 2  are s i g n i f i c a n t l y d i f f e r e n t , then i t i s p o s s i b l e  to c o r r e c t f o r t h i s component of e r r o r between a c t u a l and d i c t e d e l a s t i c deformations.  Such a n a l y s i s has  pre-  been c a r r i e d out  a t the Western F o r e s t Products Laboratory between December 19 7 8 and A p r i l  1979.  this thesis.  A d e t a i l e d d e s c r i p t i o n i s beyond the  scope of  I analysed the r e s u l t s from group number 15  34 (Hemlock) and group number 2 ( A m a b i l i s - F i r ) .  F i g u r e s 26 and  show the cumulative p r o b a b i l i t y d i s t r i b u t i o n s o f E-^ and E s p e c t i v e l y f o r Hemlock and A m a b i l i s - F i r .  re-  2  A straight line  27  was  d e r i v e d by the l e a s t squares c r i t e r i o n .to c o r r e l a t e E-^ and  .  These equations a r e : Hemlock:  E  2  = 0.7719  10~  4  10~  3  7  + 1.013  E  ±  ( p s i 10~ )  + 1.013  E  ±  ( p s i 10~ )  Rsq = 0.9951 Amabilis-Fir:  E  2  = 0.9296  7  Rsq = 0.9954 The agreement between E-^ and E , 2  f o r both s p e c i e s , i s e x c e l l e n t  and I conclude t h a t the d e s c r i b e d phenomenon, i s not a s i g n i f i cant source of e r r o r between a c t u a l and p r e d i c t e d e l a s t i c formations .  de-  I t should be noted t h a t f o r both Hemlock and .  Amabilis-Fir, E  2  i s h i g h e r than E-^, c o n t r a r y to the p r e v i o u s l y  developed h y p o t h e s i s .  T h i s may  r e s u l t from experimental e r r o r s  due to c r u s h i n g at the p o i n t s of a p p l i c a t i o n of the l o a d as w e l l as a t the supports. In view of these comments, and i f one c o n s i d e r s t h a t the experimental e r r o r s i n h e r e n t to the d e t e r m i n a t i o n of E-^ and are  E  2  n e g l i g i b l e as compared w i t h e r r o r s d e s c r i b e d i n s e c t i o n a ) ,  I conclude t h a t the d i s c r e p a n c i e s between p r e d i c t e d and  actual  e l a s t i c deformations are due to e r r o r s i n h e r e n t A h the l o n g term deformation  experiment.  In the f o l l o w i n g , e l a s t i c deformations, whenever ment i o n e d , r e f e r to the p r e d i c t e d e l a s t i c  deformations.  Creep deformations, d e f i n e d as the d i f f e r e n c e between t o t a l deformations and e l a s t i c deformations, r e f e r to the values that have been c o r r e c t e d a c c o r d i n g to the f o l l o w i n g e q u a t i o n :  35 <5 . = 6 _ S creep creep i actual  3.5.2  General Comments on Creep  6  +  i predicted  Curves  a) . F i g u r e s 2 4 - i ( i = 1,2) show t h a t the r a t e o f change of creep deformation, a t the r i g h t end o f curves, seems t o i n crease with i n c r e a s i n g value o f e l a s t i c deformation.  T h i s phe-  nomenon should be v e r i f i e d on a graph where creep deformation i s p l o t t e d a g a i n s t e l a s t i c deformation; the curve should have p o s i t i v e slope a t any p o i n t , as w e l l as p o s i t i v e c u r v a t u r e . Such a graph i s presented i n F i g u r e 28.  Creep deformation i s  expressed a t t e n d i f f e r e n t times r a n g i n g from f i v e minutes t o t h i r t e e n weeks % ten p l o t s are shown, and both s t r e s s l e v e l s are represented.  The data p o i n t s , although they are s c a t t e r e d ,  low a t r e n d c o n s i s t e n t with the above h y p o t h e s i s .  On the other  hand, i t can be seen t h a t data p o i n t s f o r the two s t r e s s occupy d i f f e r e n t r e g i o n s w i t h i n each p l o t .  fol-  levels  T h i s suggests  that  both e l a s t i c and creep deformations might depend upon the l e v e l of a p p l i e d b)  stress.  A v e r t i c a l l i n e drawn i n F i g u r e 28 from the a x i s o f  e l a s t i c deformations a t a s p e c i f i c value i n t e r c e p t s data p o i n t s f o r 4500 p s i b e f o r e i t i n t e r c e p t s data p o i n t s f o r 3000 p s i . T h i s i s t r u e , as a t r e n d , f o r any value of e l a s t i c  deformation,  as i l l u s t r a t e d f u r t h e r i n F i g u r e s 2 9 - i ( i = 1,6).  Each o f these  f i g u r e s shows creep curves f o r boards w i t h e q u i v a l e n t v a l u e s o f e l a s t i c deformation. selected:  36 mm,  S i x v a l u e s o f e l a s t i c deformations were  38 mm,  41 mm,  42 mm,  43 mm,  and 45 mm.  These  are the v a l u e s encountered both f o r 3000 p s i and 4500 p s i . These f i g u r e s show t h a t among boards w i t h e q u i v a l e n t e l a s t i c de-  36 formation and loaded to d i f f e r e n t s t r e s s l e v e l s , the ones loaded to the higher s t r e s s l e v e l appear to creep l e s s than the o t h e r s . T h i s i s p o s s i b l e , because s e v e r a l boards loaded to d i f f e r e n t s t r e s s l e v e l s can reach the same e l a s t i c deformation have d i f f e r e n t m a t e r i a l c h a r a c t e r i s t i c s  (grade e f f e c t ) and  f o r e have d i f f e r e n t moduli of e l a s t i c i t y and istics.  The e l a s t i c behaviour,  i f they there-  strength character-  as w e l l as the creep  behaviour  depend upon these variables.; c)  Among s e v e r a l boards w i t h e q u i v a l e n t e l a s t i c  deformation  and loaded to the same s t r e s s l e v e l , creep i s more pronounced i n some than others approximately  (sometimes twice as much).  These boards have  the same modulus o f e l a s t i c i t y , but they have a  d i f f e r e n t creep behaviour.  T h i s suggests  t h a t the creep d e v e l -  opment depends upon s t r e n g t h o r , e q u i v a l e n t l y , s t r e s s r a t i o .  The  estimated v a l u e s of s t r e s s r a t i o are a v a i l a b l e only f o r boards t h a t were loaded to 3000 p s i .  I emphasize, once again, t h a t f o r  t h i s reason, s t r e s s r a t i o and s t r e n g t h are e q u i v a l e n t .  This  would not be t r u e i f s t r e s s r a t i o s were computed from s e v e r a l values of a p p l i e d s t r e s s . The estimated short-term s t r e n g t h s of boards,  used i n  the computation o f the s t r e s s r a t i o s , are indeed, t h e i r strength.  These v a l u e s were o b t a i n e d by t e s t i n g the  a f t e r they were unloaded  i n Richmond Laboratory,  residual  boards,  i n a four points  l o a d i n g c o n f i g u r a t i o n , a t a constant r a t e of l o a d i n g of 5000 psi/mn  (34.5 MPa/mn) u n t i l f a i l u r e o c c u r r e d .  They are a  good e s t i m a t i o n of short-term s t r e n g t h s because the r e d u c t i o n i n s t r e n g t h over a p e r i o d of three months due  to constant l o a d -  i n g , f o r the s t r e s s l e v e l c o n s i d e r e d , i s not very  important  37 ( B a r r e t t and F o s c h i 1978).  For boards  t h a t broke w h i l e they were  loaded i n Richmond L a b o r a t o r y , the s t r e s s r a t i o had t o be timated otherwise.  es-  I w i l l not go i n t o the d e t a i l s o f these  c a l c u l a t i o n s ; these s t r e s s r a t i o s are g r e a t e r than 75% and i n t h i s a n a l y s i s there i s no need to know these v a l u e s more s p e c i fically. d)  Table X shows a v a i l a b l e v a l u e s o f s t r e s s F i g u r e s 2 4 - i ( i = 1,2)  ratios.  i n d i c a t e t h a t a few boards  ex-  h i b i t a creep curve t h a t does not f o l l o w the g e n e r a l t r e n d . These boards and 411  are 2811,  1507,2509 f o r 3000 p s i a p p l i e d  f o r 4500 p s i a p p l i e d s t r e s s .  stress,  These creep curves have  sudden changes i n s l o p e , and these changes are probably due l o c a l f a i l u r e s o c c u r r i n g i n the beams.  These boards  are r e -  presented on v a r i o u s p l o t s by i s o l a t e d p o i n t s ; they do not us anything, but c r e a t e problems i n p r e s e n t a t i o n o f graphs. these reasons, these boards  3.5.3  to  tell For  are d i s r e g a r d e d i n f u r t h e r a n a l y s e s .  Dependence of Creep Deformation  on the S t r e s s L e v e l ,  the  S t r e s s R a t i o and the Modulus o f E l a s t i c i t y In Chapter  1.4  I showed t h a t a f u n c t i o n a l e x p r e s s i o n f o r  creep deformation i s : 6-  = fit,  where the v a l u e o f x x = L/2,  c r ( t , x ) , MOE,  SR}  (space parameter) can be  s i n c e measurements o f deformation were r e c o r d e d a t the  c e n t e r o f the span o f i n d i v i d u a l boards. t i o n e d e a r l i e r , there was lifetime.  set to a constant,  The  as men-  no l o a d v a r i a t i o n d u r i n g the  l e v e l of a p p l i e d s t r e s s , a, i s , thus,  o f time and one form o f creep  Furthermore,  independent  can w r i t e the f i n a l e x p r e s s i o n f o r the  deformation:  experiment  functional  38 6c  The purpose  = f { t , a, MOE,  SR}  of the f o l l o w i n g study i s to e s t a b l i s h whether or  not dependence e x i s t s between creep deformation and the v a r i o u s parameters,  a t a given time t * .  The study of such a dependency  f o r timber beams i s complicated by the interdependence three parameters.  T h i s i s c l e a r l y i l l u s t r a t e d i n F i g u r e 28  where a l l three parameters each p l o t .  o f the  c o n t r i b u t e to the s c a t t e r w i t h i n  T h i s i s f u r t h e r i l l u s t r a t e d i n F i g u r e 30 where  f r a c t i o n a l creep, d e f i n e d as the r a t i o of creep deformation to e l a s t i c deformation, i s p l o t t e d a g a i n s t e l a s t i c  deformation.  F r a c t i o n a l creep i s expressed a t ten d i f f e r e n t times r a n g i n g from f i v e minutes t o t h i r t e e n weeks.  As i n F i g u r e 28, the e f f e c t  o f a p p l i e d s t r e s s i s c l e a r l y shown s i n c e data p o i n t s f o r the s t r e s s l e v e l s occupy d i f f e r e n t r e g i o n s w i t h i n each p l o t . cannot  say, though, whether the l e v e l of a p p l i e d s t r e s s  two  One influ-  ences both f r a c t i o n a l creep and e l a s t i c deformation or e l a s t i c deformation a l o n e .  The e f f e c t s of s t r e s s r a t i o and modulus of  e l a s t i c i t y t o g e t h e r c o n t r i b u t e t o the s c a t t e r w i t h i n each s e t o f data p o i n t s and t h e r e f o r e cannot be i s o l a t e d from one  another.  F i g u r e 31 shows graphs of f r a c t i o n a l creep p l o t t e d a g a i n s t the modulus of e l a s t i c i t y .  Once again, these  graphs  would not e x i s t i f wood were an homogeneous m a t e r i a l w i t h s t a n t value of the modulus of e l a s t i c i t y . expressed a t nine d i f f e r e n t times. a p p l i e d s t r e s s i s demonstrable the p r e v i o u s graphs.  con-  F r a c t i o n a l creep i s  The e f f e c t of the l e v e l of  to a much l e s s e r extent than i n  T h i s i s not s u r p r i s i n g i n t h a t the a b s c i s -  sa of each p o i n t (the modulus, of e l a s t i c i t y ) does not depend upon the l e v e l of a p p l i e d s t r e s s , whereas a b s c i s s a e on p r e v i o u s  39 graphs d i d , s i n c e they were e l a s t i c deformations.  Figure  31  shows t h a t , even though c r o s s e s and c i r c l e s o v e r l a p , data f o r 4500 p s i might have, as a trend, a h i g h e r o r d i n a t e those  f o r 30 00 p s i ; but t h i s phenomenon i s not  pronounced to enable a f u r t h e r c o n c l u s i o n .  points  than  sufficiently  Further i n v e s t i g a -  t i o n i s r e q u i r e d to e s t a b l i s h whether or not f r a c t i o n a l r e a l l y depends upon the l e v e l of a p p l i e d s t r e s s .  creep  Test specimens  should be s u b j e c t e d to a wider range of s t r e s s l e v e l s . On the o t h e r hand, F i g u r e 31 shows t h a t  fractional  creep increases, with d e c r e a s i n g value o f the modulus of c i t y and  the r a t e of i n c r e a s e i n c r e a s e s over time.  The s c a t t e r ,  however, i s such t h a t i t i s not p o s s i b l e , a t t h i s stage, determine whether, the i n c r e a s e i s l i n e a r or not.  elasti-  to  I t should  noted t h a t the t h i r d parameter, the s t r e s s r a t i o , i s s t i l l sent i n these p l o t s and c o n t r i b u t e s to the s c a t t e r . of determining  One  i t s e f f e c t would be to redraw each one  of  graphs f o r s p e c i a l bands of s t r e s s r a t i o s .  I considered  bands:  - 60%,  60%  0% - 30%,  - 100%.  One  30%  - 40%,  40%  - 50%,  50%  be pre-  method the five  and  might expect t h a t f r a c t i o n a l creep  increases  with i n c r e a s i n g s t r e s s r a t i o , y i e l d i n g , i e . at three months, a graph as i l l u s t r a t e d i n the f o l l o w i n g f i g u r e :  Fractional  creep 60%  - 100%  50% 40% 30% -  60% 50% 40%  0% -  30%  M.O.E.  Where l i n e s r e p r e s e n t  trends  i n the data and are not necessar-  ily straight. F i g u r e 3 2 - i ( i = 1,10) shows graphs o f f r a c t i o n a l  creep,  determined a t a p a r t i c u l a r time, p l o t t e d a g a i n s t the modulus of e l a s t i c i t y f o r the f i v e ranges o f s t r e s s r a t i o s . between f i v e minutes and t h i r t e e n weeks. the expected t r e n d s .  These graphs  support  T h i s i s b e t t e r seen i f one determines,  say a t time t = 3 months, the averages o f c o o r d i n a t e s p o i n t s from the graphs o f F i g u r e 3 2 - i . presented  Time v a r i e s  o f data  One o b t a i n s the r e s u l t s  i n Table XI f o r the f i v e ranges o f s t r e s s r a t i o s .  The  i n c r e a s e i n f r a c t i o n a l creep i s s i g n i f i c a n t f o r high values o f stress ratios strength  (greater than 60%), t h a t i s f o r m a t e r i a l with low  (lower than 5000 p s i ) .  F o r the other ranges of s t r e s s  r a t i o s , the i n c r e a s e i s r a t h e r s m a l l and does not a l l o w any conclusion. with  There i s a need t o i n v e s t i g a t e t h i s matter f u r t h e r  l a r g e r samples. I t was commonly seen t h a t low s t r e n g t h m a t e r i a l  i n the t e n s i o n zone, u s u a l l y a s s o c i a t e d w i t h  local grain dis-  t o r t i o n , while high s t r e n g t h m a t e r i a l had an i n h e r e n t l y t e n s i o n zone c a u s i n g  failed  strong  f a i l u r e to i n i t i a t e i n the compression zone.  I t i s p o s s i b l e t h a t t h i s d i f f e r e n c e i n f a i l u r e mode i s the cause of the observed phenomena f o r l o n g term deformations. As a summary, i t can be s a i d t h a t the development o f creep,  f o r sawn timber o f s t r u c t u r a l s i z e , i s i n f l u e n c e d by the  value o f the modulus o f e l a s t i c i t y . more the m a t e r i a l creeps,  The lower t h i s v a l u e , the  and the r a t e o f i n c r e a s e i n c r e a s e s over  time. Furthermore, the experimental  r e s u l t s i n d i c a t e t h a t creep  41  development i s more important f o r low s t r e n g t h  material  (that  i s , m a t e r i a l w i t h high s t r e s s r a t i o s ) , and a t h i g h e r l e v e l s o f applied (low)  stress.  I t should be r e c a l l e d , a t t h i s p o i n t ,  s t r e s s l e v e l s only were i n v e s t i g a t e d and f u r t h e r  i n v o l v i n g a wider range o f a p p l i e d  s t r e s s i s needed.  t h a t two study  42  CHAPTER 4  LINEAR VISCOELASTIC MODEL  4.1  Introduction M a t e r i a l s t h a t d i s p l a y creep, t h a t i s , an i n c r e a s i n g  deformation under s u s t a i n e d  load a r e c a l l e d v i s c o e l a s t i c  m a t e r i a l s . The c o n s t i t u t i v e equations o f these m a t e r i a l s ,  that  d e s c r i b e the r e l a t i o n between s t r e s s and s t r a i n , may be e i t h e r l i n e a r or n o n - l i n e a r . Although i t i s known t h a t b u i l d i n g m a t e r i a l s do not behave l i n e a r l y , o r do so w i t h i n l i m i t s too narrow t o cover the range of p r a c t i c a l i n t e r e s t , t h i s study attempts .to demonstrate t h a t l i n e a r i t y , because o f i t s s i m p l i c i t y , can c h a r a c t e r i z e the behaviour o f the m a t e r i a l reasonable l i m i t s , provided The  within  some assumptions are made.  behaviour of v i s c o e l a s t i c m a t e r i a l s  i n uniaxial  s t r e s s c l o s e l y resembles t h a t o f m a t e r i a l s b u i l t from d i s c r e t e e l a s t i c and v i s c o u s elements  (Fliigge 1967) . In the f o l l o w i n g ,  one  i t w i l l be assumed t h a t wood, under  case w i l l be considered;  sustained  l o a d , reaches a f i n a l value of deformation.  43  4.2 Three Parameter S o l i d The model i s r e p r e s e n t e d as shown i n the f o l l o w i n g f i g u r e , where a s p r i n g and a K e l v i n element are connected i n series:  K e l v i n element  Spring  E, E.  where: E^and E^ are moduli of e l a s t i c i t y ;  y  1  S  t  n  e  2  coefficient  of p r o p o r t i o n a l i t y between s t r e s s and s t r a i n r a t e ;  e' and e" are  s t r a i n s i n the s p r i n g and the K e l v i n element r e s p e c t i v e l y when a i s the a p p l i e d  s t r e s s . For t h i s model, one o b t a i n s the f o l l o w -  i n g s e t of equations: a  =  E e* 1  a = E  2 E  »  + y  2  B "  e = e' + e  y i e l d i n g the d i f f e r e n t i a l equation between <5 and e: J2°_  E  V  \  =  E  1 +  E  2  E  1 +  V2*  £  1 2 E E  2  E  1 +  E  2  44  In  i t s normalized a  where p^, q , n  this  derive  +  = q e  p 5  >  x  equation  a constant  the d i f f e r e n t i a l  equation  q  elastic  Fractional  c  creep,  f o r e ( t ) ; one  l  t  )  sol-  obtains:  2  parts:  )  o f e ( t ) a t t = 0;  as  follows:  -E t / (  1  "  e  2  " >  f ( t ) , i s expressed  E f (t) = _ i ( l 2 E  t = 0 and  2  i s the value  if  =  v  2  1-e E  c a n be w r i t t e n  (  a t time  o f t h e component  a £  n  parts.  0  deformation  Creep deformation  a. = a  real,  1  > E  The  o f i t scomponent  to  -%/q  i n terms o f the constants  e ( t  i t i s not possible  from the model w i t h  stress  a -H- + A e  e (t) =  or,  that:  of the constants  Applying  i s written:  P^O  the d i f f e r e n t i a l values  equation  t q ^  Q  1  inequality i s not satisfied,  positive  ving,  the d i f f e r e n t i a l  and q^ a r e such q  If  form,  -E_t/ 2 2  e  as  y  )  follows:  that i s :  45  4.3  S t r e s s Parameter The  expressions of  p a r t i c u l a r time t , the a t i o n , expressed as of a p p l i e d  tested  c  t o t a l deformation and  s t r a i n s , are  possible  at  l i n e a r l y dependent upon the  t e s t r e s u l t s r e f e r to two  graphs such as deformation p l o t t e d  s t r e s s cannot be  o b t a i n e d . The  not  test results yielded  =  f{6 )  f(t*) =  g{6 }  f(t*) =  h{E }  c  against the  e  e  ±  at v a r i o u s times t * . The re-written  expressions f o r e  by  inserting  above equations,  yielding:  = E^e  E  e  (t) = e  e  E  f ( t ) = ± -£ e 2 e  E  or E^ = ° / e  e  0  l  — (  -E,t, 2  1 - e  1  /v  / y  2  )  2  ( i -  2 e  (t) and  )  e  to  stress  graphs: 6 (t*)  a  creep deform-  m a t e r i a l behaves l i n e a r l y i n creep w i t h r e s p e c t  l e v e l s o n l y and  be  the  to d i r e c t l y v e r i f y whether or  s t r e s s . T h i s i s because the  following  f ( t ) show t h a t ,  stress.  I t i s not the  e ( t ) , e ( t ) , and  f(t)  can  into  the  level  46  f (t) = -=± 2  ( 1 - e  / y  Z  2  )  E  In the  following section, I w i l l  sions,  a p p l i e d t o a non-homogenous m a t e r i a l , y i e l d  similar  to the It  refer  test  should  results  similar a)  s t r e s s e d that these  they  trends  Creep deformation t*, e  a given  stress  coordinates straight  (e  line  proportional  c  )-  going to E^  One  can  en  and  time  value  of  y  g  l F  (E ,y -t*)  a^,  2  t h e r e i s o n l y one  the o r i g i n  a t time  2  = e E f(E ,  were c o n s t a n t e  E  2  E  e l  =  2  2  and  l o c a t e d on  e  y  2  a  c  slope i s the  as  are probably  i n the  wood, not  same d i r e c t i o n  a t a g i v e n time  as  t*:  , t*)  constant,  following equation °0  with  a m a t e r i a l such  vary  y  point  of p r o p o r t i o n a l i t y  , F w o u l d a l s o be  w o u l d a l s o be  c  s i n c e the £  1  deformation.  