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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, Ecole Nationale Superieure des Arts et Metiers, 1977 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ...APPUED SCIENCE i n THE FACULTY OF GRADUATE STUDIES (Department of C i v i l Engineering) We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA October 1979 © Robert J. M. Fouquet, 1979 In present ing th i s thes is in p a r t i a l fu l f i lment of the requirements for an advanced degree at the Univers i ty of B r i t i s h Columbia, I agree that the L ibrary sha l l make it f ree ly ava i lab le for reference and study. I fur ther agree that permission for extensive copying of th is thes is for scho lar l y purposes may be granted by the Head of my Department or by h is representat ives . It is understood that copying or pub l i ca t ion of th is thes is for f i n a n c i a l gain sha l l not be allowed without my wri t ten permission. Department of C i v i l Engineering The Univers i ty of B r i t i s h Columbia 2 0 7 5 Wesbrook Place Vancouver, Canada V6T 1W5 October 15, 1979. ABSTRACT Two sets of data are analyzed i n the the s i s . The f i r s t set was derived from the long 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 of Douglas-Fir loaded under constant bending stress to lev e l s lower than or equal to 3110 p s i (21.44 MPa). The second set was derived from the long term deforma-tions of 2 i n x 6 i n x 12 f t j o i s t s of Hemlock loaded under con-stant bending stress to levels of 3000 p s i (20.68 MPa) and 4500 p s i ("31.0 2 MPa) . The analysis shows that the creep behaviour of structur a l size beams depends upon the material 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 strength lower than 5000 p s i (34.33 MPa) appeared to creep 1.5 times more than,material with a strength higher than that l e v e l , over a three month period. In addition, the test results 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 applied stress. A method i s presented to predict the creep behaviour of a s t r u c t u r a l - s i z e specimen at discrete times over a three month period. The method consists of expressing the creep de-formation, <S , i n terms of the e l a s t i c deformation, <5 , or c e equivalently, the f r a c t i o n a l creep (f = 6* /6 ) i n terms of 8 or i n terms of the modulus of e l a s t i c i t y , e This work i s li m i t e d to the stress l e v e l s investigated and to s p e c i f i c temperature (10°C< :Q<30°C) and moisture content (8%<MC<12%) conditions. While t h i s method could be employed i n preliminary design procedures, i t has been esp e c i a l l y designed for more complex studies of the creep behaviour of structures including f l o o r systems, trusses, etc... The advantage of the method i s that i n t h i s kind, of analysis the modulus of e l a s t i c i t y of the i n d i v i d u a l components can be used. This thesis also presents a set of creep curves that cover a three year span. These creep curves show that the average t o t a l deformation of beams loaded to a stress l e v e l of 3110 p s i (21.44 MPa), at t h i s time, i s approximately 1.6 times the e l a s t i c deformation. TABLE OF CONTENTS Page ABSTRACT i i i TABLE OF CONTENTS V LIST OF TABLES v i i i LIST OF FIGURES x ACKNOWLEDGEMENTS xv CHAPTER 1. INTRODUCTION 1 1.1 O b j e c t i v e s 1 1.2 Rheology 2 1.3 F a c t o r s I n f l u e n c i n g Long Term Deformations 4 . 1.3.1 Time 5 1.3.2 S t r e s s 5 1.3.3 Temperature 8 1.3.4 Moisture Content 8 1.3.5 Volume 9 1.4 Long Term Deformation; F u n c t i o n a l Form 10 2. EXPERIMENT IN SURREY 12 2.1 P r e l i m i n a r i e s 12 2.2 M a t e r i a l Used 12 2.2.1 S e l e c t i o n o f M a t e r i a l a t M i l l S i t e 12 2.2.2 Grouping the M a t e r i a l 13 \ 2.3 P h y s i c a l Arrangement 16 v i 2.4 Deformation Measurement 17 2.4.1 Expected I n i t i a l Deflections .. 17 2.4.2 Actual I n i t i a l Deflections .... 17 2.4.3 Deflection as a Function of Time 19 2.5 Interpretations of Results 20 3. EXPERIMENT IN RICHMOND 22 3.1- M a t e r i a l i s e d ........ . 22 3.1.1 Selection of Material at M i l l Site 22 3.1.2 Grouping 22 3.2 Loading Configuration 24 3.3 Testing Procedure 25 3.4 Deformation Measurements 26 3.4.1 Expected I n i t i a l Deflections..., 26 3.4.2 Actual I n i t i a l Deflections .... 27 3.4.3 Deformation as a Function of Time 2 8 3.5 Interpretations of Results 29 3.5.1 E l a s t i c Deformations 30 3.5.2 General Comments on Creep Curves 35 3.5.3 Dependence of Creep Deformation on the Stress Level, the Stress Ratio and the Modulus of E l a s t i -c i t y 37 4. LINEAR VISCOELASTIC MODEL 42 4.1 Introduction 42 4.2 Three Parameter S o l i d 4 3 4.3 Stress Parameter 45 4.4 Comments 5 3 5. ANALYTICAL MODEL FITTED TO THE DATA 55 v i i 5.1 Model 55 5.2 Comments 57 5.3 Q u a n t i t a t i v e A n a l y s i s 58 5.3.1 Method 58 5.3.2 A n a l y s i s o f Creep Deformation a t Three Months 59 5.3.3 A n a l y s i s o f F r a c t i o n a l Creep a t Three Months 67 5.3.4 Summary 70 5.4 F i n a l R e s u l t s 72 6. CONCLUSIONS 75 BIBLIOGRAPHY 79 APPENDICES A - l MATHEMATICAL TREATMENT OF CREEP DATA CREEP DEFORMATION 17 3 A-2 MATHEMATICAL TREATMENT OF CREEP DATA CREEP DEFORMATION 178 A-3 MATHEMATICAL TREATMENT OF CREEP DATA CREEP DEFORMATION 184 B - l MATHEMATICAL TREATMENT OF CREEP DATA FRACTIONAL CREEP 189 B-2 MATHEMATICAL TREATMENT OF CREEP DATA FRACTIONAL CREEP 193 v i i i LIST OF TABLES Page Table I Starting time a f t e r loading for measuring creep 81 Table II Grouping of the material for Surrey experiment according to E-values (Batch 1) 82 Table III Grouping of the material for Surrey experiment according to E-values (Batch 2) 82 Table IV-a Data for Surrey test (10 70 psi) 83 Table IV-b Data for Surrey t e s t (1410 psi) 84 Table IV-c Data for Surrey test (.2.110 psi) 85 Table IV-d Date for Surrey test (3110 psi) 86 Table V S t a t i s t i c a l informations on the modulus of e l a s t i c i t y of boards used i n Surrey experiment 87 Table Vl-a Creep data from Surrey experiment (Group 1070 psi) 88 Table Vl-b Creep data from Surrey experiment (Group 1410 psi) 89 Table VI-c Creep data from Surrey experiment (Group 2110 psi) 90 Table Vl-d Creep data from Surrey experiment (Group 3110 psi) 91 Table VII Distance between a beam support and an empirical reference for the measurements 92 Table VIII-1 Creep data for boards loaded to \ 3110 p s i (Surrey experiment) (Board number 1115) . 9 3 Table VIII-2 Creep data for boards loaded to 3110 p s i (Surrey experiment) (Board number 1609) 94 Table VIII-3 Creep data for boards loaded to 3110 p s i (Surrey experiment) (Board number 1589) 95 i x Table VIII-4 Table VII1-5 Table VIII-6 Table VIII-7 Table VIII-8 Table IX Table X Table XI Table XII Table XIII Table XIV Creep data for boards loaded to 3110 p s i (Surrey experiment) (Board number 2672) 96 Creep data for boards loaded to 3110 p s i (Surrey experiment) (Board number 2395) 97 Creep data for boards loaded to 3110 p s i (Surrey experiment) (Board number 1896) 98 Creep data for boards loaded to 3110 p s i (Surrey experiment) (Board number 1829) 99 Creep data for boards loaded to 3110 p s i (Surrey experiment) (Board number 1419) 100 Data for Richmond experiment (Sub-group 1) 101 Data for Richmond experiment (Sub-group 2) 10 3 Fractional creep expressed at three months for f i v e ranges of stress r a t i o s . (Average values) 107 C o e f f i c i e n t s A and B of the c o r r e l a -t i o n equation between creep deforma-t i o n and e l a s t i c deformation for two stress l e v e l s 108 Coeffi c i e n t s A and B of the c o r r e l a -t i o n equation between creep deforma-tion and e l a s t i c deformation, d i s r e -gard of the stress l e v e l 109 Coef f i c i e n t s C and D of 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 , disregard of the stress l e v e l 110 X LIST OF FIGURES Page Figure 1 Stress and s t r a i n versus time d i a -grams for a specimen of a hypotheti-c a l material I l l Figure 2 Frac t i o n a l creep i n Hoop Pine i n bending 112 Figure 3-a Devices used to measure creep of wood i n bending 113 Figure 3-b The increase i n f r a c t i o n a l creep of the small and the large specimens between 1/256 day and 1 day as func-t i o n of stress i n t e n s i t y 113 Figure 4 Creep compliance versus time for 4 stress le v e l s expressed as percentr-ages of proportional l i m i t 114 Figure 5 Theoretical development of creep under intermittent loading 115 Figure 6 Creep curves for various types of temperature 116 Figure 7 Creep versus time for hardboard and plywood 117 _3 Figure 8 Creep from 10 hours to 1 hour of 1/8" fiberboard i n tension versus stress and f r a c t i o n a l stress 117 Figure 9 Sample I. Grouping of the material. 118 Figure 10 Strength d i s t r i b u t i o n of the control sample from Sample 1 119 Figure 11 Test set-up for Surrey experiment ... 120 Figure 12-1 Test set-up for Surrey experiment. Picture as an end-view 121 Figure 12-2 Test set-up for Surrey experiment. Picture as an end-view 122 Figure 13 Deflection versus time for four stress levels. 123 Figure 14 Deflection versus time for four stress l e v e l s 124 XI Figure 15 Figure 16 Figure 17 Figure 18 Figure 19 Figure 20 Figure 21 Figure 22 Figure 23 Figure 24-1 Figure 24-2 Figure 25 Figure 26 Figure 27 Figure 28 Figure 29-1 Figure 29-2 Sample I I . Grouping of material.... 125 Strength d i s t r i b u t i o n of the control sample from population II 126 Strength d i s t r i b u t i o n of the control sample from population II 127 Picture of Richmond set-up 128 Picture of Richmond set-up 12 9 Charts of load and d e f l e c t i o n v a r i a -tions with respect to time during transfer of the load 130 Example of creep curves plotted on a b i l i n e a r graph 131 Examples of creep curves plotted on a f u l l logarithmic graph and a semi logarithmic graph 132 Creep curves for boards of sub-group 1 133 Creep curves for boards of sub-group 2 loaded to 3000 p s i 137 Creep curves for boards of sub-group 2 loaded to 4500 p s i 140 Actual e l a s t i c deformations versus predicted 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 -tions 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. Hemlock 142 Cumulative p r o b a b i l i t y d i s t r i b u -tions 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. Amabilis f i r 143 Creep d e f l e c t i o n , measured at 10 d i f f e r e n t times, versus e l a s t i c deformation, for two stress l e v e l s . Raw data 144 Creep curves for boards with same value of e l a s t i c deformation (36 mm) 145 Creep curves for boards with same value of e l a s t i c deformation (38 mm) 146 x i i Figure 29-3 Figure 29-4 Figure 29-5 Figure 29-6 Figure 30 Figure 31 Figure 32-1 Figure 32-2 Figure 32-3 Figure 32-4 Figure 32-5 Figure 32-6 Figure 32-7 Creep curves for boards with same value of e l a s t i c deformation (41 mm) 147 Creep curves for boards with same value of e l a s t i c deformation (42 mm) 148 Creep curves for boards with same value of e l a s t i c deformation (4 3 mm) 149 Creep curves for boards with same value of e l a s t i c deformation (45 mm) 150 Fractional creep expressed at 10 d i f f e r e n t times versus e l a s t i c de-formation, for two stress l e v e l s . Raw data 151 Frac t i o n a l creep, expressed at 10 d i f f e r e n t times versus modulus of e l a s t i c i t y , for two stress l e v e l s . Raw data 152 Fracti o n a l creep plotted against the modulus of e l a s t i c i t y f or s p e c i f i c ranges of stress r a t i o s , at time t = 5 minutes 153 Fracti o n a l creep plotted against the modulus of e l a s t i c i t y for s p e c i f i c ranges of stress r a t i o s , at time t = 1 hour 154 Fractional creep plotted against the modulus of e l a s t i c i t y for s p e c i f i c ranges of stress r a t i o s , at time t = 10 hours 155 Fractional Creep plotted against the modulus of e l a s t i c i t y for s p e c i f i c ranges of stress r a t i o s , at time t = 2 days 156 Fractional creep plotted against the modulus of e l a s t i c i t y for s p e c i f i c ranges of stress r a t i o s , at time t = 4 days 157 Fractional Creep plotted against the modulus of e l a s t i c i t y for s p e c i f i c ranges of stress r a t i o s , at time t = 3 weeks 158 Fracti o n a l creep plotted against the modulus of e l a s t i c i t y for s p e c i f i c ranges of stress r a t i o s , at time x i i i t = 6 weeks 159 Figure 32-8 Fractional creep plotted against the modulus of e l a s t i c i t y for s p e c i f i c ranges of stress r a t i o s , at time t = 9 weeks. 160 Figure 32-9 Fractional creep plotted against the modulus of e l a s t i c i t y for s p e c i f i c ranges of stress r a t i o s , at time t = 3 months 161 Figure 32-10 Fr a c t i o n a l creep plotted against the modulus of e l a s t i c i t y for s p e c i f i c ranges of stress r a t i o s , at time t = 13 weeks 162 Figure 33 Creep deformation versus e l a s t i c de-formation. A n a l y t i c a l model f i t t e d to data for 3000 p s i 163 Figure 34 Creep deformation versus e l a s t i c de-formation. A n a l y t i c a l model f i t t e d to data for 4500 p s i 164 Figure 35 Fract i o n a l creep versus e l a s t i c de-formation. A n a l y t i c a l model f i t t e d to data for 3000 p s i 165 Figure 36 Fract i o n a l creep versus e l a s t i c de-formation. A n a l y t i c a l model f i t t e d to data for 4500 p s i 166 Figure 37 Fract 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 for 3000 p s i 167 Figure 38 Fractional 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 for 4500 p s i 168 Figure 39 Fract i o n a l creep, expressed at 10 d i f f e r e n t times, versus stress r a t i o for 1 stress l e v e l : 3000 p s i . Raw data 169 Figure 40 Creep deformation versus e l a s t i c de-formation. A n a l y t i c a l model f i t t e d to data points for 3000 and 4500 p s i . 170 Figure 41 Fract i o n a l creep versus e l a s t i c de-formation. A n a l y t i c a l model f i t t e d to data points for 3000 and 4500 p s i . 171 x i v F i g u r e 42 F r a c t i o n a l c r e e p v e r s u s m o d u l u s o f e l a s t i c i t y . A n a l y t i c a l m o d e l f i t t e d t o d a t a p o i n t s f o r 3000 a n d 4500 p s i . 172 X V ACKNOWLEDGEMENTS The author wishes to thank Prof. B. Madsen and Dr. M. Olson of the University of B r i t i s h Columbia, Department of C i v i l Engineering, whose advice and c r i t i c a l review of the work are greatly appreciated. Special acknowledgement i s hereby given to Forintek Canada Corporation, Department of Wood Engineering, for the use of i t s laboratory f a c i l i t i e s . In p a r t i c u l a r , thanks to Dr. J.D. Barrett, Dr. R.O. Foschi, and Dr. P. Gl^s, whose enthusiasm and suggestions were of great value, as was the technical assistance of Mr. B. McKinney and Mr. L. Olson. In addition, I am gratef u l to Mr. R. Grigg for his help with the computer at the Department of C i v i l Engineering, and to Ms. S. Pennant and J. F r y x e l l for t h e i r e d i t o r i a l comments and typing the manuscript. 1 CHAPTER 1 INTRODUCTION 1.1 Objectives The design of structures r e l i e s on the understanding of t h e i r basic behaviour. Such a basis includes the material properties, the behaviour of the component parts and the be-haviour of the complete structure as a unit. One requirement of the design procedure i s that expected loads be ca r r i e d safely. This condition can be f u l f i l l e d i f the strength properties of the material are known. Another requirement of the design procedure i s to provide a structure that meets functional requirements. In t h i s context, i t may be esse n t i a l that the appearance of the structure, as well as i t s component parts, be c a r e f u l l y considered. This l a t t e r consideration w i l l place deformation l i m i t s on s t r u c t u r a l com-ponents, and these are often the governing factors of design for wood. For new materials such as polymers, the design pro-cedure i s based upon an expected l i f e t i m e - that i s upon a cer-t a i n deformation by a certain time. For wood, some codes allow for long term deformations by using conservative and rather 2 crude design formulae, but f a i l to incorporate the true func-t i o n a l relationship of these long term deformations. In order to accurately predict the behaviour of wooden structures, 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. This study i s aimed at c o l l e c t i n g basic information on creep for lumber and at developing simple models which would, in an approximate way, describe the time behaviour under specif -i c environmental conditions (constant humidity and temperature). 1.2 Rheology Rheology i s the branch of science that studies the long term deformational behaviour of materials. Rheological studies have t r a d i t i o n a l l y emphasized descriptive observations more than the actual mechanisms at a molecular l e v e l . This i s not s u r p r i s -ing considering the s t r u c t u r a l complexity of wood. A complete molecular theory would be very d i f f i c u l t to develop indeed. Rheological models are often used i n such descriptive studies. This approach assumes that the deformations of mate-r i a l s may be c l a s s i f i e d as either e l a s t i c , viscous or p l a s t i c . From th i s assumption, mathematical expressions for the deformational behaviour may be derived. These expressions cor-respond to the deformation of mechanical systems consisting of springs ( e l a s t i c i t y ) , dashpots (viscosity) and f r i c t i o n elements ( p l a s t i c i t y ) . There i s great discrepancy i n the terminology used to describe these phenomena. For example Nielsen (1972) has used terms shown i n Figure 1 where deformation for a hypothetical 3 material i n a u n i a x i a l test i s plotted 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 deformations. The l a t t e r i s i n most cases neglected. As w i l l be shown i n further chapters where experimental re s u l t s are presented, i t i s extremely d i f f i c u l t to quantify the instantaneous deformation. The main reason i s that the time of application of the load has a f i n i t e value, and the time-dependent deformation takes place already before f u l l load has been transferred. Furthermore, studies by Spencer (1978) have shown how wood strength i s i n -fluenced by the rate of application of the load; i t i s not known,, however, to what extent the instantaneous deformation i s affected. In common research practice, the i n i t i a l deformation i s defined as the t o t a l deformation at a s p e c i f i e d time aft e r s t a r t -ing the loading. In Table I, Nielsen l i s t s a number of such times for various materials.. For wood, t h i s time goes up to one minute. The deformations shown i n Figure 1 are strains; i n t h i s present work, deformations w i l l refer to deflections. The i n -stantaneous deformation w i l l be c a l l e d e l a s t i c deformation and 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 the loading. After the i n i t i a l time, the deformation continues to i n -crease and i s referred to as creep deformation. I t consists of a recoverable part, the delayed e l a s t i c deformation, and an irrecoverable part, the viscous deformation. These components can only be d i f f e r e n t i a t e d by unloading the structure; but i t i s d i f f i c u l t to unload while simultaneously taking deformation measurements. Thus, I did not d i f f e r e n t i a t e between the two components of creep deformation i n t h i s study. The creep de-formation at a given time, w i l l be defined as the difference between the t o t a l deformation at t h i s p a r t i c u l a r time and the e l a s t i c deformation. Another quantity w i l l be used to characterize the creep development: f r a c t i o n a l creep, defined as the r a t i o of creep deformation to e l a s t i c deformation. 