of the a x i s , with  t * ). F o r  t h e y may  e  follow  as f o l l o w s :  Furthermore, p o i n t s are  in fact,  would  of  2  r e w r i t e the expression of  t * and c  E  expressed  ( the c o e f f i c i e n t  c  2  £ e  test  i n t h a t graphs  elastic  be  through  e (t*) If E  important  versus  i s a random v a r i a b l e ;  constant values; E^.  e  '  =  level  of F evaluated  however, E ^  expressions  deflection.  plotted  can  c  c  value  This i s not  as g r a p h s o f  e (t*) For  theoretical  been d e r i v e d from e x p e r i m e n t s ,  At a g i v e n time  graphs  whereas t h e q u a n t i t i e s d e r i v e d f r o m t h e  are d e f l e c t i o n s .  strains,had  expres-  results.  be  to s t r a i n s ,  demonstrate t h a t these  constant  at a  regardless of always h o l d s  the  true:  giv-  47  t * i s now a constant and the f u n c t i o n F depends upon the constants of the model's component p a r t s o n l y . For an homogenous m a t e r i a l , E^ i s a constant and so are E versus e  2  and y^-  A  graph of e  c  plotted  then has the f o l l o w i n g c o n f i g u r a t i o n :  f o r a given l e v e l of a p p l i e d s t r e s s . F, however, i s not constant and one may expect t h a t i t i n c r e a s e s w i t h i n c r e a s i n g value of e  g  (that i s w i t h d e c r e a s i n g value of E ^ ) . A graph of e  ing c o n f i g u r a t i o n : e  plotted against e  may then have the f o l l o w -  48  The  l i n e s t h a t a r e shown a r e t r e n d s .  T h i s i s the c o n f i g u r a t i o n  of graphs 6. = f { 6 } f o r the t e s t e d m a t e r i a l . I t i s &  w i t h the hypothesis o f l i n e a r i t y ,  since points  compatible  and  ,  shown on the graph, f o r a specimen i , can be on a s t r a i g h t l i n e going through the o r i g i n . I t i s important to note t h a t the p r e s e n t a n a l y s i s does not demonstrate t h a t wood i s a l i n e a r v i s c o e l a s t i c m a t e r i a l . T h i s could be done by t e s t i n g a specimen i a t two d i f f e r e n t s t r e s s l e v e l s and v e r i f y i n g t h a t passing  and  are on a s t r a i g h t l i n e  through the o r i g i n . T h i s a n a l y s i s does show, however,  t h a t the t e s t e d m a t e r i a l behaves i n a way t h a t i s compatible with the behaviour o f a l i n e a r v i s c o e l a s t i c m a t e r i a l sustained  under  load.  b) F r a c t i o n a l creep p l o t t e d a g a i n s t e l a s t i c At a given  deformation.  time t * , f can be expressed as f o l l o w s : 0 F(E ,y ,t*)  f(t*) = |  n  2  2  e t * i s now a c o n s t a n t and F i s the same f u n c t i o n as i n the previous has  case. F o r an homogenous m a t e r i a l , the f u n c t i o n F  a constant value;  increases  furthermore, when e  t o the same extent  increases,  g  a  n  so t h a t E = ^0 w i l l remain c o n s t a n t . e e  A graph o f f p l o t t e d a g a i n s t configuration:  e  e  then has the f o l l o w i n g  49  :  a=a  3  _ a = a  rt  V  0=0 .  2  1  s»  e  e  For a given  s t r e s s l e v e l 0 = CK , there i s o n l y one p o i n t w i t h  coordinates  e and e . The s t r e s s has t o be v a r i e d i n order t o e c  o b t a i n s e v e r a l p o i n t s . Furthermore, the p o i n t s are l o c a t e d on an h o r i z o n t a l l i n e . T h i s i s not s u r p r i s i n g because the a n a l y t i c a l e x p r e s s i o n  of f i s independent of s t r e s s . For  a m a t e r i a l such as wood, however, E^ and  i s a random v a r i a b l e  f o r a s p e c i f i c value of a p p l i e d s t r e s s , a , e Q  e  varies i n  such a way t h a t the product E-^e w i l l remain constant e  equal t o OQ. I f one assumes t h a t E one  2  and y  2  and  are c o n s t a n t s ,  can w r i t e : f (t*) = e  where K i s constant, f plotted against  e  e  f o r a s p e c i f i c value e  o f t * . A graph of  then has the c o n f i g u r a t i o n of an hyperbola,  as shown i n the f o l l o w i n g f i g u r e :  50  e  e  I f one assumes t h a t  and y  2  vary, and furthermore,  that  these values vary i n the same d i r e c t i o n as E^, the f u n c t i o n F i s no longer c o n s t a n t . F i n c r e a s e s with i n c r e a s i n g e  e  , at a  given l e v e l of a p p l i e d s t r e s s . Since I d i d not i n v e s t i g a t e t h i s r e l a t i o n s h i p i n g r e a t e r depth i t i s i m p o s s i b l e to d e f i n e the type o f v a r i a t i o n of the f u n c t i o n F. Hence, one has to allow f o r a v a r i a t i o n such t h a t a graph o f f p l o t t e d a g a i n s t e w i l l have the f o l l o w i n g c o n f i g u r a t i o n :  e  51  The  l i n e s shown i n the graph are t r e n d s ;  they might be s t r a i g h t  l i n e s as w e l l . T h i s i s the c o n f i g u r a t i o n o f graphs f = g { 6 } g  f o r the m a t e r i a l of l i n e a r i t y ,  t e s t e d . I t i s compatible w i t h the h y p o t h e s i s  since points  and  ,shown on the graph, f o r  a specimen i , can be on an h o r i z o n t a l The  a n a l y s i s , again,  line.  shows t h a t the m a t e r i a l  tested  behaves i n a way t h a t i s compatible with the behaviour o f a l i n e a r v i s c o e l a s t i c m a t e r i a l under a s u s t a i n e d  load.  c) F r a c t i o n a l creep p l o t t e d a g a i n s t modulus o f e l a s t i c i t y . At a given  time t * , f can be expressed as f o l l o w s : f (t*) = E j F f E ^ y ^ t * )  where F i s the same f u n c t i o n as d e f i n e d  before.  ous m a t e r i a l , E^ i s constant and so are y  a n c 2  F o r an homogen-  E  ^ 2*  A  t  a  9  l v e n  time t * , a graph o f f p l o t t e d a g a i n s t E^ i s then reduced t o one  point regardless  o f the a p p l i e d s t r e s s . For a m a t e r i a l  such as wood, however, E ^ i s a random v a r i a b l e . I f , i n a f i r s t stage, one assumes t h a t E  2  and y  2  are constant,  then a graph o f f p l o t t e d a g a i n s t E^ has the f o l l o w i n g configuration:  f  52  The l i n e i s s t r a i g h t , with p o s i t i v e s l o p e , and goes through the o r i g i n . I t r e p r e s e n t s a t r e n d . One can assume t h a t  and  y  2  vary, and furthermore, t h a t the v a l u e s vary i n the same d i r e c t i o n as E^. F ( t * ) then decreases with i n c r e a s i n g v a l u e of E^ and a graph of f p l o t t e d a g a i n s t E^ may  then have the f o l l o w i n g  configuration:  f  1  or,  f  depending upon the r a t e of i n c r e a s e of the f u n c t i o n F. shown are trends  and may  the graphs of F i g u r e  well describe  The  lines  the data p o i n t s shown on  31.  R e c a l l the comment made i n s e c t i o n 3.5.3 f o r 4500 p s i , i n F i g u r e  that data points  31, might have, as a t r e n d , a h i g h e r  nate than those f o r 3000 p s i .  T h i s phenomenon i s not  with the h y p o t h e s i s of l i n e a r i t y , but to r e j e c t the assumption of l i n e a r i t y .  ordi-  compatible  i s not s i g n i f i c a n t enough We  can,  therefore,  say  t h a t the behaviour of the t e s t e d m a t e r i a l i s compatible w i t h  the  behaviour of a l i n e a r v i s c o e l a s t i c m a t e r i a l , under s u s t a i n e d  load.  4.4.  Comments Traditionally,  behave, under s u s t a i n e d  linear v i s c o e l a s t i c materials  l o a d , i n a very s p e c i f i c manner t h a t  i s i n f l u e n c e d by the l e v e l of a p p l i e d s t r e s s .  The  study of such  behaviour, to date, has.not allowed f o r the v a r i a t i o n parameters of m o d e l i s a t i o n , tered.  (homogenous)  such as E^ and  Wood, however, i s not homogenous and  have a high c o e f f i c i e n t o f v a r i a t i o n . m a t e r i a l , under s u s t a i n e d  of  the  p r e v i o u s l y encouni t s p r o p e r t i e s '.  I demonstrated t h a t  this  l o a d , behaves i n a manner t h a t i s  compatible w i t h the hypothesis of l i n e a r v i s c o e l a s t i c i t y .  I  have not determined whether or not the m a t e r i a l i s l i n e a r l y viscoelastic.  Under the assumption t h a t the m a t e r i a l can  be  regarded as a l i n e a r v i s c o e l a s t i c m a t e r i a l , the f o l l o w i n g equat i o n , w r i t t e n f o r a specimen i of the m a t e r i a l a t a given t * , may  hold  true:  time  e . (t*) = E.F • e . ci 1 1 ei T h i s equation has the f o l l o w i n g g r a p h i c a l form:  The  slope o f the l i n e i s E^F^. Once the q u a n t i t y E^F^ , a t a  given time t * , i s known, i t i s p o s s i b l e to determine e^^ a l e n t l y , the v a l u e  (equiv-  6 ^ ) from a s p e c i f i c value o f e ^ ( e q u i v a i e  e n t l y , from a s p e c i f i c value o f S ^ ) • The  o b j e c t o f the f o l l o w i n g a n a l y s i s i s t o propose a meth-  od o f o b t a i n i n g these v a l u e s E.F.'s f o r the m a t e r i a l  tested.  55  CHAPTER 5  ANALYTICAL MODEL FITTED TO THE DATA  5.1  Model Graphs o f creep deformation p l o t t e d a g a i n s t e l a s t i c de-  formation, are drawn f o r two d i f f e r e n t s t r e s s l e v e l s presented at v a r i o u s times i n F i g u r e 28.  I t has been shown t h a t each o f  these graphs, drawn f o r a homogeneous m a t e r i a l s u b j e c t e d t o one l e v e l o f a p p l i e d s t r e s s , reduces t o one p o i n t .  I w i l l use data  p o i n t s on graphs from F i g u r e 2 8 f o r one s t r e s s l e v e l a t a time to e s t a b l i s h a c o r r e l a t i o n between <5 and 6 . When t h i s i s done, c e ' a value o f 6 . w i l l be o b t a i n e d from a value o f 6 . through the ci ei 3  c o r r e l a t i o n e x p r e s s i o n g i v e n a t a s p e c i f i c time. nates  (  5 e  i '  6  ci^ '  a  s  P  e  c  l  m  e  n  i'  W  1  l l  The c o o r d i -  e  t> p l o t t e d on a graph  of creep deformation versus e l a s t i c deformation and a s t r a i g h t l i n e , c h a r a c t e r i z i n g l i n e a r i t y between creep deformation and a p p l i e d s t r e s s , w i l l be drawn, w i t h s l o p e :  I f one  assumes, as d i s c u s s e d i n the p r e v i o u s chapter,  t h a t the m a t e r i a l may  be c o n s i d e r e d as a l i n e a r  viscoelastic  m a t e r i a l , then the creep behaviour of such a specimen w i l l known, a t a s p e c i f i c time, f o r other s t r e s s l e v e l s . r e l a t i o n between <5  and  6  The  be  cor-  has the equation of a p a r a b o l a ,  such as: 5 ( t ) .= A ( t , a )  6  c  The parabola goes through  E  + B(t,a ) 8  2  the o r i g i n ; the c o e f f i c i e n t s A  B depend upon the s t r e s s l e v e l as w e l l as time. F i g u r e 2 8 then y i e l d s two  and  Each graph  from  p a r a b o l a s , one per s t r e s s l e v e l .  For  one s t r e s s l e v e l , say 3000 p s i , one o b t a i n s : <5 (t) = A(t) c  6  + B(t)  E  6  2 E  At a s p e c i f i c time t * , A and B are constant v a l u e s , y i e l d i n g : 8 = A c The  6  e  + B  6  2  e  c o e f f i c i e n t s A and B have been determined  twenty times -  that i s twice f o r each of the ten times graphs are drawn - by the l e a s t squares method.  Table XII shows the v a l u e s of A and  The parabolas are p l o t t e d i n F i g u r e s 33 and 4500 p s i , r e s p e c t i v e l y . e q u a t i o n by  B.  34 f o r 3000 p s i and  D i v i d i n g the two members of the above  6 , one o b t a i n s . E  f = A + B 8 e which: i s the e q u a t i o n of a s t r a i g h t l i n e .  The  l i n e s obtained at  v a r i o u s times are drawn on the graphs of f r a c t i o n a l creep p l o t t e d a g a i n s t e l a s t i c deformation, presented i n F i g u r e s 35 and 3000 p s i and 4500 p s i , r e s p e c t i v e l y . can be expressed,  36 f o r  In the above equation, 8^  f o r a g i v e n s t r e s s l e v e l . i n terms of the  modulus of e l a s t i c i t y as f o l l o w s :  For a t y p i c a l 2 i n x 6 i n  s e c t i o n and a span o f 138 i n , one  can w r i t e : o  e  —  5.619 E  and . e  6  _ 8.42 8 E ~  f o r 3000 p s i and 4500 p s i , r e s p e c t i v e l y . the u n i t s o f E are p s i x 10  -7  In these e x p r e s s i o n s  . . and those o f <5 are m i l l i m e t r e s . e  F r a c t i o n a l creep then becomes:  f =A  5 +  -g  1 9 B  E  and f = A +  8  '  4  f  B  f o r 3000 p s i and 4500 p s i , r e s p e c t i v e l y . equations o f h y p e r b o l a s .  These e x p r e s s i o n s are  These are drawn on graphs o f f r a c t i o n a l  creep p l o t t e d a g a i n s t the modulus o f e l a s t i c i t y , , presented i n F i g u r e s 37 and 38 f o r 3000, p s i and 4500 p s i r e s p e c t i v e l y .  5.2  Comments The wide spread s c a t t e r of data p o i n t s i n the graphs o f  F i g u r e 28 makes i t d i f f i c u l t t o s e l e c t a c o r r e l a t i n g between creep deformation and e l a s t i c deformation.  function The assump-  t i o n t h a t creep deformation and e l a s t i c deformation together have zero value has y i e l d e d the s e l e c t i o n o f a p a r a b o l a going the o r i g i n .  through  T h i s curve has a simple mathematical f o r m u l a t i o n and  may w e l l r e p r e s e n t the behaviour o f a t h e o r e t i c a l l i n e a r v i s c o -  58 e l a s t i c m a t e r i a l , as d i s c u s s e d i n s e c t i o n 4.3.a.  I t leads t o a  l i n e a r r e l a t i o n s h i p between f r a c t i o n a l creep and e l a s t i c  deforma-  t i o n t h a t may again, r e p r e s e n t the behaviour o f a t h e o r e t i c a l l i n e a r v i s c o e l a s t i c m a t e r i a l , as d i s c u s s e d i n s e c t i o n 4.3.b. However, a problem  a r i s e s when f r a c t i o n a l creep i s ex-  pressed i n terms o f the modulus o f e l a s t i c i t y . has the a n a l y t i c a l form o f an h y p e r b o l a .  This expression  T h i s curve, although  i t has a shape t h a t i s compatible with the assumption  of linear  v i s c o e l a s t i c i t y , as d i s c u s s e d i n s e c t i o n 4.3.c, should be, a c c o r d i n g t o t h i s assumption, stress.  independent  o f the l e v e l o f a p p l i e d  F i g u r e s 37 and 38 show t h a t the hyperbolas, f i t t e d t o  the data p o i n t s , are not independent  o f the l e v e l o f a p p l i e d  s t r e s s s i n c e they are not i d e n t i c a l , f o r the two d i f f e r e n t levels.  stress  The a n a l y s i s , a t the present stage, has not demonstrated  whether or not t h i s dependency i s s i g n i f i c a n t .  T h i s item w i l l be  d e a l t w i t h i n s e c t i o n 5.3. 1  Note t h a t the c o e f f i c i e n t s A s of the c o r r e l a t i o n equat i o n s between e l a s t i c and creep deformations have n e g a t i v e values f o r 4500 p s i , as shown i n Table X I I .  T h i s has no p h y s i c a l mean-  i n g and i s due t o t h e . l a c k o f data points, i n the r e g i o n s o f low S 's e  as seen i n F i g u r e 28.  5.3  Quantitative Analysis  5.3.1  Method Data f o r creep deformation and f r a c t i o n a l creep i n t u r n  were t r e a t e d by the UBC BMD02R computer program adapted U.CLA-BMD documentation.  from  T h i s program.computes a sequence o f  m u l t i p l e l i n e a r r e g r e s s i o n equations i n a stepwise manner.  At  each step, one v a r i a b l e i s added t o the r e g r e s s i o n e q u a t i o n . The v a r i a b l e added i s the one which makes the g r e a t e s t r e d u c t i o n i n the e r r o r sum or squares.  E q u i v a l e n t l y , i t i s the v a r i a b l e ,  t h a t , i f i t were added,.would have the h i g h e s t F-value.  In  a d d i t i o n , v a r i a b l e s can be f o r c e d i n t o the r e g r e s s i o n e q u a t i o n . Non-forced v a r i a b l e s are a u t o m a t i c a l l y removed when t h e i r values become too low.  One may  F-  a n a l y s e , i n t u r n , creep deforma-  t i o n and f r a c t i o n a l creep, both expressed a t three months.  5.3.2  A n a l y s i s o f Creep Deformation  a t Three Months  The o r i g i n a l v a r i a b l e s t h a t are fed i n t o the computer are creep deformation  (measured a t three months), l e v e l o f a p p l i e d  s t r e s s , s t r e s s r a t i o , modulus of e l a s t i c i t y , and e l a s t i c deformation.  Three computer runs are made; i n the f i r s t run, data  p o i n t s f o r 3000 p s i only are considered; i n the other two both s t r e s s l e v e l s are c o n s i d e r e d .  a)  First  run  The o r i g i n a l v a r i a b l e s a r e : XI = 5-  (creep  X2 = SR  (stress r a t i o )  X3. = MOE  (Modulus of E l a s t i c i t y )  X4 = 6  (elastic  g  deformation)  deformation)  The v a r i a b l e s , added by t r a n s g e n e r a t i o n , a r e : X5 = X2  X3  X6  X 2  X4  X7 = X3  X4  =  runs,  60 . X8 = X9  X4  =  X 2 X 4  XlO = X3 The  2  X4  2  2  c o r r e l a t i o n equation, i n v e s t i g a t e d , i s : 6  c  = A '  1  + A (SR) + A (MOE) + A ( S ) 2  4  3  + Ac(SR)(MOE) + A ( S R ) ( 6 ) _> o e C  + A (MOE) (<5 ) + 7  + A (SR) ( 6 ) 9  where A^  ( i = 1,10)  A (6 )  e  2  g  2  e  + A (MOE) ( 6 )  e  1Q  are constant v a l u e s .  2  e  T h i s e q u a t i o n has been  s e l e c t e d because i t i n c l u d e s the simple l i n e a r r e g r e s s i o n com2 ponents, model.  as w e l l as terms i n ( S e )  ,  i n t r o d u c e d i n the  analytical  63 o b s e r v a t i o n s , out of 76 f o r 3000 p s i , are c o n s i d e r e d  because boards  t h a t have f a i l e d as w e l l as boards whose behaviour  has been e x p l a i n e d otherwise  ( s e c t i o n 3.5.2.d) are d i s r e g a r d e d .  The r e s u l t s are presented i n Appendix A - l .  Two  steps are  con-  sidered: Step 1:  V a r i a b l e X8 i s e n t e r e d .  The r e g r e s s i o n e q u a t i o n  and  the c o r r e l a t i o n c o e f f i c i e n t a r e : 6  c  =.-0.023  +  0.016  <5  2  e  R = 0.7917 Step 2:  V a r i a b l e XlO i s e n t e r e d .  The  r e g r e s s i o n equation  and the c o r r e l a t i o n c o e f f i c i e n t a r e : 6  = 22.105 + 0.038 (5 )  C  2  - 0.250(MOE)(6 )  6  2  6  R = 0.8030 where 6; and c  The 1.  6  e  are expressed  i n mm,  MOE  i n p s i x 10  f o l l o w i n g i n t e r p r e t a t i o n s can be made:  Step 2 can be i g n o r e d because the F-value a t t a c h e d t o XlO,  3.05 29, i s lower than the l i m i t  (approximately 4 f o r a sample  with 63 o b s e r v a t i o n s ) t o ensure the 95% confidence  interval.  Furthermore, the f o l l o w i n g f i g u r e shows t h a t the i n c r e a s e i n c o r r e l a t i o n c o e f f i c i e n t , from the f i r s t step t o the second, i s negligible.  R  2.  The c o r r e l a t i o n m a t r i x shows t h a t creep deformation i s  weakly dependent  on the s t r e s s r a t i o  ( v a r i a b l e X2) s i n c e the  c o r r e l a t i o n , c o e f f i c i e n t between v a r i a b l e s XI and X2 i s s m a l l (0.4019). 3.  The r e g r e s s i o n e q u a t i o n (from Step 1)resembles the equation  obtained i n s e c t i o n 5.1 f o r 3000 p s i .  T h i s i s not s u r p r i s i n g  because i n both cases computations i n v o l v e the same (least squares).  b)  Second run  The o r i g i n a l v a r i a b l e s a r e : XI = 5 X2 =  (creep deformation) (applied stress)  a  X3 = MOE  (modulus o f e l a s t i c i t y )  X4 = <$  ( e l a s t i c deformation)  e  The v a r i a b l e s , added by the t r a n s g e n e r a t i o n a r e : X5 = X2 X3  criterion  where S  and 6" are expressed i n mm, e  a i s p s i and  7  MOE i n p s i x 1 0 ~ . The f o l l o w i n g i n t e r p r e t a t i o n s can be made: 1.  The three v a r i a b l e s entered, are s i g n i f i c a n t i n the 95%  confidence i n t e r v a l s i n c e t h e i r F-value i s h i g h e r than the l i m i t required 2.  (4 a p p r o x i m a t e l y ) .  The f o l l o w i n g  efficient  f i g u r e shows the i n c r e a s e  i n c o r r e l a t i o n co-  from step t o s t e p :  0.8102  395  0.8307  Step  T"  2  I t i n d i c a t e s t h a t the c o r r e l a t i o n i s not s i g n i f i c a n t l y improved by c o n s i d e r i n g  the a p p l i e d s t r e s s and the modulus of e l a s t i c i t y ,  i n a d d i t i o n t o the e l a s t i c deformation.  The d i f f e r e n c e between  the equation given a t Step 1 and the one given a t Step 2 i s i l l u s t r a t e d i n the f o l l o w i n g  graph:  Step 1  Step 2  X6 = X2 X4 X7 = X3 X4 X8 = X 4  The  2  X9 = X2 X4  2  XlO = X3 X 4  2  c o r r e l a t i o n equation, i n v e s t i g a t e d , i s : 6  c  = A  ±  + A ( a ) + A (MOE) + A ^ i S ^ 2  3  +A (a)(MOE) + A ( a ) ( 6 ) 5  6  + A (6 ) 8  e  2  e  + A (a) (6 ) g  e  where A^ ( i = 1,10) are constant v a l u e s .  2  +A (MOE)(S ) ?  e  + A (MOE) 1Q  (S )  2  Q  91 o b s e r v a t i o n s out o f  105 f o r both 3000 p s i and 4500 p s i s t r e s s l e v e l s , are c o n s i d e r e d . As b e f o r e , boards t h a t have f a i l e d as w e l l as boards whose behaviour has been d e s c r i b e d otherwise  ( s e c t i o n 3.5.2.d) are d i s -  regarded.  i n Appendix A-2.  The r e s u l t s are presented  Three  steps are c o n s i d e r e d : Step 1:  V a r i a b l e X4 i s e n t e r e d . and  The r e g r e s s i o n e q u a t i o n  correlation c o e f f i c i e n t are: 5„ = -9.909 + 0.805 6 c e R = 0.8012  Step 2:  V a r i a b l e X2 i s e n t e r e d .  The r e g r e s s i o n e q u a t i o n  and c o r r e l a t i o n c o e f f i c i e n t a r e : <5 = -6.669 + 1.058 6 - 0.00339 a c e R = 0.8307 where &  c  Step 3:  and &  Q  are expressed  V a r i a b l e X3 i s e n t e r e d .  i n mm,  a inpsi.  The r e g r e s s i o n equation  and c o r r e l a t i o n c o e f f i c i e n t a r e : 6  c  = 37.841 + 1.954 <S  R = 0.8395  - 0.01185 a + 152.324 MOE  If  (as was done i n s e c t i o n 5.1) a r e g r e s s i o n  equation i s de-  r i v e d f o r one s t r e s s l e v e l a t a time, then a graph o f creep deformation p l o t t e d a g a i n s t  e l a s t i c deformation i n d i c a t e s c l e a r -  l y an e f f e c t o f the l e v e l o f a p p l i e d s t r e s s  (Step 2 ) .  The p r e -  sent study shows t h a t t h i s e f f e c t i s i n s i g n i f i c a n t and a r e gression  equation d e r i v e d  same, time gives 3.  once f o r both s t r e s s l e v e l s a t the  as good a c o r r e l a t i o n (Step 1 ) .  The c o r r e l a t i o n m a t r i x shows t h a t the v a r i a b l e X4 and X8  have the same c o r r e l a t i o n c o e f f i c i e n t (0.8102). X8 c o u l d have been entered i n Step 1.  In the next run v a r i a b l e  X8 w i l l be a f o r c e d v a r i a b l e and w i l l be entered  c) The  1  E i t h e r X4 o r  first.  T h i r d run  same o r i g i n a l v a r i a b l e s as w e l l as those added by t r a n s -  g e n e r a t i o n as i n the second run are c o n s i d e r e d . c o r r e l a t i o n equation i s i n v e s t i g a t e d . sented" i n Appendix A-3. Step 1:  Variable and  The same  The r e s u l t s are pre-  Two steps are c o n s i d e r e d :  X8 i s e n t e r e d .  The r e g r e s s i o n  equation  correlation c o e f f i c i e n t s are: 6 = 3.799 + 0.01125 S c e  2  R = 0.8102 where both.6  and 5 G  Step 2:  Variable and  are expressed i n mm.  C  X 9 i s entered.  The r e g r e s s i o n  equation  correlation coefficients are: 5 • = -0.0405 + 0.0239 S c e  2  - 2.5 x 1 0  _ 6  a 6 e  2  R = 0.8347 where &  c  The  and 6  e  are expressed i n mm,  a in psi.  f o l l o w i n g i n t e r p r e t a t i o n s can be made:  1.  Both X8 and X9 are s i g n i f i c a n t v a r i a b l e s i n the 95% con-  fidence  interval.  2 2. ft n o n - l i n e a r model ( t h i r d run) i n 6^ does not improve the c o r r e l a t i o n obtained w i t h a l i n e a r model (second run) i n 6 . e  3.  We can compare r e s u l t s o b t a i n e d i n the f i r s t and t h i r d  runs.  We have: Third run, f i r s t step: S„ = 3.79 + 0.0112 S c e R = 0.8102  (1)  T h i r d run, second s t e p : 2  6  2  6„ = -0.405 + 0.0239 6 - 2.5 x 1 0 ~ a 6 c e e  (2)  R = 0.8347 F i r s t run, f i r s t 6  c  step:  = 0.023 + 0.01617 <5 e  2  (3)  R = 0.7917 Expressions Equation tion  (2).  (2) and (3) are almost  i d e n t i c a l f o r a = 3000 p s i .  (1) has a higher i n t e r c e p t and a lower slope than equaT h i s i s the same i n t e r p r e t a t i o n as the second one i n  the second run, but i t a p p l i e s now t o parabolas straight lines.  instead of  I t i s r e p r e s e n t e d i n the f o l l o w i n g f i g u r e :  Step 2 4500 p s i Step 1  The  r e g r e s s i o n equation,  sented by two  p l o t t e d at the second s t e p , i s r e p r e -  parabolas,  one  per s t r e s s l e v e l .  s e n t a t i o n c l e a r l y i n d i c a t e s t h a t a graph o f &  c  <$  e  Such a r e p r e plotted  i s dependent upon the l e v e l of a p p l i e d s t r e s s .  study i n d i c a t e s t h a t o n l y one of data p o i n t s together,  parabola  against  The  present  f i t t e d t o the two  sets  g i v e s as good a c o r r e l a t i o n (Step  1).  This i s otherwise demonstrated i n a graph where the c o r r e l a t i o n c o e f f i c i e n t i s p l o t t e d a g a i n s t the number of steps  i n the  re-  gression analysis:  0.8102  0.8347  — I  Step  The  increase  i s very 3.  i n c o r r e l a t i o n c o e f f i c i e n t from Step 1 to Step 2  small.  Equation  (2) becomes: 6  c  = -0.405 + 0.0164 6  2  e  87 = -0.405 + 0.0127 & c_ e f o r 3000 p s i and  2  4500 p s i s t r e s s l e v e l s r e s p e c t i v e l y .  These  equations are compared w i t h the ones d e r i v e d i n s e c t i o n 5.1 the a n a l y t i c a l model: 8. = 0.000317 6 + 0.0161 8 ' c e e° 6  c  = -0.091 8  e  + 0.0147 6 2 e  There i s c l o s e agreement between the two  s e t s of  (3000 p s i ) (4500 p s i ) equations.  for  5.3.3  A n a l y s i s of F r a c t i o n a l Creep a t Three Months The  a n a l y s i s o f data f o r creep deformation  i n d i c a t e s that  both a l i n e a r model and a n o n - l i n e a r model can be chosen to c o r r e l a t e creep deformation to the former,  one  and e l a s t i c deformation.  With r e s p e c t  can w r i t e :  6 = A + B c  6 e  With r e s p e c t to the l a t t e r , one' can w r i t e : 6 = A + B c  6 e  2  or S  c  The  = A  6 + e  f i r s t e x p r e s s i o n was  f i r s t step.  The  B 6 e  2  o b t a i n e d i n s e c t i o n 5.3.2  second e x p r e s s i o n was  third  derived i n section  run, 5.1  a f t e r the s e l e c t i o n of a p a r a b o l i c model going through  the o r -  igin.  obtains  I f one d i v i d e s the three e x p r e s s i o n s by  s^, one  expressions f o r the f r a c t i o n a l creep presented below. f = § '4- B . o e 6 e  e  f = A + B 6 e The purpose of the a n a l y s i s of data on f r a c t i o n a l creep,  expres-  sed a t time t = 3 months, i s to see whether or not one model should be given p r e f e r e n c e . The o r i g i n a l v a r i a b l e s t h a t are f e d i n t o the computer are f r a c t i o n a l creep  (measured a t three months), l e v e l o f a p p l i e d  s t r e s s , s t r e s s r a t i o , modulus of e l a s t i c i t y , and e l a s t i c deformation.  a)  Two  runs are made.  First  run  68 Only one s t r e s s l e v e l i s c o n s i d e r e d - 3000 p s i .  The o r i g i n a l  variables are: XI = f  (fractional  X2 = SR  (stress  X3 = MOE  (modulus o f e l a s t i c i t y )  X4 .= 6  (elastic  e  creep)  ratio)  deformation)  The v a r i a b l e s , added by t r a n s g e n e r a t i o n , a r e : X5 = 1/X4 X5 = 1/X3 The c o r r e l a t i o n equation i s : f = -Aj + A (SR) + A (MOE) + A ( 6 ) 2  +  A  5  3  +  e  A  6  4  e  MOE  where A^ ( i = 1,6) are constant v a l u e s .  R e s u l t s are presented  i n Appendix B - l . The 1.  f o l l o w i n g comments can be made:  F r a c t i o n a l creep i s p o o r l y dependent on the s t r e s s  ratio  (low value o f the c o r r e l a t i o n c o e f f i c i e n t between XI and X2 i n the c o r r e l a t i o n m a t r i x ) . 2.  For one s t r e s s l e v e l - 3000 p s i -, a n o n - l i n e a r model i s  s e l e c t e d ; the r e g r e s s i o n equation and c o r r e l a t i o n  coefficients  are: f = 3.755 +  ^4^"  R = 0.5209 at the f i r s t step o f the r e g r e s s i o n a n a l y s i s .  In the equation  -7 f o r f , E i s expressed  i n p s i x 10  can be n e g l e c t e d because the F-value  and f m  %.  The second step  a t t a c h e d t o the v a r i a b l e  X5 i s lower than the l i m i t to ensure the 95% confidence 3.  interval.  The e x p r e s s i o n f o r f resembles the one obtained i n s e c t i o n 5.1  69 when the a n a l y t i c a l model was d e r i v e d .  R e c a l l the e x p r e s s i o n f o r  f t h a t was o b t a i n e d : f = 0.0317 +  b)  9  ,  ° E  6  7  Second run  The same v a r i a b l e s are kept, except X2 which i s now the l e v e l o f a p p l i e d s t r e s s , and two s t r e s s l e v e l s are c o n s i d e r e d : and 4500 p s i .  3000 p s i  The c o r r e l a t i o n e q u a t i o n i s : f = A  x  + A ( o ) + A (MOE) 2  +  A A  5  + A (-6 )  3  * + A 6 6 e  4  - i _  A  MOE  where A^ ( i = 1,6) are constant v a l u e s . sented i n Appendix B-2.  e  The r e s u l t s are pre-  The r e g r e s s i o n a n a l y s i s i s done i n two  steps: .Step  1:  V a r i a b l e X4 i s entered.  The r e g r e s s i o n equation  and c o r r e l a t i o n c o e f f i c i e n t a r e : f = 21.84 - 0.83 R = 0.4604 where f i s expressed Step 2:  i n % and 6  V a r i a b l e X2 i s e n t e r e d .  Q  i n mm.  The r e g r e s s i o n e q u a t i o n  and c o r r e l a t i o n c o e f f i c i e n t a r e : f = 30.95 + 1.54 <5  e  0.0096 a  R = 0.5419 where f i s expressed i n %, <s  e  The 1.  i n mm and a i n p s i .  following.comments can be made:  In the f i r s t run, as w e l l as i n the second one, the low c o r -  r e l a t i o n c o e f f i c i e n t s i n d i c a t e a wide-spread  s c a t t e r o f data  p o i n t s i n F i g u r e 30. 2.  The i n c r e a s e i n c o r r e l a t i o n c o e f f i c i e n t from Step 1 t o Step  70 2 i s not very l a r g e .  T h i s i s i l l u s t r a t e d i n the  following  figure:  0.5419  Step  This i n d i c a t e s t h a t the c o r r e l a t i o n between f r a c t i o n a l creep and e l a s t i c deformation i s s l i g h t l y improved when the parameter i s p r e s e n t i n the c o r r e l a t i o n e q u a t i o n .  stress  T h i s phenom-  enon i s not compatible w i t h the assumption of l i n e a r v i s c o elasticity.  However, s i n c e the i n c r e a s e i n c o r r e l a t i o n  e f f i c i e n t i s s m a l l , one  can n e g l e c t the phenomenon and  cocorrelate  f r a c t i o n a l creep and e l a s t i c deformation independently of the l e v e l of applied stress. If 6  g  The c o r r e l a t i o n e q u a t i o n i s l i n e a r .  i s expressed i n terms of the modulus o f e l a s t i -  c i t y , the r e l a t i o n between f r a c t i o n creep and MOE t i o n of an h y p e r b o l a .  has the equa-  The n o n - l i n e a r model, d e r i v e d to c o r r e l a t e  creep deformation and e l a s t i c deformation, i s p r e f e r a b l e .  5.3.4  Summary T h i s q u a n t i t a t i v e a n a l y s i s , c a r r i e d out on data f o r creep  deformation and f r a c t i o n a l creep, both expressed a t three months, indicates that: 1.  The c o r r e l a t i o n between creep deformation a r i d c e l a s t i c deforma-  t i o n , does not depend on the s t r e s s r a t i o .  T h i s r e s u l t does not  c o n t r a d i c t the comments made i n s e c t i o n 3.5 where I observed t h a t creep deformation and f r a c t i o n a l creep were h i g h e r f o r low s t r e n g t h m a t e r i a l - t h a t i s , f o r m a t e r i a l with high s t r e s s T h i s e f f e c t i s accounted  f o r i n t h a t .5  ratios.  i s expressed i n terms o f  <5 , and <$ depends upon the m a t e r i a l c h a r a c t e r i s t i c s . E  2.  e  One may c o n s i d e r the c o r r e l a t i o n between creep deformation  and e l a s t i c deformation, as w e l l as the c o r r e l a t i o n between f r a c t i o n a l creep and e l a s t i c deformation, independent applied 3.  o f the l e v e l o f  stress.  Creep deformation i s n o n - l i n e a r l y dependent upon e l a s t i c de-  formation.  T h i s r e s u l t v e r i f i e s the hypothesis made i n s e c t i o n  5.1 when the a n a l y t i c a l model was d e r i v e d .  The r e l a t i o n between  S and $ can be w r i t t e n as f o l l o w s : s c 6  C  = A + B  where A and B a r e c o n s t a n t s .  This expression i s s l i g h t l y  differ-  ent from the one o b t a i n e d i n s e c t i o n 5.1; which was: 6  C  = C 6  e  + D 6  where C and D are c o n s t a n t s .  2 e  These two e x p r e s s i o n s y i e l d very  s i m i l a r g r a p h i c a l r e p r e s e n t a t i o n s , and t h e r e f o r e e i t h e r may be used t o c o r r e l a t e creep deformation t o e l a s t i c 4.  deformation.  F r a c t i o n a l creep i s l i n e a r l y dependent upon e l a s t i c deforma-  tion.  T h i s r e s u l t , again, v e r i f i e s the h y p o t h e s i s made i n  s e c t i o n 5.1.  The r e l a t i o n between f and 6 may be w r i t t e n as e  follows: f = E + F 6  e  where E and F are c o n s t a n t s . 5.  The r e s u l t s , presented, are r e s t r i c t e d t o the s t r e s s  levels  72 investigated. These comments suggest s e c t i o n 5.1 may  be used to c o r r e l a t e creep deformation  e l a s t i c deformation, level.  5.4  t h a t the model, developed  in and  a t v a r i o u s times, r e g a r d l e s s o f the s t r e s s  T h i s i s the s u b j e c t of the next s e c t i o n .  F i n a l Results The  c o r r e l a t i o n between 6  a parabola such  and  6  has the equation o f  as: 6 (t)  = A(t) 6  c  + B(t)  e  6  2 e  where A and B, t h i s time, do. not depend, on the l e v e l o f a p p l i e d stress.  For a s p e c i f i c time t * , one 6  2  c  (t*) = A 6 + B 6 e e  obtains: .  where A and B are constant v a l u e s .  This f u n c t i o n has been  to the data p o i n t s of each graph from F i g u r e 28. shows values o f A and B, determined  i n r e g i o n s o f low  <$ 's. e  presented i n F i g u r e 40.  ranging  The A's have negative  i s due  values;  to l a c k of data p o i n t s  The parabolas are p l o t t e d on a graph Parabolas  and t h i r t e e n we:eks are almost them i s p l o t t e d .  XIII  from ten o b s e r v a t i o n s  from f i v e minutes t o t h i r t e e n weeks. t h i s has no p h y s i c a l meaning and  Table  fitted  a t three months, twelve weeks,  i d e n t i c a l ; t h e r e f o r e , only one  As b e f o r e , one  can w r i t e the e x p r e s s i o n o f  f r a c t i o n a l creep as f o l l o w s : f(t*) = A + B 5 e which i s a l i n e a r r e l a t i o n s h i p between f r a c t i o n a l creep  and  e l a s t i c deformation.  These s t r a i g h t l i n e s are p l o t t e d on a  graph shown i n F i g u r e  41.  of  73 I i n d i c a t e d i n s e c t i o n 5.1  t h a t a r e l a t i o n s h i p between  f r a c t i o n a l creep and modulus o f e l a s t i c i t y i s o b t a i n e d by expressing 6  e  i n terms of MOE.  upon the a p p l i e d s t r e s s . here.  Such an e x p r e s s i o n , however, depends Hence, t h i s method cannot be  used  Instead, hyperbolas such as: D  f (^t * ) = C + MOE r  ;  u  w i l l be d e r i v e d from graphs shown i n F i g u r e 31 by f i t t i n g  the  e x p r e s s i o n to the data p o i n t s o b t a i n e d a t d i f f e r e n t times r a n g i n g from f i v e minutes  to t h i r t e e n weeks.  The r e s u l t s f o r C and  at each time, are presented i n Table XIV. p l o t t e d on a graph shown i n F i g u r e 42.  D,  The hyperbolas are  The hyperbolas a t three  months, ((twelve weeks), and t h i r t e e n weeks are almost  identical;  t h e r e f o r e o n l y one of them i s p l o t t e d . R e c a l l i n g the reasons why  I established a correlation  between creep deformation and e l a s t i c deformation, i t i s now p o s s i b l e t o p r e d i c t the creep behaviour of a specimen  of the  m a t e r i a l under a c o n s t a n t s t r e s s of 3000 p s i or-.4500 p s i over a p e r i o d of t h r e e months.  Furthermore,  i f one assumes t h a t the  m a t e r i a l behaves as a l i n e a r v i s c o e l a s t i c m a t e r i a l , one  may  d e s c r i b e the specimen's creep behaviour a t o t h e r l e v e l s of a p p l i e d s t r e s s , as d i s c u s s e d i n s e c t i o n One may  attempt t o v e r i f y the model by a p p l y i n g i t to  t h e . r e s u l t s from the Surrey experiment. the l e v e l s of a p p l i e d s t r e s s were 1070 and 3110  4.4.  p s i ; the m a t e r i a l was  In t h i s p s i , 1410  Douglas-Fir.  to v e r i f y whether or not the model may  experiment, p s i , 2110  psi,  Thus I w i l l t r y  be u t i l i s e d under  c o n d i t i o n s broader than the ones under which i t was d e r i v e d . I w i l l use the f o l l o w i n g e q u a t i o n :  74 <S = -10.070 x 10 c where 6" and.S are expressed c e c  3  S + 14.168 x 10~ e  i n mm.  3  2  6 e  The values o f 6 , from e  s e c t i o n 2.4.3 are shown i n the f o l l o w i n g , f o r the v a r i o u s s t r e s s levels: 1070 p s i :  0.52 i n  (13.2 mm)  1410 p s i :  0.68 i n  (17.3 mm)  2110 p s i :  0.90 i n  (22.9 mm)  3110 p s i :  1.30 i n  (33.0 mm)  One o b t a i n s the f o l l o w i n g values o f t o t a l deformation a t 3 months; these values are composed with the a c t u a l v a l u e s ,  from  F i g u r e 14.  Stress  level  model  actual  psi  in  mm  in  mm  1070  0.61  15.5  0. 66  16.8  1410  0. 84  21.4  0.98  24.9  2110  1.20  30.1  1.10  27.0  3110  1.89  48.1  1.60  40.6  One can see t h a t the agreement between the two s e t s o f numbers, from an e n g i n e e r i n g viewpoint, i s e x c e l l e n t . the Surrey Laboratory experiment  The r e s u l t s  do not r e j e c t the model.  from  75  CHAPTER 6  CONCLUSIONS  The  r e s u l t s o f the f o r e g o i n g  study apply t o the s t r e s s  l e v e l s ...investigated and t o s p e c i f i c c o n d i t i o n s of temperature ,o  (10 C46 «30°°C) and moisture content leads t o the f o l l o w i n g  l.The d e f o r m a t i o n a l  (8%<MC<12%). The a n a l y s i s  conclusions:  response o f s t r u c t u r a l - s i z e timber beams  depends on t h e i r m a t e r i a l c h a r a c t e r i s t i c s . S p e c i f i c a l l y , material  with a s t r e n g t h lower than 5000 p s i (34.33 MPa) appeared,  to creep 1.5 times more than m a t e r i a l w i t h a s t r e n g t h higher that l e v e l  than  , over a three-month p e r i o d .  COMMENTS: To v e r i f y and more q u i c k l y q u a n t i f y the phenomenon d e s c r i b e d above, i t would be necessary t h a t f u t u r e i n v e s t i g a t i o n s employ sub-samples o f a l a r g e r s i z e . I t may prove advantageous to d e s c r i b e the phenomenon more s p e c i f i c a l l y , s i n c e commercial lumber presents properties .  c o e f f i c i e n t s o f v a r i a t i o n up t o 50% i n i t s strength-  76  2. The t e s t r e s u l t s support the assumption  of a l i n e a r  s h i p between the creep deformation of s t r u c t u r a l - s i z e beams and a p p l i e d  relationtimber  stress.  