1.3 Factors Influencing Long Term Deformations Creep deformation, & i s commonly expressed as a func-tion of various parameters, i n the following form: &c = f i t , a (t,x), 9 (t,x), M.C., V} where t i s the time parameter, a the stress parameter, x the space parameter (defining the location where the measurements are recorded), 9 the temperature, M.C. the moisture content and V the volume of the specimen. This may not be a t o t a l picture of the dependency, but i s a usual f i r s t approximation. It i s important to know to what extent and i n what manner creep deformation i s influenced by these parameters. This knowledge may a f f e c t the design of any long term loading experiments as well as the interpretation of experimental re-sults . I w i l l discuss, the preceding parameters one at a time, assuming the others remain constant. 1.3.1 Time For a given material, the magnitude of creep increases with time, as i l l u s t r a t e d i n the following figure: 1.3.2 Stress a) For a given material, the magnitude of creep increases with increasing stress, as i l l u s t r a t e d i n the following figure: There are several examples i n the l i t e r a t u r e that i l -l u strate t h i s tendency for wood and wood-based materials. Kingston and Clarke (1961) studied several species. 6 Specimens 3/4 i n wide (19 mm), by 7/16 i n deep (11 mm) with a span of 12 i n (305 mm) were subjected to four point bending. Figure 2 shows some of t h e i r results for Hoop Pine. The sec-ond of t h e i r graphs shows f r a c t i o n a l creep plotted against the stress expressed as a percentage of ultimate stress which i s another way of i l l u s t r a t i n g the trend. Cederberg and Danielsson .(1970) presented experimental results on Pine and Spruce, construction grade T200 and T300 (Swedish Standard) . Figure 3-a shows the experimental arrange-ment for two s i z e s : 0.8 i n x 0.8 i n x 15.8 i n (20 mm x 20 mm x 400 mm) and 2 i n x 4 i n x 118 i n (50 mm x 100 mm x 3000 mm). Figure 3-b shows the increase of f r a c t i o n a l creep (e^/e^) be-tween 1/256 day and one day as a function of stress i n t e n s i t y expressed as the r a t i o to the ultimate stress, for two d i f f e r -ent sizes. I t can be seen that the f r a c t i o n a l creep of the small specimens increases s i g n i f i c a n t l y more with the stress in t e n s i t y than that of large specimens. Nielsen (1972) argues that the influence of si 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 material such as knots and grain d i s t o r t i o n , i s not very s i g n i f -icant 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 influenced to the same extent. In contrast t h i s may have a marked.influence on the fracture stress a_., large beams being up to 50% weeker than small beams. Nakai (1978) studied eight wood-based materials. Speci-mens 2 i n (50 mm) wide by 18 i n (450 mm) long, with a thickness ranging from .32 i n (8 mm) to 1 i n (.25 mm), were subjected to four point bending. Stress levels were selected at 5/4,4/4,3/4, and 2/4 of proportional l i m i t , defined as that point at which 7 the r e l a t i o n between stress and deformation ceases to be l i n e a r . Figure 4 shows some of his r e s u l t s : the creep compliance., J, defined as the s t r a i n per unit stress, i s plotted versus time for selected stress l e v e l s , for st r u c t u r a l plywood mark 12 Ko. From these graphs, i t can be seen that the magnitude of creep increases with increasing applied stress. b) Stress influences the creep development i n a second way: the creep behaviour of a material at a< c e r t a i n moment depends upon the previous stress history. Some examples of stress h i s t o r i e s and t h e i r influence on creep deformation can be found in the l i t e r a t u r e . Unfortunately, these studies do not apply to wood d i r e c t l y . Nielsen (1972) shows the t h e o r e t i c a l devel-opment of creep under intermittent loading (Figure 5); the creep tendency i s decreased upon each new loading. Work by Lundgren (1968) on hardboard seems to confirm t h i s trend for wood-based materials. A creep experiment i s based on a constant magnitude of applied load. I did not consider load.variations such as intermittent loading 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 to a load history which i s very seldom constant; hence the long-term deformational behaviour would be better described under these actual conditions. On the other hand, constant stress loading w i l l l i k e l y y i e l d an upper bound of deformation ( possibly fatigue loading excluded ), and i t would be very d i f f i c u l t to define load h i s t o r i e s which would represent the great variety of loads on a structure. 8 1.3.3 Temperature Temperature influences creep development i n two ways: a) For a given material, the magnitude of creep increases with increasing temperature. An i l l u s t r a t i o n of t h i s point i s seen i n Figure 2 (Kingston and Clarke 1961) . Sauer and Haygreen (196 8) tested hardboard i n flexure and found that at a stress i n t e n s i t y of 30%, an increase i n tem-perature from 2 2.2 C to 33.3 C gave an increase i n creep deflec-tion of about 50%. b) Changes i n temperature during a creep experiment w i l l give r i s e to i n t e r n a l stresses i n a specimen (the i n t e r n a l stresses depend on the gradient of temperature with respect to time as well as to specimen dimensions) and thereby influence the creep. An i l l u s t r a t i o n of t h i s point i s given by Kitahara and Yukawa (1964) who studied. Chamaecyparis obtusa i n small s i z e : 0.4 i n x 0.4 i n x 16.8 i n (10 mm x 10 mm x 420 mm). Specimens were tested using four point bending at a stress l e v e l of 1370 p s i (9.45 MPa). Figure 6 shows the d i f f e r e n t types of temperature used as well as plots of deflection versus time. 1.3.4 Moisture Content A great deal of research has been c a r r i e d out to study the influence of moisture content upon creep deformation. Schniewind (1968) presents a good survey of the re s u l t s obtain-ed up to 1968. More s p e c i f i c results concerning wood i n struc-t u r a l sizes can be found i n the work by Nielsen (1972), appendix B. As a summary, one can say that moisture influences the 9 creep development i n two ways: a) A specimen i n equilibrium at a high 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 at a lower RH. b) I f a change i n the moisture content occurs during the creep experiment, i t w i l l also cause a change, most often an increase, i n the magnitude of creep. These two phenomena are i l l u s t r a t e d i n the following figures: Creep Creep deformation 6 deformation <5 Relative humidity Time t 1.3.5 Volume The specimen volume influences creep development i n two ways: a) Volume exerts an influence on various other factors such as gradient of temperature or moisture content. b) The greater the volume, the more l i k e l y the material w i l l contain flaws and, therefore, deform more e a s i l y . In tests with f u l l size members, thi s phenomenum i s d i r e c t l y accounted for; however, for a given volume there i s s t i l l a discrepancy 10 i n t h e t y p e s o f d e f e c t s p r e s e n t i n t h e m a t e r i a l , t h e i r number, s i z e , l o c a t i o n . . . C o m m e r c i a l m a t e r i a l i s g r a d e d a c c o r d i n g t o t h e k i n d a n d q u a n t i t i e s o f f l a w s a l l o w e d . N o t e t h a t t h i s e f f e c t , r e f e r r e d t o a s g r a d e e f f e c t , i s n o t d i r e c t l y a c c o u n t e d f o r i n t h e f u n c t i o n a l e x p r e s s i o n o f c r e e p d e f o r m a t i o n . 1.4 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 F o r m The p r e v i o u s s e c t i o n h a s shown t h a t t h e c r e e p d e f o r m a t i o n i s h i g h l y d e p e n d e n t upon s e v e r a l f a c t o r s a n d i t s s t u d y on a p h e n o m e n o l o g i c a l b a s i s i s t h e r e f o r e r a t h e r d i f f i c u l t . I n common p r a c t i c e , 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 some v a r i a b l e s t o c o n s t a n t v a l u e s a n d s t u d y i n g t h e d e p e n d e n c e o f c r e e p d e f o r -m a t i o n u p o n a few s e l e c t e d v a r i a b l e s . I n t h i s s t u d y , t h e s p a c e p a r a m e t e r , t e m p e r a t u r e a n d m o i s -t u r e c o n t e n t p a r a m e t e r s w i l l be c o n s t a n t . H o w e v e r , we w i l l a t t e m p t t o a c c o u n t f o r a n o t h e r f a c t o r , t h e g r a d e e f f e c t , t h a t d i d n o t a p p e a r i n t h e f u n c t i o n a l e x p r e s s i o n o f 5 , g i v e n i n t h e p r e v i o u s s e c t i o n . An e a s y way t o do t h i s i s t o c o n s i d e r two new v a r i a b l e s , t h e m o d u l u s o f e l a s t i c i t y a n d t h e s t r e n g t h . B o t h t h e s e v a r i a b l e s d e p e n d upon t h e g r a d e o f t h e m a t e r i a l : . . I n i t s f u n c t i o n a l , f o r m , c r e e p d e f o r m a t i o n c a n now be e x -p r e s s e d as f o l l o w s : & = f{t, a ( t , x ) , MOE, s t r e n g t h } o r i n t h e f o l l o w i n g way: <5c = f { t , a ( t , x ) , MOE, SR} w h e r e SR i s t h e s t r e s s r a t i o , d e f i n e d a s t h e r a t i o o f a p p l i e d s t r e s s t o s h o r t - t e r m s t r e n g t h ; MOE i s t h e m o d u l u s o f e l a s t i c i t y . I f the stress r a t i o s are computed at one l e v e l of applied stress, as w i l l be the case i n t h i s thesis, then the two expressions are equivalent i n that strength and stress r a t i o are inversely proportional. The function f w i l l be determined by f i t t i n g mathematical expressions to creep curves. A large number of mathematical ex-pressions have been applied successfully to creep (more than 30 expressions for metals, 10 for concrete), however these ex-pressions are e s s e n t i a l l y time functions. Furthermore, only a few have been applied to wood. Nielsen (1972) showed an example of time function for hardboard and plywood subject to tension. The power function e = A t b c was used, f i t t i n g the data shown i n Figure 7. Kingston and Clarke (1961) assumed that recoverable time-dependent deformation of wood w i l l , as to f i r s t approxi-mation, follow the hyperbolic sine law, as developed by Eyring's (1935) application of s t a t i s t i c a l mechanics to rate processes. This law i s expressed i n terms of thermodynamic constants. This law i s written: = K, sinh K~a dt 1 2 where K^ and ^ are experimental constants. Figure 8 shows an application of t h i s theory to fiberboard (masonite i n tension) (Lundgren 1968). The agreement between theory and experimental results i s very good. Due to time l i m i t a t i o n s I was not able to perform a refined mathematical treatment of this sort. 12 CHAPTER 2 EXPERIMENT IN SURREY 2.1 P r e l i m i n a r i e s The p r e v i o u s chapter has shown t h a t long term deforma-t i o n s depend upon s e v e r a l v a r i a b l e s such as moisture content, temperature, s t r e s s , e t c . In the experimental work presented i n t h i s t h e s i s , moisture content and temperature were approximately c o n s t a n t . The o r i g i n a l purpose of the experiment, c a r r i e d out i n Surrey, B.C., was to i n v e s t i g a t e the r e l a t i o n s h i p between ap-p l i e d s t r e s s and time f o r commercial lumber. The p h y s i c a l arrangement was designed a c c o r d i n g l y . Some deformation measurements were taken.over three years i n o r d e r to get a coarse i d e a o f creep. Although these measure-ments are not very a c c u r a t e , they c o n s t i t u t e a r a t h e r unique data s e t th a t covers a long time span. 2.2 M a t e r i a l Used 2 . 2 . 1 S e l e c t i o n of M a t e r i a l a t M i l l S i t e 13 Jo i s t s of Douglas F i r 2 i n x 6 i n x 12 f t (40 mm x 140 mm x 3600 mm) were selected i n 1973 from a single m i l l i n the i n t e r i o r of B r i t i s h Columbia. Material was selected as i t came from the planer m i l l a f t e r the material had been k i l n dried to the standard maximum 19%. This material was e s s e n t i a l l y "no. 2 and better" grade with only a few boards of no. 3 grade included. The lumber was grade marked at the m i l l , stacked and wrapped before shipment to the laboratory i n Vancouver. 2.2.2- Grouping 'tne Materia 1 The sub-sample of lumber used i n the actual experiments was selected from the i n i t i a l shipment using the following pro-cedure (Figure 9). Step 1: The Shipment of thi s material was made i n two batches; the f i r s t load arrived i n the laboratory on July 17, 1973. The load consisted of eight packages, each con-taining 128 boards, for a t o t a l of 1024 boards. The second load arrived i n the laboratory on August 16, 1973. I t consisted of 12 packages (1536 boards). Step 2: 400 boards, out of the f i r s t 1024, were randomly selected, and the rest sent to storage. 600 boards, out of the next 1536, were randomly selected and the rest sent to storage. Step 3: a) Boards were v i s u a l l y inspected to determine the weaker of the two edges. b) Moisture content readings were taken with a resistance type moisture meter. Readings were 14 taken at 3 feet, 6 feet and 9 feet from the end of the boards. c) The boards were subjected to t h i r d point load and t h e i r E-value calculated. They were con-sequently assigned to E groups depending upon the calculated E-value. Table IT and III' show the grouping for the f i r s t and second batches, respectively. The formula used for the c a l c u l a t i o n of the modulus of e l a s t i c i t y i s : . 23 P I 3 0 = 648 EI where P = 50.3 lbs (226.4 N) 1 = 138 i n (3500 mm) g 4 I = 20.8 i n 4 (8.66 x 10 mm ) for average 2 in x 6 i n section. <5 = d e f l e c t i o n measured at the centre of the span. Step 4: The boards of each E-category were ranked according to t h e i r apparent strength. (The boards were v i s u -a l l y graded for strength using slope of grain, knot size and location etc. as strength indicators (National Lumber Grades Authorities, 1970)):. After being ranked from strongest to weakest, the boards from each category were divided into four sub-groups s t a r t i n g from the lowest ranking strength. Step 5: The boards from each sub-group (for a t o t a l of 44 sub-groups) were randomly allocated to four groups. The end r e s u l t i s four groups of 250 boards each. Step 6: For each group, 10 boards out of 250 were discarded randomly. Step 7: One of the 4 groups was set aside and constituted 15 a control sample. The 3 remaining p i l e s were subdivided by assigning the f i r s t board to sub-group "a", the second board to sub-group "b", the t h i r d board to sub-group "c", the fourth board to sub-group "a", etc. The end r e s u l t i s 9 groups of 80 boards each. The boards from the control group were re-weighed, and a moisture content reading taken at the centre of the board. The dimensions were also recorded: the width and depth at the centre to the nearest 0.01 i n and the length to the nearest h a l f - i n c h . The 240 specimens were then tested to f a i l u r e , using a third-point loading configuration and a rate of loading of 5000 psi/mn (34.5 MPa/mn), which produced an average time to f a i l u r e of approx-imately one minute. Figure 10 shows the strength d i s t r i b u t i o n of the control sample, plotted using the normalized ranking method. Step 8: Four groups out of the nine f i n a l groups were ran-domly selected for the purpose of the experiment. The experimental 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 . This was done by excluding the boards that had a high modulus of e l a s t i c i t y as well as small knots and were therefore unlik e l y to f a i l during the l i f e - t i m e of the experiment. 2.3 Physical Arrangement The experiment was started on January 4, 19 75. Two hundred and nineteen boards were loaded at various stress l e v e l s , using the set-up i n Figure 11. 'Photographs of the experimental arrangement are?seen i n Figures 12-1 and 12-2. Thirty-three boards were subjected to a stress l e v e l of 1070 p s i (7.38 MPa) corresponding to 0.71 times the f i f t h percentile value on the strength d i s t r i b u t i o n of the control sample. The factor of 0.71 i s the strength reduction predicted according to the Madison hypothesis for a constant load of 7 day duration. This factor was introduced i n previous experiments designed to study the time-strength r e l a t i o n s h i p over a 7 day period, and was kept for consistancy i n these t r i a l s (Madsen, 1976). Forty-two boards were subjected to a stress l e v e l of 1410 p s i (9.72 MPa) corresponding to 0.71 times the tenth per-c e n t i l e value. Seventy-two boards were subjected to a stress l e v e l of 2110 p s i (14.55 MPa) corresponding to 0.71 times the twenty-fifth percentile value. Seventy-two boards were subjected to a stress l e v e l of 3110 p s i (21.44 MPa) corresponding to 0.71 times the f i f t i e t h percentile value. I analyzed the measurements obtained from a random sub-sample of the boards that were s t i l l unbroken i n February 1979. This sub-sample consisted of: 15 boards subjected to 1070 p s i , 16 boards subjected to 1410 p s i , 25 boards subjected to 2110 p s i , and 8 boards subjected to 3110 p s i . Tables IV a, b, c and d show" a l i s t of these boards with t h e i r E-value calculated as mentioned i n Step 3 of section 2.2.2. Table V shows the mean and standard deviation values of the modulus of e l a s t i c i t y f or the four 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 also shown. 