COMMENTS:It should be s t r e s s e d , w i t h i n a 9 5% c o n f i d e n c e i n t e r v a l , the c o r r e l a t i o n equation between f r a c t i o n a l creep and e l a s t i c deformation i n c l u d e d a term t h a t i s dependent on the l e v e l o f applied stress  ( s e c t i o n 5.3.3 second run, step 2 ) . Using  this  term,which i s incompatible with the h y p o t h e s i s o f l i n e a r i t y , d i d not s i g n i f i c a n t l y improve the p r e c i s i o n o f the c o r r e l a t i o n and was,therefore, d e l e t e d . But the appearance o f the term may suggest t h a t the r e l a t i o n between creep deformation and a p p l i e d s t r e s s i s not n e c e s s a r i l y l i n e a r f o r s t r e s s  levels  higher than 4 500 p s i (31.02 MPa). From an e n g i n e e r i n g p o i n t of view, there i s no need t o i n v e s t i g a t e t h i s f u r t h e r because the s t r e s s e s a p p l i e d i n t h i s a n a l y s i s cover  approximately  twice the range o f s t r e s s v a l u e s used i n d e s i g n . N e v e r t h e l e s s , from a s c i e n t i f i c p o i n t o f view, i t may be i n t e r e s t i n g t o expand t h i s i n v e s t i g a t i o n i n order t o compare.lumber  results  w i t h c l e a r wood r e s u l t s . F o r the l a t t e r , i t has been assumed t h a t l i n e a r i t y occurs f o r s t r e s s l e v e l s as h i g h as 80% o f u l timate s t r e s s . I t i s u n l i k e l y t h a t one would o b t a i n the same r e s u l t s with specimens o f s t r u c t u r a l  size.  3. The t e s t r e s u l t s from an experiment  i n which beams were  loaded t o s t r e s s l e v e l s up t o 3110 p s i (21.44; MPa), over a: t h r e e year p e r i o d , showed t h a t the average  constant  t o t a l defor-  mation,  a t t h i s time, i s approximately 1.6 times the average  elastic  deformation.  77  COMMENTS; The value 1.6 was d e r i v e d from t e s t s w i t h approximately constant environmental c o n d i t i o n s of temperature  and humidity.  T h i s may not be the worst case; a s t r u c t u r a l component w i l l experience s e r v i c e c o n d i t i o n s w i t h changes i n temperature and humidity.  4. A method t o p r e d i c t the creep behaviour o f s t r u c t u r a l s i z e timber beams a t d i s c r e t e times over a t h r e e month p e r i o d has been presented. T h i s p r e d i c t i o n c o n s i s t s o f u s i n g a p a r a b o l i c equation t o express the deformation, 8- , a t a g i v e n time, i n terms o f the e l a s t i c deformation, 6 . S i m i l a r l y , the f r a c t i o n a l creep, f = f u n c t i o n of 6  g  c  /6 ,  a  n  be expressed as a l i n e a r  or an h y p e r b o l i c f u n c t i o n of the modulus of  elasticity. COMMENTS: The p r e d i c t i o n o f creep deformation i s made f o r d i s c r e t e p o i n t s , r a t h e r than f o r a continuous spectrum. E x p r e s s i n g these r e s u l t s i n the form o f a mathematical over time might  d e s c r i p t i o n of creep  seem more e l e g a n t and s o p h i s t i c a t e d . But, from  an e n g i n e e r i n g p o i n t o f v i e w , p o i n t estimates such as the curves shown i n F i g s . 40, 41, and 42 a r e s u f f i c i e n t t o d e s c r i b e the long-term d e f o r m a t i o n a l behaviour,of the beams. In a d d i t i o n , t h i s p r e d i c t i o n method can be used f o r d e s i g n purpose, though  even  i t was developed e s p e c i a l l y f o r the complex a n a l y s i s  of the behaviour of assemblies of s t r u c t u r a l components under s u s t a i n e d l o a d , i n c l u d i n g f l o o r systems and t r u s s e s . T h e of the method developed i s t h a t  advantage  , i n t h i s type o f a n a l y s i s ,  the modulus o f e l a s t i c i t y o f the i n d i v i d u a l components can be used.  78  5. The method developed  l e d t o p r e d i c t i o n s of creep  deformation  at three months c o n s i s t e n t w i t h experimental data, w i t h engineering  tolerable  accuracy.  COMMENTS: The method was developed beams. N e v e r t h e l e s s , i t appears D o u g l a s - f i r beams as w e l l .  u s i n g data from Hemlock  t o p r e d i c t the behaviour of  79  BIBLIOGRAPHY 1. B a r r e t t , J . D . and Foschi,R.O., "Duration of Load and Probab i l i t y of F a i l u r e i n Wood. P a r t I . M o d e l l i n g Creep Rupture." Canadian J o u r n a l o f C i v i l E n g i n e e r i n g . V o l . 5. No. 4. p. 505-514, 1978. 2. Cederberg,A.M. and Danielsson,H., "Creep experiments on Wood". Unpublished Master T h e s i s . Lund., 197 0. 3. Clouser,W.S., "Creep o f Small Wooden Beams under Constant Bending Load". F o r e s t Products L a b o r a t o r y . F o r e s t S e r v i c e , US Dept. o f A g r i c u l t u r e . Report no. 2150. 25p. Madison, 1959. 4. Curry,W.T., "The Modulus of E l a s t i c i t y and Modulus of R i g i d i t y of S t r u c t u r a l Timber". Proceedings Wood E n g i n e e r i n g Group, IUFRO. Blokhus, Denmark. Paper no. 10, V o l . no. 2,197 6. 5. Flugge,W., " V i s c o e l a s t i c i t y " . B l a i s d e l l P u b l . Co., London,1967. 6. Kingston,R.S.T. and Clarke,L.N., "Some Aspects o f the R h e o l o g i c a l Behaviour of Wood. I and I I . " A u s t r a l i a n J o u r n a l of A p p l i e d S c i e n c e , no. 12. p.211,240. 1961. 7. K i t a h a r a , K . and Yukawa,K., "The I n f l u e n c e o f the Change of Temperature on Creep i n Bending." J . Japan Wood Res. Soc. V o l . 10. no. 5. p.169-174. 1974. 8. Lundgren,S.A., "Hardwood as C o n s t r u c t i o n M a t e r i a l - a V i s c o - e l a s t i c body." Holz a l s R. and W. V o l . 1/no. 1. p. 19-23. J a n . 1957. 9. Madsen,B. and B a r r e t t , J . D . , "Time-Strength R e l a t i o n s h i p f o r Lumber." The U n i v e r s i t y of B r i t i s h Columbia- Dept. o f C i v i l E n g i n e e r i n g . S t r u c t u r a l Research S e r i e s . Report no. 13. 1976. 10. Nakai,T., "Bending Creep T e s t s i n Wood-Based M a t e r i a l s . " F i r s t I n t e r n a t i o n a l Conference on Wood F r a c t u r e . B a n f f , Canada. 1978. 11. N a t i o n a l Lumber Grades A u t h o r i t y , " S t a n d a r d Grading Rules f o r Canadian Lumber." Vancouver, Canada. 1970. 13. N i e l s e n , A . , "Rheology o f B u i l d i n g M a t e r i a l s . " I n s t , f o r Byggnadsteknik. LTH. B u l l . 3. 57p. Lund. 19 68. 14. N i e l s e n , A . , "Rheology o f B u i l d i n g M a t e r i a l s . " N a t i o n a l Swedish C o u n c i l f o r B u i l d i n g Research. Stockholm, Sweden. Document D6. 1972.  80  15. Pierce,C.B., "The W e i b u l l D i s t r i b u t i o n and the D e t e r m i n a t i o n of i t s Parameters f o r A p p l i c a t i o n to Timber Strength Data." B u i l d i n g Research E s t a b l i s h m e n t . P r i n c e s Risborough Labora t o r y . P r i n c e s Risborough. U.K. Report CP26/76. 1976. 16. Sauer,D.J. and Haygreen,J.G., " E f f e c t s of S o r p t i o n on the F l e x u r a l Creep Behaviour o f Hardboard." F o r e s t Products J o u r n a l . V o l . 18/ no. 10. p.57-63. Oct. 1968. 17. Schniewind,A.P., "Recent P r o g r e s s i n the Study o f the Rheology o f Wood." Wood and Technology. V o l . 2. p.188-206. 1968. 18. Spencer,R., "Rate o f Loading E f f e c t i n Bending f o r D o u g l a s - F i r Lumber. Proceedings, F i r s t I n t e r n a t i o n a l Conference on Wood F r a c t u r e . B a n f f , Canada. 1978. 11  19. Sugiyama,H., "The Creep D e f l e c t i o n o f Wood Subjected to Bending Under Constant Loading." T r a n s a c t i o n s o f A r c h i t e c t u r a l I n s t i t u t e o f Japan. No. 55. p.60-70. 1957.  Time  Material  Reference Sellevold  (1969)  Cement paste  0.20 sec.  Perkitny  (1965)  Wood  3  sec.  Lundgren  (1968)  Wood-based products  3.6  sec.  C e l l u l a r concrete  2-4  sec.  Cement paste  5  sec.  Concrete  5  sec.  Wood f  1  min.  1  min.  1  min.  Nielsen Ruetz  (1968b) (1966)  G l a n v i l l e a c c . t o Evans E r i c k s o n and Sauer  (195 8)  (1969)  Bach  (1972)  Wood ^  Chow  (1970)  Furniture  panels  Table I . ' S t a r t i n g time a f t e r l o a d i n g f o r measuring creep. From.Nielsen  (1972).  82  Group  Deflection 3  ( i n x 10~ ) El  S<145  E2  Number o f Boards  E-values 6  ( p s i x 10 ) 1.56<E  89  146<S<160  1.41<E<1.55  59  E3  161<6<172  1.31<E<1.40  58  E4  173<6<200  1.13<E<1.30  99  E5  201<6  E<1.12  95  Table I I .  Group  Grouping o f the m a t e r i a l f o r Surrey experiment a c c o r d i n g t o E-values (Batch 1 ) .  Deflection 3  ( i n x 10~ )  E-values  Number o f Boards 6  ( p s i x 10 )  El  6<145  E2  146<6<160  1.41<E<1.55  88  E3  161<S<172  1.31<E<1.40  74  E4  173<6<200  1.13<E<1.30  135  E5  201<6<230  .98<E<1.12  90  E6  231<6  E< .97  94  Table I I I .  1.56<E  119  Grouping o f the m a t e r i a l f o r Surrey experiment a c c o r d i n g t o E-values (Batch 2 ) .  Board no.  Width (in)  Depth (in)  2224 2487 2527 2569 2350 1673 2480 1830 1266 1202 1149 2498 1779 2570 1523  1.485 1.496 1.500 1.496 1.500 1.486 1.495 1.481 1.484 1.513 1.508 1.500 1.490 1.497 1.500  5.455 5.492 5.484 5.463 5.510 5.450 5.494 5.390 5.490 5.504 5.494 5. 460 5.386 5.480 5.469  E-value ( p s i x 10 )  I n i t i a l Deflection  r  1.8139 1.1825 1.4596 1.3341 1.1380 1.2854 1.0959 2.0280 0 .9577 1.2605 1.0507 1.2878 1.6092 1.2982 1.0862  Predicted (in) 0.37 0.55 0.45 0.50 0.57 0.53 0.60 0.35 0.69 0.51 0.62 0.52 0.41 0.51 0.61  Table IV-a. Data f o r Surrey t e s t  .• - ••  Measured (in) North South Average 1.09 0.97 1.12 0.8.1 1.06 -0.42 -0.46 -1.05 -0.34 0.42 0.47 0.55 0.10 0.40 0.30  -0.03 0.08 0.01 -0.09 -0.26 0.87 1.02 1.02 1.27 0.77 0.81 -0.24 -0.69 -0.24 -0.54  (1070 p s i ) .  0.53 0.53 0.57 0.36 0.40 0.22 0.28 0. 0.47 0.59 0.64 0.16 -0 .29 0.08 -0.12  Error (%) 42.7 -5.2 25.6 -27.8 -29.6 • -57.1 -53.1 -104.4 -32.7 16.5 3.6 -69.7 -172.3 -84.0 -119.5  Board no.  1800 1979 2616 1431 2641 2500 2238 2516 2474 1781 1838 1989 1772 1389 1697 12 26  Width (in)  1.500 1.500 .1.500 1.480 1.495 1.490 1.499 1.500 . 1.513 1.505 1.505 1.512 1.500 1.500 1.515 1.470  Depth (in)  E-value ( p s i x 10 )  5.5 5.498 5.465 5.435 5.425 5.515 5.485 5.475 5.493 5.460 5.445 5.508 5.475 5.460 5.483 5.285  Table IV-b.  1.0287 1.3607 1.0486 1.3856 1.2341 1.1086 1.1544 1.4034 1.1872 1.4552 1.7858 1.7040 1.6124 1.1577 1.2807 1.2224  Initial Predicted (in) 0.83 0.63 0.83 0.65 0.73 0.77 0.75 0.62 0.72 0.60 0.49 0 .50 0.54 0.75 0.67 0.81  Data f o r Surrey t e s t  Deflection  Measured North South 1.81 0.45 1.63 1.41 0.20 0.59 0.37 0 .55 0.14 0.11 0.96 0.12 1.83 1.16 1.13 0.95  0.99 0.69 0.63 0.79 0.88 0.49 0.99 1.09 1.04 0.50 0 .12 0.65 0.44 0.67 0.58 0.64  (1410 p s i ) .  (in) Average  Error (%)  1.40 0.57 1.13 1.10 0.54 0.54 0.68 0.82 0.59 0.31 0.54 0.39 1.14 0.82 0.86 0.79  -67 .9 9.7 -35.5 -69.2 25.7 30.1 9.3 -32.4 17.9 49.2 -9.5 22.6 -110.5 -7.6 -27.8 1.5  Board no.  Width (in)  Depth (in)  1632 2326 2148 2387 1213 1462 2053 2050 1211 1192 1931 1868 1894 2348 1700 1822 2664 2621 2441 1982 1300 2218 1128 2161 1637  1.473 1.503 1.500 1.485 1.490 1.500 1.497 1.477 1.497 1.494 1.505 1.485 1.510 1.479 1.509 1.460 1.480 1.494 1.495 1.503 1.500 1.480 1.511 1.580 1.480  5.445 5.476 5.485 5.450 5.472 5.450 5.468 5.345 5.45 5.5 5.375 5.479 5.470 5.434 5.500 5.350 5.340 5.45 5.513 5.465 5.490 5.452 5.491 5.440 5.455  E-value ( p s i x 10 )  Table IV-- c .  Initial  fi  1.8981 1.8142 1.4402 1.8333 1.6857 1.8293 1.4754 1.5706 1.5605 1.4230 1.7229 1.3721 1.8117 1.8570 1.2182 1.2517 2.2373 1.3377 1.2344 1.1945 1.2322 1.3973 1.0654 1.3020 1.3162  Predicted (in) 0.71 0.72 0.90 0.72 0.78 0.72 0.89 0.90 0.85 0.8 8 0.79 0.95 0.71 0.73 1.05 1.14 0.64 0.99 1.04 1.09 1.05 0.96 1.20 0.97 1.01  Data f o r Surrey t e s t  Deflection  Measured (in) North South Average 0.74 1.29 0.84 1.09 0.99 1.29 0.59 1.06 1.06 1.21 0.69 0.79 0.34 0.69 0.19 0.59 0.49 0.44 1.20 1.50 1.50 1.45 1.76 1.51 1.51  0.64 0. 33 0 .78 1.04 0.78 0.64 0.94 1.23 1.26 1.18 0.94 1.28 1.28 0.98 1.78 1.93 0.78 0.83 0.71 1.23 1.11 1.06 1.23 0.73 1.31  (.2110 p s i ) .  0.69 0.81 0.81 1.06 0. 89 0.96 0.76 1.14 1.16 1.19 0.81 1.04 0.81 0.84 0.99 1.26 0.64 0.64 0.96 1.36 1.30 1.25 1.49 1.12 1.41  Error (%) 3.1 -13.3 9.6 -46.2 -14.0 -33.5 14.0 -26.7 -37.0 -35.7 -2.1 -8.6 -13.6 -15.2 5.7 -10.3 -0.3 35.6 7.6 -24.9 -24.7 -31.4 -24.5 -15.9 -39.3  Board no.  Width (in)  Depth (in)  1115 1609 1585 2672 2395 1896 1829 1419  1.513 1.495 1.481 1.400 1.500 1.465 1.472 1.486  5.468 5.423 5.460 5.323 5.500 5.423 5.425 5.425  E-value,( p s i x 10 )  2.0950 1.6767 1.7723 1.8031 1.1680 1.2674 1.5364 1.3876  Table IV-d.  Initial Predicted (in) 0.91 1.18 1.10 1.24 1.62 1.59 1.31 1.43  Data f o r Surrey t e s t  Deflection  Measured North South 0.86 1.10 1.48 1.33 1.53 0.98 2.0 1.71  0.55 1.36 0.87 1.10 1.70 1.15 0 .56 0.86  (3110 p s i ) .  (in) Average 0.71 1.23 1.17 1.21 1.62 1.06 1.28 1.28  Error (%) 22.5 -4.3 -6.5 1.8 0.1 33.1 1.9 10.3  Group  Modulus o f E l a s t i c i t y Mean p s i x 10  STD p s i x 10  1.3258 1.3206 1.5232 1.5883  0.2948 0.2272 0.2906 0.3087  g  1070 1410 2110 3110  psi psi psi psi  Table V.  g  C.V %  22.2 17.2 19.1 19 .4  S t a t i s t i c a l i n f o r m a t i o n s on the Modulus of E l a s t i c i t y o f boards used i n Surrey Experiment.  88  DATE  1070 PSI DEFLECTION (IN)  DAY 05 08 12 18 22 26 02 22 15 12 17 19 18 22 22 03 10 27 27 03 27 28 09 29 26 03 01  MONTH 01 01 01 01 01 01 02 02 03 04 05 07 10 11 12 01 02 03 04 06 07 09 11 12 11 05 02  YEAR  HOURS  19 75 1975 1975 1975 1975 1975 1975 1975 1975 1975 1975 1975 1975 1975 1975 1976 1976 1976 1976 1976 1976 19 76 1976 1976 1977 1978 1979  16:00 0. 19 :30 75.50 15:00 167. 12:30 308.50 412. 20:00 15:00 503. 660.50 16 :30 1135.75 11:45 1700.25 12:15 11:40 2359.67 11:15 3211.25 18 :00 , 4730. 11:00 6895. 11:10 7723.16 12 :00 8456. 8710.75 9596. 10736. 11468. 12356. 13664. 15152. 16160. 17372. 25328. 29144. 35660.  TABLE V l - a .  TIME (HRS)  MEAN  STD  0.29 0.29 0.32 0.31 0.34 0.35 0.37 0.37 0.40 0.44 0.46 0.48 0.50 0.50 0.46 0.51 0.52 0.52 0.54 0.54 0.51 0.53 0.56 0.55 0.62 0.61 0.64  0.30 0.31 0.29 0.31 0. 30 0.31 0.33 0.33 0.35 0. 36 0.37 0.36 0.34 0.35 0. 33 0.36 0.33 0.33 0.31 0.31 0.31 0. 30 0.30 0.30 0.30 0.29 0. 30  Creep data from Surrey experiment (Group 1070 p s i ) .  CVv (%) 103.4 106.9 90.6 100. 88.2 88.6 89.2 89 .2 87.5 81.8 80.4 75.0 68. 70. 71.7 70.6 63.5 63.5 57.4 57.4 60.8 56.6 53.6 54.5 48.4 47.5 46 .9  89  DATE  1410 PSI DEFLECTION (IN)  DAY  MONTH  YEAR  HOURS  TIME (HRS)  MEAN  STD  04 08 12 18 22 26 02 22 15 12 17 19 18 22 22 03 10 27 27 03 27 28 09 29 26 03 01  01 01 01 01 01 01 02 02 03 04 05 07 10 11 12 01 02 03 04 06 07 09 11 12 11 05 02  1975 1975 1975 1975 1975 1975 1975 1975 1975 1975 1975 1975 1975 1975 19 75 1976 1976 1976 1976 1976 1976 1976 1976 1976 1977 1978 1979  16 :00 19:30 15:00 12:30 20 :00 15 :00 16:30 11:45 12:15 11:40 11:15 18:00 11:00 11:10 12:00  0. 99.50 191. 332.50 436. 527. 684.50 1159.75 1724.25 2383.67 3235.25 4754. 6919. 7747.20 8480. 8734.75 9620. 10760. 11492. 12380. 13688. 15176. 16184. 17396. 25352. 29168. 35684.  0.60 0.78 0.80 0.78 0 .83 0.84 0. 85 0.87 0.93 0.98 1.00 1.05 0.03 1.05 1.0 8 1.10 1.09 1.10 1.13 1.12 1.10 1.10 1.11 1.12 1.15 1.15 1.17  0.40 0.32 0.34 0.33 0.35 0.34 0.34 0. 38 0.39 0.40 0.40 0. 36 0.35 0.36 0.39 0. 39 0.39 0.39 0. 39 0.38 0.35 0.33 0.33 0. 31 0.31 0. 32 0.32  TABLE V l - b .  CV (%)  Creep data from Surrey experiment (Group 1410 p s i ) .  67. 41. 42.5 42.3 42.2 40.5 40. 43.7 41.9 40.8 40. 34.3 34. 34.3 36.1 35.5 35.8 35.5 34.5 33.9 31.8 30/ 29.7 27.7 27. 27.8 27.4  90  2110 PSI  DATE  DEFLECTION (IN) DAY 04 08 12 18 22 26 02 22 15 12 17 19 18 22 22 03 10 27 27 03 27 28 09 29 26 03 01  MONTH 01 01 01 01 01 01 02 02 03 04 05 07 10 11 12 01 02 03 04 06 07 09 11 12 11 05 02  YEAR  HOURS  TIME (HRS)  MEAN  STD  1975 1975 1975 1975 1975 1975 1975 1975 1975 1975 1975 1975 1975 1975 1975 1976 1976 1976 1976 1976 1976 1976 1976 1976 1977 1978 1979  16:00 19:30 15:00 12:30 20:00 15:00 16 :30 11:45 12:15 11:40 11:15 18 :00 11:00 11:10 12:00  0. 99.50 191. 332.50 436. 527. 684.50 1159.75 1724.25 2383.67 3235.25 4754. 6919. 7747.20 8480. 8734.75 9620. 10760. 11492. 12380. 13688. 15176. 16184. 17396. 25352. 29168. 35684.  1.02 1.05 1.06 1.05 1.10 1.10 1.12 1.11 1.14 1.16 1.20 1.22 1.29 1.31 1. 31 1. 31 1.29 1.32 1.31 1.34 1.32 1.36 1. 37 1.39 1.45 1.43 1.47  0.25 0.26 0.27 0.28 0.28 0.29 0. 31 0.30 0. 32 0.32 0.33 0.32 0.33 0.34 0. 33 0.33 0.33 0.35 0.34 0. 31 0. 34 0. 33 0.34 0. 34 0.34 0.33 0. 36  TABLE V I - c .  CV (%)  Creep data from Surrey experiment (Group 2110 p s i ) .  24.5 24.8 25.5 26.7 25.5 26 .4 27.7 27. 28.1 27.6 27.5 26.2 25.6 26. 25.2 25.2 25.6 26.5 26. 23.1 25.8 24.3 24.8 24.5 23.4 23.1 24.5  91  DATE  3110 PSI DEFLECTION (IN)  DAY 05 08 12 18 22 26 02 22 15 12 17 19 18 22 22 03 10 27 27 03 27 28 09 29 26 03 01  MONTH 01 01 01 01 01 01 02 02 03 04 05 07 10 11 12 01 02 03 04 06 07 09 11 12 11 05 02  YEAR  HOURS  TIME (HRS)  MEAN  1975 1975 1975 1975 1975 1975 1975 19 75 1975 1975 19 75 1975 1975 1975 19 75 1976 19 76 19 76 1976 1976 1976 1976 1976 19 76 1977 1978 1979  12:00 19:30 15:00 12:30 20:00 15:00 16:30 11:45 12:15 11:40 11:15 18:00 11:00 11:10 12:00  0. 79.50 171. 312.50 416. 507. 664.50 1139.75 1704.25 2363.67 3215.25 4734. 6899. 7727.16 8460. 8714.75 9600. 10740. 11472. 12360. 13668. 15156. 16164. 17376. 25332. 29148. 35664.  1.20 1.26 1.26 1.28 1.31 1.30 1.31 1.33 1.44 1.48 1.51 1.56: 1.62 1.64 1.64 1.66 1.67 1.69 1.72 1.75 1.71 1.73 1.75 1.80 1. 83 1. 81 1.87  TABLE V l - d .  STD 0.25 0.27 0.28 0.26 0.28 0.29 • 0.30 0.31 0.32 0.35 0.35 0.33 0.34 0 .33 0.34 0. 35 0 .36 0. 36 0.35 0.31 0.35 0.36 0.36 0. 35 0.35 0.33 0.37  Creep data from Surrey experiment (Group 3110 p s i ) .  CV (%) 20.8 21.4 22.2 20.3 21.4 22.3 22.9 23.3 22.2 23.6 23.2 21.2 21. 20.1 20.7 21.1 21.3 21.3 20.3 17.7 20.5 20.8 20.6 19.4 19.1 18.2 19.8  BOARD  NUMBER  1115 1609 1585 2,6 72 2395 1896 1829 1419  TABLE V I I .  y  Y' S  n (in)  (in)  2.91 3.35 3.43 3.23 3.23 2.83 3.15 3.31  3.15 2.76 3.27 2.95 2.95 3.15 3.11 3.11  D i s t a n c e b e t w e e n a beam s u p p o r t and an e m p i r i c a l r e f e r e n c e f o r the measurements.  TIME (hrs) 0. 79.50 171. 312.50 416. 507. 664.50 1139.75 1704.25 2363.67 3215.25 4734. 6899. 7727.16 8460. 8714.15 9600. 10740. 11472. 12360. 13668. 15156. 16164. 17376. 25332. 29148. 35664 .  <5._ n (in) 2. 05 1.95 2.05 2. 2. 2. 2. 2. 1.90 1. 85 1.75 1.75 1.70 .1.70 1.70 1.70 1.75 1.70 1.60 1.40 1.65 1.60 1.60 1.50 1.50 1. 50 1.55  TABLE V I I I - 1 .  6 s (in) 2.60 2.65 2.55 2.45 2.50 2.50 2.50 2.50 2.35 2.40 2.40 2.30 2.30 2.25 2.30 2.20 2. 2.20 2.15 2.20 2.25 2.25 2.20 2.20 2.10 2.15 2.10  f n (in) 0.86 0.96 0.86 0.91 0.91 0.91 0.91 0.91 1.01 1.06 1.16 1.16 1.21 1.21 1.21 1.21 1.16 1.21 1.31 1.41 1.26 1.31 1.31 1.41 1.41 1.41 1.36  f s (in) 0.55 0.50 0.60 0.70 0.65 0.65 0.65 0.65 0 . 80 0.75 0.75 0.85 0.85 0.90 0.85 0.95 1.15 0.95 1. 0.95 0.90 0.90 0.95 0.95 1.05 1. .105  f average (in) 0.71 0.73 0.73 0.81 0.78 0.78 0.78 0.78 0.91 0.91 0.96 1.01 1.03 1.06 1.03 1.08 1.16 1.08 1.16 1.18 1.08 1.11 1.13 1.18 1.23 1.21 1.21  Creep data f o r boards loaded t o 3110 p s i (Surrey experiment) (Board number 1115).  TIME ((Kara) 0. 79.50 171. 312.50 416. 507. 664.50 1139.75 1704.25 2363.67 3215.25 4734. 6899. 7727.16 8460. 8714.75 9600. 10740. 11472 12360. 13668. 15156. 16164. 17376. 25332. 29148. 35664.  6\ n (in) 2.25 2.20 2.20 2.25 2.20 2.20 2.20 2.15 2.15 2 . 05 2.05 2.05 2.05 2.05 2. 2.05 2. 2. , 1.95 1.95 1.95 1.95 1.90 1.90 1. 85 1.90 1. 85  6  f  s (in) 1.40 1.25 1. 30 1.25 1.20 1.20 1.25 1.15 1. 0.95 0.95 0. 85 0.85 0.85 0.85 0.75 0.80 0.80 0.75 0.80 0.75 0. 85 0.90 0.85 0.90 0.85 0. 80  TABLE V I I I - 2 .  n (in) 1.10 1.15 1.15 1.10 1.15 1.15 1.15 1.20 1.20 1.30 1.30 1.30 1.30 1.30 1.35 1.20 1.35 1.35 1.40 1.40 1.40 1.50 1.45 1.45 1.50 1.45 1.50  f s (in) 1.36 1.51 1.46 1.51 1.56 1.56 1.51 1.61 1.76 1. 81 1.81 1.91 1.91 1.91 1.91 2.01 1.96 1.96 2.01 1.96 2.01 1.91 1.86 1.91 1.8 6 1.91 1.96  f average (in) 1.23 1.33 1.31 1. 31 1.36 1.36 1.33 1.41 1.48 1.56 1.56 1.61 1.61 1.61 1.63 1.66 1.66 1.66 1.71 1.68 1.71 1.66 1.66 1.68 1.68 1.68 1.73  Creep data f o r boards loaded t o 3110 p s i (Surrey experiment) (Board number 1609).  TIME (hrs)  6.. n (in)  0. 79.50 171. 312.50 416. 507. 664.50 1139.75 1704.25 2363.67 3215.25 4734. 6899. 7727.16 8460. 8714.75 9600. 10740. 11472. 12360. 13668, 15156. 16164. 17376. 25332. 29148. 35664.  1.95 1.80 1.80 1. 80 1.80 1.80 1.80 1. 85 1. 80 1.80 1.80 1. 80 1.70 1.65 1.70 1.75 1.70 1.70 1. 70 1.55 1.65 1.65 1.60 1.50 1. 35 1. 30 1.35  6\ s (in)  TABLE V I I I - 3 .  2.40 2.45 2.40 2.40 2.40 2.50 2.45 2.40 2.30 2.30 2.30 2.10 2.05 2.05 2.05 2.05 2.05 2. 1.85 1.90 2. 1.90 2. 1.90 1.90 1.95 1.95  f n (in) 1.48 1.63 1.63 1.63 1.63 1.63 1.63 1.58 1.63 1.63 1.63 1.63 1.73 1.78 . 1.73 1.68 1.73 1.73 1.73 1.88 1.78 1.78 1. 83 1.93 2.08 2.13 2.08  f s (in) 0.87 0.92 0.87 0.87 0.87 0 .77 0. 82 0.87 0.97 0.97 0.97 0 .17 1.22 1.22 1.22 1.22 1.22 1.27 1.42 1.37 1.27 1.37 1.27 1.37 1.37 1. 32 1. 32  f average (in) 1.18 1.28 1.25 1.25 1.25 1.20 1.23 1.23 1.30 1.30 1.30 1.40 1.48 1.50 1.48 1.45 1.48 1.50 1.58 1.63 1.53 1.58 1.55 1.65 1.73 1.73. 