2.4 Deformation Measurement 2.4.1 Expected I n i t i a l Deflections The loading configuration i s represented i n the follow-ing f i gure: Where deformations are measured. -(—5 i n 46 i n 46 .in 138 i n 5 in — f -46 i n 144 i n .x I used the simple homogeneous beam theory to predict the i n i t i a l d e f l e c t i o n at x = 5 i n from- the point of applica-tion of the load. I used E-values as well as cross-sectional dimensions shown i n Tables IV a, b, c and d i n the calcu l a t i o n s . Deflection due to shear was neglected. Tables IV a, b, c and d show these values of i n i t i a l d e f l e c t i o n . 2.4.2 Actual I n i t i a l Deflections A scheme of measurements that were done i s represented 18 as follows: 138 i n NORTH SOUTH s i Empirical references 6 and c5 were measured within half an hour of each other, at n s various times during the t e s t i n g period, with a higher frequency i n the beginninq. 6 . and 6 . are the f i r s t values available of ^ = n i s i 6 and <5 . Y and Y were measured once as c h a r a c t e r i s t i c s of n s n s the test set-up. C a l l i n g f . and f . the actual i n i t i a l deflec-* ^ nx s i tions of a board at i t s north and south ends, one obtains the following: f . = Y • - 5 . nx n n i ' s i ' S S I The r e s u l t s for f . and f ., as well as t h e i r average hif . + f .) nx s i y nx s i are presented i n Tables IV a, b, c and d f o r the four stress l e v e l s . Error on i n i t i a l d e f l e c t i o n , defined as the r a t i o of the difference between measured and predicted deflections to the predicted d e f l e c t i o n i s also shown. 2.4.3 Deflection as a Function of Time If we define the def l e c t i o n of one group (defined by i t s stress level) as the average def 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 of each group at various times throughout the experiment. Standard deviations as well as c o e f f i c i e n t s of variations are also shown. As an example of the calculations that have yielded these r e s u l t s , consider the group of eight beams loaded at 3110 p s i . Table VII shows Y and Y for each board, as character-n ' s i s t i c s of the test set-up at the s p e c i f i c location of the board. Table V I I I - i (i = 1,8) shows the measured 6 n and 6 , the calculated f and f and the average %(f + f ) for a given board i over the course of the experiment. Unfortunately, i t was not possible to measure y n and y s for every board i n the three groups. In such a case I used the average values of available Y ' S and v 1s to estimate Y„ and Y . 1 n ' s 'n 1 s The steps y i e l d i n g the results presented i n Tables VI a, b, and c (groups 1070 p s i , 1410 p s i , 2110 psi) are available from the University of B r i t i s h Columbia, Department of C i v i l Engineering. The raw data presented i n Tables VI a, b, c and d are plotted i n Figure 13. Points ( i = 1,4) represent the expected i n i t i a l de-f l e c t i o n s for the four groups: M* (0.1 hr, 0.52 in) (13.2 mm) M^ (0.1 hr, 0.68 in) (17.3 mm) M3 (0.1 hr, 0.90 in) (22.9 mm) 20 (0.1 hr, 1.30 in) (33.0 mm) Time t = 0.1 hr was chosen as the o r i g i n of times. Points (i = 1,4) represent actual i n i t i a l deflections for the four groups: MJ (1 hr, 0.29 in) ( .7.4 mm) 2 M Q (1 hr, 0.60 in) (15.2 mm) MQ (1 hr, 1.02 in) (25.9 mm) MQ (1 hr, 1.20 in) (30.5 mm) Time t = 1 hr was chosen a r b i t r a r i l y since I did not know when the f i r s t measurements were taken. 2.5 Interpretations of Results As mentioned i n the previous section, the physical arrangement of t h i s experiment introduced l i m i t a t i o n s on the type of deformations, recorded, as well as t h e i r accuracy. For these reasons, the results presented, re f e r to a deformation defined as the average of the deflections of i n d i v i d u a l boards, measured at a convenient location (5 inches away from the point of application of the load). I did not measure actual values of e l a s t i c deformations. Therefore, I assumed that points ( i = 1,4), shown i n Figure 13, refer to the actual values of e l a s t i c deformation. I t i s then possible to redraw the four creep curves, shown i n Figure 13, from these points M ^ ( i = 1,4). The redrawn curves are shown i n Figure 14. I assumed that any experimental error remained constant over time. Trend l i n e s have been drawn as well . They indicate that the rate of change of creep deformation seems to increase with increasing applied stress. I t should be stressed that the presentation of the re-sults on a semi-logarithmic graph accentuates the slope of the curve i n the upper region of the X-axis (time). Had the curves been plotted on a b i l i n e a r graph, they would be very f l a t indeed, ind i c a t i n g that any substantial increase in deformation takes place over a long period of time. 22 CHAPTER 3 EXPERIMENT IN RICHMOND 3.1 Material Used 3.1.1 Selection of Material at M i l l Site Hem-fir j o i s t s , 2 i n x 6 i n x 12 f t (40 mm x 140 mm x 3600 ram), were selected i n August 1977 from a m i l l i n the Greater Vancouver area. (30% of the boards were Amabilis-Fir; 70% were Hemlock.) 2947 pieces were selected as they came from the planer m i l l a f t e r they had been k i l n dried to the standard maximum 19%. This material was "no. 2 and better" grade. 3.1.2 Grouping After the boards had arrived i n the Laboratory, they were numbered, re-graded by a COFI lumber inspector; species i d e n t i f i c a t i o n chips were taken. A l l pieces were dried at the laboratory to a moisture content ranging between 8% and 12%. Figure 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 species were separated, and badly warped pieces were removed. Step 2: The boards were subjected to t h i r d point load and the i r E-value calculated. They were subsequently ranked according to these values. The f i r s t 18 boards were then randomly allocated to 18 d i f f e r e n t groups. The next 18 boards were simi-l a r l y allocated to the 18 groups. This resulted i n 18 Hemlock groups and 5 Amabilis-Fir groups s t a t i s -t i c a l l y s i m i l a r within each species with regard to the value of the modulus of e l a s t i c i t y . Step 3: Random selection of six groups. Groups no. 2 for Amabilis-Fir and no. 15 for Hemlock were used to investigate E-values (Chapter 3.5.1). Groups no. 2, 6, 8 and 9 for Hemlock were used i n the creep experiment. Figures 16 and 17 show the strength d i s t r i b u t i o n of the control sample (Group no. 17). The boards of this group were broken at a constant rate of loading of 5000 psi/mn (34.5 MPa/mn) that produced an average time to f a i l u r e of approx-imately one minute. In Figure 16, two curves were f i t t e d to the raw data; the s o l i d l i n e represents a three parameter Weibull d i s t r i b u t i o n ; the dashed l i n e represents a two parameter Weibull d i s t r i b u t i o n . In Figure 17, a lognormal d i s t r i b u t i o n was f i t t e d to the data points. These three d i s t r i b u t i o n s have been widely used to approximate data on various properties of wood. In pa r t i c u l a r , the three parameter Weibull d i s t r i b u t i o n has been put forward as being a good representation for timber strength data (Pierce 1976). This i s v e r i f i e d i n the present work; Figures 16 and 17 show that a better f i t i s obtained with the three parameter Weibull d i s t r i b u t i o n . The r e l a t i o n between the cumulative p r o b a b i l i t y of f a i l u r e , F, and the applied stress, a, i s then written as f o l l o w s : r F = 1 - exp 1.987 /a - 1428\ \6156 / where a i s expressed i n p s i . 3.2 Loading Configuration 46 i n 46 i n 46 i n i t " 'h A four points loading configuration i s represented i n the above figure. The span has a constant value of 138 i n and the loads are applied at the two t h i r d points. I computed the value of F for each loaded beam such that the beam was subjected to a maximum stress of either 3000 p s i (20.68 MPa) or 4500 p s i (31.02 MPa). The maximum stress that occurs between the t h i r d point i s constant i n thi s region and i s due to bending only. 25 These two stress levels were derived from the f i f t h and twentieth percentiles of the strength d i s t r i b u t i o n of the control sample (Group no. 17) seen i n Figures 16 and 17. From one of these values of maximum stress, the force F i for the in d i v i d u a l board i i s computed from the following equation: F• = b.h. 2 °^ aii 1 i x where: 2L a max i s the maximum stress b. i s the width of the board l h^ i s the depth of the board L i s the span (138 in) (3505 mm) 3.3 Testing Procedure The experiment was i n i t i a t e d by the Western Forest Products Laboratory, Structure D i v i s i o n , i n Richmond, B.C., i n early 1978. Group numbers 2, 6, 8 and 9 (Hemlock) (Figure 15) con-taining 100 beams each were used. The deformation measurements of a randomly selected sub-sample of boards were taken over a three month period with a higher frequency of measurements i n the beginning. This experiment was carried out i n two stages. The f i r s t group of 97 boards (79 of 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 obtained two to three hours after the f u l l load had been transferred to the beams. I w i l l refer to this group of 97 boards as sub-group 1. Sub-group 2 consisted of 105 boards (76 of which belonged to group number 2; 11 to group number 6 and 18 to group number 9), loaded i n November 1978. Deformation informations were obtained, t h i s time, from the beginning of load transfer. The beams were stored i n the Western Forest Products Laboratory for about one year p r i o r to transfer to a laboratory i n Richmond where they were loaded. They had by then reached a stable l e v e l of moisture content (8% to 12%) and ambient temperature. These conditions were kept approximately constant during the Richmond Laboratory experiments. The test set-up i s indicated by the photographs i n Figures 18 and 19. The experimental data for sub-groups 1 and 2 are indicated i n Tables IX and X. The board number, group number, cross sectional dimen-sions, l e v e l of applied stress and modulus of e l a s t i c i t y are shown for 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 predict i n i t i a l d e f l e c t i o n of a beam at the centre of i t s span. For the loading configuration presented i n section 3.2, one obtains: 23 F . L 3 ' s i = 648 E.I. l l where E^ i s the modulus of e l a s t i c i t y of a s p e c i f i c board i . Tables IX and X show the predicted i n i t i a l d e flection for sub-groups 1 and 2, respectively. 3.4; 2 Actual I n i t i a l Deflections a) Sub-group 1 For sub-group 1 the actual i n i t i a l d e f l e c t i o n refers to the f i r s t deformation measurement obtained. I t i s not the e l a s t i c deformation, defined as the deformation of a beam immediately a f t e r f u l l load has been transferred to i t (the load transfer takes about one minute). The deformation was recorded by r u l e r , to the nearest millimetre. Table IX shows these values of i n i t i a l d eflections. b) Sub-group 2 For the boards of t h i s sub-group greater care was taken i n monitoring the d e f l e c t i o n at the centre of the span, from time t = 0, defined as the time when load transfer was started. The rate of loading was such that f u l l load was tra n s f e r -red i n approximately one minute. Charts of load and d e f l e c t i o n variations with respect to time were recorded while the load was transferred and up to the moment when the d e f l e c t i o n reached a stable value. A measurement of the d e f l e c t i o n was then taken with a r u l e r , to the nearest millimetre. Close agreement was reached between t h i s value and the value read, on the chart. This value was the d e f l e c t i o n at time t = 4 minutes. I estimated the deflec-tions at t = 2 minutes and t = 1 minute by in t e r p o l a t i n g from this reference point on the chart. The def 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 at the centre of i t s span, shown i n Table X. Figure 20 shows a chart of load and d e f l e c t i o n variations with respect to time for board number 2673 from group number 2. Points M^ , M 2 and refer to the deformations at times t = 4 minutes (0.067 hr), t = 2 minutes (0.033 hr), and t = 1 minute 28 (0.017 h r ) . 3.4.3 Deformation as a Function of Time The defl e c t i o n of the beams at t h e i r half-span was r e -corded regularly over a period of three months. A numerical description of these data points i s not presented i n t h i s thesis, but i s available from the University of B r i t i s h Columbia, Depart-ment of C i v i l Engineering. Curves can be presented i n three ways: defl e c t i o n versus time plotted on a b i l i n e a r graph, deflection versus time plotted on a semi-logarithmic graph, and defl e c t i o n versus time plotted on a f u l l - l o g a r i t h m i c graph. An example of a b i l i n e a r graph i s given i n Figure 21. Seven creep curves are shown. Deflection in millimetres i s plotted against time i n hours for boards number 1513, 2498, 2238, 2207, 2497, 2687 and 2756, selected at random. Values of applied stress, e l a s t i c deformation and f i n a l measured deformation are shown. This method of presenting creep curves i s representative of the true evolution of the deformation i n time, but i s not very p r a c t i c a l i n that any substantial increase i n deformation takes place over a long period of time. Examples of semi- and f u l l - l o g a r i t h m i c plots are shown i n Figure 22. In the lower graph, def l e c t i o n i s s t i l l plotted on a linear scale i n millimetres while time i s plotted on a l o g a r i t h -mic scale i n hours. I have presented curves for the same seven boards as i n the previous Figure. In the top graph, both d e f l e c -tion and time are plotted on a logarithmic scale. The advantage of such a graph i s that an expression for the creep deformation such as (also c a l l e d power function), i s transformed to a straight l i n e Log e c = Log A + b log t in a log-log p l o t . This expression has been widely used to de-scribe the time dependence of creep deformation. Although we can f i n d some examples of i t s application to wood-based mater-i a l s (Sugiyama 1957; Clouser 1959; Nielsen 1968), the power func-tion has been mainly used for building materials other than wood. An example of the application of the power function i s shown i n Figure 7. Figure 22 shows l i n e a r i t y up to time t = 200 hours, but one cannot extrapolate further, and more information i s needed on the creep deformation at times greater than 5000 hours. I chose the semi-logarithmic configuration to present the data because of i t s c l a r i t y ; the rate of change of deformation with respect to time i s c l e a r l y shown as the slope of the curve. Figure 2 3 shows creep curves for the boards of sub-group 1. For each creep curve, three numbers are shown. The number at the l e f t end of the curve i s the f i r s t available value of deforma-tion Cs ); the number at the r i g h t end of the curve i s the l a s t measured value of deformation (6 f c) and the t h i r d number i s board number (_BN) . Figure 24-i ( i = 1,2) shows creep curves f o r the boards of sub-group 2 that were loaded to 3000 p s i , and 4500 p s i respectively. The creep curves have been positioned according to increasing value of e l a s t i c deformation. 3.5 Interpretations of Results 30 3.5.1 E l a s t i c Deformations In the following section, I w i l l address myself to the resu l t s obtained for sub-group 2 since the actual i n i t i a l de-formations for boards of sub-group 1 are not known, as discussed in. section 3.4.2. Tables^ X-i ( i = 1,3) show that there i s a discrepancy between expected and actual e l a s t i c deformations. This i s better seen i n Figure 25, where actual e l a s t i c deforma-tions are plotted against expected e l a s t i c deformation for the two- stress l e v e l s . Regression l i n e s have been f i t t e d to the two sets of data points according to the least squares c r i t e r i o n . As shown i n Figure 25 t h e i r equation as well as the squares of 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 0*4378 + 1.073 ^p r e cj£ ct e cj Rsq = 0.9127 and 6 = . n = 2.957 + 1.017 6 . , actual predicted Rsq = 0.8950 for 3000 p s i and 4500 p s i respectively. In these equations, 6 , and 6 ^ are expressd i n millimetres. I f 6 . .. actual predicted * actual and 5 p r e c j i c t e d 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 of a regression equation such as « . , = A + B { , . . , actual predicted would be 0 and 1 respectively. A graph of 6 , plotted against ^predicted w o u l d show a straight l i n e at 45 between the p o s i t i v e sides of the axes. A c o e f f i c i e n t A with non-zero value indicates a v e r t i c a l displacement of the straight l i n e ; i t represents a discrepancy that i s constant throughout the range of e l a s t i c deformations. A c o e f f i c i e n t B. greater that 1 indicates 31 a rotation of the straight l i n e around the o r i g i n of the axes. It indicates a discrepancy that increases with increasing value of e l a s t i c deformation. This discrepancy results from experi-mentation, either i n a physical sense or from a t h e o r e t i c a l view-point. There are several pot e n t i a l sources of t h i s discrepancy, a) F i r s t l y , there are two main types of errors inherent to the measurements. The f i r s t type w i l l be repeated throughout the experiment l i f e t i m e : the error of the measuring device that reads to the nearest millimetre. This includes the error due to the positioning of the measuring device at each recording time. The second type of error i s due to the bending of the supporting s t e e l structure. This error occurs once only and w i l l be reproduced at each c a l c u l a t i o n of the d e f l e c t i o n . Furthermore, i t depends upon the l e v e l of applied stress, as i l l u s t r a t e d i n the following figure: m T 2 Vi w ////////////////'// After loading 7777777777777777777/ Before Loading The recorded deformation i s 8, such that 6 = %2 - l1 I t includes a component b due to the bending of the supporting structure. This component has been estimated and can be as high as 2.5 to 3 mm, for the highest l e v e l of applied stress. 32 b) Secondly, i t has been established that the presence of de-fects such as knots and zones of i n c l i n e d grain causes l o c a l i z e d reduction i n modulus of e l a s t i c i t y (Curry 1976) . Consequently, since gross deformations are used i n the calculations, the c a l -culated values of the modulus of e l a s t i c i t y depend on the type of loading, on the number and size of defects present, and t h e i r location within the st r u c t u r a l member. S p e c i f i c a l l y , I have hypothesized that a value of the modulus of e l a s t i c i t y computed from a deformation recorded at the t h i r d points of a member i n a t h i r d point loading configuration might lead to an under e s t i -mation of the maximum deformation of the member. This i s i l l u s -trated i n the following figure: 46 i n 46 i n 46 i n th e o r e t i c a l deflected shape actual deflected shape The valuei^o-f the modulus of e l a s t i c i t y , for a s p e c i f i c beam, i s computed from the following equation: 8 - 5 F i L 3 162 E.I. l l where 6^ i s the deflection expressed as the average of the de-f l e c t i o n of the two t h i r d points. The value E^, obtained i n a s p e c i f i c test (section 3.1.2, step 2), i s then used to compute the e l a s t i c deformation of the beam under f u l l transfer of the 33 load i n the duration of load experiment. The equation used i s : 23 F.L 3 6 = e 648 E.I. 1 l Such a procedure involves the hypothesis that the beam, under load, assumes a t h e o r e t i c a l deflected shape. This hypothesis may not be correct and the previous figure i l l u s t r a t e s an ex-ample where the o r e t i c a l and actual deflected shapes do not match even though they have common t h i r d points. An easy way to deter-mine whether or not t h i s phenomenon gives r i s e to a component of error between actual and predicted e l a s t i c deformations i s pre-sent i n the following. Subject a beam to loads applied at i t s t h i r d points and record the deflection at these t h i r d points as well as at the half point. The l a t t e r measurement i s used to compute E 2 i n the following equation: 2.3F..L3 8 = 1 — 2 648 E 0 I . 2 l The average of the f i r s t measurements i s used to compute E^ i n the following equation: 5 F ± L 3  Sl/3 = — ~ 162 E.I. 1 l If E^ and E 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 possible to correct for t h i s component of error between actual and pre-dicted e l a s t i c deformations. Such analysis has been c a r r i e d out at the Western Forest Products Laboratory between December 19 7 8 and A p r i l 1979. A detailed description i s beyond the scope of t h i s thesis. I analysed the r e s u l t s from group number 15 34 (Hemlock) and group number 2 (Amabilis-Fir). Figures 26 and 27 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 of E-^  and E 2 r e -spectively for Hemlock and Amabilis-Fir. A straight l i n e was derived by the least squares c r i t e r i o n .to correlate E-^  and . These equations are: Hemlock: E 2 = 0.7719 10~ 4 + 1.013 E± (psi 10~7) Rsq = 0.9951 Amabilis-Fir: E 2 = 0.9296 10~ 3 + 1.013 E± (psi 10~7) Rsq = 0.9954 The agreement between E-^  and E 2 , for both species, i s excellent and I conclude that the described phenomenon, i s not a s i g n i f i -cant source of error between actual and predicted e l a s t i c de-formations . I t should be noted that for both Hemlock and . Amabilis-Fir, E 2 i s higher than E-^ , contrary to the previously developed hypothesis. This may r e s u l t from experimental errors due to crushing at the points of application of the load as well as at the supports. In view of these comments, and i f one considers that the experimental errors inherent to the determination of E-^  and E 2 are n e g l i g i b l e as compared with errors described i n section a), I conclude that the discrepancies between predicted and actual e l a s t i c deformations are due to errors inherent Ah the long term deformation experiment. In the following, e l a s t i c deformations, whenever men-tioned, r e f e r to the predicted e l a s t i c deformations. Creep deformations, defined as the difference between t o t a l deformations and e l a s t i c deformations, refer to the values that have been corrected according to the following equation: 35 <5 . = 6 _ S + 6 creep creep i actual i predicted 3.5.2 General Comments on Creep Curves a) . Figures 24-i ( i = 1,2) show that the rate of change of creep deformation, at the right end of curves, seems to i n -crease with increasing value of e l a s t i c deformation. This phe-nomenon should be v e r i f i e d on a graph where creep deformation i s plotted against e l a s t i c deformation; the curve should have pos i t i v e slope at any point, as well as posit i v e curvature. Such a graph i s presented i n Figure 28. Creep deformation i s expressed at ten d i f f e r e n t times ranging from f i v e minutes to thirteen weeks % ten plots are shown, and both stress l e v e l s are represented. The data points, although they are scattered, f o l -low a trend consistent with the above hypothesis. On the other hand, i t can be seen that data points for the two stress l e v e l s occupy d i f f e r e n t regions within each pl o t . This suggests that both e l a s t i c and creep deformations might depend upon the l e v e l of applied stress. b) A v e r t i c a l l i n e drawn i n Figure 28 from the axis of e l a s t i c deformations at a s p e c i f i c value intercepts data points for 4500 p s i before i t intercepts data points for 3000 p s i . This i s true, as a trend, for any value of e l a s t i c deformation, as i l l u s t r a t e d further i n Figures 29-i ( i = 1,6). Each of these figures shows creep curves for boards with equivalent values of e l a s t i c deformation. Six values of e l a s t i c deformations were selected: 36 mm, 38 mm, 41 mm, 42 mm, 43 mm, and 45 mm. These are the values encountered both for 3000 p s i and 4500 p s i . These figures show that among boards with equivalent e l a s t i c de-36 formation and loaded to d i f f e r e n t stress l e v e l s , the ones loaded to the higher stress l e v e l appear to creep less than the others. This i s possible, because several boards loaded to d i f f e r e n t stress l e v e l s can reach the same e l a s t i c deformation i f they have d i f f e r e n t material c h a r a c t e r i s t i c s (grade effect) and there-fore have d i f f e r e n t moduli of e l a s t i c i t y and strength character-i s t i c s . The e l a s t i c behaviour, as well as the creep behaviour depend upon these variables.; c) Among several boards with equivalent e l a s t i c deformation and loaded to the same stress l e v e l , creep i s more pronounced i n some than others (sometimes twice as much). These boards have approximately the same modulus of 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. This suggests that the creep devel-opment depends upon strength or, equivalently, stress r a t i o . The estimated values of stress r a t i o are available only for boards that were loaded to 3000 p s i . I emphasize, once again, that for th i s reason, stress r a t i o and strength are equivalent. This would not be true i f stress r a t i o s were computed from several values of applied stress. The estimated short-term strengths of boards, used i n the computation of the stress r a t i o s , are indeed, t h e i r residual strength. These values were obtained by te s t i n g the boards, a f t e r they were unloaded i n Richmond Laboratory, i n a four points loading configuration, at a constant rate of loading of 5000 psi/mn (34.5 MPa/mn) u n t i l f a i l u r e occurred. They are a good estimation of short-term strengths because the reduction i n strength over a period of three months due to constant load-ing, for the stress l e v e l considered, i s not very important 37 (Barrett and Foschi 1978). For boards that broke while they were loaded i n Richmond Laboratory, the stress r a t i o had to be es-timated otherwise. I w i l l not go into the d e t a i l s of these calculations; these stress r a t i o s are greater than 75% and i n this analysis there i s no need to know these values more speci-f i c a l l y . Table X shows available values of stress r a t i o s . d) Figures 24-i ( i = 1,2) indicate that a few boards ex-h i b i t a creep curve that does not follow the general trend. These boards are 2811, 1507,2509 for 3000 p s i applied stress, and 411 for 4500 p s i applied stress. These creep curves have sudden changes i n slope, and these changes are probably due to l o c a l f a i l u r e s occurring i n the beams. These boards are re-presented on various plots by i s o l a t e d points; they do not t e l l us anything, but create problems i n presentation of graphs. For these reasons, these boards are disregarded i n further analyses. 3.5.3 Dependence of Creep Deformation on the Stress Level, the Stress Ratio and the Modulus of E l a s t i c i t y In Chapter 1.4 I showed that a functional expression for creep deformation i s : 6- = f i t , c r(t,x), MOE, SR} where the value of x (space parameter) can be set to a constant, x = L/2, since measurements of deformation were recorded at the center of the span of i n d i v i d u a l boards. Furthermore, as men-tioned e a r l i e r , there was no load v a r i a t i o n during the experiment l i f e t i m e . The l e v e l of applied stress, a, i s , thus, independent of time and one can write the f i n a l expression for the functional form of creep deformation: 38 6 c = f{t, a, MOE, SR} The purpose of the following study i s to est a b l i s h whether or not dependence exists between creep deformation and the various parameters, at a given time t * . The study of such a dependency for timber beams i s complicated by the interdependence of the three parameters. This i s c l e a r l y i l l u s t r a t e d in Figure 28 where a l l three parameters contribute to the scatter within each p l o t . This i s further i l l u s t r a t e d i n Figure 30 where f r a c t i o n a l creep, defined as the r a t i o of creep deformation to e l a s t i c deformation, i s plotted against e l a s t i c deformation. Fractional creep i s expressed at ten d i f f e r e n t times ranging from f i v e minutes to thirteen weeks. As i n Figure 28, the e f f e c t of applied stress i s c l e a r l y shown since data points for the two stress l e v e l s occupy d i f f e r e n t regions within each p l o t . One cannot say, though, whether the l e v e l of applied stress i n f l u -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 alone. The e f f e c t s of stress r a t i o and modulus of e l a s t i c i t y together contribute to the scatter within each set of data points and therefore cannot be i s o l a t e d from one another. Figure 31 shows graphs of f r a c t i o n a l creep plotted against 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 material with con-stant value of the modulus of e l a s t i c i t y . F r a c t i o n a l creep i s expressed at nine d i f f e r e n t times. The e f f e c t of the l e v e l of applied stress i s demonstrable to a much lesser extent than i n the previous graphs. This i s not surprising i n that the abscis-sa of each point (the modulus, of e l a s t i c i t y ) does not depend upon the l e v e l of applied stress, whereas abscissae on previous 39 graphs did, since they were e l a s t i c deformations. Figure 31 shows that, even though crosses and c i r c l e s overlap, data points for 4500 p s i might have, as a trend, a higher ordinate than those for 30 00 p s i ; but t h i s phenomenon i s not s u f f i c i e n t l y pronounced to enable a further conclusion. Further investiga-t i o n i s required to establish whether or not f r a c t i o n a l creep r e a l l y depends upon the l e v e l of applied stress. Test specimens should be subjected to a wider range of stress l e v e l s . creep increases, with decreasing value of the modulus of e l a s t i -c i t y and the rate of increase increases over time. The scatter, however, i s such that i t i s not possible, at t h i s stage, to determine whether, the increase i s l i n e a r or not. I t should be noted that the t h i r d parameter, the stress r a t i o , i s s t i l l pre-sent i n these plots and contributes to the scatter. One method of determining i t s e f f e c t would be to redraw each one of the graphs for s p e c i a l bands of stress r a t i o s . I considered f i v e bands: 0% - 30%, 30% - 40%, 40% - 50%, 50% - 60%, and 60% - 100%. One might expect that f r a c t i o n a l creep increases with increasing stress r a t i o , y i e l d i n g , ie. at three months, a graph as i l l u s t r a t e d i n the following figure: On the other hand, Figure 31 shows that f r a c t i o n a l F ractional creep 60% - 100% 50% - 60% 40% - 50% 30% - 40% 0% - 30% M.O.E. Where li n e s represent trends i n the data and are not necessar-i l y s t r a i g h t . Figure 32-i ( i = 1,10) shows graphs of f r a c t i o n a l creep, determined at a p a r t i c u l a r time, plotted against the modulus of e l a s t i c i t y for the f i v e ranges of stress r a t i o s . Time varies between f i v e minutes and thirteen weeks. These graphs support the expected trends. This i s better seen i f one determines, say at time t = 3 months, the averages of coordinates of data points from the graphs of Figure 32-i. One obtains the r e s u l t s presented i n Table XI for the f i v e ranges of stress r a t i o s . The increase 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 for high values of stress r a t i o s (greater than 60%), that i s for material with low strength (lower than 5000 p s i ) . For the other ranges of stress r a t i o s , the increase i s rather small and does not allow any con-clusion. There i s a need to investigate t h i s matter further with larger samples. It was commonly seen that low strength material f a i l e d i n the tension zone, usually associated with l o c a l grain d i s -t o r t i o n , while high strength material had an inherently strong tension zone causing f a i l u r e to i n i t i a t e i n the compression zone. It i s possible that t h i s difference i n f a i l u r e mode i s the cause of the observed phenomena for long term deformations. As a summary, i t can be said that the development of creep, for sawn timber of s t r u c t u r a l s i z e , i s influenced by the value of the modulus of e l a s t i c i t y . The lower th i s value, the more the material creeps, and the rate of increase increases over time. Furthermore, the experimental results indicate that creep 41 development i s more important f o r low strength material (that i s , material with high stress r a t i o s ) , and at higher levels of applied stress. I t should be r e c a l l e d , at th i s point, that two (low) stress l e v e l s only were investigated and further study involving a wider range of applied stress i s needed. 42 CHAPTER 4 LINEAR VISCOELASTIC MODEL 4.1 Introduction Materials that display creep, that i s , an increasing deformation under sustained load are c a l l e d v i s c o e l a s t i c materials. The constitutive equations of these materials, that describe the r e l a t i o n between stress and s t r a i n , may be either linear or non-linear. Although i t i s known that building materials do not behave l i n e a r l y , or do so within l i m i t s too narrow to 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 that l i n e a r i t y , because of i t s s i m p l i -c i t y , can characterize the behaviour of the material within reasonable l i m i t s , provided some assumptions are made. The behaviour of v i s c o e l a s t i c materials i n uniaxial stress c l o s e l y resembles that of materials b u i l t from discrete e l a s t i c and viscous elements (Fliigge 1967) . In the following, one case w i l l be considered; i t w i l l be assumed that wood, under sustained load, reaches a f i n a l value of deformation. 43 4.2 Three Parameter S o l i d The model i s represented as shown i n the following figure, where a spring and a Kelvin element are connected i n series : Spring E. Kelvin element E, where: E^and E^ are moduli of e l a s t i c i t y ; y 2 1 S t n e c o e f f i c i e n t of proportionality between stress and s t r a i n rate; e' and e" are strains i n the spring and the Kelvin element respectively when a i s the applied stress. For t h i s model, one obtains the follow-ing set of equations: a = E 1e* 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: VJ2°_ = E 1 E 2 £ V 2 * \ E 1 + E 2 E 1 + E 2 E 1 + E 2 44 I n i t s n o r m a l i z e d f o r m , t h e d i f f e r e n t i a l e q u a t i o n i s w r i t t e n : a + p 1 5 = q Q e t q ^ w h e r e p^, q n , and q ^ a r e s u c h t h a t : q x > P^O I f t h i s i n e q u a l i t y i s n o t s a t i s f i e d , i t i s n o t p o s s i b l e t o d e r i v e t h e d i f f e r e n t i a l e q u a t i o n f r o m t h e m o d e l w i t h r e a l , p o s i t i v e v a l u e s o f t h e c o n s t a n t s o f i t s component p a r t s . A p p l y i n g a c o n s t a n t s t r e s s a. = a n a t t i m e t = 0 and s o l -v i n g , t h e d i f f e r e n t i a l e q u a t i o n f o r e ( t ) ; one o b t a i n s : a -%/q e ( t ) = -H- + Ae 1  q 0 o r , i n t e r m s o f t h e c o n s t a n t s o f t h e component p a r t s : e ( t > 1-e 2 v2 ) E l E 2 The e l a s t i c d e f o r m a t i o n i s t h e v a l u e o f e ( t ) a t t = 0; t h a t i s : C r e e p d e f o r m a t i o n c a n be w r i t t e n a s f o l l o w s : a -E t / £ c ( t ) = i f ( 1 " e " 2 > F r a c t i o n a l c r e e p , f ( t ) , i s e x p r e s s e d a s f o l l o w s : E - E _ t / f ( t ) = _ i ( l - e 2 y2 ) E 2 45 4.3 Stress Parameter The expressions of e ( t ) , e c ( t ) , and f ( t ) show that, at a p a r t i c u l a r time t, the t o t a l deformation and the creep deform-ation, expressed as s t r a i n s , are l i n e a r l y dependent upon the l e v e l of applied stress. It i s not possible to d i r e c t l y v e r i f y whether or not the tested material behaves l i n e a r l y i n creep with respect to stress. This i s because the test r e s u l t s refer to two stress l e v e l s only and graphs such as deformation plotted against stress cannot be obtained. The t e s t r e s u l t s yielded the following graphs: 6 c ( t * ) = f { 6 e ) f ( t * ) = g{6 e} f ( t * ) = h{E±} at various times t * . The expressions for e (t) and f(t) can be re-written by i n s e r t i n g = E ^ e e or E^ = ° 0 / e e into the above equations, y i e l d i n g : E l - E , t , e (t) = e e — ( 1 - e 1 /v2 ) E 2 f ( t ) = ± -£ ( i - e 2 / y 2 ) e e E 2 46 f (t) = -=± ( 1 - e Z / y 2 ) E 2 In the f o l l o w i n g s e c t i o n , I w i l l demonstrate t h a t these expres-s i o n s , a p p l i e d to a non-homogenous m a t e r i a l , y i e l d graphs s i m i l a r to the t e s t r e s u l t s . I t should be s t r e s s e d t h a t these t h e o r e t i c a l e x p r e s s i o n s r e f e r t o s t r a i n s , whereas the q u a n t i t i e s d e r i v e d from the t e s t r e s u l t s are d e f l e c t i o n s . T h i s i s not important i n t h a t graphs of s t r a i n s , h a d they been d e r i v e d from experiments, would f o l l o w s i m i l a r trends as graphs of d e f l e c t i o n . a) Creep deformation p l o t t e d versus e l a s t i c deformation. At a g i v e n time t * , ec can be expressed as f o l l o w s : e c ( t * ) = £ e E l F ( E 2 , y 2 - t * ) For a given s t r e s s l e v e l a^, t h e r e i s o n l y one p o i n t w i t h c o o r d i n a t e s (e ' e c ) - Furthermore, p o i n t s are l o c a t e d on a s t r a i g h t l i n e going through the o r i g i n of the a x i s , with slope p r o p o r t i o n a l to E^ ( the c o e f f i c i e n t of p r o p o r t i o n a l i t y i s the value of F e v a l u a t e d a t time t * ). 