1.70  Creep data f o r boards loaded to 3110 p s i (Surrey experiment) (Board number 15 85).  TIME (hts) 0. 79.50 171. 312.50 416. 507. 664.50 1139.75 1704.25 2363.67 3215.25 4734. 6899 . 7727.16 8460. 8714.75 9600. 10740. 11472. 12360. 13668. 15156. 16164. 17 376. 25332. 29148. 35664.  6 n (in) 1.90 1.85 1.80 1.80 1.80 1.80 1.75 1.80 1.65 1.70 1.65 1.60 1.45 1.50 1.50 1.45 1.45 1.45 1.45 1.45 1.45 1.45 1.45 1.40 1.45 1.50 1.30  <5 s (in) 1.85 1. 70 1.70 1.65 1.60 1.60 1.60 1.55 1.50 1.35 1.35 1. 35 1.35 1. 30 1.25 1.25 1.25 1.25 1.25 1.25 1.25 1.25 1.35 1.25 1.30 1.20 1.20  TABLE V I I I - 4 .  f n (in) 1.33 1.38 1.43 1.43 1.43 1.43 1.48 1.43 1.58 1.53 1.5 8 1.63 1.78 1.73 1.73 1. 78 1.78 1.78 1.78 1.78 1.78 1.78 1.78 1. 83 1.78 1.73 1.93  f  f  1.10 1.25 1.25 1.30 1.35 1.35 1.35 1.40 1.45 1.60 1.60 1.60 1.60 1.65 1.70 1.70 1.70 1.70 1.70 1.70 1.70 1.70 1.60 1.70 1.65 1.75 1.75  average (in) 1.22 1.32 1.34 1.37 1. 39 1.39 1.42 1.42 1.52 1.57 1.59 1.62 1.69 1.69 1.72 1.74 1.74 1.74 1.74 1.74 1.74 1.74 1.69 1.77 1.72 1.74 1.84  s (in)  Creep data f o r boards loaded t o 3110 p s i (Surrey experiment) Board number 2672.  TIME (hrs) 0. 79.50 171. 312.50 416. 507. 664.50 1139.75 1704.25 2363.67 3215.25 4734. 6899. 7727.16 8460. 8714.75 9600 . 10740. 11472. 12360. 13668. 15156. 16164. 17376. 25332. 29148. 35664.  6. n (in)  6, s (in)  1.70 1.65 1.65 1.60 1.60 1.55 1.50 1.50 1.35 1.35 1. 30 1.30 1.20 1.25 1.25 1.15 1.20 1.10 1.20 1.20 1.20 1.15 1.15 1.15 1.10 1.35 1.10  TABLE V I I I - 5 .  1.25 1.10 1.05 1.05 0.95 1. 0.95 0.85 0.70 0.50 0.50 0.50 0.50 0.45 0.50 0.40 0 .40 0.40 0.20 0.40 0.40 0.40 0.40 0.40 0.35 0.30 0.20  f n (in) 1.53 1.58 1.58 1.63 1.63 1.68 1.73 1.73 1.88 1. 88 1.93 1.93 2.03 1.98 1.98 2.08 2.03 2.13 2.03 2.03 2.03 2.08 2.08 2.08 2.13 1.88 2.13  f  f  1.70 1.85 1.90 1.90 2. 1.95 2. 2.10 2.25 2.45 2.45 2.45 2.45 2.50 2.45 2.55 2.55 2.55 2.75 2.55 2.55 2.55 2.55 2.55 2.60 2.65 2.75  average (in) 1.62 1.72 1.74 1.77 1.83 1.82 1. 87 1.92 2. 07 2.17 2.19 2.19 2.24 2.24 2.22 2.32 2.29 2.34 2.39 2.29 2 .29 2. 30 2.31 2.31 2.37 2.27 2.44  s (in)  Creep data f o r boards loaded t o 3110 p s i (Surrey experiment) (Board Number 2 395).  TIME (Hrs) 0. 79.50 171. 312.50 416. 507. 664.50 1139.75 1704.25 2363.67 3215.25 4734 . 6899. 7727.16 8460. 8714.75 9600. 10740. 11472. 12360. 13668. 15156. 16164. 17376. 25332. 29148. 35664 .  6  <5  n  (in) 1.85 1.60 1.80 1.80 1.70 1.70 1.70 1.70 1.55 1.45 1.30 1.30 1.20 1.20 1.15 1.15 1.10 1.05 1.05 0.95 1. 0.95 0.90 0.85 0.80 0.90 0.75  TABLE V I I I - 6 .  s (in) 2. 1.95 1.90 1.85 1.80 1.85 1.90 1. 80 1. 60 1.55 1. 60 1.50 1.40 1.25 1. 30 1. 30 1.30 1.25 1.20 1.15 1.20 1.05 0.95 0.95 0.90 0.85 0.90  f n (in) 0.98 1.23 1.03 1.03 1.13 1.13 1.13 1.13 1.28 1.38 1.53 1.53 1.63 1.63 1.68 1.68 1.73 1.78 1.78 1.88 1.83 1.88 1.93 1.98 2.03 1.93 2.08  f s (in) 1.15 1.20 1.25 1.30 1.35 1.30 1.25 1. 35 1.55 1.60 1.55 1.65 1.75 1.90 1.85 1.85 1.85 1.90 1.95 2.00 1.95 2.10 2.20 2.20 2.25 2.30 2.25  f average (in) 1.07 1.22 1.14 1.17 1.24 1.22 1.19 1.24 1.42 1.49 1.54 1.59 1.69 1.77 1.77 1.77 1.79 1.84 1.86 1.94 1.89 1.99 2.07 2 .09 2.14 2;12 2.17  Creep data f o r boards loaded to 3110 p s i (Surrey experiment) (Board number 1896).  TIME (hrs) 0. 79.50 171. 312.50 416. 507. 664.50 1138.75 1704.25 2363.67 3215.25 4734. 6899. 7727.16 8460 . 8714.75 9600. 10740. 11472. 12360. 13668. 15156. 16164. 17376. 25332. 29148. 35664.  6  n (in) 1.15 1.15 1.15 1.15 1.10 1.15 1.15 1.15 1.15 1.05 1. 1. 0.95 0.90 0.95 0.90 0. 85 0.90 0.90 0.85 0. 85 0.80 0.70 0.70 0.65 0.65 0.55  TABLE V I I I - 7 .  6  s (in) 2.55 2.60 2.60 2.60 2.55 2 .60 2.55 2.55 2.50 2.50 2.45 2.45 2. 40 2.35 2.35 2. 30 2.40 2.35 2.35 2.25 2.25 2.35 2.20 2.15 2.15 2.15 2.10  f n (in) 2. 2. 2. 2. 2.05 2. 2. 2. 3. 2.10 2.15 2.15 2.40 2.25 2.20 2 .25 2. 30 2.25 2.25 2.30 2.30 2.35 2.45 2.45 2.50 2.50 2.60  f s (in) 0.56 0 .51 0.51 0.51 0.56 0.51 0.56 0.56 0.61 0.61 0.66 0.66 0. 71 0.76 0.76 0. 81 0.71 0.76 0.76 0.86 0.86 0.76 0.91 0.96 0.96 0.96 1.01  f average (in) 1.28 1.26 1.26 1.26 1.31 1.26 1.28 1.28 1.31 1.36 1.41 1.41 1.46 1.51 1.48 1.53 1.51 1.51 1.51 1.58 1.58 1.56 1.68 1.71 1.73 1.73 1.81  Creep data f o r boards loaded to 3110 p s i (Surrey experiment) (Board number 1829).  100 TIME (hrs) 0. 79.50 171. 312.50 416. 507. 664.50 1139.75 1704.25 2363.67 3215.25 4 734. 6899. 7727.16 8460. 8714.75 9600. 10740. 11472. 12360. 1366 8. 15156. 16164. 17376. 25332. 29148. 35664.  n (in)  6_ s (in)  1.60 1.55 1.50 1.50 1.60 1.50 1.50 1.55 1.45 1.45 1.35 1. 30 1.25 1.25 1.20 1.25 1.25 1.20 1.20 1.15 1.1.5 1.10 1.05 1. 1. 1. 0.95  2.25 2.25 2.20 2.20 2.10 2.10 2.10 2.15 1.95 2. 1.95 1.80 1.65 1.60 1.55 1.60 1.60 1.55 1.55 1.50 1.50 1.50 1.50 1.35 1.35 1.30 1. 35  TABLE V I I I - 8 .  f : rt  (in) 1.71 1. 76 1.81 1.81 1. 71 1.81 1.81 1.76 1.86 1.86 1.96 2.01 2.06 2.06 2.11 2.06 2.06 2.11 2.11 2.16 2.16 2.21 2.26 2.31 2.31 2.31 2 . 36  f  s (in) 0 .86 0.86 0.91 0.91 1.01 1.01 1.01 0.96 1.16 1.11 1.16 1.31 1.46 1.51 1.56 1.51 1.51 1.56 1.56 1.61 1.61 1.61 1.61 1.76 1.76 1. 81 1.76  f average (in) 1.29 1.31 1. 36 1.36 1.36 1.41 1.41 1.36 1.51 1.49 1.56 1.66 1.76 1.79 1.84 1.79 1.79 1.84 1.84 1.89 1.89 1.91 1.94 2.04 2.04 2.06 2.06  Creep data f o r boards loaded t o 3110 p s i (Surrey experiment) (Board number 1419).  101  TABLE IX. Data f o r Richmond experiment  (Sub-group 1 ) .  BOARD GROUP WIDTH DEPTH STRESS MOE PREDICTED ACTUAL NUMBER NUMBER (IN) (IN) (PSI) (PSI x 10" ) 6 . 6 . 653 1395 1101 158 400 305 733 1318 803 1188 765 770 764 679 313 116 7 1030 .472 2768 2853 97 269 1.6 78 1680 1679 2276 2299 1582 1557 2727 2699 2855 1820 1897 44 62 997 519 2212 2304 923 545 1892 318 2124 371 2399 14 2503  8 8 8 8 2 2 8 8 8 8 2 2 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 2 2 2 8 8 8 2 8 8 2 8 2 8 2 8  1.50 1.50 1.51 1.50 1.50 1.50 1.46 1.52 1.49 1.49 1.50 1.50 1.49 1.52 1.51 1.50 1.52 1.51 1.50 1.46 1.51 1.54 1.52 1.52 1.49 1.52 1.52 1.48 1.53 1.52 1.51 1.47 1.52 1.51 1.48 1.52 1.48 1.50 1.54 1.50 1.49 1.53 1.51 1.50 1.52 1.54 1.54 1.50 1.50  5.29 5.36 5.36 5.40 5.44 5.41 5.36 5.43 5.32 5.29 5. 35 5.36 5.40 5. 38 5.43 5.38 5.48 5.44 5. 39 5.29 5. 37 5.48 5.47 5.43 5.40 5.47 5.47 5.38 5.45 5.4 3 5.36 5. 33 5.49 5.40 5.28 5.40 5. 36 5.38 5.47 5.45 5.41 5.35 5.37 5.38 5.43 5.48 5.54 5.40 5.41  3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 300 0 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000  0.19321 0.18169 0.22467 0.24719 0.20620 0.18078 0.24649 0.16931 0.21008 0.21692 0.17492 0.17047 0.22097 0.19176 0.21805 0.20472 0.15912 0.17032 0.18417 0.23657 0.21339 0.15667 0.18374 0.19753 0.20089 0.145 39 0.13042 0.25928 0.17430 0.187 27 0.22925 0.25048 0.14046 0.20879 0.20410 0.19237 0.15440 0.19485 0.21970 0.18362 0.19561 0.175 87 0.16809 0.22852 0.17105 0.15535 0.15867 0.16127 0.16627  l  l  30.2 31.8 25.7 23.1 27.4 31.5 23.4 35.6 27.7 26 .9 33.0 33.8 25.9 30.0 26 .2 27.9 35.6 33.3 31.2 24 .6 26.9 36.1 30.7 28. 7 28.4 38.9 43.4 22.1 32.5 30 .5 25.1 23.1 40.1 27.4 28.7 29.7 37.3 29.5 25.7 31.0 29 .2 32. 8 34.3 25.1 33.3 36 .3 35.1 35.6 34.3  35.1 39.1 22.1 24 .9 29.0 35.1 24.9 36.1 29 .0 31.0 39.9 38.1 30.0 31.0 30 . 0 30 .0 42.9 36.1 26.9 25.9 27 .9 41.9 35 .1 31.0 31.0 41.9 48 .0 23.1 33.0 34 .0 26 .9 23.1 45.0 20.1 31.0 33.0 46.0 33.0 29 .0 33.0 34.0 38.1 37.1 29.0 38.1 39 .1 46.0 39.9 41.9  102 Table IX continued.  BOARD GROUP WIDTH DEPTH STRESS MOE PREDICTED ACTUAL NUMBER NUMBER (IN) (IN) 6. (PSI) (PSI x 10 )• 6 . l l ?  2526 2131 2026 2557 2037 2745 2331 1587 1593 1583 2616 393 2587 1904 404 2406 76 392 355 324 1016 1076 298 2644 1963 2144 2046 2031 2819 1231 1353 2350 2327 1289 1439 1254 1323 1290 940 1482 1526 1313 1510 1432 1472 1111 1533 52  8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 2 2 2 2 2 2 2 8 8 8 8 8 8 9 8 8 8 8 8 8 8 8 9 8 8 8 8 8 8 8 8 8  1.51 1.51 1.51 1.50 1.50 1.47 1.52 1.49 1.52 1.50 1.50 1.53 1.52 1.50 1.50 1.51 1.52 1.52 1.49 1.48 1.51 1.48 1.51 1.49 1.51 1.50 1.51 1.51 1.50 1.49 1.51 1.51 1.54 1.50 1.52 1.48 1.49 1.50 1.47 1.51 1.49 1.51 1.50 1.51 1.52 1.47 1.49 1.40  5.43 5.41. 5.44 5.25 5.45 5.27 5.43 5. 37 5.45 5.43 5.43 5.42 5.48 5.40 5.38 5.38 5.46 5.43 5.40 5.48 5.44 5.32 5.41 5.37 5.43 5.36 5.46 5.44 5.44 5.33 5.47 5.36 5.50 5.36 3000 5. 32 5.40 5.47 5.42 5.44 5.45 5.40 5.40 5.36 5.48 5.31 5.44 5.28  3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000  0.17636 0.17130 0.21631 0.19621 0.20622 0.21334 0.16783 0.23281 0.21160 0.15984 0.20564 0.19370 0.16527 0.24347 0.17790 0.13917 0.18339 0.23904 0.19402 0.21492 0.20385 0.18630 0.23115 0.19964 0.17791 0.18606 0.16060 0.18151 0.22696 0.22277 0.20482 0.21075 0.19 7 9.9 0.22444 . 0.16791 0.20898 0.20103 0.18615 0.21854 0.17780 0.20026 0.17308 0.15520 0.23383 0.20214 0.23811 0.19037 0.24321  32.3 33. 3 26.2 30.0 27.4 27.4 34.0 24.6 26.9 35.6 27.7 29.5 34 .0 23.6 32.3 41.4 31.0 23.9 29.5 26 .2 27.9 31.2 24.6 28.7 32.0 31.0 35. 3 31.2 25.1 25.9 27.7 27.4 28.4 25.7 33.5 27.7 28.4 30.2 26.2 32.0 28.2 33.0 36.8 24.6 27.9 24 .4 29.7 24.1  37.1 39.1 29.0 35.1 32.0 32.0 38.1 25.9 30.0 43.9 31.0 32.0 38.1 25.9 34.0 45.0 34.0 26.9 33.0 27.9 32.0 32.0 25.9 31.0 47.0 33.0 34.0 43.9 30.0 29 .0 29.0 27.9 31.0 29.0 39.1 33.0 31.0 35.1 31.0 38.1 25.9 39.1 42.9 29. 0 32.0 26.9 31.0 27.9  TABLE X. BOARD NUMBER  1634 2239 1918 569 1653 1947 2284 2803 2059 1057 2072 2057 146 1463 1386 1934 1978 572 1973 2491 1961 2748 2778 2519 2498 2003 225 2509 1405 2653  WIDTH GROUP (in) NUMBER :  2 2 2 2 2 2 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2  1.50 1.53 1.51 1.51 1.52 1.52 1.52 1.52 1.50 1.48 1.52 1.53 1.53 1.52 1.57 1.47 1.47 1.48 1.51 1.49 1.50 1.51 1.53 1.54 1.52 1.52 1.50 1.51 1.50 1.50  Data f o r Richmond experiment DEPTH (in)  5.43 5.48 5.47 5.35 5.45 5.43 5.46 5.36 5.26 5.36 5.45 5.46 5.45 5.44 5.46 5.29 5.36 5.26 5.48 5.28 5.33 5.43 5.43 5.50 5.45 5.47 5.33 5.26 5.50 5.44  STRESS (psi)  3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000  MOE (psi) (E+07) 0.14820 0.14967 0.17669 0.11039 0.17262 0.14529 0.16553 0.19817 0.21369 0.25249 0.18256 0.17007 0.22289 0.19123 0.16437 0.22677 0.22245 0.19094 0.14893 0.21889 0.19566 0.19079 0.21550 0.21313 0.20910 0.19513 0.21807 0.17863 0.20971 0.18004  (Sub-group 2 ) . PREDICTED i (mm) 38.4 37.7 32.0 52.3 32.8 39.2 34.2 29.1 27.1 22.8 31.1 33.3 25.4 29.7 34.4 25.8 25.9 30.8 37.9 26.7 29.6 29.8 26.4 26.4 27.1 29.0 26.6 32.9 26.8 31.6  ACTUAL i (mm)  S.R. (%)  40.0 43.0 36.0 36.0 43.0 35.0 31.0 28.0 25.0 29.0 35.0 29.0 35.0 37.0 29.0 28.0 32.0 41.0 28.0 31.0 34.0 28.0 28.0 30.0 31.0 31.0 36.0 30.0 33.0  COMMENTS  F.O.L. F.O.L. F.O.L. F.O.L. F.O.L. 29.7 55.0 26.1 25.0 21.7 73.5 43.4 24.3 62.7 62.6 27.0 52.8 40.8 32.8 32.6 27.0 53.8 38.3 48.8 44.4 37.7 45.7 85.7 42.0 71.5  TABLE X c o n t i n u e d . BOARD NUMBER  2363 2713 555 2811 2857 2561 905 1589 2207 2220 2725 1793 2499 2623 2210 2279 1663 837 2522 2655 2884 2673 745 946 1513 1165 1365 1357 41 1207  GROUP NUMBER  2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2  WIDTH (in)  DEPTH (in)  STRESS (psi)  1.54 1.51 1.51 1.53 1.50 1.46 1.52 1.47 1.52 1.53 1.51 1.51 1.51 1.51 1.55 1.51 1.51 1.50 1.52 1.48 1.51 1.50 1.48 1.50 1.51 1.52 1.52 1.52 1.50 1.51  5.48 5.47 5.39 5.45 5.39 5.35 5.46 5.32 5.41 5.48 5.38 5.48 5.41 5.48 5.52 5.46 5.37 5.46 5.49 5.41 5.47 5.41 5.32 5.43 5.45 5.45 5.48 5.45 5.41 5.45  3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000  MOE (psi) (E+07) 0.16436 0.15738 0.15825 0.18457 0.25064 0.16711 0.19932 0.23282 0.13144 0.22346 0.16854 0.19734 0.22809 0.16743 0.14580 0.23933 0.16165 0.20099 0.17772 0.18831 0.17134 0.20796 0.21628 0.19818 0.26006 0.18844 0.20311 0.24560 0.19298 0.19435  PREDICTED i (mm) 34.3 35.9 36.2 30.7 22.9 34.6 28.4 25.0 43.5 25.2 34.1 28.6 25.0 33.7 38.4 23.6 35.6 28.2 31.7 30.3 33.0 27.5 26.9 28.7 21.8 30.1 27.8 23.1 29.6 29.2  ACTUAL i (mm)  S.R.  38.0 39.0 41.0 30.0 24.0 26.0 33.0 29.0 45.0 27.0 35.0 32.0 25.0 37.0 42.0 26.0 37.0 32.0 36.0 33.0 38.0 27.0 27.0 32.0 24.0 32.0 31.0 25.0 32.0 33.0  35.9 62.2 50.2 87.8 22.8 27.9 59.3 27.0 58.3 41.9 59.0 34.6 24.1 31.7 74.7 35.9 35.1 42.5 44.7 80.5 59.5 23.7 27.6 56.6 23.7 35.7 47.0 22.9 43.9 40.6  COMMENTS  TABLE X c o n t i n u e d . BOARD NUMBER  1337 762 2318 345 2352 2238 1332 2282 1786 2800 1620 1671 1602 2756 2482 2497 2067 2469 1969 2687 2642 2826 2549 2489 410 1923 2600 2807 942 932  GROUP NUMBER  2 2 2 2 2 2 2 2 6 6 6 6 6 6 6 6 6 6 6 9 9 9 9 9 9 9 9 9 9 9  WIDTH (in)  DEPTH (in)  STRESS (psi)  1.50 1.52 1.51 1.51 1.50 1.54 1.53 1.52 1.52 1.53 1.53 1.51 1.51 1.48 1.50 1.50 1.52 1.48 1.47 1.50 1.49 1.51 1.47 1.48 1.50 1.49 1.49 1.54 1.51 1.50  5.48 5.40 5.34 5.43 5.48 5.51 5.50 5.47 5.48 5.47 5.49 5.39 5.44 5.27 5.34 5.41 5.47 5.36 5.22 5.41 5.39 5.47 5.31 5.30 5.43 5.38 5.42 5.47 5.44 5.37  3000 3000 3000 3000 3000 3000 3000 3000 4500 4500 4500 4500 4500 4500 4500 4500 4500 4500 4500 4500 4500 4500 4500 4500 4500 4500 4500 4500 4500 4500  MOE (psi) (E+07) 0.20886 0.21830 0.20712 0.17978 0.17523 0.18611 0.15512 0.16186 0.18372 0.20467 0.17038 0.18369 0.21890 0.17380 0.19486 0.24174 0.21157 0.18681 0.21544 0.19991 0.19422 0.23949 0.22664 0.23473 0.22932 0.21067 0.20568 0.17893 0.19825 0.21665  PREDICTED i (mm) 27.0 26.2 27.9 31.7 32.2 30.1 36.2 34.9 46.0 41.4 49.6 46.8 38.9 50.6 44.5 35.4 40.1 46. 3 41.2 42.9 44.3 35.4 38.5 37.3 37.2 40.9 41.6 43.0 43.0 39.8  ACTUAL i (mm)  S.R.  30.0 30.0 31.0 35.0 35.0 36.0 41.0 39.0 47.0 44.0 52.0 48.0 43.0 55.0 47.0 36.0 45.0 50.0 45.0 47.0 46.0 38.0 42.0 41.0 41.0 44.0 45.0 47.0 47.0 45.0  49.7 58.6  COMMENTS  F.O.L. 36.6 F.O.L. F.O.L. F.O.L. -  -- • -  TABLE X c o n t i n u e d . BOARD NUMBER  2147 2838 827 1280 418 1507 2536 2321 1047 1584 244 411 2568 2008 2068  GROUP NUMBER  9 9 2 2 2 2 2 2 2 2 9 9 9 9 9  WIDTH (in)  DEPTH (in)  STRESS (psi)  1.49 1.50 1.51 1.50 1.51 1.50 1.48 1.52 1.50 1.51 1.50 1.52 1.48 1.51 1.50  5.25 5.46 5.42 5.36 5.43 5.48 5.39 5.42 5.41 5.38 5.39 5. 35 5.40 5.47 5.46  4500 4500 : 3000 3000 3000 3000 3000 3000 3000 3000 4500 4500 4500 4500 4500  MOE (psi) (E+07)  0.21364 0.21513 0.23702 0.19070 0.17845 0.17628 0.23947 0.18708 0.22140 0.17286 0.16900 0.22179 0.19119 0.17730 0.19110  PREDICTED i (mm)  41.3 39.5 24.1 30.2 31.9 32.0 23.9 30.5 25.8 33.2 50.9 39.1 44.9 47.8 44.4  ACTUAL i (mm)  45.0 45.0 26.0 34.0 35.0 35.0 26.0 32.0 26.0 37.0 54.0 44.0 51.0 54.0 51.0  S.R.  -  29.3 28.8 52.0 51.3 37.0 31.6 31.5 36.4  --  COMMENTS  STRESS RATIO (%)  STRENGTH (psi)  FRACTIONAL CREEP (average) a t 3 months  NUMBER OF DATA POINTS  (%)  0<SR<30  S_>10000 B  43.3  17  30<SR<40  10000>6 >7500 B  47.8  16  40<SR<50  7500>6 >6000 B  45.3  13  50<SR<60  6000><5 >5000 a  55.4  12  60<SR<100  5000><S >3000  62.3  6  T5  D  TABLE XI.  F r a c t i o n a l creep expressed a t three months f o r f i v e ranges o f s t r e s s r a t i o s . (Average values.)  3000 PSI TIME  A  4500 PSI B  3  (x 10" )  A 3  (x 10~ )  B 3  (x 10" )  1/mm  3  (x 10" ) 1/mm  5 min  0. 672  0.484  -13.908  0.647  1 hour  0.758  0.973  -27.566  1.368  10 hours  0.789  2.175  -20.856  2.069  2 days  1.391  3.497  12.079  2.485  4 days  1.792  4.378  -4.553  3.789  3 weeks  2.261  9 .169  -62.659  8.731  6 weeks  1.623  13.515  -91.492  12.554  9 weeks  0.393  15.377  -119.367  14.694  3 months  0. 317  16.136  -90.597  14.726  0.219  16.309  -105.590  15.206  13 weeks  TABLE X I I .  C o e f f i c i e n t s A and B o f the c o r r e l a t i o n equation between creep deformation and e l a s t i c deformation f o r two s t r e s s l e v e l s .  3000 PSI and 4500 PSI TIME  A  B 3  (x 10" )  3  (x 10" ) 1/mm  5 min  -0.089  0. 399  1 hour  -0.227  0.844  10 hours  -1.297  1. 855  2 days  -1.658  3.093  4 days  -1.259  3.997  3 weeks  -4.438  8.133  6 weeks  -8.226  11.789  9 weeks  -9.679  13.447  -10.070  14.168  -1.0.115  14.320  3 months 13 weeks  TABLE X I I I .  C o e f f i c i e n t s A and B o f the c o r r e l a t i o n equation between creep deformation and e l a s t i c deformation, d regard o f the s t r e s s l e v e l  3000 PSI and 4500 PSI D  C  TIME  2  (x 10" )  5  (psi x 10 )  5 min  -2.439  0 .731  1 hour  -2.786  1.094  10 hours  -1.949  1.634  2 days  -1.842  2.432  4 days  -2.863  3. 202  3 weeks  -2.830  6.028  6 weeks  0.658  7.925  9 weeks  4.920  8.248  3 months  6 .060  8.512  6.067  8.615  13 weeks  TABLE XIV.  C o e f f i c i e n t s C and D o f the c o r r e l a t i o n equation between f r a c t i o n a l creep and the modulus of e l a s t i c i t y , d i s r e g a r d o f the stress l e v e l .  STRESS  FIGURE 1.  S t r e s s and s t r a i n versus time diagrams f o r a specimen o f a h y p o t h e t i c a l m a t e r i a l . From N i e l s e n (1972).  112  STRESS  FIGUEE 2.  F r a c t i o n a l creep i n bending. Specimens S t r e s s as percentage s t r e n g t h a t 21.5 C. and C l a r k e (1961).  Hoop Pine i n i n small s i z e . of ultimate From Kingston  FRACTIONAL CREEP, .12-1  (  (1/256 -1 day) 1  1—  1  1  1  • .10 *' .08  .OA  o  • o  _  1000  \  M  )  9  -IP50  f  0  o  Symbols Small Large  .02 500  -  ^  «  •  ()  10  o  • 20  • *  •  °  •  65%RH  80%RH  Trend  •  •  30 AO 50 STRESS INTENSITY,  60 76 6 / 6 , •/. B  '  FIGURE 3-b.  FIGURE 3-a.  n  °  o o  500  /  o  '  1000  A  1  .06  3000  •  D e v i c e s u s e d t o measure c r e e p o f wood i n b e n d i n g . Two d i f f e r e n t specimen s i z e s . A l l dimensions i n mm. C e d e r b e r g and D a n i e l s s o n (1970) .  The i n c r e a s e i n f r a c t i o n a l c r e e p o f t h e s m a l l and t h e l a r g e s p e c i m e n s between 1/256 day and 1 d a y as f u n c t i o n o f stress intensity. Data from C e d e r b e r g and D a n i e l s s o n , r e p l o t t e d by N i e l s e n ( 1 9 7 2 ) . ;  0  1—  1  i — I  1  2  3  1  4  1  I  l  5  6  7  i  8  l  9  t  10  Time (10 hr) 3  FIGURE 4.  Creep compliance versus time f o r 4 s t r e s s l e v e l s express as percentages o f p r o p o r t i o n a l l i m i t . From Nakai (1978)  115  to /\\ LO LU cn i— 0 0  —> TIME  Virgin  creep  curve  Recovery  curve Reloading  curve  TIME  FIGURE 5.  T h e o r e t i c a l development o f creep under i n t e r m i t t e n t l o a d i n g . From N i e l s e n (1972).  116  NO.  TYPE 20°C Const. (120 ran). 30°C 40°C 50°C 20°C (10 mn) 50 °C 30 °C ( 50 °C 50 °C 40 °C 4 0 mn )— 20°C 50 C ' 40 mn 30°C ) - 50 °C 40 mn 40 °C ( ) 50°C 30 °C (30 mn)- 50 °C 30°C (50 mn)- 50°C 50 C (5 mn) - 40°C  1 2 3 4 5 6 7  II  8  6  9 10 11 12 13  (110 mn) ( " ) ( " ) (90 mn)  No. (  5  )  (125 mn) 6 7  i  O  £  e i—  o  o  LU o . _) U. LU Q  120  A  10  50  100  0  •> TI M E FIGURE 6.  20  ( mn )  Creep curves f o r v a r i o u s types o f temperature. From K i t a h a r a and Yukawa (19 64).  C R E E P S T R A I N . E , W-S  C R E E P 10"  c  2000 i  h-lh.M-S  •  '  •  •  •  *-  1/8 x 50 K 500 mm  1500 169 10' -sinh (0.0U8<j) 6  I  1000  500  0 TIME, hours  010  B  FIGURE 7.  Creep v e r s u s time f o r hardboard and plywood. From Lundgren (1968 r e p l o t t e d by N i e l s e n (1968a).  0  25  50  75  100 125  150 175 200 k p / c m *  6  10  20  30  AO  60  50  70  80 %  _3  F i g u r e 8.  Creep from 10 hours to 1 hour o f 1/8" f i b e r b o a r d i n t e n s i o n versus s t r e s s , a , and versus f r a c t i o n a l s t r e s s , a/a_,. Data from Lundgren (1968) r e p l o t t e d by N i e l s e n (1972) . (Masonite a t 65% RH).  118  Steps  FIGURE 9.  Sample I.  Grouping o f the m a t e r i a l .  0.0  20.0  40.0  FIGURE 10.  60.0  60.0  100.0  FAILURE STRESS (PSI)  120.0 „  (XlO  2  )  140.0  160.0  Strength d i s t r i b u t i o n o f the c o n t r o l sample from Sample I . Raw data; three parameter Weibull d i s t r i b u t i o n .  160.0  200  120  PULLEYS  SIDE VIEW OF TEST  SET-UP  SPACER  LEAD WEIGHT  END VIEW OF TEST FIGURE 11.  Test set-up  SET-UP  f o r Surrey experiment.  FIGURE 12-1.  Test set-up f o r Surrey experiment. P i c t u r e as an end-view.  FIGURE 12-2.  Test set-up f o r Surrey experiment. P i c t u r e as an end-view.  123  CM  i to  s -M  LU  .M  0  -|—> 10" 1  0  -M  i i mii[—i i 1111111—i i | n i i i | — I I | MIM|—I I I I I — I 4 10° 4 10 1 4 10 2 4 10 3 4 10" TIME (HOURS) (LOG SCALE)  FIGURE 13.  I  I 4  D e f l e c t i o n versus time f o r four s t r e s s l e v e l s . Raw data from Surrey experiment.  10 5  124  10"  i 11 m i | — i i 111 n i | — i i 111ni|—i i 111 — i i 111 4 10° 4 10 1 4 10 2 4 10 3 4 10 4 TIME (HOURS) (LOG SCALE)  FIGURE 14.  I  I | Ml!l|  4  D e f l e c t i o n versus time f o r four s t r e s s l e v e l s . C o r r e c t e d data from Surrey experiment.  10 5  125  Steps —  ©  -  -<—'  to  c CO £ o O; Q.  —  o LO  •  1 (100  spc.)  Gr o u p  2  ( 1 0 0 spc . )  Group  3  (100  spc.)  Group  4 (100  spc.)  Group  5 (100  s p c.  Grou p  6  (1 0 0  spc.)  Group  7  (100  spc  Grou p 8  (100  spc.")  Group  ( 100  spc.)  G r o u p 10 ( 1 0 0  spc.)  Group  11 ( 1 0 0  spc.)  Group  12 ( 1 0 0  spc.)  Group  13 ( 1 4 0  spc.)  Group  14 ( 1 4 0  spc.)  9 4  spc.  11  spc.  O  o  o _i  O  LU o  Group  c o  3  cn  c  9  |  G r o u p 1 5 ( 1 4 0 s p c .)  a  ~\ 79  spc"  18  spc.  13 9  spc.  o  —•  >-  G r o u p 16  (1 40 s p c.)  G r o u p 17  (140 spc.)  G r o u p 18  (140  CL  . £ a cr  ^Control  sample  spc.)  to a*  cr  Group  1 ( 1 4 0 s p c .)  LL. <j,  ' £ LT) — — — i  o a.  —  CL  Grou p 2  a o  DO LO  .a . c£  FIGURE 15.  (1 4 0 s p c . )  Group 3 ( 1 4 0  13 7  spc.  spc.)  Group  4 { 1 4 0 sp c.)  Group  5 ( 1 40 s p c . )  Sample I I . Grouping of the  material.  FIGURE 16.  S t r e n g t h d i s t r i b u t i o n of the c o n t r o l sample from p o p u l a t i o n I I . Raw data; two and three parameter Weibull d i s t r i b u t i o n s . 1  y-  0.0  15.0  30.0  FIGURE 17,  45.0  60.0  75.0  STRENGTH (PSI)  90.0  „  (XlO  2  )  105.0  120.0  135.0  Strength d i s t r i b u t i o n o f the c o n t r o l sample from p o p u l a t i o n I I . Raw data; lognormal d i s t r i b u t i o n .  150.  129  FIGURE 19.  P i c t u r e of Richmond t e s t  set-up.  :pt c ^ Cs-KD 1/1  }  TliEe  2^  NO  . 3 srs  2,7-/5 f&  f 9i>  ±1.1.  JoX  Z5^  on  .1  FIGURE 2 0 .  Charts of load and d e f l e c t i o n v a r i a t i o n s with r e s p e c t t o time during t r a n s f e r of the load.  FIGURE 21.  Example of creep curves p l o t t e d on a b i - l i n e a r graph. Data from the Richmond experiment.  132  22.  Examples of creep curves p l o t t e d on a f u l l l o g a r i t h m i c graph ( t o p ) , and s e m i - l o g a r i t h m i c graph (bottom). Data from Richmond experiment.  133  3000  PSI  Applied  stress  Groups  no.  8  8,  6  (mm)  6  70,  (mm)  60 3  z  I I0"«  '  I ' I ""I  3  5  10"  1 3  I  l'»U  i  5  IIC  1 3  I  i  11»i|  S 10'  1 3  | i | nii|  S 10'  1 3  i  i  |nn|  5 10*  r3  | i | Tin)  S 10'  TIME (HRS) (LOG SCALE)  FIGURE 23.  Creep curves f o r boards of sub-group Richmond experiment.  1.  3000 PSI Applied stress G r o u p s no  8 & 2  BN  (mm)  36 1557  33.  29  I  I0"«  '  I I M'»|  1  | t -| > 1111  3 5 10-' 3 S 10*  1  U 32  | I | MII|  3 5 10"  1 1  HHII'I  3 5 10'  TIME (HRS) (LOG SCflLEJ  FIGURE 23 continued.  1  | I |llll|  3 5 10*  1  | I | llll|  3 5 10'  135  , ' 10-«  '  ' ''""i 3 S 10-'  1  I 1 1 ""I 3 S 10*  '  I 3  1  I ""I  5 10'  • I ' I MM| r | 11 " i i | 1 | i | iitii 3 5 10' 3 5 10" 3 5 n.  TIME (HRS) (LOG SCALE)  FIGURE 23 continued.  3000 PSI Applied stress Groups  no. 8 & 2 91  1  '  ' '  3  '  5 10-  i •' " " l  3 5 JO*  1  1  1  <" I '  3 S JO'  3 5 JO'  1 - | I | ITTT)  TIME (HRSJ (LOG SCflLEJ  FIGURE 23 continued.  3 5 10-  6  r.T I I Hill  3  510'  1  (mm)  137 3000 P S I Applied stress G r o u p  1  10-  ' ' 3  FIGURE 24-1.  1  1  S  10-  I I ""I S 10* 1  3  ' I 3  I  n o .  |ini| 10'  5  7  1 | i I mil 3  5  0»  r  TIME (HRS) (LOG SCALE) T  I  3  i I run 5 0.  1 i i mm 3  5  0-  Creep curves f o r boards o f sub-group 2 loaded to 3000 p s i . Richmond experiment.  138 3000 PSI Applied stress Group  ^ 'j'j'ii ' m  1  no  2  'J'S z * m  'j'j'z 3  TIME (HRS) (LOG SCALE)  FIGURE 24.1 c o n t i n u e d .  5  "  1 3  5 ,0  '  3000 PSI Applied stress Group no  TIME  2  (HRS) (LOG scflLE)  FIGURE 24.1 continued.  5 ,0  *  3 s  140 (mm)  (mm)  UJ  o  o Si  to 4500  p s i  .  R  l  c  h  m  o  n  d  experiment,  3000 psi : 0. 4 37 8 • 1.07 3 • 6 Rsq = 0.9127  6 = Q  i 0.0  1 10.0  ~i 20.0  p  (mm) $  x  Q  :  3000  x  ;  4500 PS]  i  1  30.0  40.0  PS)  i 50.0  PREniCTED ELASTIC DEFORMATION (MM)  FIGURE 25.  1 60  A c t u a l e l a s t i c deformations versus predicted e l a s t i c deformations. 2 s t r e s s l e v e l s . Data from Richmond experiment.  o  0.10  0.12  0.U MODULUS  0.1 6 O F  0.1 8 0.20 ELASTICITY  0 22 ( psi » 10  7  0 24 )  0 26  FIGURE 26. Cumulative p r o b a b i l i t y d i s t r i b u t i o n s o f the modulus o f e l a s t i c i t y obtained by two d i f f e r e n t methods. Hemlock.  0.2 8  0.30  o  0  10  0.12  0 U MODULUS  FIGURE 27.  0 16 0.18 OF E L A S T I C I T Y  0 20 ( psi  0.2 2 « 10  0 24 7  0.26  0.28  0 30  )  Cumulative p r o b a b i l i t y d i s t r i b u t i o n s of the modulus of e l a s t i c i t y obtained by two d i f f e r e n t methods. A m a b i l i s f i r .  144  1 00 8 hrs ( 6 weeks)  tr  t= 1 5 1 2 h r s (9 w e e k s )  «1 %  O  X  oft ec so  0.0  0  13.0 20.0 30.0 ... 4.0  50.0 CO.O  1  t = 504 hrs ( 3 w e e Us)  0  0  oo ? i %X  OO ^ o da  - 1 —  0.0 10.0 ... » «  ~  © X  K  30.0  20.0  40.0 50.0 60.0  * X  ,  °6 (^oo?' o o  t = 20 1 6 hrs ( 3 months)  B-i  0.0  10.0  20.0  30.0 40.0 50.0 60.0 t = 96 hrs X ** X X  £ E  K K *  3.0 13.0  V  47  —I •—I 1 1 20.0 30.0 40.0 0 0 t =5 4. 8 hrs  Ro  o  nO  I—  <P  jpo  o  3  O LU  —r-  0.0 13.0  _J Ll_  i •  20.0  eo.o  40.0  IO.O  o.o  40.0  20.0  t= 21 8 4 h r s ( 13 w e e k s )  t = 10hrs  LU Q  FO.O SO.O  a. UJ LU  or o  0.0 10.0  30.0  40.0 SO.O 60.0  o •  t= 1 hr  X  1  3000 PSI 4300 PSI  s  X  0.0 13.0  .20.0  30.0 Time  aa -  &n< x  3 CT  40.0 50.0 60.D  "  x  x  JtXx  o „o_x*x  t = 5 mn  j o ^"6 o  —I 10.0  30.0  40.0 50.0 60.0  ^ELASTIC  FIGURE 28.  0  10.0 20.0  DEFORMATION  I—  40.0  (mm)  Creep d e f l e c t i o n , measured a t 10 d i f f e r e n t times, versus e l a s t i c deformation, f o r 2 s t r e s s l e v e l s . Raw d a t a .  a a  a—i MOE.  Board no. a. 03  3Io  (x10  2509  0.1 7 8 6 3  2 238  0,1 8611  2 52 2  0.17772  2497  0.24174  1 6 53  0.1 72 62  7  PSI)  S.R. (%) 85.6  /  4 4.7  -  LU  2 5 0 9 ( 3000  PSI)  2238 ( 3000  PSI)  2522(3000  PSI)  2497 (4500  PSI)  1653 ( 3 0 0 0  PSI)  36 m m LU  a rvi  a  <  10-'  I  3  1  I  1  5  "'I  10-'  I  I  1  3  FIGURE 29.1.  I • • *"I  5  10°  1  I I I Mil|  3 5  10'  1  I I I Mll|  3  5 10'  1  TIME (HRS) ( L O G SCflLEJ  l l l llll|  3  5 10*  1  I I 1 I lll[  3 5 10*  Creep curves f o r boards with same value of e l a s t i c deformation (36 mm).  Ul  Board no. o a. 03  M.O.E. (*10  2 363 2 S84 2 82 6  7  PS!)  SR.  (7o)  3 5.9 59. 5  0.1643 6 017134 0.23949  2363 ( 3000 PSI)  3Co ID  2884 (3000 PSI) 282 6 (4500 PSI ) 38 m m LU Q  o o  —i—| H " l 10-* 3 5 10-» 1  r  —I  1  3  H 11"| 5 10°  TIME FIGURE 29.2.  1—| r| int| 1—| i 11ni] 1—| i 11 inr 3 5 10' 3 5 10* 3 5 10'  (HRS)  ' i i i"ni  3 5 10*  (LOG S C A L E )  Creep curves f o r boards with same value of e l a s t i c deformation  (38 mm).  o— Board no.  M O E . (*10  1973 555 13 32 2489 41 0  a. cc  PSI)  7  0.14893 0.1 5825 0.1 551 2 0. 23 473 0 22932  SR. (%) 32.8 50. 2 1973 (3000 PSI ) —  -  555 (3000  PSI)  1332 (3000 PSI) 24 89 ( 4500 PSI) c_)  a  LU  -  41 0 ( 4500 PS I )  4 1 mm  U_ LU  a o  a a '  10 *  1  | i | mi | 3 5 10->  FIGURE 29.3.  1—| i |  3  5  fi • | 10°  T  1—| i | in»|  3  5  10'  1—| n; '"H  3  5  10*  TIME (HRS) (LOG SCALE)  ""1 10*  1—| H  3 5  1  3  — I 5  1  "1' 1 10 M1 4  Creep curves f o r boards with same value of e l a s t i c deformation (41 mm).  o—• Board no. 22 1 0 254 9  o.  M O E . (*10  7  PSI)  S.R. (%)  0.1 4 580 0.22 66 4  74.7-  2210(3000 PSI) 254 9 (4 500 PSI)  42 mm LU LU O  o. ro  a  i-| Tnq 3 5 10->  — I — |  a 10-*  1—| H iur|  3  5 10°  TIME FIGURE 2 9.4.  1—| i 11 ni|  3  5 10'  1—| n " 1 ,T  3  5 10*  —I ' \ ""I 3 5 10»  1  1—| H  3  "H 5 10» 1  (HRS) (LOG SCALE!  Creep curves f o r boards with same value o f e l a s t i c deformation (42  Board  no.  19 A 7 1602  o a_ 03  M.O.E. (*10  7  PSI)  S. R. (%) 29.7  O.U 5 2 9 0. 21 890  —  194 7 (3000 PSI) 1602 (4500 PS I )  H o '—  ID  43 mm  LU LL. LU  a a a. CM  io-* FIGURE  - i — | ' i TTiiq 3 5 10->  29.5.  1—i i~| i n i | 3 5 10°  1—| i | 3 5  10"  1—| l | MIT[ 3 5 10*  TIME (HRS) (LOG SCALE)  1—| i 11 n i | 3 5 10' 10»  1—i H in-q 3 5 10  d  Creep curves f o r boards with same v a l u e s o f e l a s t i c deformation (43  o a.  B o a r d no. 1969 22 07 21 4 7 2600 2067 2 838 932  a a. OD  2Eo  MOE.  (K1  0  7  PSI)  S. R. (V.)  0.21 544 0 1314 4 0. 2 1 3 6 4 0.20568 0.21 1 57 0.21513 0.216 65  1969 (4500 PSI )  58. 3  2207 ( 3 0 0 0 PSI) 214 7 ( 4 500 PSI )  -  2 6 0 0 ( 4 5 0 0 PSI) 2 06 7 ( 4 5 0 0 P S I )  45  mm  2 8 3 8 ( 4 5 0 0 PS I)  LU  9 3 2 (45 00  9-  I  Li_ LU  PSI)  a  a a'  10-*  | i | r T111 3 5 10-'  1—| i | nn| 3 5 10°  1—| i | nif| 3 5 10'  1—|  3  5  r\ nnj 10*  3  1—| i T mi] 1—| i 11 nq 5 10» 3 5 10*  TIME (HRS) (LOG SCALE) FIGURE 29.6 .  C r e e p c u r v e s f o r b o a r d s w i t h same v a l u e s o f e l a s t i c d e f o r m a t i o n  O  (45 mm)  151  t-100 8 hrs I 6 w ee ks )  8) O  P-  6  .  o »  t = 1 5 1 2 hrs ( 9 weeks )  • »»T  x «  e  *  - i — 30.0  a  *  t= 504 hrs (3 weeks)  8T  18 • -  o  oe»" 1  0  ~I— :o.o  -I 10.0  «x *  eo J B " ^ o x ^ " O O » „ X X O * 8 O  R-  "  0  - i — 40.0  50.0  60.0  0  —1 20.0  30.0  t = 2016 h r s  40.0  SO.O  ( 3 mont  60.0  hs)  t = 96hrs 0  S  RX*  10.0  8  8-  so.o  20.0  D_ LU LU  x "„ „ x * *  a.o  t= 48 hrs  or  o  o «f»o  »,  0  &'ff»Vfc X X X X 13.0  o  X  o  30.0  0.0  xx « x x 60.0  40.0  —I 20,0  0.0  t = 10hrs 0  — I—  R-i  <  :-  or U- _  0  50.0  60.1  —\ t = 2184 hrs SO.O ( 13 w e e k s ) Qt  MOD K l  „ t 4500 PSI — i — 10.0  —r 20.0  90.0 .  40.0  t = 1 hr  A —I— 10.0  20.0  X.O  Time -I— 13.0  20.0  SO.O  ) 40.0  SO.O SO.O  oP  X x  ,«6"<5  fS^BX^x  1 30.0  40.0  O.O  t = 5 mn  40.0  SO.O  ^.ELASTIC  FIGURE 30.  I  60.0  —1— 10.0  DEFORMATION  50.0  (mm)  F r a c t i o n a l creep expressed a t 10 d i f f e r e n t times v e r s u s e l a s t i c deformation, f o r 2 s t r e s s l e v e l s . Raw d a t a .  9  t = 50 A hrs ( 3 weeks )  a" X  ° M  «  X  «  as  » «  «  e a — I — 0.13  e  — I —  —I  0.16  —>  a»  1.2  _  K  a a  •>  t = 1008 hrs ( 6 weeks)  *  i " x o«  X  o  a —I  — i —  I  0.22  O.M  0.26  —I 0.21  3.16  t: 96 hrs •*> CL Ixl LU  CH  1  o.i  1  o.i2  o.i4  * « ! • % & 4 | ^ a" a"  •„ a ~1 a"a«61a-ea* a e1 1 P a" « 1 o.ie a.it 0.2 0.22 0.24 0.26 1  r2.2  *  a *  < z o  0.1  0.12  0.14  J.l»  0.10  1  0.2  a  p  1  0.22  1  0.24  1  0.»  0.21  cr  0.1  T  0.12  0.14  -—i  1  0.16  0.(1  i 0.2  —^ 0.22  1  •  B  _a  a  0.1  t= 1 0 hrs  1  0.24  M  1  oi  <  t = 1512 hrs ( 9 weeks )  * aa  /•/ 0 8  1  0.26  oT»a  Iu4  oT»  iui  o^a  0T22  a 0.1  0.12  * a •»•« |MIH — l a ••» 0.14 0.11 0.11 0.2 1  aw "na-) a • 0.22 0.24 0.21  iTa  Zu  r~ 0.1  1.1  t = 2184 hrs ( 13 weeks )  t= 1 hr 1  OJ  • 1  % 1  Zn  _ a ,  •a a 1  Zn  o>t  x  "  t= 48 hrs  <_>  1  (Lo  #  *  —  M  o'.io .  a  0  ,  ,  0.21  0.1  lime t: 5mn  a 9  ,B  0.1  0 12  , " ~,«mr*»^v.w.u*. •, i II 0.11 0.2 0.22 0.24  0.14  M.0. E. ( psi * 1 0 ) 7  FIGURE 31.  0.20  0.20  3.1  01  -i—  0.12 .  F r a c t i o n a l creep, expressed a t 10 d i f f e r e n t e l a s t i c i t y , f o r 2 s t r e s s l e v e l s . Raw data.  1 il.)4  1  0.16  1 0.1»  1  0.2  1  O.a  1  0.24  times versus modulus of  r  0.21  1  0.21  1  1.1 Ul  to  153  6 0 %<S.R.< 100 %  c E IT)  5 0 %<S.R.< 6 0 %  0.1  -i0.12  r  0.14  O ©DO -0i.16  B  0.18  r-  0.?  0.72  0.24  0.76  0.76  0.3  D_ LU UJ  4 0°/o<S.R.< 5 0 %  rr  0.1  0.12  .  0.16  0.14  0.2  0.72  0.24  ~ 1 0.26  1  0.78  10.3  30%<S.R.< A 0 %  I—  O <  rr LL 0.1  -i0.12  1  0.14  0r O O QD  0.16  0.18  0.2  -e*a 0.22  q  0.24  r0.26  0 %<S.R.< 3 0 %  — i — 0.12  M.O. E . ( ps i  1 0  FIGURE 32.1. F r a c t i o n a l creep p l o t t e d a g a i n s t the modulus of e l a s t i c i t y f o r s p e c i f i c ranges of s t r e s s r a t i o s , a t time= 5 minutes.  154  60 % < S . R . < 100 %  0.1  1  1  1  1  — i  1  1  1  1  1  0.12  0.14  0.16  0.10  0.2  0.22  0.24  0.2S  0.26  0.3  5 0 %<S.R.< 6 0 % O  B-  e °o a  -I Q_ LU LU  0.1  0.12  1 0.14  1  0.16  ° i °—<«i  " i  1  1  1  0.1B  0.72  0.74  0.76  0.76  0.7  1 0.3  4 0 °/.<S.R. < 5 0 7o  CC  O 0.1  o  O  1  1  1  1  0.12  0.14  0.16  0.16  »  I " 0.2  " — I — 0.22  1 0.24  1 0.26 -  1  1  0.26  0.3  I—  3 0 % <S.R. < 40 V.  O  <  *  CC o  LL.  A  0.1  1 0.12  1 0.14  0  a  r « — " — i 0.16 0.10  =*i—" 0.2  1° 0.72  i 0.24  1 0.26  1 0.20  1 0.3  0 % < S . R . < 3 0 7o  Ba ! — » 0.1  0.12  0.14  0.16  0.16  0.2  0.2}  ii—a_ * — »  rn » , »1 0.24  0.2S  I 0.20  1 0.3  ^ M . O . E. ( p s i x 1 0 ) 7  FIGURE 32.2.  F r a c t i o n a l creep p l o t t e d a g a i n s t the modulus of e l a s t i c i t y f o r s p e c i f i c ranges o f s t r e s s r a t i o s , a t time= 1 hour.  155  6 0 %<S.R.< 1 00 %  oo ° 5 0 %<S.R.< 6 0 %  1 0.1  0.12  1 0.14  o 1  00  0.16  1  *  0.18  1  1  1  1  1  0.72  0.24  0.26  0.28  0.3  l  0.?  LU LU  CC  4 0 %<S.R.< 50 %  O I  <  e  1  B  0.1  0.12  1  0.14  1  0.16  ©©o  feo  1  Q  1  0.18  1  0.22  0.2  1  1  0.24  0.26  1  0.28  I  0.3  I—  O  < Ct  *  3 0 % <S.R< 4 0 %  e  o  LL  -i  0.1  0.12  1  0.14  _fl a_ r — — " — i 0.16  Is  1  0.18  .0.2  0.22  1  1  0.24  0.26  1  0.20  1  0.3  0 % <S.R.< 3 0 % o . .  1  0-1  0.12  i  3.14  „ °  ° e„  8  o, „  1  1  i  1  1  1  1  1  0.16  0.16  0.2  0.23  0.24  0.26  0.21  0.3  7  _^ M. 0. E. ( ps i x 1 0  )  FIGURE 32.3. F r a c t i o n a l creep p l o t t e d a g a i n s t the modulus of e l a s t i c i t y f o r s p e c i f i c ranges o f s t r e s s r a t i o s , a t time = 10 hours.  156  6 0 °/,<S.R.< 100 % »1  Ul  1  0.1  0.12  1  0.14  1  I  0.16  1  0.16  0.2  1  0.22  1  1  0.24  0.26  1  0.26  1  0.3  CO  5 0 % < S . R . < 60 %  —i 0.12  1  0.14  i 0.16  1  1  0.16  0.2  1  0.22  1  1  0.24  0.26  r— 0.26  -i 0.3  0_  LU UJ  cr o <  4 0 % <S.R. < 5 0 %  1  0.1  0.12  1  0.14  1  1  0.16  1  0.16  0.2  o  A  1  1  0.24  0.26  3 0 %<S.R.<  < cr. L L  1  0.22  O  — i — 3.12  1  0.3  40 %  0 ° '  -|  —r— 0.14  1  0.2B  —I  —I  0.24  0.16  0.26  0 °/o<S.R.< 30 %  —i  0.12  1  0.14  a  1—"  0.16  1  0.16  p. a  1  0.2  e 8 • a a a a i  0.22  • 1  0.24  a 1  0.26  1  0.20  1  0.1  _^ M . O . E . ( p s i x 1 0 ) 7  FIGURE 32.4.  F r a c t i o n a l creep p l o t t e d a g a i n s t the modulus of e l a s t i c i t y f o r s p e c i f i c ranges of s t r e s s r a t i o s , a t time = 2 days.  157  6 0 7.<S.R. < 10 0 %  1/1  —i  0.12  1  1  1  1  1  1  1  1  1  0.14  0.16  0.1B  0.2  0.22  0.24  0.26  0.28  0.3  5 0%<S.R. < 60 %  Ol  ©  •  0.1  i  0.12  1  0.14  1  0.16  ©a  1  0.18  1  1  0.22  0.2  1Z. ? >  1  0.24  0.26  0.28  0.3  Q.  4 0%<S.R. < 5 0 %  LU LU  CC O  T3  o oo  < z  o — i—  „ i-  3 0 %<S.R. < 40 %  o  a  CC  0 °/.< S.R. < 3 0 %  » ©„ — i  1  0.14  0.16  ^  FIGURE 32.5.  1  0.16  1  0.2  0  °„ 1  0.23  8  ©  o  1  0.24  M.O.E. ( p s i x l 0  © 1  7  0.26  1  3.26  1  0.3  )  F r a c t i o n a l creep p l o t t e d a g a i n s t the modulus of e l a s t i c i t y f o r s p e c i f i c ranges o f s t r e s s r a t i o s , a t time = 4 days.  158  © o  B-  6 0 7o<S.R. < 1 0 0 % —i 0.12  1  0.14  1  0.16  1  0.16  1  1  0.2  1  1  0.24  0.??  0.76  1  0.26  1  0.3  50°/.<S.R. < 6 0 %  R-  0.1  0.12  0.14  —I  0.16  I  1—  1  0.16  0.2  I  0.22  I  0.24  0.76  I  0.2S  1  0.3  in  4 0%<S.R.< 50 %  2 e 8  "S.  0.1  __0.12  —1 0.14  ~I  °  —I  0.16  l> X  0.2  Q_ UJ  0.26  0.3  30 % < S . R . < 4 0 %  LU  cr 0  oo  —I—  0.12  i—  —1  0.14  1  0.16  „  1  2.16  1  0.2  1  "I  0.22  1  0.24  0.26  r 0.;  < cr (JL  A 0%<S.R.<  °° R-  IT 0.1  e e  o  i 0.12  1  0.14  1  0.16  1  0.16 .  1  0.2  1  1  0.22  0.24  3 07.  o i 0.26  1  0.26  1  0.3  ^ M. 0. E. ( p s i x 1 0 ) 7  FIGURE 32.6.  F r a c t i o n a l creep p l o t t e d a g a i n s t the modulus of e l a s t i c i t y f o r s p e c i f i c ranges o f s t r e s s r a t i o s , a t time = 3 weeks.  l/J  4 0%<S.R.< 5 0 %  oo  o  e  e  a  3 0 %<S.R.< /,0 V. CL  8i  o'.i2  LU UJ  CC o  TAA  oTii  TAB  an  a - 1 — 0.12  I  1  1  1  1  014  0.16  0.16  0.2  0.22  I  0.24  1 0.26  1 0.20  0.1  0T22  0T24  ' o'.ie  0T20  0.1  tTm  0.1  50%<S.R.< 6 0 %  9 i  < o  0T2  •>  °. R  I—  o <  o'.i  _  CC  0TT3  77*  Zit  oTn  7A»  OTM  oTJs  LL  A  0 % <S. R.< 3 0 %  ^  60%<S.R.< 1 0 0 % — i  0.12  1 0.14  1 0.16  1 0.10  1 0.2  J > M. 0. E. (psi FIGURE 32.7.  1  1  0.22  x  0.24  1 0.26  1 0.26  —1 0.12  1 0.14  1 0.1B  1 0.18  i 0.?  1 0.22  1 0.24  1 0.28  r 0.28  10 )  F r a c t i o n a l creep p l o t t e d against the modulus o f e l a s t i c i t y f o r s p e c i f i c ranges o f s t r e s s r a t i o s , a t time = 6 weeks.  1 0.3  40 %< SR. < 50 7, —i  0.14  —  1  0.16  1  0.16  1  1  0.22  0.2  1  0.24  1  0.26  1  0.26  1  5 0 7,<SR < 607.  0.3  a  S i  D_ LU LU  LT.  ° a  4-  a  Ri  O  * i  —i  30 7. < S.R. < 40 7.  0.1  0.12  —I 0.14  1  0.16  1  0.16  1  1  1  0.22  0.2  <  0.24  1  0.26  1  0.20  0.12  1  0.14  1  0.16  1  0.18  1  0.2  1  0.22  1  0.24  1  0.26  1  0.28  n 0.  I  0.3  0 7. <S.R. < 307.  LL.  A «i  o.i  6 0 7.<S.R. < 100 7.  Ri  Ri  o^ia  o^  Z* Zit Zi > M.OE. (  FIGURE 32.8.  ps1  Zn Z»  Zx  Zn  0.3  "Zi  Zii Zi*  oae  ZTa Zi Zn Zu Zx Zx  v 10 ) 7  F r a c t i o n a l creep p l o t t e d a g a i n s t the modulus o f e l a s t i c i t y f o r s p e c i f i c ranges o f s t r e s s r a t i o s , a t time = 9 v/eeks.  0.  • •  30 %  GO  ~-  1  r  ~  &  < S.R. < AO  7!i2  %  77A  o»  O'.IB o'u :  AOVo<S.R.<  77i  7.a q'.M  oTn  0T24  IKS  oTa  1  0 V. <S.R.  1  0.12  1  0.14  1  0.16  1  1  0.1B  0.2  .  1  1  0.22  0.24  77,  77*  (psi  x 1 0  )  1  0.28  1  I  0.28  0.3  < 30 V.  60 V. <S.R.  si  Zi  %  0.3  0.1  \> M O E .  0.3  o1.a  5 0 % < S . R . < 60  O  oTia  "io'.»  50V,  77> 77t  °t  772 77A  77*  7\T Ii  7^  F r a c t i o n a l c r e e p p l o t t e d a g a i n s t the modulus o f e l a s t i c i t y f o r ranges of s t r e s s r a t i o s , a t time = 3 months.  77* specific  <  100%  77,7, i—•  H  1  8  ••  ««  0  40%<S.R.<50% 50 % < SR.< 60 %  si —  i  —  0.13  a  L»  1^16  TAB o l  o!?? ir?4  1  o ^ O'.TB .  o.s  3 0 °/.< S.R.<40%  si —i  0.12  o o  1  1  1  1  0.14 0.16 0.18 0.2  1  1  1  1  ~i  0.22 0.24 0.26 0.20 0.3 6 0 %<S.R.<100 %  —  i  —1  —  0.12  1  0.1B  0.3  1  1  1  1  8i 1  3.22 0.24 0.25 0.28 0.3 0 %<  S.R.< 3 0 %  o'.i2 i  0.1  1  1  1  1  1  0.12 0.14 0.16 0.10 0.2  1  1  1  1  0T14  0T16  OTIB  0T2  0T22  0T24  O~K  0T20 O .S  i  0.22 0.24 0.26 0.28 0.3  _ > M . 0 . E . ( psi * 1 0 ) 7  FIGURE 32.10.  F r a c t i o n a l creep p l o t t e d a g a i n s t the modulus of e l a s t i c i t y f o r s p e c i f i c ranges of s t r e s s r a t i o s , a t time = 13 weeks.  163  APPLIED  S T R E S S = 3 000  psi  IME 3 months 9 weeks 6 weeks  3 weeks  4 days 2 d ay s 10  hours  1 hour 5 minutes 10.0  20.0 EL AS  FIGURE 33.  C  30.0 40.0 50.0 60.0 D E F O R M A TI 0 N ( mm )  Creep deformation versus e l a s t i c deformation. A n a l y t i c a l model f i t t e d to data f o r 3000 p s i .  164  APPLIE D  STRESS  = 4500  psi  IM E  3 months 9 weeks 6 weeks  3 weeks  10.0  2 0.0 ELASTI C  FIGURE 34.  30.0 40.0 DEFORMATION  50.0 ( mm )  Creep deformation versus e l a s t i c deformation. A n a l y t i c a l model f i t t e d to data f o r 4500 p s i .  APPLI E D  S T R E S S = 3 000  psi  TIME 3 months 9 week s 6 weeks  3 w eek s  days days 10  hours  1 hour 5 minutes 0.0  FIGURE 35.  10.0  20.0 E LASTI C  30.0 40,0 50.0 D E F O R M A TION ( m m  60.0  F r a c t i o n a l creep versus e l a s t i c deformation. A n a l y t i c a l model f i t t e d to data f o r 3000 p s i .  FIGURE 36.  F r a c t i o n a l creep versus e l a s t i c deformation. A n a l y t i c a l model f i t t e d t o data f o r 4500 p s i .  APPLIED  STRESS  =  3 000  psi  TIME 3 9 6 3 4 2  months w ee k s w ee k s weeks days day s  .10 h o u r s 1 hour 5 mn 0.10  0.12  0.U MODULUS  0.16 OF  0.2 0 0.18 E L A S T I C ITY  0.2 2 0.2 4 ( p s i * 10 ) 7  FIGURE 37. F r a c t i o n a l creep versus modulus of e l a s t i c i t y . A n a l y t i c a l model f i t t e d to data f o r 3000 p s i .  0.2 6  0.28  0.30  APPLIED  STRESS  = 450 0  psi  "IME 3 9 6 3 4 2  months weeks wee ks week s day s day s  10 h o u r s 1 hour 5 mn 0.10  0.12  0.14 MODULUS  0.1 6 OF  0.18 E L A ST I C  0.20 TY  0.24  0.22 psi  *  10  7  )  FIGURE.38. F r a c t i o n a l creep versus modulus of e l a s t i c i t y . A n a l y t i c a l model f i t t e d t o data f o r 4500 p s i .  026  0 .28  0.30  169 t= 1 0 0 8 h r s (6weeks)  *1  •  B  t= 1512 hrs (9 w e e k s )  a  #a e  *  a  a  .