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 ; E 2 and y 2 are probably not constant v a l u e s ; i n f a c t , they may vary i n the same d i r e c t i o n as E^. One can r e w r i t e the e x p r e s s i o n of e c a t a g i v e n time t * : e c ( t * ) = e e E 1 f ( E 2 , y 2 , t*) I f E 2 and y 2 were constant , F would a l s o be constant a t a g i v -en time t * and e c would a l s o be c o n s t a n t , r e g a r d l e s s of the value of c g s i n c e the f o l l o w i n g e q u a t i o n always h o l d s t r u e : £ e E l = °0 47 t* i s now a constant and the function F depends upon the constants of the model's component parts only. For an homogenous material, E^ i s a constant and so are E2 and y^- A graph of e c plotted versus e then has the following configuration: for a given l e v e l of applied stress. F, however, i s not constant and one may expect that i t increases with increasing value of e g (that i s with decreasing value of E^). A graph of e plotted against e may then have the follow-ing configuration: e 48 The l i n e s that are shown are trends. This i s the configuration of graphs 6. = f { 6 & } f o r the tested material. I t i s compatible with the hypothesis of l i n e a r i t y , since points and , shown on the graph, for a specimen i , can be on a straight l i n e going through the o r i g i n . I t i s important to note that the present analysis does not demonstrate that wood i s a l i n e a r v i s c o e l a s t i c material. This could be done by tes t i n g a specimen i at two d i f f e r e n t stress l e v e l s and v e r i f y i n g that and are on a straight l i n e passing through the o r i g i n . This analysis does show, however, that the tested material behaves i n a way that i s compatible with the behaviour of a l i n e a r v i s c o e l a s t i c material under sustained load. b) F r a c t i o n a l creep plotted against e l a s t i c deformation. At a given time t*, f can be expressed as follows: f ( t * ) = | 0 n F ( E 2 , y 2 , t * ) e t* i s now a constant and F i s the same function as i n the previous case. For an homogenous material, the function F has a constant value; furthermore, when e g increases, a n increases to the same extent so that E = ^0 w i l l remain constant. e e A graph of f plotted against e e then has the following configuration: 49 : a = a 3 _ rt a = a 2 V 0=0 . 1 s» e e For a given stress l e v e l 0 = CK , there i s only one point with coordinates e and e . The stress has to be varied i n order to e c obtain several points. Furthermore, the points are located on an horizontal l i n e . This i s not surprising because the a n a l y t i c a l expression of f i s independent of stress. For a material such as wood, however, E^ i s a random variable and for a s p e c i f i c value of applied stress, a Q , e e varies i n such a way that the product E-^ee w i l l remain constant and equal to OQ. If one assumes that E 2 and y 2 are constants, one can write: f (t*) = -ee where K i s constant, for a s p e c i f i c value of t * . A graph of f plotted against e e then has the configuration of an hyperbola, as shown i n the following figure: 50 e e If one assumes that 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 function F i s no longer constant. F increases with increasing e e , at a given l e v e l of applied stress. Since I did not investigate th i s r e l a t i o n s h i p i n greater depth i t i s impossible to define the type of v a r i a t i o n of the function F. Hence, one has to allow for a v a r i a t i o n such that a graph of f plotted against e e w i l l have the following configuration: 5 1 The l i n e s shown i n the graph are trends; they might be straight l i n e s as well. This i s the configuration of graphs f = g { 6 g } for the material tested. I t i s compatible with the hypothesis of l i n e a r i t y , since points and ,shown on the graph, for a specimen i , can be on an horizontal l i n e . The analysis, again, shows that the material tested behaves i n a way that i s compatible with the behaviour of a li n e a r v i s c o e l a s t i c material under a sustained load. c) Fractional creep plotted against modulus of e l a s t i c i t y . At a given time t * , f can be expressed as follows: f (t*) = E j F f E ^ y ^ t * ) where F i s the same function as defined before. For an homogen-ous material, E^ i s constant and so are y 2 a n c^ E2* A t a 9 l v e n time t*, a graph of f plotted against E^ i s then reduced to one point regardless of the applied stress. For a material such as wood, however, E^ i s a random varia b l e . I f , i n a f i r s t stage, one assumes that E 2 and y 2 are constant, then a graph of f plotted against E^ has the following configuration: f 52 The l i n e i s straight, with posi t i v e slope, and goes through the o r i g i n . I t represents a trend. One can assume that and y 2 vary, and furthermore, that the values vary i n the same d i r e c t i o n as E^. F(t*) then decreases with increasing value of E^ and a graph of f plotted against E^ may then have the following configuration: f 1 or, f depending upon the rate of increase of the function F. The l i n e s shown are trends and may well describe the data points shown on the graphs of Figure 31. Recall the comment made i n section 3.5.3 that data points for 4500 p s i , i n Figure 31, might have, as a trend, a higher o r d i -nate than those for 3000 p s i . This phenomenon i s not compatible with the hypothesis of l i n e a r i t y , but i s not s i g n i f i c a n t enough to r e j e c t the assumption of l i n e a r i t y . We can, therefore, say that the behaviour of the tested material i s compatible with the behaviour of a l i n e a r v i s c o e l a s t i c material, under sustained load. 4.4. Comments T r a d i t i o n a l l y , l i n e a r v i s c o e l a s t i c materials (homogenous) behave, under sustained load, i n a very s p e c i f i c manner that i s influenced by the l e v e l of applied stress. The study of such behaviour, to date, has.not allowed for the v a r i a t i o n of the parameters of modelisation, such as E^ and previously encoun-tered. Wood, however, i s not homogenous and i t s properties '. have a high c o e f f i c i e n t of v a r i a t i o n . I demonstrated that t h i s material, under sustained load, behaves i n a manner that i s compatible with 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 material i s l i n e a r l y v i s c o e l a s t i c . Under the assumption that the material can be regarded as a l i n e a r v i s c o e l a s t i c material, the following equa-tion , written for a specimen i of the material at a given time t*, may hold true: e . (t*) = E.F • e . c i 1 1 e i This equation has the following graphical form: The slope of the l i n e i s E^F^. Once the quantity E^F^ , at a given time t * , i s known, i t i s possible to determine e^^ (equiv-al e n t l y , the value 6 ^ ) from a s p e c i f i c value of e e^(equivai-ently, from a s p e c i f i c value of S ^ ) • The object of the following analysis i s to propose a meth-od of obtaining these values E.F.'s for the material tested. 55 CHAPTER 5 ANALYTICAL MODEL FITTED TO THE DATA 5.1 Model Graphs of creep deformation plotted against e l a s t i c de-formation, are drawn for two d i f f e r e n t stress l e v e l s presented at various times i n Figure 28. It has been shown that each of these graphs, drawn for a homogeneous material subjected to one le v e l of applied stress, reduces to one point. I w i l l use data points on graphs from Figure 2 8 for one stress l e v e l at a time to es t a b l i s h a corr e l a t i o n between <5 and 6 . When th i s i s done, c e ' a value of 6 . w i l l be obtained from a value of 6 . through the c i e i 3 c o r r e l a t i o n expression given at a s p e c i f i c time. The coordi-nates ( 5 e i ' 6 c i ^ ' a s P e c l m e n i ' W 1 l l t>e plotted on a graph of creep deformation versus e l a s t i c deformation and a straight l i n e , characterizing l i n e a r i t y between creep deformation and applied stress, w i l l be drawn, with slope: If one assumes, as discussed i n the previous chapter, that the material may be considered as a l i n e a r v i s c o e l a s t i c material, then the creep behaviour of such a specimen w i l l be known, at a s p e c i f i c time, for other stress l e v e l s . The cor-r e l a t i o n between <5 and 6 has the equation of a parabola, such as: 5 c(t) .= A(t,a ) 6 E + B(t,a ) 8 2 The parabola goes through the o r i g i n ; the c o e f f i c i e n t s A and B depend upon the stress l e v e l as well as time. Each graph from Figure 2 8 then y i e l d s two parabolas, one per stress l e v e l . For one stress l e v e l , say 3000 p s i , one obtains: <5c(t) = A(t) 6 E + B(t) 6 E 2 At a s p e c i f i c time t * , A and B are constant values, y i e l d i n g : 8 = A 6 + B 6 2 c e e The c o e f f i c i e n t s A and B have been determined twenty times -that i s twice for each of the ten times graphs are drawn - by the l e a s t squares method. Table XII shows the values of A and B. The parabolas are plotted i n Figures 33 and 34 for 3000 psi and 4500 p s i , respectively. Dividing the two members of the above equation by 6 E , one obtains. f = A + B 8 e which: i s the equation of a straight l i n e . The lines obtained at various times are drawn on the graphs of f r a c t i o n a l creep plotted against e l a s t i c deformation, presented i n Figures 35 and 36 for 3000 p s i and 4500 p s i , respectively. In the above equation, 8^ can be expressed, for a given stress l e v e l . i n terms of the modulus of e l a s t i c i t y as follows: For a t y p i c a l 2 i n x 6 i n section and a span of 138 i n , one can write: 5.619 o — and e E . _ 8.42 8 6e E ~ for 3000 p s i and 4500 p s i , respectively. In these expressions -7 . . the units of E are p s i x 10 and those of <5 are millimetres. e Fractional creep then becomes: f = A + 5 - g 1 9 B E and f = A + 8 ' 4 f B for 3000 psi and 4500 p s i , respectively. These expressions are equations of hyperbolas. These are drawn on graphs of f r a c t i o n a l creep plotted against the modulus of e l a s t i c i t y , , presented i n Figures 37 and 38 for 3000, p s i and 4500 p s i respectively. 5.2 Comments The wide spread scatter of data points i n the graphs of Figure 28 makes i t d i f f i c u l t to select a c o r r e l a t i n g function between creep deformation and e l a s t i c deformation. The assump-ti o n that creep deformation and e l a s t i c deformation together have zero value has yielded the selection of a parabola going through the o r i g i n . This curve has a simple mathematical formulation and may well represent the behaviour of a t h e o r e t i c a l l i n e a r visco-58 e l a s t i c material, as discussed i n section 4.3.a. I t leads to a li n e a r relationship between f r a c t i o n a l creep and e l a s t i c deforma-tion that may again, represent the behaviour of a th 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 material, as discussed i n section 4.3.b. However, a problem arises when f r a c t i o n a l creep i s ex-pressed i n terms of the modulus of e l a s t i c i t y . This expression has the a n a l y t i c a l form of an hyperbola. This curve, although i t has a shape that i s compatible with the assumption of l i n e a r v i s c o e l a s t i c i t y , as discussed i n section 4.3.c, should be, ac-cording to t h i s assumption, independent of the l e v e l of applied stress. Figures 37 and 38 show that the hyperbolas, f i t t e d to the data points, are not independent of the l e v e l of applied stress since they are not i d e n t i c a l , for the two d i f f e r e n t stress l e v e l s . The analysis, at the present stage, has not demonstrated whether or not thi s dependency i s s i g n i f i c a n t . This item w i l l be dealt with i n section 5.3. Note that the c o e f f i c i e n t s A 1s of the cor r e l a t i o n equa-tions between e l a s t i c and creep deformations have negative values for 4500 p s i , as shown i n Table XII. This has no physical mean-ing and i s due to the.lack of data points, i n the regions of low S e's as seen i n Figure 28. 5.3 Quantitative Analysis 5.3.1 Method Data for creep deformation and f r a c t i o n a l creep i n turn were treated by the UBC BMD02R computer program adapted from U.CLA-BMD documentation. This program.computes a sequence of multiple l i n e a r regression equations i n a stepwise manner. At each step, one variable i s added to the regression equation. The variable added i s the one which makes the greatest reduction i n the error sum or squares. Equivalently, i t i s the variable, that, i f i t were added,.would have the highest F-value. In addition, variables can be forced into the regression equation. Non-forced variables are automatically removed when t h e i r F-values become too low. One may analyse, i n turn, creep deforma-t i o n and f r a c t i o n a l creep, both expressed at three months. 5.3.2 Analysis of Creep Deformation at Three Months The o r i g i n a l variables that are fed into the computer are creep deformation (measured at three months), l e v e l of applied stress, stress 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 deforma-tio n . Three computer runs are made; i n the f i r s t run, data points for 3000 p s i only are considered; i n the other two runs, both stress levels are considered. a) F i r s t run The o r i g i n a l variables are: XI = 5- (creep deformation) X2 = SR (stress ratio) X3. = MOE (Modulus of E l a s t i c i t y ) X4 = 6 g ( e l a s t i c deformation) The variables, added by transgeneration, are: X5 = X2 X3 X 6 = X 2 X4 X7 = X3 X4 60 . X8 = X4 2 X 9 = X 2 X 4 2 XlO = X3 X4 2 The correlation equation, investigated, i s : 6 c = A 1 + A 2(SR) + A3(MOE) + A 4 ( S ) ' + Ac(SR)(MOE) + A C(SR)(6 ) _> o e + A7(MOE) (<5e) + A g ( 6 e ) 2 + A 9(SR) ( 6 e ) 2 + A 1 Q(MOE) ( 6 e ) 2 where A^ ( i = 1,10) are constant values. This equation has been selected because i t includes the simple l i n e a r regression com-2 ponents, as well as terms i n ( S e ) , introduced i n the a n a l y t i c a l model. 63 observations, out of 76 for 3000 p s i , are considered because boards that have f a i l e d as well as boards whose behaviour has been explained otherwise (section 3.5.2.d) are disregarded. The results are presented i n Appendix A - l . Two steps are con-sidered: Step 1: Variable X8 i s entered. The regression equation and the co r r e l a t i o n c o e f f i c i e n t are: 6 = . - 0 . 0 2 3 + 0.016 <5 2 c e R = 0.7917 Step 2: Variable XlO i s entered. The regression 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 are: 6 = 22.105 + 0.038 (5 ) 2 - 0.250(MOE)(6 ) 2 C 6 6 R = 0.8030 where 6;c and 6 e are expressed i n mm, MOE i n psi x 10 The following interpretations can be made: 1. Step 2 can be ignored because the F-value attached to XlO, 3.05 29, i s lower than the l i m i t (approximately 4 for a sample with 63 observations) to ensure the 95% confidence i n t e r v a l . Furthermore, the following figure shows that the increase i n corr 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 to the second, i s n e g l i g i b l e . R 2. The co r r e l a t i o n matrix shows that creep deformation i s weakly dependent on the stress r a t i o (variable X2) since the correlation, c o e f f i c i e n t between variables XI and X2 i s small (0.4019). 3. The regression equation (from Step 1)resembles the equation obtained i n section 5.1 for 3000 p s i . This i s not surprising because i n both cases computations involve the same c r i t e r i o n (least squares). b) Second run The o r i g i n a l variables are: XI = 5 (creep deformation) X2 = a (applied stress) X3 = MOE (modulus of e l a s t i c i t y ) X4 = <$e ( e l a s t i c deformation) The variables, added by the transgeneration are: X5 = X2 X3 where S and 6"e are expressed i n mm, a i s p s i and MOE i n p s i x 10~ 7. The following interpretations can be made: 1. The three variables 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 since t h e i r F-value i s higher than the l i m i t required (4 approximately). 2. The following figure shows the increase i n c o r r e l a t i o n co-e f f i c i e n t from step to step: 0.8102 0.8307 T" 2 395 Step It indicates that the correlation i s not s i g n i f i c a n t l y improved by considering the applied stress and the modulus of e l a s t i c i t y , i n addition to the e l a s t i c deformation. The difference between the equation given at Step 1 and the one given at Step 2 i s i l l u s t r a t e d in the following graph: Step 1 Step 2 X6 = X2 X4 X7 = X3 X4 X8 = X4 2 X9 = X2 X4 2 XlO = X3 X4 2 The correlation equation, investigated, i s : 6 c = A ± + A 2 ( a ) + A3(MOE) + A ^ i S ^ +A 5 (a)(MOE) + A 6 ( a ) ( 6 e ) +A ?(MOE)(S e) + A 8 ( 6 e ) 2 + A g ( a ) ( 6 e ) 2 + A 1 Q(MOE) (SQ)2 where A^ ( i = 1,10) are constant values. 91 observations out of 105 for both 3000 p s i and 4500 p s i stress l e v e l s , are considered. As before, boards that have f a i l e d as well as boards whose be-haviour has been described otherwise (section 3.5.2.d) are d i s -regarded. The results are presented i n Appendix A-2. Three steps are considered: Step 1: Variable X4 i s entered. The regression equation and c o r r e l a t i o n 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: Variable X2 i s entered. The regression equation and c o r r e l a t i o n c o e f f i c i e n t are: <5 = -6.669 + 1.058 6 - 0.00339 a c e R = 0.8307 where &c and &Q are expressed i n mm, a i n p s i . Step 3: Variable X3 i s entered. The regression equation and c o r r e l a t i o n c o e f f i c i e n t are: 6 c = 37.841 + 1.954 <S - 0.01185 a + 152.324 MOE R = 0.8395 If (as was done i n section 5.1) a regression equation i s de-rived for one stress l e v e l at a time, then a graph of creep deformation plotted against e l a s t i c deformation indicates c l e a r -l y an e f f e c t of the l e v e l of applied stress (Step 2). The pre-sent study shows that 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 re-gression equation derived once for both stress levels at the same, time gives as good a co r r e l a t i o n (Step 1). 3. The co r r e l a t i o n matrix shows that the variable 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). Either X4 or X8 could have been entered i n Step 1. In the next run variable X8 w i l l be a forced variable and w i l l be entered f i r s t . c ) 1 Third run The same o r i g i n a l variables as well as those added by trans-generation as i n the second run are considered. The same corr e l a t i o n equation i s investigated. The results are pre-sented" i n Appendix A-3. Two steps are considered: Step 1: Variable X8 i s entered. The regression equation and c o r r e l a t i o n c o e f f i c i e n t s are: 6 = 3.799 + 0.01125 S 2 c e R = 0.8102 where both.6 and 5 are expressed i n mm. G C Step 2: Variable X9 i s entered. The regression equation and c o r r e l a t i o n c o e f f i c i e n t s are: 5 • = -0.0405 + 0.0239 S 2 - 2.5 x 1 0 _ 6 a 6 2 c e e R = 0.8347 where &c and 6 e are expressed i n mm, a i n p s i . The following interpretations can be made: 1. Both X8 and X9 are s i g n i f i c a n t variables i n the 95% con-fidence i n t e r v a l . 