•  8  I  I  49.0  00.0  .  e  ' V . »*  1  I  00.0  a*  a  100.0 if  e  t = 50 4 hrs ( 3 week s ) *  * a 3.0  •  A*  9  I  1  30.0  1  40.0  00.0  I  1  00.0  100.0  OJ  a S e a e _ .  3.0  r  i  30.0  40.0  1  1  00.0  80.0  t= 201 6 hrs ( 3 months)  1  !"» '  aa  t = 96hrs .•o  l 0.0  30.0  a at?  _e  %  a  e  ee »oi « * Oo* oo  a  e  a  a  a  |  1  1  1  40.0  80.0  00.0  100.0  a  i.  t= 48 hrs  D_ LU LU  i  i  30.0  CC  1  40.0  1  oo.o  1  ao.o  IOO.O  8  O o.o  ao.o  4D.o  ao.o  m.o  IOO.O  t = 21 84 hrs ( 1 3 weeks )  t = 10 hrs  < Z o -r  30.0  4D.o  ao.o  <  cc  ao.o  100.0  t = 1 hr o.o  30.0  40.0  ao.o  Time i.o  40.0  n.o  ao.o  STRESS  aa  Si  t= 5 mn ao.o  o  ioo.o  ioo.o  3.0  1  30.0  1  40.0  1  ro.o (I)  1  oo.o  R A T I O (V.)  FIGURE 39. F r a c t i o n a l c r e e p , e x p r e s s e d a t 10 d i f f e r e n t t i m e s , v e r s u s s t r e s s r a t i o , f o r 1 s t r e s s l e v e l : 3000 p s i . Raw d a t a .  :  ioo.o  170  APPLIED  STRESSES  . 3000 psi , 4 500  psi  M E  3 months 9 weeks 6 weeks  3 weeks  4 days 2 day s 10 h o u r s 1 hour 5 minutes 0.0  1 0.0 2 0.0 3 0.0 4 0.0 ELASTIC D E F O R M A' ION  FIGURE 40.  5 0.0 (mm)  60.0  Creep deformation versus e l a s t i c deformation. A n a l y t i c a l model f i t t e d to data p o i n t s f o r 3000 and 4500 p s i .  APPLIE D  3000  ps i , A 5 00 p s i  TIME  13 w e e k s 3 months 9 weeks 6 weeks  3 weeks  A days 2 10 1 5 10.0  41.  2 0.0 ELASTIC  d ay s hours hour minutes  30.0 AO.O 50.0 60.0 DEFORMATION ( mm )  F r a c t i o n a l creep versus e l a s t i c deformation. A n a l y t i c a l model f i t t e d t o data p o i n t s f o r 3000 and 4500 p s i .  APPLIED  STRESSES  TIME  . 3 0 0 0 p s i , A 5 00 p s i  3 months 9 weeks 6 weeks 3 weeks 4 days 2 days 10 h o u r s 1 hour 5 mn 0.10  0.12  O U  0 16  MODULUS  FIGURE 42.  0.1 8 OF  0.20 ELASTICITY  0.28  0 (psi  » 10' ]  F r a c t i o n a l creep versus modulus of e l a s t i c i t y . A n a l y t i c a l model f i t t e d to data p o i n t s f o r 3000 and 4500 p s i .  ^ to  APPENDIX A - l  MATHEMATICAL TREATMENT OF CREEP by BMD:02R COMPUTER PROGRAM.  CREEP DEFORMATION  I N I T I A L VARIABLES:  XI = 6  c X2 = SR X3 = MOE X4 = <S  e ADDED VARIABLES:  X5 = X2 X3 X6  = X2 X4  X7  = X3 X4  X8  =  X9  = X2  X4  2  XlO  = X3  X4  2  X4  2  1 BMD02R - STEPWISE REGRESS I ON - REVISED MARCH 27, 1973 HEALTH SCIENCES COMPUTING FACILITY, UCLA PR GBL EM C30E " " CRPOEF NUMBER OF CASES 63 NUMBER OF ORIGINAL VARIABLES 4 NUMBER OF VARIABLES ADDED 6 TOTAL NUHJER OF VARIABLES 10 NUMBER OF SUB-PROBLEMS 1 THE VARI A3 LE FORMAT IS (10X.4F8.3 t  4 5  6  ~T~ 8 9 TO 11 12 13 14 15 16 17 18 19 20 21 22 23 24  25  ~  26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52"" 53 54 55 56 57 58 59 60  '  VARIABLE MEAN 1 14.40954 2 41.34435 3 19.73979 4 29.54172 S 797.24268 6 1247.42139 7 570.56250 8 892.55151 9 38453.046 88 13 16849.933 59 1 CORRELATION MATRIX 0 VARIABLE 1 2 NUMBER  STANDARD DEVIATION 5. 72 505 14.25918 2.90453 4.4892 f 242.95103 543 .12256 6.60479 280.28857 21553.41797 2527.26147 3  1 1 .000 0.4019 -0.7358 2 1. 000 -0.4635 3 1.000 4 5 6 7 8 9 10 1 SUB-PROBLM 1 DEPENDENT VARIABLE 1 MAXIMUM NUMBER OF STWS 20 F-LEVEL FOR INCLUSION 3.0C0000 F-LEVEL FOR DELETION 3.000000 0.001000 TOLERANCE LEVEL  ~ S T E P NUiBER 1 VARIABLE ENTERED MULTIPLE R STD. ERIOR (JF EST. ANALYSIS OF VARIANCE REGRESSION  4 0.7785 0.4133 -0.9810 1.000  5  6  7  0.9591E-01 0.5923 0.9041 0.9431 -0.7057E-01 -0.6913 0.4453E-02 0.6713 1.000 0 .7135 l.OOu  -0.22 66 -0.3542 0.2557 -0.1897 -0 .2 7 75 -0.34 79 1.000  8 0.7917 3.5264 DF 1  SUM JF SQUAKES 1273.578  MfcM SQUARE 12 73.57a  F RAT 1 1C2.417  0  8 0.7917 0. 6856 0.8377 0.3944 -0.9597 -0.7880 0.9954 0.7978 -U.1590E--01 0.5338 0.6615 0.9708 - 0 . 1897 - 0 . 3204 i.ooo 0.7994 1.000  9  13 0.7693 0.3909 -0.9732 0.9972 -0.1651E-C1 0.6521 -0.1162 0.9925 0.7819 1.000  f  61 62 63 6* 65 66 67 68 69 7 0 ~ 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120  RESIDUAL  61  758.548  VARIABLES  VARIABLE  -0.02306 0.01617  (CONSTANT 8  EQUATION  IN  COEFFICIENT  12.435  VAKIAuLEi NUT IN  STO. ERkOR  VARlAbcE  F TO REMOVE  0.03160  102.4169  (2)  .  *  STEP N U M B E R  OF  0.15968 0.13960 -0.16322 0.177O0 0.14*77 -0.12734 0.14355 -0.22004  0. 8 4 4 5 0.0789 0.0091 0 . 9 9 97 0.5625 0.9640 0. 3 6 0 9 0.0149  1.5699 1.1925 1.6423 1.9543 1.3767 U.9B90 1.2623 3.0529  (2) (21 (2) (21 12) (21 (2) (21  VARIANCE OF 2 60  S U N OF S Q U A R E S 1310.305 721.821  IN  COEFFICIENT  VARI ABLE  (CONSTANT 8 10  V A R I A B L E S NOT IN VARIABLE  F TC REMOVE  1 0.01289 0.00143  8.9351 3.0529  (21 12 I  \ . .  • F - L E V E L OA T O L E R A N C E 1 SUMMARY TABLE STEP NUMBER  (NSUF^ICIENT  FOR FURTHER  VARIABLE ENTERED REMOVEO  1 2 OF RESIDUALS Y  8 10 "  MULTIPLE R  Y  "  '  PART I A L  2 . 3 4 5 6 7 9  CORK.  0.16271 0.00934 0.01780 0.18073 0.14281 0 .00661 0.11310  EQUATION TOLERANCE  0.8445 0.0504 0 . 0 0 34 0.9997 0.5612 0.6145 0.3520  F  TO E N T E R  1.6044 0.005 2 0.0187 1.9921 1.2284 0.0026 0.7645  (21 (21 (2) (21 (21 (21 (21  COMMUTATION  0.7917 0.8030 '  F RATIO 54.458  EQUATION  S T O . ERROR  22.10471 0.03852 -0.00250  MEAN SQUARE 655.153 12.030  •  CASE  2 3 4 5 6 7 9 10  0.8030 3.4685  EST.  VARIABLES  LIST  • "•<  10  REGRESSION RES10UAL  1  f TO ENTER ,  2  ENTERED  MULTIPLE R STO. ERIOR O F ANALYSIS  EQUATION  TOLERANCE  I  *  VARIABLE  PARTIAL CORK.  "  " "  RSQ  INCREASE IN RSC  0.6267 0.6448  0.6267 O.Oldl  F VALUE TO C M ER OK R E M O V E 102. 4 1 6 9 3.0529  INOE  NUHbEK OF VARIABLES  1 2  Ih  f  <•  ?  121 lot lc£  123 124 125 126 127 128 129 130 131 132  153  134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150  151 152  153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174  ITS 176  177 "178" 179 180  XI  NUMBER  1  1 2 ' 3 4 5 6 7 ""8 9 10 11 1 2 13 14 1 5 1 6 1 7 1 8 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 3 5 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 CASE  "  11  32. 0000 21. 0000 12. 5790 14.0000 7. 5800 18.5820 16.0000 12.5830 14. 5840 19.5860 11.0000 11.0000 14.0000 30. 6400 12.0000 14.0000 11.0000 11.0000 12. 0000 13.6460 12.0000 12.0000  14. 0000  13.0000 11.0000 6.6540 13.6540 16.6550 25.0000 26.0000 9.6590 8.6600 13.0000 9.0000  27.7lflo  12.0000 21.7110 12.0000 11.0000 19. 0000 23.0000 11.7180 16.0000 8.0000 16. 7230 23. 7240 14.0010 12.0000 7.0000 9.0000 9. ooor, 9.0000 15.0000 12.0000 10. 5100 Y  COMPUTED  25.5395 18.8129 12.8225 1C.8858 9.3148 15. 2861 17.7308 11.0126 13.9572 19.1C98 10.0880 10.6657 13.4317 24.0110 1C.5651 13.0258 13.9868 11.4386 11.8870 12.0334 13.5486 10.8398 12.1641 9.9072 14.4913 10.9442 16.9747 19.1341 21.0990 20.7952 9.5106 18.2693 13.0310 9.8922 32.8641 11.0992 17.9612 13.3194 10.5408 18.3726 25.2173 10.2235 19. 7644 12.8505 16.2267 14.2800 17.4663 11.9866 10.9323 13.0595 9. 5440 14.3754 12.7002 9.9469 13.6323 Y  RESIDUAL  6.4605 2.1871 -0.2435 3.1142 -1.7348 3.2954 -1.7308 0.626(1 0.4762 0.9120 0.3343 0.5683 6.62 90 1.4349 0.9742 -2.9868 -0.4386 0.1130 1.6126 -1.5486 1.1602 1.8359 3.0928 -3.4913 -4.2902 -3.3207 -2.4791 3.9010 5.2048 0.1484 -9.6093 -C.0310 -0.8922 -5.1541 0.9008 3.7498 -1 .3194 0.4592 0.6274 -2.2173 1.4945 -3.7644 -4.8505 0.4963 9.4440 -3.4653 0.0134 -3.9323 -4.0595 -0.5440 -5.3754 2.2998 2.0531 -3.1223  X( 81 1514.5242 1169.2295 846.5190 755.9746 521.4375 964.7859 110 7.62 52 6 47.2441 " 682.5059 1185.7690 663.6802 671.7947 946.8540 1433.8337 715.0278 878.1741 889.8879 697.4883 695.0603 735.4402 838.3337 706.9751 717.9187 573.2668 928.9082 665.6912 U04.2993 1177.3130 1288.7383 1312.6855 523.3115 1194.8774 806.3896 622.5525 1888.7720 636.9060 1161.6501 816.6445 627.2517 1134.4775 1474.4832 559.3220 1267.4309 793.0415 1003.2419 920.2124 1087.2844 754.5454 721, 4055 824 ?236 47^.4575 905 .1)281 770.9509 533.0554 876.2190  X( 10 1 22295.1C16 19354.2461 16775.4688 161*4.4219 13165.7695 1 7613.1250 1HH37. 3628 14426.4219 16876. 1602 19490.4805 15050.2734 14944. C664 18079.2227 21 354.0820 15651 .2461 17182.34 77 16978.1641 15030.8711 14813.8203 15378.0547 16358.4063 15417.0078 15055.4648 13728.0234 17378.0039 14738.3984 19088.9102 19350.3203 20282. 1602 20 773.2 461 13116.2773 19967.5938 16072.9609 14494.2656 24826.0156 142 32.2969 19578.4531 16115.6563 14306.9844 18994.5508 21497.9609 13366.2500 20*88.0117 159 39.3398 17d29.6172 17328.5117 18o29.5J13 I5t.9l.6273 I5o02.!3i>25 16344.3672 12364.'461 17063. 7<S56 15658.Y8S2 13091 ,b359 16909.2 773  -  CTl  (  " . (  1  f f F l N U M B E R X( 1 1 C O M P U T E D R E S I L U A L X ( 8) 182 183 56 13.5110 13.5821 -0.0711 851.2974 8 4 " 5 7 6.0000 10.1448 -4.1448 578.7395 185 58 16.0000 13. 7865 2.2135 914.C942 186 59 10. 5650 15.9597 -5.J94/ 1017.1636 60 10.0000 12. 1646 -2.1646 729.0542 188 61 14.CC00 11. 1142 2.8854 687.2788 189 62 16.5680 11.7989 4.7691 780.7554 1 9 0 " " 6 3 2 2 . 0 0 0 0 " 1 6 . 6 8 0 2 ~ 5 . 3 1 9 8 ~ 1 0 35.7454 191 192 T93 194 F I N I S H CARD ENCOUNTEREO 195 PROGRAM TERMINATED END O f F I L E  XI 10) 10544.9609 1 1717.^013 17431.7813 18151.2852 15227.0234 15003.29o9 161 71.007b Idl49.3i.94'  "  ~  *  "  "  "  "  178  APPENDIX A-2  MATHEMATICAL TREATMENT OF CREEP DATA by BMD:02R COMPUTER PROGRAM.  CREEP DEFORMATION  INITIAL VARIABLES:  XI = 6 c X2 = a X3 = MOE X4 = 6  ADDED VARIABLES:  X5 = X2 X3 X6 = X2 X4 X7 = X3 X4 X8 = X 4  2  X9 = X2 X 4  2  XlO = X3 X 4  2  1 BHD02R - STEPWISE REGRESSION - REVISED MARCH 27, 1973 HEALTH SCIENCES COMPUTING FACILITY. UCLA  I 2 3 •  5  >  -  6 7 8 9  10 11 12 13 14 15  rt>  17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34  PROBLEM CQOE CRPDEF 91 NUMBER OF CASES 4 NUMBER OF ORIGINAL VARIABLES NUMBER OF VAR IABLES ADDED 6 TOTAL NUM1ER JF VARIABLES 10 NJMBER OF SUB-PROBLEMS 1 THE VAR 1 A3 LE FORMAT IS < 10X ,4F8.31-  VARIABLE 1 2 1 4 5 6 7 8 9 13 1 CORRELATION 0 VARIABLE NUMBER  36  1 2 3 4 5  37  6  38  7 8  35  39  40 41 42 43 44 45 46 47 48  MEAN 17.15468 34.61537 19.90491 33.61629 690.73291 1205.94775 659.26001 1186.76831 44135.88672 22970.50391 MATRIX 1 2  1 .000  — <  ST ANOARO DEV IAMON 7.52491 6.96142 2.68173 7.57275 173.17403 506.50879 134.05516 541.76416 28639.96875 9687.78906 3  0.5502 1.000  4  -0.5258 0.8102 0.9290E-01 0.8116 1.000 -0.4916 1.000  6  5 0.1673 0.85B2 0.5a 29 0.4022 1.000  0.7143 0.9520 -0.1957 0.9480 0.6643 1.000  9  10 1 SU B-PR0BL1 1 DEPENDENT VARIABLE 1 MAXIMUM HUMBER OF STEPS 20 F-LEVEL FOR INCLUSION 3.000000 F-LEVEL FOR DELETION 3.000000 TOLERANCE LEVEL 0.001000  49  50 51 52 53  54 55 56 57 58  STEP NU1BER 1 VARIABLE ENTERED  MULTIPLE R  STO. ER<OR OF EST.  ANALYSIS OF VARIANCE  59  60  REGRESSION  4  0.8102  4.4355  DF 1  SUM OF SQUARES 3345.24 8  MEAN SQUARE 3345.248  F RATIO 170.039  7 0.54 91 0 .99 80 0.10 57 0.80 54 0.db34 0.9479 1.0 00  a 0. 8102 0.B191 -0.4539 0.9948 0.419a 0.9547 0.8133 1.000  9 0. 7481 0.9142 -0.2611 0.96 32 0.5924 0.9924 0. 9098 0.9766 1 .000  13 0.7147 0.9529 -0.1880 0.9458 0.6691 0.9994 0.9509 0.9527 0.9915 1.000  c  61  ?  65 66 67 68 69  RESIDUAL  62 63  ....  7  89  1750.939  VARIABLES VARIABLE  IN  COEFFICIENT  19.673  >  VARlAULtS NO I IN  EQUATION  STD. ERROR  F TO REMOVE  VARIABLE  CORK.  TOLERANCE  F TO ENTER .  \  * -9.90912 0.80S08  (CONSTANT  4  Q  71  1 0.35174  170.0387  (2) .  2 3 5 6 7 8 9 10  72  73 74 75  76 77 78 79 80 81 82 83 84 85 86  PARTIAL  EQUATION  •  STEP  NUiBER  0. 3414 0.7584 0.8382 0.1013 0.3513 0.0105 0.0723 0.1054  -0.24985 -0.29561 -0.28851 -0.29792 0.C7083 -0.20439 -0.27138  9.57<rB 5.8593 6.4263 7.9900 8.5710 0.4437 3.8364 6.9963  (21  (2)  (21  (21 (21 (2) (21 (21  2  VARIABLE  ENTERED  MULTIPLE  R  STO.  -6.3l325~  2 0.8307  ER3.0R O F E S T .  4.2361  87  88 89 90 91 92 93 94 95 96 97 98 99 100 101  ANALYSIS  OF  VARIANCE DF 2 88  REGRESSION RESIDUAL  S U M OF S Q U A R E S 3517.C64 1579.123  VARIABLES VARIABLE  IN  COEFFICIENT  MEAN S Q U A R E 1756.532 17.945  F RATIO 97.998  EQUATION S T D . ERROR  V A R I A B L E S NOT F  TO REMOVE  VARIABLE  2 4  -0.33971 1.05 8 5 2  0.10978 0.10092  9.5748 110.0098  (2) . (21 .  3 5 6  7  *  8 9 10  *  108  118 119 120  TOLERANCE  F  TO E N T E R  -6 . 66 9 6 6 t  (CONST ANT  102  117  CORR.  EQUATION  •  103 104 105 106 107 109 110 111 112 113 114 115 116  PART 1 A L  IN  STEP NUIBER 3 VARIABLE ENTERED  3  MULTIPLE R STD. ERROR O F E S T . ANALYSIS  OF  0.8395 4.1563  VARIANCE  REGRESSION RESIDUAL  DF 3 87  SUM UF SQUARES 3591.830 1504.358  MEAN S Q U A R E 1197.276 17.291  F RATIO 69.241  0.21759 0.15467 0.10829 0.13418 0.14277 0.11757 0.17190  0. 0498 0.0097 0.0035 0.0040 0.0101 0.0208 0.0048  4.3238 2.1361 1.0324 1.5950 1.8102 1.2195 2.6490  (21 (21  W\ (21 (21 121 (21  r  s. r  •  121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 U 5 146 147 148 149 150 151 152 153 15* 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 " 179 180  VARIABLES IN EQUATION VARIA3t.fi  COEFFICIEM  STD. ERROR  -37.84108 1 -1.18513 1.52324 1.95439  (CONSTANT 2 3 4  VARIABLES NUT F TO REMOVE  •  VARIABLE  PARTIAL CURR.  7.9389 (2 1 . 4.3238 (21 . 19.5442 (21 .  5 6 7 8 9 10  IN EQUATION TOLERANCE  F TO ENTER  < 0.42061 0.73254 0.44208  •  0.01700 -0.01062 0.05456 -0.05699 -0.03677 0.04362  0.0056 0.0025 0.0034 0.0035 0.0117 0.0028  0. 0249 0.0097 0.2568 0.2603 0.1164 0. 1640  (21 (21 (21 121 12) 121  F-LEVEL OR TOLERANCE INSUFFICIENT FOR FURTHER COMPUTATION 1 SUMMARY TABLE STEP NUMBER  1  1 2 3 LIST OF «. ESiOUALS CA SE NU1 BER 1 2 3  ~  VARIABLE ENTERED REMOVED  4 5 6 7 8 9 10 1 1 12 13 1 4 15 I 6 17 18 19 20 2 1 ~ 22 23 24 2 5 26 27 " 28 29  Y XI  It  32.0000 21. 0 0 0 0 12.5790 14. 0000 7. 5800 18.5820 16. 0 0 0 0 12.5830 14.5840 19.5860 11.0000 11.0000 14. 0000 30.6400 12.0000 14.0000 11.0000 11. 0 0 0 0 12.0000 13.6460 12.0000 12.0000 14. OCOO 13.0000 ll.OCOC 6. 6540 13.6540 16. 6550 25. 0000  MULTIPLE RSQ  INC REASE IN RS C  0.8102 0.8307 0.8395  0.6564 0.6901 0.7048  0.6564 0.0337 0.0147  Y COMPUTED  RESIDUAL  XI 4)  XI 2 )  XI 31  25.2957 18.6478 13.6542 12.8912 9.6939 15.1188 17.5550 10.2783 13.7932 18.9421 11.4966 11.1454 15.8284 23.2957 12.2078 14.3253 13.9684 11.0464 1C.5955 11.4572 ~"12.9155 11.7877 10.9149 9.8761 14.6678 10.7549 17.8822 18.7002 20.7386  6.7043 2.3522 -1.0752 1.1088 -2.1134 3.4632 -1.5550 2.3047 0.7908 0.6439 -0.4966 -0.1454 -1.8284 7.3443 -0.2078 -0.3253 -2.9684 -0.0464 1.4045 2.1888 -0.9155 0.2123 3.0851 3.1239 -3.6678 -4.1009 -4.2282 -2.0452 4.2614  39.1730 34.1940 29.0950 2 7.4950 22.8350 31.0610 33.2810 25.4410 29.7070 34.4350 25.7620 25.9190 30.7710 37.8660 26.7400 29.6340 29.8310 26.4100 26.3640 27.1190 23.9540 26.5890 26.7940 23.9430 30.4780 25.8010 33.2310 34.3120 35. 3990  30.0000 30.0000 30.0000 30.0000 30.000C 30.0000 30.0000 30.0000 30 . 0 000 30.0000 30.0000 30.0000 30.0000 30.0000 30.0000 30.0000 30.C000 30.0000 30. OCOO 30.0000 30.0000 30.0000 30.000U 30.0000 30. it000 30.0000 30.0000 30.0000 3 0 . J JOO  14.5290 16.5530 19.8 170 21.3690 25.2490 16.2560 17.0070 22.2890 19.1230 16.4370 22.6 776 22.2450 19.0 94C 14.8930 21.8890 19.5 660 19.0790 21.5 500 21.3130 <!0.9100 19.5130 tl.-.OJO 2 0. v 710 23.9't70 18.7 0 S 0 22.1 iGO 17.2U6C l c * . )60 15. .' -80  R  4 2 3  '  F VALUE TO ENTER OR REMOVE  170.0387 9.5748 4. 3238  NUMBER OF IKOfc VARIABLES IN  1 2 3  f  s r  181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 20 3 204  30 3 1 32 33 34 3 5 36 37 3 8 39 40 41 42 43 44  13. ccob  ~~  ~~ «>s" 46 47 48 49 50  "ST 52 53 54  20$  206 207 208 209 210 211 212 213 214 21S 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 " 212 233 234 235 236 237 238 239 240  26.OCOO 9. 65 90 8.6600  55  9. OOUO 27. 7100 12.0000 21.7110 12.0000 11.0000 19. 0000 23.0000 11. 7180 16.0000 8.0000 16. 72 30 23. 7240 14.0010 12.0000 7.0000 9.0000 9. 0000 9.0000 15.0000 12.0000 10. 5100  21.5200 9.4922 19.6174 12.4652 10.8332 31. 5645 9.9664 18.8893 12.5154 10.2964 17.9366 23.8607 9.2821 20.8066 12.2583 15.5795 14.5757 17.1484 11.9676 12.0427 12.9189 8.8339 14.1206 11.8093 9.1388 13.8526  4.4800 0.1668 -10.9574 0.5348 -1.8332 -3.8545 2.0336 2.821r -0.5154 0.7036 1.0634 -0.8607 2.4359 -4.8066 -4.2583 1.1435 9.1483 -3.1474 0.0324 -5.0427 -3.9189 0.1661 -5.1206 3.1907 2.8612 -3.3426  36.2310 22.8760 34.5670 28.39/0 24.9510 43.4600 25.2370 34.0830 28.5770 25.0450 33.6820 38.3990 23.6500 35.6010 28.1610 31.6740 30.3350 32.9740 27.4690 26.8590 28.7180 21 .8050 30.0920 27.7660 23.0880 29.6010  30.0000 30.0000 30.0000 30.0000 30.0000 30.0000 30.000C 30.0000 30.0C0C 30.00uu 30.0000 30.OCOO 30.0000 30.0000 30.0000 30.0000 30.0000 30.0000 30.0000 30.0000 3o.00oG 30.0000 30.0000 30.0000 30.0000 30.0000  15.8250 25.0640 16.7 110 IV.932o 23.2 820 13.1440 22.3460 lb.8540 19.7 340 22.8090 16.7430 14.5 800 23.9330 16.1650 20.0 990 17.7 720 18.8310 17.1340 20.7960 21.6280 19.8180 26.0060 18.8440 20.3110 24.5600 19.2980  Y COMPUTED  RESIDUAL  XI 41  X( 2)  X( 3 1  13.2326 9.7257 14.7424 16.1188 11.1900 11.0938 12.7641 16.1950 26.8012 20.9279 31.6370 28.3133 18.2525 34.2159 25.5782 14.9264 19.3405 27. 7644 22.2050 23.0485 24.9570 14.4664 18.6321 17.4078 16.5173 20.8505 21.4277 28.6495 23.0321  0.2784 -3. 7257 1.2576 -5.553d -1.1900 2.9062 3.8039 5.8050 -5.6522 -2.7859 -2.4960 -2.3723 -0.1055 -0.0609 6.6268 1.0736 -0.3405 10.4436 6.7950 3.9515 2.9900 -0.4664 -0.6321 -0.4073 -5.5173 4.0325 0.5123 -6.7105 -8.0321  29.1770 24.0570 30.2340 31.8930 27.0010 26.2160 27.9420 32.1830 46.0440 41.4060 49.5580 46.8200 38.9280 50.6110 44.5500 35.4460 40.0560 46.2960 41.2200 42.8620 44.2820 35.3860 38.5190 37.2620 37.2280 40.8980 41.5830 47.3630 42.9830  30.0000 30.0000 30.0000 30.0000 30.OCOO 30.0000 30.0000 30.OCOO 45.0CC0 45.0000 45.COO0 45.0000 45.0000 45.0000 45.0000 45.0000 45.000C 45.0000 45.0000 45.0000 45.00CO 45.CC0C 45.00CO 45.0000 45.0000 45.0000 45.0000 45.0000 45.00GJ  19.4350 23.7020 19.0700 17.8450 20.8 860 21.B300 20.7120 17.5230 18.3720 20.4670 17.0380 18.3690 21.8 900 17.3800 19.4 860 24.1740 21.1570 18.6810 21.5 440 19.9910 lv.4220 23.9490 22.66*0 23.4730 22.9320 21.0b80 20.5b80 17.8930 I9.b250  1 CASE NIH BER 56 57 58 59 60  bl  62 63 64 65 66 67 68 69 " 70 71 72 73 74 75 7 6 " 77 78 79 80 81 8 2 "" 83 84  Y X(  l l  13.51 10 6. 0000 16.0000 10. 5650 10.0000 14.0000 16. 56 80 22.0000 21.1490 18. 1420 29. 1410 25.9410 18. 1470 34.1550 32.2050 16. 0000 19.0000 38.2080 29. 0000 27.0000 27.9470 14.0000 18.0000 17.0000 11.OCOO 24.8830 21.9400 21.9390 15. 0000  85 86 87 88 89 90 9 1  241 2*2 243 244 24S 246 247 248 249 "256" 251 252 ENO  bt  12.00C0 23.0000 30.4980 38.4960 28. 9940 23.4920 17.0000  F I N I S H CARD ENCOUNTERED PROGRAM T E R M I N A T E D H I E  19.70 20 34.0300 25. 7033 29.2513 24.7671 22.1458 18.7278  -7.7020 •11.0300 4.7947 9.2447 4.2269 1.3463 -1.7278  39.8450 5Q.89J0 44.9000 47.7980 44.4280 41.3300 39.4650  45.000C 45.0000 45.00C0 4S.0GCC 45.00C0 45.OCOO 45. OCOO  21.6650 16.9000 19.1190 17.7 300 19.1100 21.3640 21.5130  APPENDIX A-3  MATHEMATICAL TREATMENT OF CREEP DATA by BMD:02R COMPUTER PROGRAM.  CREEP DEFORMATION  INITIAL VARIABLES:  XI = <5 c X2 = a X3 = MOE x4  = a  e ADDED VARIABLES:  X5 = X2 X3 X6 = X2 X4 X7 = X3 X4 X8 = X 4  2  X9 = X2 X 4 XlO  = X3 X 4  (forced 2  2  variable)  1 6MD02R HEALTH  TOll  - S T E P W I S E R E G R E S S I O N - R E V I S E D MARCH SCIENCES COMPUTING F A C I L I T Y , UCLA  27, 1973  PROBLEM C3DE CRPDEF NUMBER O F C A S E S 91 NUMBER OF O R I G I N A L VARIABLES 4 NUMBER OF V A R I A B L E S ADDED 6 TOTAL NUM3ER OF V A R I A B L E S 10 NUMBER O F S U B - P R C B L E H S 1 THE VARIA3LE FORMAT I S IT0X.4F8.3r  12 13 14 15 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 46 ^ 9 50 51 53 54 5  2  ~5T"  56 57 "58 59 60  VARIABLE 1 i 3 4 5 6 7 8 9 10  STANOARD  OEV I A T I O N 7.52491 6.96142 2.68173 7.57275 173.17403 506.50879 134.05516 541.76416 28639.96875 9687.78906  MEAN 17.15468 34.61537 19.90491 33.61629 690.73291 1205.94775 6 5 9 . 26001 1186.76831 44135.88672 22970.50391  10  VARIABLE NUMBER 0.5502 1 1.000 2 1.000 3 4 5 6 7 8 9 10 1 SUB-PROBLM 1 DEPENDENT V A R I A B L E MAXIMUM NUMBER OF STEPS F-LEVEL FOR INCLUSION F-LEVEL FOR DELETION TOLERANCE LEVEL  -  -0.5258 C.8102 0.9290E-01 0.8116 1.000 v -0.4916 1.000  0.1673 0.8582 0.5829 0.4022 1.000  0.7143 0.9520 -0.1957 0.S480 0.6643 1.000  0 . 54 91 0 . 9 9 80 0 . 1 0 57 0 . 8 0 54 0 . 86 34 0.9479 1.000  0.8102 0.8191 -0.4539 0.9948 0.4198 0.954 7 0.8133 1.000  0.7481 0.9142 - 0 . 2611 0.9632 0.5924 0.9924 0.9098 0.9766 1.000  0.7147 0.9529 -0.1B80 0.9458 0.6691 0.9994 0.9509 0.9527 0. 