2 2. ft non-linear model (third run) i n 6^ does not improve the cor r e l a t i o n obtained with a l i n e a r model (second run) i n 6 . e 3. We can compare results obtained 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 Third run, second step: 6„ = -0.405 + 0.0239 6 2 - 2.5 x 10~ 6a 6 2 c e e R = 0.8347 F i r s t run, f i r s t step: 6 = 0.023 + 0.01617 <5 2 c e (1) (2) (3) R = 0.7917 Expressions (2) and (3) are almost i d e n t i c a l for a = 3000 p s i . Equation (1) has a higher intercept and a lower slope than equa-tion (2). This i s the same interpretation as the second one i n the second run, but i t applies now to parabolas instead of straight l i n e s . I t i s represented i n the following figure: Step 2 4500 p s i Step 1 The regression equation, plotted at the second step, i s repre-sented by two parabolas, one per stress l e v e l . Such a repre-sentation c l e a r l y indicates that a graph of &c plotted against <$e i s dependent upon the l e v e l of applied stress. The present study indicates that only one parabola f i t t e d to the two sets of data points together, gives 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 plotted against the number of steps i n the re-gression analysis: 0.8102 0.8347 — I Step The increase 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 i s very small. 3. Equation (2) becomes: 6 = -0.405 + 0.0164 6 2 c e 87 = -0.405 + 0.0127 & 2 c_ e for 3000 p s i and 4500 p s i stress l e v e l s respectively. These equations are compared with the ones derived i n section 5.1 for the a n a l y t i c a l model: ° (3000 psi) 2 8. = 0.000317 6 + 0.0161 8 ' c e e 6 = -0.091 8 + 0.0147 6 c e e (4500 psi) There i s close agreement between the two sets of equations. 5.3.3 Analysis of F r a c t i o n a l Creep at Three Months The analysis of data for creep deformation indicates that both a l i n e a r model and a non-linear model can be chosen to cor-re l a t e creep deformation and e l a s t i c deformation. With respect to the former, one can write: 6 = A + B 6 c e With respect to the l a t t e r , one' can write: 2 6 = A + B 6 c e or S = A 6 + B 6 2 c e e The f i r s t expression was obtained i n section 5.3.2 t h i r d run, f i r s t step. The second expression was derived i n section 5.1 afte r the selection of a parabolic model going through the or-i g i n . I f one divides the three expressions by s^, one obtains expressions for 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 analysis of data on f r a c t i o n a l creep, expres-sed at time t = 3 months, i s to see whether or not one model should be given preference. The o r i g i n a l variables that are fed into the computer are f r a c t i o n a l creep (measured at three months), l e v e l of applied stress, stress 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 deforma-ti o n . Two runs are made. a) F i r s t run 68 Only one stress l e v e l i s considered - 3000 p s i . The o r i g i n a l variables are: XI = f (f r a c t i o n a l creep) X2 = SR (stress ratio) X3 = MOE (modulus of e l a s t i c i t y ) X4 .= 6 e ( e l a s t i c deformation) The variables, added by transgeneration, are: X5 = 1/X4 X5 = 1/X3 The c o r r e l a t i o n equation i s : f = -Aj + A 2(SR) + A3(MOE) + A 4(6 e) + A5 + A6 MOE e where A^ ( i = 1,6) are constant values. Results are presented i n Appendix B - l . The following comments can be made: 1. Fractional creep i s poorly dependent on the stress r a t i o (low value of the co 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 co r r e l a t i o n matrix). 2. For one stress l e v e l - 3000 p s i -, a non-linear model i s selected; the regression equation and cor r e l a t i o n c o e f f i c i e n t s are: f = 3.755 + 4^^ " R = 0.5209 at the f i r s t step of the regression analysis. In the equation -7 for f, E i s expressed i n p s i x 10 and f m %. The second step can be neglected because the F-value attached to the variable X5 i s lower than the l i m i t to ensure the 95% confidence i n t e r v a l . 3. The expression for f resembles the one obtained i n section 5.1 69 when the a n a l y t i c a l model was derived. Recall the expression for f that was obtained: f = 0.0317 + 9 , ° 6 7 E b) Second run The same variables are kept, except X2 which i s now the l e v e l of applied stress, and two stress levels are considered: 3000 p s i and 4500 p s i . The co r r e l a t i o n equation i s : f = A x + A 2(o) + A3(MOE) + A 4(-6 e) + A * + A - i _ A5 6 A6 MOE e where A^ (i = 1,6) are constant values. The res u l t s are pre-sented i n Appendix B-2. The regression analysis i s done i n two steps: . S t e p 1: Variable X4 i s entered. The regression equation and c o r r e l a t i o n c o e f f i c i e n t are: f = 21.84 - 0.83 R = 0.4604 where f i s expressed i n % and 6 Q i n mm. Step 2: Variable X2 i s entered. The regression equation and c o r r e l a t i o n c o e f f i c i e n t are: f = 30.95 + 1.54 <5e 0.0096 a R = 0.5419 where f i s expressed i n %, <s e i n mm and a i n p s i . The following.comments can be made: 1. In the f i r s t run, as well as i n the second one, the low cor-r e l a t i o n c o e f f i c i e n t s indicate a wide-spread scatter of data points i n Figure 30. 2. The increase i n co 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 70 2 i s not very large. This i s i l l u s t r a t e d i n the following figure: 0.5419 Step This indicates that 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 stress parameter i s present i n the c o r r e l a t i o n equation. This phenom-enon i s not compatible with the assumption of l i n e a r visco-e l a s t i c i t y . However, since the increase i n c o r r e l a t i o n co-e f f i c i e n t i s small, one can neglect the phenomenon and correlate 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. The c o r r e l a t i o n equation i s l i n e a r . If 6 g i s expressed i n terms of the modulus of 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 has the equa-t i o n of an hyperbola. The non-linear model, derived to correlate creep deformation and e l a s t i c deformation, i s preferable. 5.3.4 Summary This quantitative analysis, carried out on data for creep deformation and f r a c t i o n a l creep, both expressed at three months, indicates that: 1. The c o r r e l a t i o n between creep deformation aridcelastic deforma-t i o n , does not depend on the stress r a t i o . This r e s u l t does not contradict the comments made i n section 3.5 where I observed that creep deformation and f r a c t i o n a l creep were higher for low strength material - that i s , for material with high stress r a t i o s . This e f f e c t i s accounted for i n that .5 i s expressed i n terms of <5 E, and <$e depends upon the material c h a r a c t e r i s t i c s . 2. One may consider the cor r e l a t i o n between creep deformation and e l a s t i c deformation, as well as the co 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 of the l e v e l of applied stress. 3. Creep deformation i s non-linearly dependent upon e l a s t i c de-formation. This r e s u l t v e r i f i e s the hypothesis made i n section 5.1 when the a n a l y t i c a l model was derived. The r e l a t i o n between S and $ can be written as follows: s c 6 C = A + B where A and B are constants. This expression i s s l i g h t l y d i f f e r -ent from the one obtained i n section 5.1; which was: 6 C = C 6 e + D 6 e 2 where C and D are constants. These two expressions y i e l d very s i m i l a r graphical representations, and therefore either may be used to correlate creep deformation to e l a s t i c deformation. 4. Fractional creep i s l i n e a r l y dependent upon e l a s t i c deforma-ti o n . This r e s u l t , again, v e r i f i e s the hypothesis made i n section 5.1. The r e l a t i o n between f and 6 may be written as e follows: f = E + F 6 e where E and F are constants. 5. The r e s u l t s , presented, are r e s t r i c t e d to the stress l e v e l s 72 investigated. These comments suggest that the model, developed i n section 5.1 may be used to correlate creep deformation and e l a s t i c deformation, at various times, regardless of the stress l e v e l . This i s the subject of the next section. 5.4 F i n a l Results The c o r r e l a t i o n between 6 and 6 has the equation of a parabola such as: 6 c(t) = A(t) 6 e + B(t) 6 e 2 where A and B, t h i s time, do. not depend, on the l e v e l of applied stress. For a s p e c i f i c time t * , one obtains: 6 (t*) = A 6 + B 6 2 . c e e where A and B are constant values. This function has been f i t t e d to the data points of each graph from Figure 28. Table XIII shows values of A and B, determined from ten observations ranging from f i v e minutes to thirteen weeks. The A's have negative values; this has no physical meaning and i s due to lack of data points i n regions of low <$e's. The parabolas are plotted on a graph presented i n Figure 40. Parabolas at three months, twelve weeks, and thirteen we:eks are almost i d e n t i c a l ; therefore, only one of them i s plotted. As before, one can write the expression of f r a c t i o n a l creep as follows: 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 straight l i n e s are plotted on a graph shown i n Figure 41. 73 I indicated i n section 5.1 that a rel a t i o n s h i p between f r a c t i o n a l creep and modulus of e l a s t i c i t y i s obtained by expres-sing 6 e i n terms of MOE. Such an expression, however, depends upon the applied stress. Hence, t h i s method cannot be used here. Instead, hyperbolas such as: f ( t * ) = C + D  r ^ ; u MOE w i l l be derived from graphs shown i n Figure 31 by f i t t i n g the expression to the data points obtained at d i f f e r e n t times ranging from f i v e minutes to thirteen weeks. The results for C and D, at each time, are presented i n Table XIV. The hyperbolas are plotted on a graph shown i n Figure 42. The hyperbolas at three months, ((twelve weeks), and thirteen weeks are almost i d e n t i c a l ; therefore only one of them i s plotted. Recalling the reasons why I established a c o r r e l a t i o n between creep deformation and e l a s t i c deformation, i t i s now possible to predict the creep behaviour of a specimen of the material under a constant stress of 3000 p s i or-.4500 p s i over a period of three months. Furthermore, i f one assumes that the material behaves as a l i n e a r v i s c o e l a s t i c material, one may describe the specimen's creep behaviour at other l e v e l s of applied stress, as discussed i n section 4.4. One may attempt to v e r i f y the model by applying i t to the.results from the Surrey experiment. In t h i s experiment, the lev e l s of applied stress were 1070 p s i , 1410 p s i , 2110 p s i , and 3110 p s i ; the material was Douglas-Fir. Thus I w i l l t r y to v e r i f y whether or not the model may be u t i l i s e d under conditions broader than the ones under which i t was derived. I w i l l use the following equation: 74 <S = -10.070 x 10 c 3 S + 14.168 x 10~ 3 6 2 e e where 6" and.S are expressed i n mm. The values of 6 , from c e c e section 2.4.3 are shown i n the following, for the various stress l e v e l s : 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 obtains the following values of t o t a l deformation at 3 months; these values are composed with the actual values, from Figure 14. Stress l e v e l model actual p s i i n mm i n 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 that the agreement between the two sets of numbers, from an engineering viewpoint, i s excellent. The results from the Surrey Laboratory experiment do not r e j e c t the model. 75 CHAPTER 6 CONCLUSIONS The r e s u l t s of the foregoing study apply to the stress lev e l s ...investigated and to s p e c i f i c conditions of temperature (10,oC46 «30°°C) and moisture content (8%<MC<12%). The analysis leads to the following conclusions: l.The deformational response of s t r u c t u r a l - s i z e timber beams depends on t h e i r material 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 , mater-i a l with a strength lower than 5000 p s i (34.33 MPa) appeared, to creep 1.5 times more than material with a strength higher than that l e v e l , over a three-month period. COMMENTS: To v e r i f y and more quickly quantify the phenomenon described above, i t would be necessary that future investigations employ sub-samples of a larger s i z e . I t may prove advantageous to describe the phenomenon more s p e c i f i c a l l y , since commercial lumber presents c o e f f i c i e n t s of v a r i a t i o n up to 50% i n i t s strength-properties . 76 2. The tes t results support the assumption of a l i n e a r r e l a t i o n -ship between the creep deformation of s t r u c t u r a l - s i z e timber beams and applied stress. COMMENTS:It should be stressed, within a 9 5% confidence 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 included a term that i s dependent on the l e v e l of applied stress (section 5.3.3 second run, step 2). Using t h i s term,which i s incompatible with the hypothesis of l i n e a r i t y , did not s i g n i f i c a n t l y improve the precision of the co r r e l a t i o n and was,therefore, deleted. But the appearance of the term may suggest that the r e l a t i o n between creep deformation and applied stress i s not necessarily l i n e a r for stress l e v e l s higher than 4 500 p s i (31.02 MPa). From an engineering point of view, there i s no need to investigate t h i s further because the stresses applied i n t h i s analysis cover approximately twice the range of stress values used i n design. Nevertheless, from a s c i e n t i f i c point of view, i t may be int e r e s t i n g to expand t h i s investigation i n order to compare.lumber res u l t s with clear wood r e s u l t s . For the l a t t e r , i t has been assumed that l i n e a r i t y occurs for stress l e v e l s as high as 80% of u l -timate stress. I t i s unl i k e l y that one would obtain the same results with specimens of st r u c t u r a l s i z e . 3. The test r e s u l t s from an experiment i n which beams were loaded to stress l e v e l s up to 3110 p s i (21.44; MPa), constant over a: three year period, showed that the average t o t a l defor-mation, at t h i s time, i s approximately 1.6 times the average e l a s t i c deformation. 77 COMMENTS; The value 1.6 was derived from tests with approximately constant environmental conditions of temperature and humidity. This may not be the worst case; a str u c t u r a l component w i l l experience service conditions with changes i n temperature and humidity. 4. A method to predict the creep behaviour of s t r u c t u r a l -size timber beams at discrete times over a three month period has been presented. This prediction consists of using a para-b o l i c equation to express the deformation, 8- , at a given time, i n terms of the e l a s t i c deformation, 6 . Si m i l a r l y , the f r a c t i o n a l creep, f = /6 , c a n be expressed as a lin e a r function of 6 g or an hyperbolic function of the modulus of e l a s t i c i t y . COMMENTS: The prediction of creep deformation i s made for d i s -crete points, rather than for a continuous spectrum. Expressing these r e s u l t s i n the form of a mathematical description of creep over time might seem more elegant and sophisticated. But, from an engineering point of view,point estimates such as the curves shown i n Figs. 40, 41, and 42 are s u f f i c i e n t to describe the long-term deformational behaviour,of the beams. In addition, t h i s prediction method can be used for design purpose, even though i t was developed es p e c i a l l y for the complex analysis of the behaviour of assemblies of structural components under sustained load, including f l o o r systems and trusses.The advantage of the method developed i s that , i n thi s type of analysis, the modulus of e l a s t i c i t y of the ind i v i d u a l components can be used. 78 5. The method developed led to predictions of creep deformation at three months consistent with experimental data, with tolerable engineering accuracy. COMMENTS: The method was developed using data from Hemlock beams. Nevertheless, i t appears to predict the behaviour of Douglas-fir beams as well. 79 BIBLIOGRAPHY 1. Barrett,J.D. and Foschi,R.O., "Duration of Load and Proba-b i l i t y of Fa i l u r e i n Wood. Part I. Modelling Creep Rupture." Canadian Journal of C i v i l Engineering. Vol. 5. No. 4. p. 505-514, 1978. 2. Cederberg,A.M. and Danielsson,H., "Creep experiments on Wood". Unpublished Master Thesis. Lund., 197 0. 3. Clouser,W.S., "Creep of Small Wooden Beams under Constant Bending Load". Forest Products Laboratory. Forest Service, US Dept. of Agriculture. 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 Rigid-i t y of Structural Timber". Proceedings Wood Engineering Group, IUFRO. Blokhus, Denmark. Paper no. 10, Vol. 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 Publ. Co., London,1967. 6. Kingston,R.S.T. and Clarke,L.N., "Some Aspects of the Rheological Behaviour of Wood. I and I I . " Australian Journal of Applied Science, no. 12. p.211,240. 1961. 7. Kitahara,K. and Yukawa,K., "The Influence of the Change of Temperature on Creep in Bending." J. Japan Wood Res. Soc. Vol. 10. no. 5. p.169-174. 1974. 8. Lundgren,S.A., "Hardwood as Construction Material- a Vi s c o - e l a s t i c body." Holz als R. and W. Vol. 1/no. 1. p. 19-23. Jan. 1957. 9. Madsen,B. and Barrett,J.D., "Time-Strength Relationship for Lumber." The University of B r i t i s h Columbia- Dept. of C i v i l Engineering. Structural Research Series. Report no. 13. 1976. 10. Nakai,T., "Bending Creep Tests i n Wood-Based Materials." F i r s t International Conference on Wood Fracture. Banff, Canada. 1978. 11. National Lumber Grades Authority,"Standard Grading Rules for Canadian Lumber." Vancouver, Canada. 1970. 13. Nielsen,A., "Rheology of Building Materials." Inst, for Byggnadsteknik. LTH. B u l l . 3. 57p. Lund. 19 68. 14. Nielsen,A., "Rheology of Building Materials." National Swedish Council for Building Research. Stockholm, Sweden. Document D6. 1972. 80 15. Pierce,C.B., "The Weibull D i s t r i b u t i o n and the Determination of i t s Parameters for Application to Timber Strength Data." Building Research Establishment. Princes Risborough Labor-atory. Princes Risborough. U.K. Report CP26/76. 1976. 16. Sauer,D.J. and Haygreen,J.G., "Effects of Sorption on the Flexural Creep Behaviour of Hardboard." Forest Products Journal. Vol. 18/ no. 10. p.57-63. Oct. 1968. 17. Schniewind,A.P., "Recent Progress i n the Study of the Rheology of Wood." Wood and Technology. Vol. 2. p.188-206. 1968. 18. Spencer,R., "Rate of Loading E f f e c t i n Bending for Douglas-Fir Lumber.11 Proceedings, F i r s t International Conference on Wood Fracture. Banff, Canada. 1978. 19. Sugiyama,H., "The Creep Deflection of Wood Subjected to Bending Under Constant Loading." Transactions of Architec-t u r a l I n s t i t u t e of Japan. No. 55. p.60-70. 1957. Reference Material Time Sellevold (1969) Cement paste 0.20 sec. Perkitny (1965) Wood 3 sec. Lundgren (1968) Wood-based products 3.6 sec. Nielsen (1968b) C e l l u l a r concrete 2-4 sec. Ruetz (1966) Cement paste 5 sec. G l a n v i l l e acc. to Evans (195 8) Concrete 5 sec. Erickson and Sauer (1969) Wood f 1 min. Bach (1972) Wood ^ 1 min. Chow (1970) Furniture panels 1 min. Table I. ' Starting time after loading for measuring creep. From.Nielsen (1972). 82 Group Deflection (in x 10~3) E-values (psi x 10 6) Number of Boards E l S<145 1.56<E 89 E2 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 . Grouping of the material for Surrey experiment according to E-values (Batch 1). Group Deflection (in x 10~3) E-values (psi x 10 6) Number of Boards E l 6<145 1.56<E 119 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 . Grouping of the material for Surrey experiment according to E-values (Batch 2). Board Width Depth E-value r I n i t i a l Deflection .• - •• no. (in) (in) (psi x 10 ) Predicted Measured (in) Error (in) North South Average (%) 2224 1.485 5.455 1.8139 0.37 1.09 -0.03 0.53 42.7 2487 1.496 5.492 1.1825 0.55 0.97 0.08 0.53 -5.2 2527 1.500 5.484 1.4596 0.45 1.12 0.01 0.57 25.6 2569 1.496 5.463 1.3341 0.50 0.8.1 -0.09 0.36 -27.8 2350 1.500 5.510 1.1380 0.57 1.06 -0.26 0.40 -29.6 1673 1.486 5.450 1.2854 0.53 -0.42 0.87 0.22 • -57.1 2480 1.495 5.494 1.0959 0.60 -0.46 1.02 0.28 -53.1 1830 1.481 5.390 2.0280 0.35 -1.05 1.02 0. -104.4 1266 1.484 5.490 0 .9577 0.69 -0.34 1.27 0.47 -32.7 1202 1.513 5.504 1.2605 0.51 0.42 0.77 0.59 16.5 1149 1.508 5.494 1.0507 0.62 0.47 0.81 0.64 3.6 2498 1.500 5. 460 1.2878 0.52 0.55 -0.24 0.16 -69.7 1779 1.490 5.386 1.6092 0.41 0.10 -0.69 -0 .29 -172.3 2570 1.497 5.480 1.2982 0.51 0.40 -0.24 0.08 -84.0 1523 1.500 5.469 1.0862 0.61 0.30 -0.54 -0.12 -119.5 Table IV-a. Data for Surrey test (1070 p s i ) . Board Width Depth E-value I n i t i a l Deflection no. (in) (in) (psi x 10 ) Predicted Measured (in) Error (in) North South Average (%) 1800 1.500 5.5 1.0287 0.83 1.81 0.99 1.40 -67 .9 1979 1.500 5.498 1.3607 0.63 0.45 0.69 0.57 9.7 2616 .1.500 5.465 1.0486 0.83 1.63 0.63 1.13 -35.5 1431 1.480 5.435 1.3856 0.65 1.41 0.79 1.10 -69.2 2641 1.495 5.425 1.2341 0.73 0.20 0.88 0.54 25.7 2500 1.490 5.515 1.1086 0.77 0.59 0.49 0.54 30.1 2238 1.499 5.485 1.1544 0.75 0.37 0.99 0.68 9.3 2516 1.500 5.475 1.4034 0.62 0 .55 1.09 0.82 -32.4 2474 . 1.513 5.493 1.1872 0.72 0.14 1.04 0.59 17.9 1781 1.505 5.460 1.4552 0.60 0.11 0.50 0.31 49.2 1838 1.505 5.445 1.7858 0.49 0.96 0 .12 0.54 -9.5 1989 1.512 5.508 1.7040 0 .50 0.12 0.65 0.39 22.6 1772 1.500 5.475 1.6124 0.54 1.83 0.44 1.14 -110.5 1389 1.500 5.460 1.1577 0.75 1.16 0.67 0.82 -7.6 1697 1.515 5.483 1.2807 0.67 1.13 0.58 0.86 -27.8 12 26 1.470 5.285 1.2224 0.81 0.95 0.64 0.79 1.5 Table IV-b. Data for Surrey test (1410 p s i ) . Board Width (in) Depth (in) E-value f i (psi x 10 ) I n i t i a l Deflection no. Predicted Measured (in) Error (in) North South Average (%) 1632 1.473 5.445 1.8981 0.71 0.74 0.64 0.69 3.1 2326 1.503 5.476 1.8142 0.72 1.29 0. 33 0.81 -13.3 2148 1.500 5.485 1.4402 0.90 0.84 0 .78 0.81 9.6 2387 1.485 5.450 1.8333 0.72 1.09 1.04 1.06 -46.2 1213 1.490 5.472 1.6857 0.78 0.99 0.78 0. 89 -14.0 1462 1.500 5.450 1.8293 0.72 1.29 0.64 0.96 -33.5 2053 1.497 5.468 1.4754 0.89 0.59 0.94 0.76 14.0 2050 1.477 5.345 1.5706 0.90 1.06 1.23 1.14 -26.7 1211 1.497 5.45 1.5605 0.85 1.06 1.26 1.16 -37.0 1192 1.494 5.5 1.4230 0.8 8 1.21 1.18 1.19 -35.7 1931 1.505 5.375 1.7229 0.79 0.69 0.94 0.81 -2.1 1868 1.485 5.479 1.3721 0.95 0.79 1.28 1.04 -8.6 1894 1.510 5.470 1.8117 0.71 0.34 1.28 0.81 -13.6 2348 1.479 5.434 1.8570 0.73 0.69 0.98 0.84 -15.2 1700 1.509 5.500 1.2182 1.05 0.19 1.78 0.99 5.7 1822 1.460 5.350 1.2517 1.14 0.59 1.93 1.26 -10.3 2664 1.480 5.340 2.2373 0.64 0.49 0.78 0.64 -0.3 2621 1.494 5.45 1.3377 0.99 0.44 0.83 0.64 35.6 2441 1.495 5.513 1.2344 1.04 1.20 0.71 0.96 7.6 1982 1.503 5.465 1.1945 1.09 1.50 1.23 1.36 -24.9 1300 1.500 5.490 1.2322 1.05 1.50 1.11 1.30 -24.7 2218 1.480 5.452 1.3973 0.96 1.45 1.06 1.25 -31.4 1128 1.511 5.491 1.0654 1.20 1.76 1.23 1.49 -24.5 2161 1.580 5.440 1.3020 0.97 1.51 0.73 1.12 -15.9 1637 1.480 5.455 1.3162 1.01 1.51 1.31 1.41 -39.3 Table IV--c. Data for Surrey test (.2110 psi) . Board Width Depth E-value,- I n i t i a l Deflection no. (in) (in) (psi x 10 ) Predicted Measured (in) Error (in) North South Average (%) 1115 1.513 5.468 2.0950 0.91 0.86 0.55 0.71 22.5 1609 1.495 5.423 1.6767 1.18 1.10 1.36 1.23 -4.3 1585 1.481 5.460 1.7723 1.10 1.48 0.87 1.17 -6.5 2672 1.400 5.323 1.8031 1.24 1.33 1.10 1.21 1.8 2395 1.500 5.500 1.1680 1.62 1.53 1.70 1.62 0.1 1896 1.465 5.423 1.2674 1.59 0.98 1.15 1.06 33.1 1829 1.472 5.425 1.5364 1.31 2.0 0 .56 1.28 1.9 1419 1.486 5.425 1.3876 1.43 1.71 0.86 1.28 10.3 Table IV-d. Data f o r Surrey test (3110 p s i ) . Group Modulus of E l a s t i c i t y Mean g STD g C.V psi x 10 ps i x 10 % 1070 p s i 1.3258 0.2948 22.2 1410 p s i 1.3206 0.2272 17.2 2110 p s i 1.5232 0.2906 19.1 3110 p s i 1.5883 0.3087 19 .4 Table V. S t a t i s t i c a l informations on the Modulus of E l a s t i c i t y of boards used i n Surrey Experiment. 88 DATE 1070 PSI DAY MONTH YEAR HOURS TIME (HRS) DEFLECTION (IN) MEAN STD CVv (%) 05 01 19 75 16:00 0. 0.29 0.30 103.4 08 01 1975 19 :30 75.50 0.29 0.31 106.9 12 01 1975 15:00 167. 0.32 0.29 90.6 18 01 1975 12:30 308.50 0.31 0.31 100. 22 01 1975 20:00 412. 0.34 0. 30 88.2 26 01 1975 15:00 503. 0.35 0.31 88.6 02 02 1975 16 :30 660.50 0.37 0.33 89.2 22 02 1975 11:45 1135.75 0.37 0.33 89 .2 15 03 1975 12:15 1700.25 0.40 0.35 87.5 12 04 1975 11:40 2359.67 0.44 0. 36 81.8 17 05 1975 11:15 3211.25 0.46 0.37 80.4 19 07 1975 18 :00 , 4730. 0.48 0.36 75.0 18 10 1975 11:00 6895. 0.50 0.34 68. 22 11 1975 11:10 7723.16 0.50 0.35 70. 22 12 1975 12 :00 8456. 0.46 0. 33 71.7 03 01 1976 8710.75 0.51 0.36 70.6 10 02 1976 9596. 0.52 0.33 63.5 27 03 1976 10736. 0.52 0.33 63.5 27 04 1976 11468. 0.54 0.31 57.4 03 06 1976 12356. 0.54 0.31 57.4 27 07 1976 13664. 0.51 0.31 60.8 28 09 19 76 15152. 0.53 0. 30 56.6 09 11 1976 16160. 0.56 0.30 53.6 29 12 1976 17372. 0.55 0.30 54.5 26 11 1977 25328. 0.62 0.30 48.4 03 05 1978 29144. 0.61 0.29 47.5 01 02 1979 35660. 0.64 0. 30 46 .9 TABLE Vl-a. Creep data from Surrey experiment (Group 1070 p s i ) . 89 DATE 1410 PSI DAY MONTH YEAR HOURS TIME (HRS) DEFLECTION (IN) MEAN STD CV (%) 04 01 1975 16 :00 0. 0.60 0.40 67. 08 01 1975 19:30 99.50 0.78 0.32 41. 12 01 1975 15:00 191. 0.80 0.34 42.5 18 01 1975 12:30 332.50 0.78 0.33 42.3 22 01 1975 20 :00 436. 0 .83 0.35 42.2 26 01 1975 15 :00 527. 0.84 0.34 40.5 02 02 1975 16:30 684.50 0. 85 0.34 40. 22 02 1975 11:45 1159.75 0.87 0. 38 43.7 15 03 1975 12:15 1724.25 0.93 0.39 41.9 12 04 1975 11:40 2383.67 0.98 0.40 40.8 17 05 1975 11:15 3235.25 1.00 0.40 40. 19 07 1975 18:00 4754. 1.05 0. 36 34.3 18 10 1975 11:00 6919. 0.03 0.35 34. 22 11 1975 11:10 7747.20 1.05 0.36 34.3 22 12 19 75 12:00 8480. 1.0 8 0.39 36.1 03 01 1976 8734.75 1.10 0. 39 35.5 10 02 1976 9620. 1.09 0.39 35.8 27 03 1976 10760. 1.10 0.39 35.5 27 04 1976 11492. 1.13 0. 39 34.5 03 06 1976 12380. 1.12 0.38 33.9 27 07 1976 13688. 1.10 0.35 31.8 28 09 1976 15176. 1.10 0.33 30/ 09 11 1976 16184. 1.11 0.33 29.7 29 12 1976 17396. 1.12 0. 31 27.7 26 11 1977 25352. 1.15 0.31 27. 03 05 1978 29168. 1.15 0. 32 27.8 01 02 1979 35684. 1.17 0.32 27.4 TABLE Vl-b. Creep data from Surrey experiment (Group 1410 p s i ) . 90 DATE 2110 PSI DAY MONTH YEAR HOURS TIME (HRS) DEFLECTION (IN) MEAN STD CV (%) 04 01 1975 16:00 0. 1.02 0.25 24.5 08 01 1975 19:30 99.50 1.05 0.26 24.8 12 01 1975 15:00 191. 1.06 0.27 25.5 18 01 1975 12:30 332.50 1.05 0.28 26.7 22 01 1975 20:00 436. 1.10 0.28 25.5 26 01 1975 15:00 527. 1.10 0.29 26 .4 02 02 1975 16 :30 684.50 1.12 0. 31 27.7 22 02 1975 11:45 1159.75 1.11 0.30 27. 15 03 1975 12:15 1724.25 1.14 0. 32 28.1 12 04 1975 11:40 2383.67 1.16 0.32 27.6 17 05 1975 11:15 3235.25 1.20 0.33 27.5 19 07 1975 18 :00 4754. 1.22 0.32 26.2 18 10 1975 11:00 6919. 1.29 0.33 25.6 22 11 1975 11:10 7747.20 1.31 0.34 26. 22 12 1975 12:00 8480. 1. 31 0. 33 25.2 03 01 1976 8734.75 1. 31 0.33 25.2 10 02 1976 9620. 1.29 0.33 25.6 27 03 1976 10760. 1.32 0.35 26.5 27 04 1976 11492. 1.31 0.34 26. 03 06 1976 12380. 1.34 0. 31 23.1 27 07 1976 13688. 1.32 0. 34 25.8 28 09 1976 15176. 1.36 0. 33 24.3 09 11 1976 16184. 1. 37 0.34 24.8 29 12 1976 17396. 1.39 0. 34 24.5 26 11 1977 25352. 1.45 0.34 23.4 03 05 1978 29168. 1.43 0.33 23.1 01 02 1979 35684. 1.47 0. 36 24.5 TABLE VI-c. Creep data from Surrey experiment (Group 2110 p s i ) . 91 DATE 3110 PSI DAY MONTH YEAR HOURS TIME (HRS) DEFLECTION (IN) MEAN STD CV (%) 05 01 1975 12:00 0. 1.20 0.25 20.8 08 01 1975 19:30 79.50 1.26 0.27 21.4 12 01 1975 15:00 171. 1.26 0.28 22.2 18 01 1975 12:30 312.50 1.28 0.26 20.3 22 01 1975 20:00 416. 1.31 0.28 21.4 26 01 1975 15:00 507. 1.30 0.29 22.3 02 02 1975 16:30 664.50 1.31 • 0.30 22.9 22 02 19 75 11:45 1139.75 1.33 0.31 23.3 15 03 1975 12:15 1704.25 1.44 0.32 22.2 12 04 1975 11:40 2363.67 1.48 0.35 23.6 17 05 19 75 11:15 3215.25 1.51 0.35 23.2 19 07 1975 18:00 4734. 1.56: 0.33 21.2 18 10 1975 11:00 6899. 1.62 0.34 21. 22 11 1975 11:10 7727.16 1.64 0 .33 20.1 22 12 19 75 12:00 8460. 1.64 0.34 20.7 03 01 1976 8714.75 1.66 0. 35 21.1 10 02 19 76 9600. 1.67 0 .36 21.3 27 03 19 76 10740. 1.69 0. 36 21.3 27 04 1976 11472. 1.72 0.35 20.3 03 06 1976 12360. 1.75 0.31 17.7 27 07 1976 13668. 1.71 0.35 20.5 28 09 1976 15156. 1.73 0.36 20.8 09 11 1976 16164. 1.75 0.36 20.6 29 12 19 76 17376. 1.80 0. 35 19.4 26 11 1977 25332. 1. 83 0.35 19.1 03 05 1978 29148. 1. 81 0.33 18.2 01 02 1979 35664. 1.87 0.37 19.8 TABLE Vl-d. Creep data from Surrey experiment (Group 3110 p s i ) . BOARD NUMBER y n ( i n ) Y ' S ( i n ) 1115 2.91 3.15 1609 3.35 2.76 1585 3.43 3.27 2,6 72 3.23 2.95 2395 3.23 2.95 1896 2.83 3.15 1829 3.15 3.11 1419 3.31 3.11 TABLE V I I . D i s t a n c e b e t w e e n a beam s u p p o r t a n d an e m p i r i c a l r e f e r e n c e f o r t h e m e a s u r e m e n t s . TIME <5._ 6 f f f (hrs) n (in) s (in) n (in) s (in) average (in) 0. 2. 05 2.60 0.86 0.55 0.71 79.50 1.95 2.65 0.96 0.50 0.73 171. 2.05 2.55 0.86 0.60 0.73 312.50 2. 2.45 0.91 0.70 0.81 416. 2. 2.50 0.91 0.65 0.78 507. 2. 2.50 0.91 0.65 0.78 664.50 2. 2.50 0.91 0.65 0.78 1139.75 2. 2.50 0.91 0.65 0.78 1704.25 1.90 2.35 1.01 0 . 80 0.91 2363.67 1. 85 2.40 1.06 0.75 0.91 3215.25 1.75 2.40 1.16 0.75 0.96 4734. 1.75 2.30 1.16 0.85 1.01 6899. 1.70 2.30 1.21 0.85 1.03 7727.16 .1.70 2.25 1.21 0.90 1.06 8460. 1.70 2.30 1.21 0.85 1.03 8714.15 1.70 2.20 1.21 0.95 1.08 9600. 1.75 2. 1.16 1.15 1.16 10740. 1.70 2.20 1.21 0.95 1.08 11472. 1.60 2.15 1.31 1. 1.16 12360. 1.40 2.20 1.41 0.95 1.18 13668. 1.65 2.25 1.26 0.90 1.08 15156. 1.60 2.25 1.31 0.90 1.11 16164. 1.60 2.20 1.31 0.95 1.13 17376. 1.50 2.20 1.41 0.95 1.18 25332. 1.50 2.10 1.41 1.05 1.23 29148. 1. 50 2.15 1.41 1. 1.21 35664 . 1.55 2.10 1.36 .105 1.21 TABLE VIII-1. Creep data for boards loaded to 3110 psi (Surrey experiment) (Board number 1115). TIME 6\ 6 f f f ((Kara) n s n s average (in) (in) (in) (in) (in) 0. 2.25 1.40 1.10 1.36 1.23 79.50 2.20 1.25 1.15 1.51 1.33 171. 2.20 1. 30 1.15 1.46 1.31 312.50 2.25 1.25 1.10 1.51 1. 31 416. 2.20 1.20 1.15 1.56 1.36 507. 2.20 1.20 1.15 1.56 1.36 664.50 2.20 1.25 1.15 1.51 1.33 1139.75 2.15 1.15 1.20 1.61 1.41 1704.25 2.15 1. 1.20 1.76 1.48 2363.67 2 . 05 0.95 1.30 1. 81 1.56 3215.25 2.05 0.95 1.30 1.81 1.56 4734. 2.05 0. 85 1.30 1.91 1.61 6899. 2.05 0.85 1.30 1.91 1.61 7727.16 2.05 0.85 1.30 1.91 1.61 8460. 2. 0.85 1.35 1.91 1.63 8714.75 2.05 0.75 1.20 2.01 1.66 9600. 2. 0.80 1.35 1.96 1.66 10740. 2. , 0.80 1.35 1.96 1.66 11472 1.95 0.75 1.40 2.01 1.71 12360. 1.95 0.80 1.40 1.96 1.68 13668. 1.95 0.75 1.40 2.01 1.71 15156. 1.95 0. 85 1.50 1.91 1.66 16164. 1.90 0.90 1.45 1.86 1.66 17376. 1.90 0.85 1.45 1.91 1.68 25332. 1. 85 0.90 1.50 1.8 6 1.68 29148. 1.90 0.85 1.45 1.91 1.68 35664. 1. 85 0. 80 1.50 1.96 1.73 TABLE VIII-2. Creep data for boards loaded to 3110 p s i (Surrey experiment) (Board number 1609). TIME (hrs) 6.. n (in) 6\ s (in) f n (in) f s (in) f average (in) 0. 1.95 2.40 1.48 0.87 1.18 79.50 1.80 2.45 1.63 0.92 1.28 171. 1.80 2.40 1.63 0.87 1.25 312.50 1. 80 2.40 1.63 0.87 1.25 416. 1.80 2.40 1.63 0.87 1.25 507. 1.80 2.50 1.63 0 .77 1.20 664.50 1.80 2.45 1.63 0. 82 1.23 1139.75 1. 85 2.40 1.58 0.87 1.23 1704.25 1. 80 2.30 1.63 0.97 1.30 2363.67 1.80 2.30 1.63 0.97 1.30 3215.25 1.80 2.30 1.63 0.97 1.30 4734. 1. 80 2.10 1.63 0 .17 1.40 6899. 1.70 2.05 1.73 1.22 1.48 7727.16 1.65 2.05 1.78 1.22 1.50 8460. 1.70 2.05 . 1.73 1.22 1.48 8714.75 1.75 2.05 1.68 1.22 1.45 9600. 1.70 2.05 1.73 1.22 1.48 10740. 1.70 2. 1.73 1.27 1.50 11472. 1. 70 1.85 1.73 1.42 1.58 12360. 1.55 1.90 1.88 1.37 1.63 13668, 1.65 2. 1.78 1.27 1.53 15156. 1.65 1.90 1.78 1.37 1.58 16164. 1.60 2. 1. 83 1.27 1.55 17376. 1.50 1.90 1.93 1.37 1.65 25332. 1. 35 1.90 2.08 1.37 1.73 29148. 1. 30 1.95 2.13 1. 32 1.73. 35664. 1.35 1.95 2.08 1. 32 1.70 TABLE VIII-3. Creep data for boards loaded to 3110 p s i (Surrey experiment) (Board number 15 85). TIME (hts) 6 n (in) <5 s (in) f n (in) f s (in) f average (in) 0. 1.90 1.85 1.33 1.10 1.22 79.50 1.85 1. 70 1.38 1.25 1.32 171. 1.80 1.70 1.43 1.25 1.34 312.50 1.80 1.65 1.43 1.30 1.37 416. 1.80 1.60 1.43 1.35 1. 39 507. 1.80 1.60 1.43 1.35 1.39 664.50 1.75 1.60 1.48 1.35 1.42 1139.75 1.80 1.55 1.43 1.40 1.42 1704.25 1.65 1.50 1.58 1.45 1.52 2363.67 1.70 1.35 1.53 1.60 1.57 3215.25 1.65 1.35 1.5 8 1.60 1.59 4734. 1.60 1. 35 1.63 1.60 1.62 6899 . 1.45 1.35 1.78 1.60 1.69 7727.16 1.50 1. 30 1.73 1.65 1.69 8460. 1.50 1.25 1.73 1.70 1.72 8714.75 1.45 1.25 1. 78 1.70 1.74 9600. 1.45 1.25 1.78 1.70 1.74 10740. 1.45 1.25 1.78 1.70 1.74 11472. 1.45 1.25 1.78 1.70 1.74 12360. 1.45 1.25 1.78 1.70 1.74 13668. 1.45 1.25 1.78 1.70 1.74 15156. 1.45 1.25 1.78 1.70 1.74 16164. 1.45 1.35 1.78 1.60 1.69 17 376. 1.40 1.25 1. 83 1.70 1.77 25332. 1.45 1.30 1.78 1.65 1.72 29148. 1.50 1.20 1.73 1.75 1.74 35664. 1.30 1.20 1.93 1.75 1.84 TABLE VIII-4. Creep data for boards loaded to 3110 p s i (Surrey experiment) Board number 2672. TIME (hrs) 6. n (in) 6, s (in) f n (in) f s (in) f average (in) 0. 1.70 1.25 1.53 1.70 1.62 79.50 1.65 1.10 1.58 1.85 1.72 171. 1.65 1.05 1.58 1.90 1.74 312.50 1.60 1.05 1.63 1.90 1.77 416. 1.60 0.95 1.63 2. 1.83 507. 1.55 1. 1.68 1.95 1.82 664.50 1.50 0.95 1.73 2. 1. 87 1139.75 1.50 0.85 1.73 2.10 1.92 1704.25 1.35 0.70 1.88 2.25 2. 07 2363.67 1.35 0.50 1. 88 2.45 2.17 3215.25 1. 30 0.50 1.93 2.45 2.19 4734. 1.30 0.50 1.93 2.45 2.19 6899. 1.20 0.50 2.03 2.45 2.24 7727.16 1.25 0.45 1.98 2.50 2.24 8460. 1.25 0.50 1.98 2.45 2.22 8714.75 1.15 0.40 2.08 2.55 2.32 9600 . 1.20 0 .40 2.03 2.55 2.29 10740. 1.10 0.40 2.13 2.55 2.34 11472. 1.20 0.20 2.03 2.75 2.39 12360. 1.20 0.40 2.03 2.55 2.29 13668. 1.20 0.40 2.03 2.55 2 .29 15156. 1.15 0.40 2.08 2.55 2. 30 16164. 1.15 0.40 2.08 2.55 2.31 17376. 1.15 0.40 2.08 2.55 2.31 25332. 1.10 0.35 2.13 2.60 2.37 29148. 1.35 0.30 1.88 2.65 2.27 35664. 1.10 0.20 2.13 2.75 2.44 TABLE VIII-5. Creep data for boards loaded to 3110 p s i (Surrey experiment) (Board Number 2 395). TIME 6 <5 f f f (Hrs) n (in) s (in) n (in) s (in) average (in) 0. 1.85 2. 0.98 1.15 1.07 79.50 1.60 1.95 1.23 1.20 1.22 171. 1.80 1.90 1.03 1.25 1.14 312.50 1.80 1.85 1.03 1.30 1.17 416. 1.70 1.80 1.13 1.35 1.24 507. 1.70 1.85 1.13 1.30 1.22 664.50 1.70 1.90 1.13 1.25 1.19 1139.75 1.70 1. 80 1.13 1. 35 1.24 1704.25 1.55 1. 60 1.28 1.55 1.42 2363.67 1.45 1.55 1.38 1.60 1.49 3215.25 1.30 1. 60 1.53 1.55 1.54 4734 . 1.30 - 1.50 1.53 1.65 1.59 6899. 1.20 1.40 1.63 1.75 1.69 7727.16 1.20 1.25 1.63 1.90 1.77 8460. 1.15 1. 30 1.68 1.85 1.77 8714.75 1.15 1. 30 1.68 1.85 1.77 9600. 1.10 1.30 1.73 1.85 1.79 10740. 1.05 1.25 1.78 1.90 1.84 11472. 1.05 1.20 1.78 1.95 1.86 12360. 0.95 1.15 1.88 2.00 1.94 13668. 1. 1.20 1.83 1.95 1.89 15156. 0.95 1.05 1.88 2.10 1.99 16164. 0.90 0.95 1.93 2.20 2.07 17376. 0.85 0.95 1.98 2.20 2 .09 25332. 0.80 0.90 2.03 2.25 2.14 29148. 0.90 0.85 1.93 2.30 2;12 35664 . 0.75 0.90 2.08 2.25 2.17 TABLE VIII-6. Creep data for boards loaded to 3110 p s i (Surrey experiment) (Board number 1896). TIME 6 6 f f f (hrs) n (in) s (in) n (in) s (in) average (in) 0. 1.15 2.55 2. 0.56 1.28 79.50 1.15 2.60 2. 0 .51 1.26 171. 1.15 2.60 2. 0.51 1.26 312.50 1.15 2.60 2. 0.51 1.26 416. 1.10 2.55 2.05 0.56 1.31 507. 1.15 2 .60 2. 0.51 1.26 664.50 1.15 2.55 2. 0.56 1.28 1138.75 1.15 2.55 2. 0.56 1.28 1704.25 1.15 2.50 3. 0.61 1.31 2363.67 1.05 2.50 2.10 0.61 1.36 3215.25 1. 2.45 2.15 0.66 1.41 4734. 1. 2.45 2.15 0.66 1.41 6899. 0.95 2. 40 2.40 0. 71 1.46 7727.16 0.90 2.35 2.25 0.76 1.51 8460 . 0.95 2.35 2.20 0.76 1.48 8714.75 0.90 2. 30 2 .25 0. 81 1.53 9600. 0. 85 2.40 2. 30 0.71 1.51 10740. 0.90 2.35 2.25 0.76 1.51 11472. 0.90 2.35 2.25 0.76 1.51 12360. 0.85 2.25 2.30 0.86 1.58 13668. 0. 85 2.25 2.30 0.86 1.58 15156. 0.80 2.35 2.35 0.76 1.56 16164. 0.70 2.20 2.45 0.91 1.68 17376. 0.70 2.15 2.45 0.96 1.71 25332. 0.65 2.15 2.50 0.96 1.73 29148. 0.65 2.15 2.50 0.96 1.73 35664. 0.55 2.10 2.60 1.01 1.81 TABLE VIII-7. Creep data for boards loaded to 3110 p s i (Surrey experiment) (Board number 1829). 100 TIME (hrs) n (in) 6_ s (in) f : rt (in) f s (in) f average (in) 0. 1.60 2.25 1.71 0 .86 1.29 79.50 1.55 2.25 1. 76 0.86 1.31 171. 1.50 2.20 1.81 0.91 1. 36 312.50 1.50 2.20 1.81 0.91 1.36 416. 1.60 2.10 1. 71 1.01 1.36 507. 1.50 2.10 1.81 1.01 1.41 664.50 1.50 2.10 1.81 1.01 1.41 1139.75 1.55 2.15 1.76 0.96 1.36 1704.25 1.45 1.95 1.86 1.16 1.51 2363.67 1.45 2. 1.86 1.11 1.49 3215.25 1.35 1.95 1.96 1.16 1.56 4 734. 1. 30 1.80 2.01 1.31 1.66 6899. 1.25 1.65 2.06 1.46 1.76 7727.16 1.25 1.60 2.06 1.51 1.79 8460. 1.20 1.55 2.11 1.56 1.84 8714.75 1.25 1.60 2.06 1.51 1.79 9600. 1.25 1.60 2.06 1.51 1.79 10740. 1.20 1.55 2.11 1.56 1.84 11472. 1.20 1.55 2.11 1.56 1.84 12360. 1.15 1.50 2.16 1.61 1.89 1366 8. 1.1.5 1.50 2.16 1.61 1.89 15156. 1.10 1.50 2.21 1.61 1.91 16164. 1.05 1.50 2.26 1.61 1.94 17376. 1. 1.35 2.31 1.76 2.04 25332. 1. 1.35 2.31 1.76 2.04 29148. 1. 1.30 2.31 1. 81 2.06 35664. 0.95 1. 35 2 . 36 1.76 2.06 TABLE VIII-8. Creep data for boards loaded to 3110 p s i (Surrey experiment) (Board number 1419). 101 TABLE IX. Data for Richmond experiment (Sub-group 1). BOARD GROUP WIDTH DEPTH STRESS MOE PREDICTED ACTUAL NUMBER NUMBER (IN) (IN) (PSI) (PSI x 10" ) 6 . l 6 . l 653 8 1.50 5.29 3000 0.19321 30.2 35.1 1395 8 1.50 5.36 3000 0.18169 31.8 39.1 1101 8 1.51 5.36 3000 0.22467 25.7 22.1 158 8 1.50 5.40 3000 0.24719 23.1 24 .9 400 2 1.50 5.44 3000 0.20620 27.4 29.0 305 2 1.50 5.41 3000 0.18078 31.5 35.1 733 8 1.46 5.36 3000 0.24649 23.4 24.9 1318 8 1.52 5.43 3000 0.16931 35.6 36.1 803 8 1.49 5.32 3000 0.21008 27.7 29 .0 1188 8 1.49 5.29 3000 0.21692 26 .9 31.0 765 2 1.50 5. 