9915 1.000  1 20 3.000000 3.000000 0.001000  STEP~~NU>4BERI" VARIABLE ENTERED  MULTIPLE R STD.  ERIOR  ANAL Y S I S  0.8102 4.4355 OF E S T .  O F V A R I A NC E  DF  1 REGRESSION  SUM OF SQUARES 3345.228  ME >N SQUAKE 33-.3.228  f RATIO 170.035  CO  s  RESIDUAL  61 62 63 64 65 66 67 6.8 69  89  1750.960  ...  19.674  VARIABLES I N EQUATION COEFFICIENT  VAR I A8L E  STO. ERROR  VARIABLES NUT IN EQUATION VARIABLE  F TC REMOVE  F TO ENTER , \  (CONSTANT 8  70  71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102  3 . 79 9 57 1 0.01125  0.00086  170.0356 (3) .  •  STEP NUMBER 2 VARIABLE ENTERED  2 3 4 5 6 7 9 10  -0.33735 -C.302o5 0.07091 -0.32491 -0.33947 -0.32224 -0.3423B -0.32111  MULTIPLE R STO. ERROR Of EST.  11.3012 8.6731 0.4448 10.36o5 11.4621 10.1968 11.6852 10.1167  121 (21 (2) (21 (21 (21 (21 (21  0.8347 4.1910  ANALYSIS OF VARIANCE REGRESSION RESIDUAL  OF 2 88  SUM OF SQUARES 3550.477 1545.711  MEAN SQUARE 1775.238 17.565  F RATIO 101.067  VARIABLES IN EQUATION COEFFICIENT  VARIABLE  0.3290 0. 7940 0. 0105 0.8238 0.0886 0.3385 0.0462 0.0924  9  (CONSTANT 8 9  STO. ERROR  -0.40476 1 0.02392' -0.00025  0.03379 0.00007  VARIABLES NOT IN PARTIAL CORK.  VARIABLE  F TO REMOVE  34.7443 "(31 . 11.6852 (2 1 .  103  104 105 106 107 108 109 110 Ul 112 113 114 115 116 117 118 119 120  TOLERANCE  PARTIAL COKk.  *  2 3 4 5 6 7 10  -0.05685 0.07979 -0.06BU4 -0.01431 -0.05562 -0.01683 -0.00256  EQUATION TOLERANCE  0. 0465 0.0753 0.0090 0.1035 0.0105 0.0498 0.0115  F TO ENTER  0.2821 0.5574 0.4047 0.0178 0.2700 0.0246 0.0006  (21 (21 (21 (2) (21 (21 (2)  f-LEVEL OR TOLERANCE INSUFFICIENT FOR FURTHER CoHPuTAflON 1 SUMMARY TtBLE STEP NUMBER  1 L  1 2 IST OF  CASE  V A R I A B L E M U L I I P L E ENTERED REMOVED R  R  E Y  S  I  8 9 D  U Y  A  L  S  '  0.8102 C.8347 " ' "  "  RSQ 0.6564 0.6967 " " ~  INCREASE IN KS C  0.6564 0.0403  F VALUE TO tMIEK UK REMOVE  170.0356 11.6652  NUMBER OF INOE VARIABLES lit  1 2  (  TU NUMBER XI II CCPPUTED RESIDUAL XI b I XI V) 122 123 1 32.OCOO 25.0071 6.9929 1534.5242 460J5.7227 ~ " " " ~ 1 2 4 " 2 2 1 . 0 0 0 0 " 1 8 . 9 5 7 8 ~ ~ 2 . 0 4 2 2 U i > 9 . ' 2 2 9 5 3 5 0 7 6 .862 8 125 3 12. 57S0 13. 6137 -1 .0347 846.5190 25395.57C3 126 4 14.0000 12.1143 1.885/ 755.9746 22679.2383 / IT7 5 7.5800 8.2303 -0.6503 521.4375 15643.1250 128 6 18.5820 15.5722 3.0098 964.7859 28943.5742 129 7 16.0000 17.9376 -1.9376 1107.6252 33228.7539 130 "8 12.5830 10i3137 2.269 3 ~~ 647.2441 19417.3242 131 9 14.5840 14.2096 0.3744 Bd2.5059 26475.1758 • 132 10 19.5860 19. 2317 0.3543 1185.7690 35573.0703 TJ3 TT 11.0000 10.5859 0.4141 663.6602 19910.4023 134 12 11.0000 10.7202 0.279a 671.7947 20153.8398 135 13 14.0000 15.2752 -1.2752 946^8540 28405.6172 £ j ' 3 0 . 6 4 0 0 " 2 3 . 3 3 97 " 7.J003 """1433.833 7 4 3 0 1 5 . 0 1 1 7 137 15 12.0000 11.4362 0.5638 715.0278 21450.8320 138 16 14.0000 14. 1379 -0.1379 878.1741 26345.2188 139 IT 11.0000 14.3319 -3.3319 889.8879 26696.6367 140 18 11.0000 11. 1457 -0.1457 697.4883 20924.6484 141 19 12. 0000 _ 1 1 .J.055 0^8945 695.0603 20851.8086 142 20 13.6460 11.7742 1.871 d 7 35.4402 22063.2031" 143 21 12.0000 13.4781 -1.4781 838.3337 25150.0117 144 22 12.0000 11.3028 0.6972 706.9751 21209.2500 R 5 2~3 14.0000 11.4841 2.5159 717.9187 21537.5586 146 24 13.0000 9.0886 3.9114 573.2668 17198.0039 147 25 11 .0000 14. 9781 -3.9781 928.9082 278Q7.2461 I48~~ T 6 6T6540 10.6192 -3.9652 665.6912 19970.7344 149 27 13.6540 17.8825 -4.2285 1104.2993 33120.9766 150 28 16.6550 19.0917 -2.4367 1177.3130 35319.3B67 T5T 5T9 25.0000 20.9369 4.0631 1288.7383 36662.1484 152 30 26.0000 21.3334 4.6666 1312.6855 39380.5664 _ 1531 31 9.6590 8^2613 1^?J 523.3115 15699.3438 " 154 32 8.6600 19.3825 -10.7225 1194.8774 35846.3203 155 33 13.0000 12.9491 0.0509 806.3896 24191.68/5 156 34 9.0000 9.9048 -0.9048 622.5525 18676.5742 157 3T5 2 7 . 710O 30.8735 -3.1635 1888.7720 56663.1563 158 36 12.0000 10.1425 1.8575 636.9060 19107.1797 159 37 21.7110 18.8323 llol.6SOI 34849.5039 T6"0 38 12.0000 13.1190 =Trri90 816.6445 24499.3359 161 39 11.0000 9.9826 1.0174 627.2517 18817.5508 162 40 19.0000 18.3823 0.6177 1134.4775 34034.3242 : I63" 41 23.0000 24.0128 -1.0128 1474.4832 44234.4922 164 42 11.7180 8.8577 2.8603 559.3220 16779.6602 165 43 16.0000 20.5840 -4.5840 1267.4309 36022.9258 166 44 8.0000 12.7281 " -4.7281 793.0415 23 791.2422 167 45 16.7230 16.2090 0.5140 1003.2419 30097.2578 168 46 23. 7240 14. 8341 8.8899 920.2124 27606.3711 l6"9" 4"T 14.0010 11.6008 -3.5998 1087.2844 32618.5313 170 48 12.0000 12.0906 -0.0906 754.5454 22636.3594 171 49 7.0000 11.5418 -4.5413 72K4055 21642.1641 172 50 9.0000 13.2528 " -4.2528 824.7236 24 741.7070 173 51 9.0000 7.4689 1.5311 475.4575 14263.7227 174 52 9.0000 14.5909 -5.5909 905.5281 27165.8398 TU 51 15.0000 12.3623 2.6377 770.9509 23128.5273 176 54 12.0000 8.4227 3.5773 533.0554 15991.6602 177 55 10.5100 14. 1055 -3.5955 8/6.2190 26286.5664 17* 1 " ' ~ " ~ " " 179 1 3  A  7  .  180  CASE  V  V  f  181 182 183  184  185 186  187 188 189 ~ ""190 191 192 19) 194 195 196 197 198 199 200 201 202 203 204 20 S 206 207 208 209 210 211 212 213 214 215 216 217 218 219  220 221 222 223  NUMBER 56 57 58 59 60 61 62 3 64 65  6  66 67 68 ' 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 fl4 85 86 87 88 89 90 91  XI  11  13. 5110 6.0000 16.0000 10.5650 10. 0000 14.0000 16.5680 22.0000 1490 18. 1420  21.  ' "34.1550 32.2050 16.0000 19.0000 38.20 80 29.0000 27.0000  27.9470 14.0000  18.0000 17.0000 11.0000 24.88 30  21.9400 21.9390  15.0000 12.0000  23.0000 30.49 80 38.4960 28.9940  23.4920  17.0000  RESIOUAL  13.6928 9.1792 14. 7327 16.4396 11.6685 10. 9767 12.5246 16.7473 26.9011 21.6771  -0.1813 -3.1792 1.267J -5.8746 -1.6685 3.0233 4.0434 5.2527 -5.7521 -3.5351  27.8292 19.1131 3 2.5865 ~ 25. 1578 15.7776 20.2607 27.2008 21.4792 23.2574 24.8512 15.7229 18.7052 17.4783 17.4456  -1.8882 -0.9661 1.5685 7.0472 0.2224 -1.2607 11.0072 7.5208 3.7426 3.0958 -1.7229 -0.7052 -0.4783  31.2280  29.1410 25.9410 18.1470  COMPUTED  "  21.1386  -2.0870  -6.4456 3.7444 0.073/  21.8663 28.4879  -6.5489  20.0435 32.9512  -9.9512  23.3912  25.5610 29.0211 25.0180 21.5961  19.6554  -8.3912 -8.0435  4.9370 9.4749 3.9760 1.8959 -2.6554  XI  81  851.2974 578.7395 914.0942 1017.1636 729.0542 687.2788 780.7554 1035.7454 2120.0503 17 14.4573  2455.9951  2192.1130 15 15.3687 2561.4724 " 1984.7026 1256.4187 1604.4829 2143.3201 1699.0834 1837.1509 1960.8953 1252.1689 1483.7131 1388.4561 1385.9236 1672.6458 1729.1453 2243.2542 1847.5383 1587.6240 2589.7920 2016.0093 2284.6492 1973.8464 1708.1689 1557.4858  XI  9)  25538. 9 U C 17362.1836 27 4 22 . 8 2 4 2 30 514.9063 21871.6250 20ul8.3633 23422.6602 31072.3594 954C2.2500 77150.5625 110519.7500 98645.0625 68192.43 75 115266.2500 ~~ ' 89311.5625 56538.8398 72201 .6875 96449.3750 76458.9375 82671 .7500 88240.2500 56347.6016 66767.0625 62400.5195 62366. 75269.0000 77811.5000 10094b.3750 83139.1875 71443.0625 1 U 5 4 0 . O250 9072o.3750 102809.1875 88823.C625 76867.5625 70086.8125  -  —<  55d6  FINISH CARD ENCOUNTERED PROGRAM TERMINATED  END OF F I L E  •  00 00  189  APPENDIX B - l  MATHEMATICAL TREATMENT OF CREEP DATA by BMD:0 2R COMPUTER PROGRAM.  FRACTIONAL CREEP  INITIAL VARIABLES:  XI = f X2 = SR X3 = MOE X4 = 6  ADDED VARIABLES:  X5 = 1/X4 X6 = 1/X3  f  1  ! t  • 10 II La IS 14 Is 16 IT is 19 20 21 22 2S 24 2. 26 2T 28 29 SO 31 32 3S 14 SS 36 ST 36 39 «0 41 4B 43 44  «5  06 «T 48 49 50 31 52 51  1 B M D 0 2 R • STFPMlSf RFGRfSblfiN - R E V I S E D MARCH 2 7 # HEALTH S C I E N C F S C O M P U T I N G F A C I L I T Y * UCLA  197J  FRACRr PR08I.FM COCF 6i NUMBER OF C A S F S 4 N U M B E D OF OnlGlMAt VAHIABl.F.S 2 NUMBER OF V » R I A B L F s ADOtD TOTAL NUMBER tlF V A R I A B L E S b N U M B E R OF S U B - P R O B L E M S 1 THE VARtABLF FORMAT I S (tOX,4F8.IN-  VARIABLE 1 2 3 4 5 6 1 CORRELATION 0 VARIABLE  STANDARD D E V I A T I O N 13,07664 14,25918 2,90453 4.46927 0,00501 0,00802  MEAN 47.7732? 41,344J« 19.7J979 B9.54l7g 0,0345* 0,05l8» MATRIX I  3  3  •  1  6  i  NUMBER  1 2 3 4  1,1)00  •0.4794 •0.4635 1.000  0.3549 I.000  0,5111 0,4113 •0,9810 1.000  5  6 1 8U8-PR0RLM 1 DEPENDENT VARIABLE MAKIMIJM NUMBER OF S T E P S F»LEVEL FOB INCLUSION F-LEVEL FOR DELETION TOLERANCE i'EVEL  STFP NUMOFR 1 VARIABLE FNTEdED  nF  0,5209 0,4127 •0.9831 0,997« •0,9799 1,000  1 20 3,000000 J,000000 0.001000  6  MULTIPLE 0 STD. F R R O P O F E S T . ANALYSIS  •0.4720 •0.4433 0.9969 •0,9831 1.000  g,9209 1 1. 2 5 3 9  VARIANCE  ("FGRtSSION RFSIDllAL  OF 1 kl  SUM  OF SQUARES 2876.53b 7725.691  MEAN SQUARE 2876,535 126.651  F RATIO 22,712  54 55  56 57 58 59 60  VARTAHLES VARIAHLF  COEFFICIENT  IN  t •  EQUATION STD. E R R O R  VARIABLES VARIABLt  F To R E M O V E , • -1 .  PARTIAL  NOI  COKR,  IN  EQUATION TOLENANCL  F  TU fcNTEH  s.  3.7«iS?0 ) 849.712J2 tTB.JObJO  (CONSTANT 6  62 6J 64 69 66 67 68 69 70 71 72 7J 74 75 76 7? 78 79 80 81 82 81 84 85 86 81 8* 89  22. 7121 (2)  !  i  •  5  0,16810 0,21091 -0.I05U9 0,22506  1  4  STEP  NUHBFK  STO.  95 9* 9» 98 99  ANALYSIS nF  0,5554 11.0546  VARIANCE  OF 2 60  RrORtSSION RFSIDUAL  VARIABLF  IN EQUATION  COEFFICIENT  'k  F RATIO 179  VARIABLES NOT IN EQUATION  •  STB, ERROR  F TO REMOVE  -16J .55901 ) 1406.18091 2S2S .05811 878.91479 2J95 .14380  (CONSTANT  MEAN SQUARE 1634.980 122.204  SUM OF SQUARES 3269.961 7312.266  VARIABLES  F-LEVEL OR TnLERANCE INSUFFICIENT-FOR 1 SUMMARY T M l E  3. 2194 (2) 7, 4263 (2)  .  VARIABLE  , ,  2 3 4  PARTIAL CORR.  0,19886 0,0094? •0.01702  TOLERANCE  0,6014 0.00S2 O.0044  101 103 10J  loa  to«j 106 107 10ft 109 110  til  112  HI  114  ns  116 117  its  119 120  1  1 2  LIST OF RESIDUALS CASE  NUMBER 1 2 J 4 5 » 7 6 9 10 11 12 li 14  INCREASE IN RSQ  0,5209 0,5554  0,2711 0,3084  0 .2713 0 .0371  Y COMPUTED  RESIDUAL  X( 6)  65.7016 54.9210 44.0220 40.2902 U\.7927 as.8677 53.0845 ^3.0725 46,6217 •45.4278 19.9981 41,4561 43.8753 61.8956  15.9884 6.4910 -0,7870 10.6288 -8,5987 10,9561 -5.0095 6,3865 2.4693 1.4492 2,7009 6.98J9 1.6227 17.0194  ENTERED  V  v( l l Hi.6900 A t . 41/10 43.2J50 50.9190 i*.lQ40 S9.8240 U8.07S0 ,j4.4590 (19.0910 S6.8T70 H2.6990 fl2.4600 45.4980 AO.9150  REMOVED  6 5  F TO ENTER  2,4292 (2) 0,0053 (2) 0,0)7} (2)  FURTHER COMPUTATION  RSQ  VARIABLE  STEP  NUMBER  too  <  -  91  92 9) 9«  (2) (2) (2) 12)  S  R  ERROR OF E S T .  90  1,7491 2,7912 0,6700 1,2194  2  VARIABLE FNTERED MULTIPLE  0.6128 O.Oiil 0,0052 0,0197  R  MULTIPLE  0,0688 0.0604 0,0505 0,0468 0,0196 0,0548 0,0558 0,0449 0,0521 0,0606 0,0441 0,0450 0,0524 0,0671  F VALUE TO ENIEH QN REMOVE 22,7121 1,2194  NUMBER OF INDfc VARIABLES IN 1  2  X( 51 0 .0255 0 .0292 0 ,0144 0 ,0164 0 .0418 0 .0122 0 .0300 0 .0141 0 ,0117 0 ,0290 0 .0188 0 ,0186 0 .0125 0 .02o4  —<  121 122 121 .24  125  <  126 12? 128 129 130 131 132 133 131 I3S "136 I3T 138 139 l«o Ml 1MB 145 11 . 145 146 141 148 149 150 151 152 15J 154 159 156 15T 156 159 160 161  IS 16 17 16 19 20 21 22 23 2(1 25 26 27 26 29 30 31 32 33 34 33 36 37 38 S4 40 41 42 43 44 45 46 47 48 49 50 Si S2 53 •4 55  /I4.H770 /,7.2a .0 .6.8750 at .6510 /|5.5|70 .0.3190 11.4450 45.1120 .2.2510 S4.2960 .6.0910 . 5 7«80 at.0840 aS.Saoo ..9.6400 71.7*10 d2.2?40 . 5.0S20 45.7790 16.0710 fcl.7600 47.5490 63.7020 al.9910 43.9?20 .6.4090 .9.8*70 49.5470 ' q4.9420 >R.4o80 .2,7970 .8.2060 42.4h20 43.6860 96.0620 .1.3.90 41.2760 i»9.9o«0 S4.0740 5t.«740 15.5070 p  1 16* 164 CASE V NUMBER X( 1) «*5 166 56 16? 46.3070 168 57 >4 9410 58 169 S2.92Q0 59 ^3.1270 170 60 17.01.0 >J 172 61 S3. 4021) 62 S9.29J0 173 63 *.H.lS90 174 175 176 177 178 FINISH CARD PNCOllNTfREn PROGRAM TERMINATED 179 END OF FILE  40.2184 41,9952 4t».5S7b 41,1187 44.5213 44.0229 46.3273 41,1657 44,8182  .... 4.43,2 47,2517 42,4119 50.9256 55.6992 58.9116 57,4369 42,2949 9?.7567 45.4561 40,4368 76,7190 41.6000 52.9791 46.1023 42,1907 54.4086 66,4212 43.2012 55.4800 45.2023 $0.6687 46.6097 52.7465 43.46B4 41.1206 49.1543 44.2507 47.1696 49.2330 43.2432 49.7901  4.6566 3.2478 •9,6826 ... , j 0.9957 6.2961 •4,6823 3.9661 7.4128 '12.4588 •11,1607 •16.6219 •9,8366 -7,1592 10,7264 1471101 •0.0709 •27.7067 0.3229 •4.3656 •12.9590 1,9490 • 11.1229 •4,1111 1,7111 2.0064 •6.5262 6,1457 •10.5180 •16,7941 1.9261 31,4003 -10.2845 0,2206 •15.0569 •13.6153 -2!«747 •17.4616 6,7910 6,7306 •10.2631 t  l  4  6  7  7  -  0,0457 0,0511 0.0524 0,0464 0,0469 0,0476 0,0512 0,0459 0,0477 o.oate 0,0515 0.0452 0,0579 0,0608 0,0635 0,0b32 0,0399 0,0598 0,6502 0,0430 0,0761 0,0448 0,0591 0,0507 0,0418 0,0597 0,0686 0,0418 0,0619 0,0496 0,0561 0,0531 0,0584 0,0481 0,0462 0,0505 0.0JB5 0.0511 0,0492 0,0407 0,0516  0,0174 0,0317 0.0135 0,0179 0.0179 0,0169 0,0145 0,0176 0.0171 0.0418 0,0128 0.0188 0,0101 0,0291 0.0279 0,0276 0,0417 0.0289 0.0152 0.0401 0,0210 0,0196 0.0291 0.0150 0,0199 0,0297 Oj0260 0,0421 0,0281 0.0155 0,0116 0,0110 0,0101 0,0164 0,0172 0.0148 0,0459 0,0112 0.0160 0,0411 0,0136  u a  r  T  r  COMPUTED  RESIDUAL  46.1539 42.3717 45.4894 49,7704 44.3611 42.4001 4*.3776 51.5240  0.1511 •17.4107 7.4106 •l6.o414 -7,5261 11 .0019 16.9154 16.6150  X( 6) 0,0515 0.0422 0,0524 0.0560 0,0479 0.0456 0.0481 0.0571  X( 5) 0,0343 0.0416 0,0331 0,0114 0.0170 0,0181 0,0158 0.0511  \  193  APPENDIX B-2  MATHEMATICAL TREATMENT OF CREEP DATA by BMD:02R COMPUTER PROGRAM.  FRACTIONAL CREEP  INITIAL VARIABLES:  XI = f X2 = a X3 = MOE X4 = 6 e  ADDED VARIABLES:  X5 = 1/X4 X6 = 1/X3  r  1 2 43 5  \ r  6  '  "  7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 " 52 53 54 55 56 57 58"' 59 60  1 BMD02R - STEPWISE REGRESSION - REVISED M A R C H 27, 19/3 HEALTH SCIENCES COMPUTING FACILITY, UCLA PROBLEM C3 0E FRACRP NUMBER OF CASES 91 NUMBER OF ORIGINAL VARIABLES 4 NUMBER OF VARIABLES AOOEO 2 TOTAL NUM3ER OF VARIABLES 6 NUMBER OF SUB-PROBLEMS 1 THE VAR I A3 LE FORMAT IS 1 10X.4F8.3I  VARIABLE 1 2 3 4 5 6 1 CORRELATION 0 VARIABLE NUMBER  MEAN 49.63556 34.61537 19.90491 33.61629 0.03121 0.05119 MATRIX 1  STANDARD DEVIATION 13. 59948 6.96142 2.68173 7.57275 0.00671 0.00728  2  3  4  1 1 .000 0.2066 -0.4425 0.4604 21.000 0.9290E-01 0.8116 3 1.000 -0.4916 4 1.000 5 6 1 1 SUB-PROBLM DEPENDENT VARIABLE 1 MAXIMUM NUMBER OF STEPS 20 F-LEVEL FOR INCLUSION 3.000000 F-LEVEL FOR DELETION 3.000000 TOLERANCE LEVEL 0.001000  STEP NIN BER 1 VARIABLE ENTERED  0.4558. -0.1256 -0.9840 0.4673 -0.5345 1.000  0.4604 12.1403  VARIANCE  REGRESSION RESIDUAL  OF 1 89  VARIABLES VARIABLE  -0.4498 -C.7546 0.5684 -0.97/9 1.000  6  4.  MULTIPLE R STO. ERROR OF EST. ANALYSIS OF  5  COEFFICIENT  SUM UF SQUARES 352 7.766 13117.367  MEAN  SQUARE 3527.766 147.3b6  F  RATIO  23.936  I N EQUATION STD.ERROR  VAKIABLLi  FTC  REMOVE  V AKIABLE  PARTIAL  WOT  CORK.  IN  EQUATIUN TOLERANCE  F TO ENTER  I—  1  c  61 62 63 64 65 66 6. 68 69 TO 71 72 73 74 75 76 77 78 79 80 81 82 83  *  '  0.16899  23.9355 (2 ) .  2  -0.32206  •  3 5 6  -0.2797C G.00248 0.30672  0. 3414 0.7584 0. 0436 0.7817  10.18tI 7.4690 O.O0U5 9.1384  121 12) 121 (2) — — C  STEP N U . 8 E R 2 VARIABLE ENTERED  2  MULTIPLE R STD. ER-OR OF EST. ANALYSIS  OF  0.5419 11.5585  VARIANCE  REGRESSION RESIDUAL  DF 2 88  VARIABLES  a.  85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112" 113 114 US 116 117 118 119 120  \  21.84 328 I 0.82675  (CONSTANT 4  VARIABLE  SUM OF SQUARES 4888.359 11756.773  MEAN SQUARE 24.4.180 133.600  IN EQUATION  COEFFICIENT  F RATIO 18.295  VARIABLES  •  STD. ERROR  F TC REMOVE  VAKIAbLE  NOT IN EQUATION  PART IAL CURR.  TOLERANCE  F TO ENTER  *  30.95923 1 -0.95595 1.53 994  (CONSTANT 2 4  0.29955 0.27537  13.1841 (21 . 31.2729 (2) .  3 5 6  0.13031 0.10168 -0.03991  0.0498 0. 0402 0.0352  1.5028 121 0.9089 (21 0.13B8 U l  F-LEVEL OR TOLERANCE INSUFFICIENT FOR FURTHER COMPUTATION 1 SUMMARY TABLE STEP NUMBER  1  1 2 L IST OF  CASE NUMBER 1 2 3 4 5 6 7 8 9 10 11 12 13 14  VARIABLE ENTERED REMOVED  4 2  INCREASE IN RSC  MULTIPLE R  RSQ  0.2119 0.0817  0.4604 0.5419  0.2119 0.2937  Y COMPUTED  RESIDUAL  X I 41  XI 2)  62.6046 54.9373 47.0851 44.6212 37.4451 50.1127 53.5313 41.4582 48.0276 55. 3084 41.9525 42.1943 49.6661 60.5919  19.0854 6.4767 -3.8501 6.2978 -4.2511 9.7113 -5.4563 B.JOOd 1.0634 1.5686 0.7465 0.2457 -4.1681 20.3231  39.1730 34.1940 29.0950 27.4950 22.8350 31.0610 33.2810 25.4410 29.7070 34.4350 25.7620 25.9190 30.7710 37.8660  30.0000 30.0000 30.0000 30.0000 30.0000 30.CC0C 30.0000  .  .ESIDUALS Y XI 11 81.6900 61.4140 43.2350 50.9190 33.1940 59.8240 48.0750 49.4590 49.0910 56.8770 42.6990 42.4400 45.4980 80.9150  30.0000 30.0000 30.0000  30.CCO0 30.0000 30.0000  30.CC0C  F VALUE TO ENTER OR REMOVE  23.9355 10. 1841  NUMBER OF INOE VARIABLES IN  1 2  f  ?  •  121 122 123 " ~ " 12V 125 126 12* 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 ~ 1 7 2 ' 173 174 175 176 177 178 179 180  15 16 1 7 18 19 20 21 22 23 24" 25 26 27 28 29 3 0 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55  "  44.8770 47.2430 36.8750 41. 6510 45. 5170 50.3190 41.4450 45.1320 52.2510 54.2960 36.0910 25.7880 41.0890 48.5400 69.6400 71.7610 42.2240 25.0520 45.7790 36.0710 63. 7600 47.5490 63.7020 41.9910 43.9220 56.4090 59.8970 49.5470 44.9420 28.4080 52.79 70 78.2060 42.4620 43.6860 26. 06 20 31.3390 41.2760 29.9080 54.0240 51.9740 35.50 70  43.4586 47.9152 48.2185 42.9504 42.8796 44.0422 46.8680 43.2261 43. 5417 "39.1514 49.2149 42.0126 53.4543 55.1190 57.5629 58.0741 37.5083 55.5117 46.0103 40.7036 69.2063 41.1441 54.7663 46.2874 40.8484 54.1488 61.4127 38.7002 57. 1039 45.6468 51.0566 48.9947 53.0585 44.5812 43.6418 46.5046 35.8590 48.6204 45.0386 37.8347 47.8643  1.4184 -0.6722 -11.3435 -1.2994 2.6374 6.2768 -5.4230 1.9059 8.7C93 15.1443 -13.1239 -16.2246 -12.3653 -6.5790 12.0771 13.6869 4.7157 -30.4597 -0.2313 -4.6326 -5.4463 6.4049 3.9357 -4.2964 3.0736 2.2602 -1.5157 10.8468 -12.1619 -17.2388 1 .7404 29.2113 -10.5965 -0.8952 -17.5793 -15.1656 5.4170 -18.7124 8.9854 14.1393 -12.3573  26.7400 29.6340 24.8310 26.4100 26.3640 27.1190 28.9540 26.589C 26.7940 23.9430 30.4780 25.8010 33.2310 34.3120 35.8990 ^6.2310 22.8760 34.5670 28.3970 24.9510 43.4600 25.2370 34.0830 28.5770 25.0450 33.6820 38.3990 23.6500 35.6010 28.1610 31.6740 30.3350 32.9740 27.4690 26.8590 28.7180 21.8050 30.0920 27.7660 23.06 80 29.6010  30.0000 30.0000 30. 0 0 0 0  3u.0OuG 30.CC00 30.0000 30.0000  3C.C000 3O.OOU0 30.0000 30.0000 30.0000  30.0000 30.0000 30.0000  30.0doc  30.0000 30.0000 30.OOOC 30.0000 30.0000 30.0000 30.0000 30.0030 30.0000 30.0000 30.GCQG 3C.000C 30.0000 30.0000 30.0000 30.0000 30.0000 30.0000 30.0000 30.0000 30 . 0 000 30.0000 30.0000 30.0000 .30.0000  1 CASE NUMBER 56 57 58 59 60 61"""" 62 63 64 65 66 -'-"-6-7' 66 69  7 XI 11 46.3070 24.9410 52.9200 33.1270 37.0350 53.4020 59.2930 68. 3590 45.9330 43.8150 58. 8010 55.4050 46.6170 67.4850  Y CCMPUTEO  RESIDUAL  XI 41  47.2114 39.3273 48.8391 51.3939 43.8605 ~42.6517 45.3096 51.8405 58.8462 51.7040 64.2576 60.0412 47.8880 65.8791  -0.9044 -14.3860 4.0909 -18.2669 -6.8255 10.7503 13.9834 16.5185 -12.9132 -7.8890 -5.4566 -4.6362 -1.2710 1.60 54  29.1770 24.0570 30.2340 31.8930 27.0010 ~26.2160. 27.9420 32.1830 46.0440 41.4060 49.5580 46.8200 3d.9280 50.bl10  XI 2) 30.OOOC 30.0000 30.0000 30.0000 30.0000 30.0000 30.0000 30.0000  45.U000 45.COCO 45.0000 45.OCOO 45. CO C O 45.0JC0  (  ~ '  '  "  181 70 72.2890 56.5456 15.7434 .4.5500 45.0GC0 182 71 45.1390 42.5260 2.6130 35.4460 45.O0CO 183 72 47.4340 49.6251 -2.1911 40.0560 45.C00U 1 8 4 " 7 3 8 2 . 5 3 0 0 5 9 . 2 3 4 3 " 2 3 . 2 9 5 7 ' ' " 4 6.29oO " " 4 5 . C 0 G G 185 74 70.3540 51.4176 Id.9364 41.2200 45.00C0 186 75 62.9920 53.9462 9.0459 42.8620 45.CC 00 TB7 76 63.1120 56.1329 6.9791 44.2820 45.CO CO 188 77 39.5630 42.4336 -2.8706 35.3860 45.0000 189 78 46.7300 47.2582 -0.5282 38.5190 45.OCOO 1 W '"79 "45.6230 45.3225 '" 0 . 3 0 0 5 3 7 . 2 6 2 0 4 5 . 0 0 0 0 191 80 29.5480 45. 2702 -15. 7222 37.2280 45.00C0 192 81 60.8420 5C.9217 9.9203 40.8980 45.U0CC I9T 82 52.7620 51.9766 0.7854 41.5830 45.00CO 194 83 46.3210 60.8774 -14.5564 47.3630 45.CO00 195 84 34.8970 54.1325 -19.2355 42.9830 45.00CG _ 30.1170 49.3002 -19.1832 39.8450 45.000u 197 86 45.1950 66.3088 -21.1138 50.8900 45.CO0O 198 87 67.9230 57.0845 10.8385 44.9000 45.0000 Vn 8"B 80.5390 61.5473 18.9917 47.7980 45.00C0 200 89 65.2610 56.3577 8.9033 44.42b0 45.0000 201 90 56 .84 00 51.5870 5.2530 41.3300 45.OOP. 202 91" 43.0760 4877150" -5.6390 39.4650 45700 CO 203 204 r  w  g  g  2o? 206 FINISH CARD ENCOUNTERED 207 PROGRAM TERMINATED END OF FILE  

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