35 3000 0.17492 33.0 39.9 770 2 1.50 5.36 3000 0.17047 33.8 38.1 764 8 1.49 5.40 3000 0.22097 25.9 30.0 679 8 1.52 5. 38 3000 0.19176 30.0 31.0 313 8 1.51 5.43 3000 0.21805 26 .2 30 . 0 116 7 8 1.50 5.38 3000 0.20472 27.9 30 .0 1030 8 1.52 5.48 3000 0.15912 35.6 42.9 .472 8 1.51 5.44 3000 0.17032 33.3 36.1 2768 8 1.50 5. 39 3000 0.18417 31.2 26.9 2853 8 1.46 5.29 3000 0.23657 24 .6 25.9 97 8 1.51 5. 37 3000 0.21339 26.9 27 .9 269 8 1.54 5.48 3000 0.15667 36.1 41.9 1.6 78 8 1.52 5.47 300 0 0.18374 30.7 35 .1 1680 8 1.52 5.43 3000 0.19753 28. 7 31.0 1679 8 1.49 5.40 3000 0.20089 28.4 31.0 2276 8 1.52 5.47 3000 0.145 39 38.9 41.9 2299 8 1.52 5.47 3000 0.13042 43.4 48 .0 1582 8 1.48 5.38 3000 0.25928 22.1 23.1 1557 8 1.53 5.45 3000 0.17430 32.5 33.0 2727 8 1.52 5.4 3 3000 0.187 27 30 .5 34 .0 2699 8 1.51 5.36 3000 0.22925 25.1 26 .9 2855 8 1.47 5. 33 3000 0.25048 23.1 23.1 1820 8 1.52 5.49 3000 0.14046 40.1 45.0 1897 8 1.51 5.40 3000 0.20879 27.4 20.1 44 2 1.48 5.28 3000 0.20410 28.7 31.0 62 2 1.52 5.40 3000 0.19237 29.7 33.0 997 2 1.48 5. 36 3000 0.15440 37.3 46.0 519 8 1.50 5.38 3000 0.19485 29.5 33.0 2212 8 1.54 5.47 3000 0.21970 25.7 29 .0 2304 8 1.50 5.45 3000 0.18362 31.0 33.0 923 2 1.49 5.41 3000 0.19561 29 .2 34.0 545 8 1.53 5.35 3000 0.175 87 32. 8 38.1 1892 8 1.51 5.37 3000 0.16809 34.3 37.1 318 2 1.50 5.38 3000 0.22852 25.1 29.0 2124 8 1.52 5.43 3000 0.17105 33.3 38.1 371 2 1.54 5.48 3000 0.15535 36 .3 39 .1 2399 8 1.54 5.54 3000 0.15867 35.1 46.0 14 2 1.50 5.40 3000 0.16127 35.6 39.9 2503 8 1.50 5.41 3000 0.16627 34.3 41.9 102 Table IX continued. BOARD NUMBER GROUP NUMBER WIDTH (IN) DEPTH (IN) STRESS (PSI) MOE ? (PSI x 10 )• PREDICTED 6 . l ACTUAL 6 . l 2526 8 1.51 5.43 3000 0.17636 32.3 37.1 2131 8 1.51 5.41. 3000 0.17130 33. 3 39.1 2026 8 1.51 5.44 3000 0.21631 26.2 29.0 2557 8 1.50 5.25 3000 0.19621 30.0 35.1 2037 8 1.50 5.45 3000 0.20622 27.4 32.0 2745 8 1.47 5.27 3000 0.21334 27.4 32.0 2331 8 1.52 5.43 3000 0.16783 34.0 38.1 1587 8 1.49 5. 37 3000 0.23281 24.6 25.9 1593 8 1.52 5.45 3000 0.21160 26.9 30.0 1583 8 1.50 5.43 3000 0.15984 35.6 43.9 2616 8 1.50 5.43 3000 0.20564 27.7 31.0 393 8 1.53 5.42 3000 0.19370 29.5 32.0 2587 8 1.52 5.48 3000 0.16527 34 .0 38.1 1904 8 1.50 5.40 3000 0.24347 23.6 25.9 404 8 1.50 5.38 3000 0.17790 32.3 34.0 2406 8 1.51 5.38 3000 0.13917 41.4 45.0 76 2 1.52 5.46 3000 0.18339 31.0 34.0 392 2 1.52 5.43 3000 0.23904 23.9 26.9 355 2 1.49 5.40 3000 0.19402 29.5 33.0 324 2 1.48 5.48 3000 0.21492 26 .2 27.9 1016 2 1.51 5.44 3000 0.20385 27.9 32.0 1076 2 1.48 5.32 3000 0.18630 31.2 32.0 298 2 1.51 5.41 3000 0.23115 24.6 25.9 2644 8 1.49 5.37 3000 0.19964 28.7 31.0 1963 8 1.51 5.43 3000 0.17791 32.0 47.0 2144 8 1.50 5.36 3000 0.18606 31.0 33.0 2046 8 1.51 5.46 3000 0.16060 35. 3 34.0 2031 8 1.51 5.44 3000 0.18151 31.2 43.9 2819 8 1.50 5.44 3000 0.22696 25.1 30.0 1231 9 1.49 5.33 3000 0.22277 25.9 29 .0 1353 8 1.51 5.47 3000 0.20482 27.7 29.0 2350 8 1.51 5.36 3000 0.21075 27.4 27.9 2327 8 1.54 5.50 3000 0.19 7 9.9 28.4 31.0 1289 8 1.50 5.36 3000 0.22444 25.7 29.0 1439 8 1.52 3000 3000 . 0.16791 33.5 39.1 1254 8 1.48 5. 32 3000 0.20898 27.7 33.0 1323 8 1.49 5.40 3000 0.20103 28.4 31.0 1290 8 1.50 5.47 3000 0.18615 30.2 35.1 940 9 1.47 5.42 3000 0.21854 26.2 31.0 1482 8 1.51 5.44 3000 0.17780 32.0 38.1 1526 8 1.49 5.45 3000 0.20026 28.2 25.9 1313 8 1.51 5.40 3000 0.17308 33.0 39.1 1510 8 1.50 5.40 3000 0.15520 36.8 42.9 1432 8 1.51 5.36 3000 0.23383 24.6 29. 0 1472 8 1.52 5.48 3000 0.20214 27.9 32.0 1111 8 1.47 5.31 3000 0.23811 24 .4 26.9 1533 8 1.49 5.44 3000 0.19037 29.7 31.0 52 8 1.40 5.28 3000 0.24321 24.1 27.9 TABLE X. Data for Richmond experiment (Sub-group 2). BOARD GROUP WIDTH DEPTH STRESS MOE PREDICTED ACTUAL S.R. COMMENTS NUMBER NUMBER : (in) (in) (psi) (psi) (E+07) i (mm) i (mm) (%) 1634 2 1.50 5.43 3000 0.14820 38.4 40.0 F.O.L. 2239 2 1.53 5.48 3000 0.14967 37.7 43.0 F.O.L. 1918 2 1.51 5.47 3000 0.17669 32.0 36.0 F.O.L. 569 2 1.51 5.35 3000 0.11039 52.3 F.O.L. 1653 2 1.52 5.45 3000 0.17262 32.8 36.0 F.O.L. 1947 2 1.52 5.43 3000 0.14529 39.2 43.0 29.7 2284 2 1.52 5.46 3000 0.16553 34.2 35.0 55.0 2803 1 1.52 5.36 3000 0.19817 29.1 31.0 26.1 2059 2 1.50 5.26 3000 0.21369 27.1 28.0 25.0 1057 2 1.48 5.36 3000 0.25249 22.8 25.0 21.7 2072 2 1.52 5.45 3000 0.18256 31.1 29.0 73.5 2057 2 1.53 5.46 3000 0.17007 33.3 35.0 43.4 146 2 1.53 5.45 3000 0.22289 25.4 29.0 24.3 1463 2 1.52 5.44 3000 0.19123 29.7 35.0 62.7 1386 2 1.57 5.46 3000 0.16437 34.4 37.0 62.6 1934 2 1.47 5.29 3000 0.22677 25.8 29.0 27.0 1978 2 1.47 5.36 3000 0.22245 25.9 28.0 52.8 572 2 1.48 5.26 3000 0.19094 30.8 32.0 40.8 1973 2 1.51 5.48 3000 0.14893 37.9 41.0 32.8 2491 2 1.49 5.28 3000 0.21889 26.7 28.0 32.6 1961 2 1.50 5.33 3000 0.19566 29.6 31.0 27.0 2748 2 1.51 5.43 3000 0.19079 29.8 34.0 53.8 2778 2 1.53 5.43 3000 0.21550 26.4 28.0 38.3 2519 2 1.54 5.50 3000 0.21313 26.4 28.0 48.8 2498 2 1.52 5.45 3000 0.20910 27.1 30.0 44.4 2003 2 1.52 5.47 3000 0.19513 29.0 31.0 37.7 225 2 1.50 5.33 3000 0.21807 26.6 31.0 45.7 2509 2 1.51 5.26 3000 0.17863 32.9 36.0 85.7 1405 2 1.50 5.50 3000 0.20971 26.8 30.0 42.0 2653 2 1.50 5.44 3000 0.18004 31.6 33.0 71.5 TABLE X continued. BOARD GROUP WIDTH DEPTH STRESS MOE PREDICTED ACTUAL S.R. COMMENTS NUMBER NUMBER (in) (in) (psi) (psi) (E+07) i (mm) i (mm) 2363 2 1.54 5.48 3000 0.16436 34.3 38.0 35.9 2713 2 1.51 5.47 3000 0.15738 35.9 39.0 62.2 555 2 1.51 5.39 3000 0.15825 36.2 41.0 50.2 2811 2 1.53 5.45 3000 0.18457 30.7 30.0 87.8 2857 2 1.50 5.39 3000 0.25064 22.9 24.0 22.8 2561 2 1.46 5.35 3000 0.16711 34.6 26.0 27.9 905 2 1.52 5.46 3000 0.19932 28.4 33.0 59.3 1589 2 1.47 5.32 3000 0.23282 25.0 29.0 27.0 2207 2 1.52 5.41 3000 0.13144 43.5 45.0 58.3 2220 2 1.53 5.48 3000 0.22346 25.2 27.0 41.9 2725 2 1.51 5.38 3000 0.16854 34.1 35.0 59.0 1793 2 1.51 5.48 3000 0.19734 28.6 32.0 34.6 2499 2 1.51 5.41 3000 0.22809 25.0 25.0 24.1 2623 2 1.51 5.48 3000 0.16743 33.7 37.0 31.7 2210 2 1.55 5.52 3000 0.14580 38.4 42.0 74.7 2279 2 1.51 5.46 3000 0.23933 23.6 26.0 35.9 1663 2 1.51 5.37 3000 0.16165 35.6 37.0 35.1 837 2 1.50 5.46 3000 0.20099 28.2 32.0 42.5 2522 2 1.52 5.49 3000 0.17772 31.7 36.0 44.7 2655 2 1.48 5.41 3000 0.18831 30.3 33.0 80.5 2884 2 1.51 5.47 3000 0.17134 33.0 38.0 59.5 2673 2 1.50 5.41 3000 0.20796 27.5 27.0 23.7 745 2 1.48 5.32 3000 0.21628 26.9 27.0 27.6 946 2 1.50 5.43 3000 0.19818 28.7 32.0 56.6 1513 2 1.51 5.45 3000 0.26006 21.8 24.0 23.7 1165 2 1.52 5.45 3000 0.18844 30.1 32.0 35.7 1365 2 1.52 5.48 3000 0.20311 27.8 31.0 47.0 1357 2 1.52 5.45 3000 0.24560 23.1 25.0 22.9 41 2 1.50 5.41 3000 0.19298 29.6 32.0 43.9 1207 2 1.51 5.45 3000 0.19435 29.2 33.0 40.6 TABLE X continued. BOARD GROUP WIDTH DEPTH STRESS MOE PREDICTED ACTUAL S.R. COMMENTS NUMBER NUMBER (in) (in) (psi) (psi) (E+07) i (mm) i (mm) 1337 2 1.50 5.48 3000 0.20886 27.0 30.0 49.7 762 2 1.52 5.40 3000 0.21830 26.2 30.0 58.6 2318 2 1.51 5.34 3000 0.20712 27.9 31.0 345 2 1.51 5.43 3000 0.17978 31.7 35.0 F.O.L. 2352 2 1.50 5.48 3000 0.17523 32.2 35.0 36.6 2238 2 1.54 5.51 3000 0.18611 30.1 36.0 F.O.L. 1332 2 1.53 5.50 3000 0.15512 36.2 41.0 F.O.L. 2282 2 1.52 5.47 3000 0.16186 34.9 39.0 F.O.L. 1786 6 1.52 5.48 4500 0.18372 46.0 47.0 -2800 6 1.53 5.47 4500 0.20467 41.4 44.0 -1620 6 1.53 5.49 4500 0.17038 49.6 52.0 -1671 6 1.51 5.39 4500 0.18369 46.8 48.0 -1602 6 1.51 5.44 4500 0.21890 38.9 43.0 - • 2756 6 1.48 5.27 4500 0.17380 50.6 55.0 -2482 6 1.50 5.34 4500 0.19486 44.5 47.0 -2497 6 1.50 5.41 4500 0.24174 35.4 36.0 -2067 6 1.52 5.47 4500 0.21157 40.1 45.0 -2469 6 1.48 5.36 4500 0.18681 46. 3 50.0 -1969 6 1.47 5.22 4500 0.21544 41.2 45.0 -2687 9 1.50 5.41 4500 0.19991 42.9 47.0 -2642 9 1.49 5.39 4500 0.19422 44.3 46.0 -2826 9 1.51 5.47 4500 0.23949 35.4 38.0 -2549 9 1.47 5.31 4500 0.22664 38.5 42.0 -2489 9 1.48 5.30 4500 0.23473 37.3 41.0 -410 9 1.50 5.43 4500 0.22932 37.2 41.0 -1923 9 1.49 5.38 4500 0.21067 40.9 44.0 -2600 9 1.49 5.42 4500 0.20568 41.6 45.0 -2807 9 1.54 5.47 4500 0.17893 43.0 47.0 -942 9 1.51 5.44 4500 0.19825 43.0 47.0 -932 9 1.50 5.37 4500 0.21665 39.8 45.0 -TABLE X continued. BOARD GROUP WIDTH DEPTH STRESS MOE PREDICTED ACTUAL S.R. COMMENTS NUMBER NUMBER (in) (in) (psi) (psi) (E+07) i (mm) i (mm) 2147 9 1.49 5.25 4500 0.21364 41.3 45.0 2838 9 1.50 5.46 4500 : 0.21513 39.5 45.0 -827 2 1.51 5.42 3000 0.23702 24.1 26.0 29.3 1280 2 1.50 5.36 3000 0.19070 30.2 34.0 28.8 418 2 1.51 5.43 3000 0.17845 31.9 35.0 52.0 1507 2 1.50 5.48 3000 0.17628 32.0 35.0 51.3 2536 2 1.48 5.39 3000 0.23947 23.9 26.0 37.0 2321 2 1.52 5.42 3000 0.18708 30.5 32.0 31.6 1047 2 1.50 5.41 3000 0.22140 25.8 26.0 31.5 1584 2 1.51 5.38 3000 0.17286 33.2 37.0 36.4 244 9 1.50 5.39 4500 0.16900 50.9 54.0 -411 9 1.52 5. 35 4500 0.22179 39.1 44.0 -2568 9 1.48 5.40 4500 0.19119 44.9 51.0 -2008 9 1.51 5.47 4500 0.17730 47.8 54.0 -2068 9 1.50 5.46 4500 0.19110 44.4 51.0 STRESS RATIO STRENGTH FRACTIONAL CREEP NUMBER OF (%) (psi) (average) at 3 months (%) DATA POINTS 0<SR<30 S_>10000 B 43.3 17 30<SR<40 10000>6T5>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><SD>3000 62.3 6 TABLE XI. Fractional creep expressed at three months for five ranges of stress r a t i o s . (Average values.) 3000 PSI 4500 PSI TIME A B A B (x 10" 3) (x 10~ 3) 1/mm (x 10"3) (x 10" 3) 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 13 weeks 0.219 16.309 -105.590 15.206 TABLE XII. Coe f f i c i e n t s A and B of the corr e l a t i o n equation between creep deformation and e l a s t i c deformation for two stress l e v e l s . 3000 PSI and 4500 PSI TIME A B (x 10" 3) (x 10" 3) 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 3 months -10.070 14.168 13 weeks -1.0.115 14.320 TABLE XIII. C o e f f i c i e n t s A and B of 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 of the stress l e v e l TIME 3000 PSI C (x 10" 2) and 4500 PSI D (psi x 10 5) 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 13 weeks 6.067 8.615 TABLE XIV. Coeff i c i e n t s C and D of the co 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 , disregard of the stress l e v e l . STRESS FIGURE 1. Stress and s t r a i n versus time diagrams for a specimen of a hypothetical material. From Nielsen (1972). 112 S T R E S S FIGUEE 2. Fractional creep i n Hoop Pine i n bending. Specimens i n small s i z e . Stress as percentage of ultimate strength at 21.5 C. From Kingston and Clarke (1961). 3000 1000 500 500 _ 1000 50 -IP-\ M ) 9 f ' FIGURE 3-a. Devices used t o measure creep of wood i n bending. Two d i f f e r e n t specimen s i z e s . A l l dimensions i n mm. Cederberg and Dani e l s s o n (1970) . FRACTIONAL CREEP, (1/256 -1 day) .12-1 ( 1 — 1 1 1 1 • .10 *' • A .08 1 / .06 o ' ° n - ^ « • .OA o o o • o • * • ° o • .02 Symbols 65%RH 80%RH Trend Small o • 0 ( Large • • ) 10 20 30 AO 50 60 76 STRESS INTENSITY, 6/6 B, •/. FIGURE 3-b. 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 smal l and the la r g e specimens between 1/256 day and 1 day as 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 . Data from Cederberg 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 (1972). ; 1— 1 i — I 1 1 I l i l t 0 1 2 3 4 5 6 7 8 9 10 Time (103hr) FIGURE 4. Creep compliance versus time for 4 stress levels express as percentages of proportional l i m i t . From Nakai (1978) 115 to /\\ LO LU cn i— 0 0 —> TIME V i r g i n creep c u r v e R e c o v e r y c u r v e R e l o a d i n g c u r v e T IME FIGURE 5. Theoretical development of creep under intermittent loading. From Nielsen (1972). 116 NO. 1 20°C 2 30°C 3 40°C 4 50°C 5 20°C 6 30 °C 7 40 °C 8 20°C 9 30°C 10 40 °C 11 30 °C 12 30°C 13 50 TYPE Const. I I (120 ran). (10 mn) ( 50 °C 50 °C 50 °C (110 mn) ( " ) ( " ) 4 0 mn i O £ e i — o LU _) U. LU Q A ( (30 mn)-(50 mn)-C (5 mn) ) — ' 40 mn ) - 50 °C 40 mn ) 50°C - 50 °C - 50°C - 40°C 50 6C (90 mn) ( ) (125 mn) o o . 10 50 100 0 20 No. 5 6 7 120 •> TI ME ( mn ) FIGURE 6. Creep curves for various types of temperature. From Kitahara and Yukawa (19 64). C R E E P S T R A I N . E c , W-S TIME, hours FIGURE 7. Creep versus time for hardboard and plywood. From Lundgren (1968 replotted by Nielsen (1968a). C R E E P 10" h - l h . M - S 2 0 0 0 i • ' • • • *-1500 1000 5 0 0 B 1/8 x 50 K 500 mm I 169 10'6-sinh (0.0U8<j) 0 0 25 5 0 75 100 125 150 175 200 k p / c m * 010 6 10 2 0 30 AO 50 6 0 70 8 0 % _ 3 Figure 8. Creep from 10 hours to 1 hour of 1/8" fiberboard i n tension versus stress, a , and versus f r a c t i o n a l stress, a/a_,. Data from Lundgren (1968) replotted by Nielsen (1972) . (Masonite at 65% RH). 1 1 8 Steps FIGURE 9. Sample I. Grouping of the material. 0.0 20.0 40.0 60.0 60.0 100.0 120.0 „ 140.0 160.0 160.0 200 FAILURE STRESS (PSI) (XlO2 ) FIGURE 10. Strength d i s t r i b u t i o n of the control sample from Sample I. Raw data; three parameter Weibull d i s t r i b u t i o n . 120 PULLEYS SIDE VIEW OF TEST SET-UP SPACER LEAD WEIGHT END VIEW OF TEST SET-UP FIGURE 11. Test set-up for Surrey experiment. FIGURE 12-1. Test set-up for Surrey experiment. Picture as an end-view. FIGURE 12-2. Test set-up for Surrey experiment. Picture as an end-view. 123 CM i to s L U .M -M 0 -M 0-|—> i i m i i [ — i i 1111111—i i | n i i i | — I I | MIM|—I I I I I — I I I 10"1 4 1 0 ° 4 10 1 4 10 2 4 10 3 4 10" 4 10 5 T I M E ( H O U R S ) ( L O G S C A L E ) FIGURE 13. Deflection versus time for four stress l e v e l s . Raw data from Surrey experiment. 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 4 102 4 10 3 4 10 4 T I M E ( H O U R S ) ( L O G S C A L E ) I I | Ml!l| 4 10 5 FIGURE 14. Deflection versus time for four stress l e v e l s . Corrected data from Surrey experiment. 125 S t e p s — © -G r o u p 1 ( 1 00 s p c . ) to c CO £ o O; — Q. o LO o _ i O LU o cr LL. <j, ' £ LT) — — o — i a. — CL DO LO Gr oup 2 ( 1 0 0 spc . ) 9 4 s p c . -<—' • O o c o 3 cn c CL . £ a cr to a* a o .a . c£ G r o u p 3 ( 1 0 0 s p c . ) G r o u p 4 ( 1 0 0 s p c . ) G r o u p 5 ( 1 00 sp c. G r o u p 7 ( 1 0 0 s p c G r o u p 10 (100 spc. ) G r o u p 11 ( 1 0 0 spc.) G r o u p 12 ( 1 0 0 spc.) G roup 13 (14 0 spc.) G r o u p 14 (14 0 spc.) G r o u p 18 (140 spc . ) G r o u p 1 (14 0 s p c .) G r o u p 2 (1 4 0 spc. ) G r o u p 3 ( 1 4 0 spc.) G r o u p 4 { 1 4 0 sp c.) G r o u p 5 ( 1 40 spc. ) G r o u p 6 (1 00 s pc . ) 11 s p c . G r o u p 8 ( 1 0 0 spc.") | ~\ 79 s p c " G r o u p 9 ( 100 spc . ) 18 s pc . G r o up 15 (140 s p c .) 13 9 s p c . a o —• >- G r o u p 16 (1 40 s p c.) G r o u p 17 (140 s p c . ) ^ C o n t r o l s a m p l e 13 7 s p c . FIGURE 15. Sample I I . Grouping of the material. FIGURE 16. Strength d i s t r i b u t i o n of the control sample from population I I . Raw data; two and three parameter Weibull d i s t r i b u t i o n s . y-1 0.0 15.0 30.0 45.0 60.0 75.0 90.0 „ 105.0 120.0 135.0 150. STRENGTH (PSI) (XlO 2 ) FIGURE 17, Strength d i s t r i b u t i o n of the control sample from population II. Raw data; lognormal d i s t r i b u t i o n . 129 FIGURE 19. Picture of Richmond test set-up. :pt c ^ Cs-KD 1/1} 2^ TliEe NO . 3 srs 2,7-/5 f& JoX f Z 5 ^ o n 9i> ±1.1. . 1 FIGURE 20. Charts of load and deflection variations with respect to time during transfer of the load. FIGURE 21. Example of creep curves plotted on a b i - l i n e a r graph. Data from the Richmond experiment. 132 22. Examples of creep curves plotted on a f u l l logarithmic graph (top), and semi-logarithmic graph (bottom). Data from Richmond experiment. 133 6 (mm) z 3000 P S I Appl ied s t ress G r o u p s no. 8 8, 7 0 , 60 3 6 (mm) I ' I ' I ""I 1 I i l ' » U 1 I i 11»i| 1 | i | n i i | 1 i i | n n | r- | i | Tin) I0"« 3 5 1 0 " 3 5 IIC 3 S 10' 3 S 10' 3 5 10* 3 S 10' TIME (HRS) (LOG SCALE) FIGURE 23. Creep curves for boards of sub-group 1. Richmond experiment. (mm) 3000 PSI Applied stress Groups no 8 & 2 33. BN 1 5 5 7 3 6 29 U 32 I ' I I M ' » | 1 | t -| > 1111 1 | I | MII| 1 1 HHII'I 1 | I | l l l l | 1 | I | l l l l | I0"« 3 5 10-' 3 S 10* 3 5 10" 3 5 10' 3 5 10* 3 5 10' TIME (HRS) (LOG SCflLEJ FIGURE 23 continued. 135 , ' ' ' ' ' " " i 1 I 1 1 ""I ' I 1 I ""I • I ' I MM| r | 11 " i i | 1 | i | i i t i i 10-« 3 S 10-' 3 S 10* 3 5 10' 3 5 10' 3 5 10" 3 5 n . TIME (HRS) (LOG SCALE) FIGURE 23 continued. 3000 PSI Applied stress Groups no. 8 & 2 9 1 6 1 (mm) 1 ' ' ' ' i • ' " " l 1 1 1 <" I ' 1 - | I | ITTT) r . T I I H i l l 3 5 10- 3 5 JO* 3 S JO' 3 5 JO' 3 5 10- 3 5 1 0 ' TIME (HRSJ (LOG SCflLEJ FIGURE 23 continued. 137 3000 PSI App l ied st ress G r o u p n o . 7 1 ' ' 1 1 I 1 I " " I ' I I | i n i | 1 | i I m i l r I i I run 1 i i m m 10- 3 S 10- 3 S T 10* 3 5 10' 3 5 0» 3 5 0. 3 5 0-TIME (HRS) (LOG SCALE) FIGURE 24-1. Creep curves for boards of sub-group 2 loaded to 3000 p s i . Richmond experiment. 138 3000 PSI A p p l i e d s t r e s s Group no 2 ^ 'j'j'ii ' m 'J'S1 z * m 'j'j'z 1 TIME (HRS) (LOG SCALE) 3 5 " 3 5 ,0' FIGURE 24.1 continued. 3000 PSI A p p l i e d s t r e s s Group no 2 TIME (HRS) (LOG scflLE) 5 ,0* 3 s FIGURE 24.1 continued. (mm) U J o o S i (mm) 140 to 4500 p s i . R l c h m o n d experiment, 3 0 0 0 p s i : 6 Q= 0. 4 37 8 • 1.07 3 • 6 p ( m m ) Rsq = 0.9127 x $ Q : 3000 PS) x ; 4500 PS] i 1 ~i i 1 i 1 0.0 10.0 20.0 30.0 40.0 50.0 60 PREniCTED ELASTIC DEFORMATION (MM) FIGURE 25. Actual e l a s t i c deformations versus predicted e l a s t i c deformations. 2 stress l e v e l s . Data from Richmond experiment. o 0.10 0.12 0 . U 0.1 6 0.1 8 0.20 0 2 2 0 24 0 2 6 0.2 8 0.30 M O D U L U S O F E L A S T I C I T Y ( p s i » 1 0 7 ) FIGURE 26. Cumulative probability 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. Hemlock. o 0 10 0 . 12 0 U 0 1 6 0 .18 0 20 0 .2 2 0 24 0 . 2 6 0.28 0 30 M O D U L U S O F E L A S T I C I T Y ( p s i « 1 0 7 ) FIGURE 27. Cumulative probability 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. Amabilis f i r . 144 0 . 0 10.0 £ E I — O L U _ J Ll_ L U Q a. U J L U or o 3.0 13.0 0.0 13.0 0.0 10.0 0.0 13.0 — I 10.0 t r 1 00 8 h r s ( 6 w e e k s ) oft ec % O X « 1 s o 0 0.0 13.0 20.0 30.0 ... 4.0 50.0 CO.O t = 504 hrs ( 3 wee Us) - ... » « * ~ © X K X °6 (^oo?' ,o o 20.0 30.0 40.0 50.0 60.0 t = 96 h r s X ** X X K K * —I •—I 1 1 20.0 30.0 40.0 50.0 t = 48 hrs —r-20.0 i • 40.0 t = 1 0 h r s 30.0 40.0 SO.O 60.0 . 2 0 . 0 t= 1 hr X &n< x 0.0 10.0 B-i R-30.0 40.0 50.0 60.D T i m e t = 5 mn 30.0 40.0 50.0 60.0 0 t= 1512 h r s (9 w e e k s ) 1 o o 0 0 ? i % X OO ^  o da - 1 — 20.0 30.0 40.0 50.0 60.0 t = 20 1 6 hrs ( 3 m o n t h s ) 47 V o <P o j p o o nO 3 eo.o o.o IO.O 20.0 40.0 FO.O SO.O t= 21 84 hrs ( 13 wee ks) o • 3000 PSI X 1 4300 PSI s a a - " x x 3 CT JtXx o „o_x*x j o "^6 o 10.0 20.0 I— 40.0 ^ E L A S T I C D E F O R M A T I O N ( m m ) FIGURE 28. Creep de f l e c t i o n , measured at 10 d i f f e r e n t times, versus e l a s t i c deformation, for 2 stress l e v e l s . Raw data. a a a — i a . 03 3 I o L U L U a r v i Board no. M O E . ( x 1 0 7 PSI) S.R. (%) 2509 0.1 7 8 6 3 85 . 6 2 238 0,1 8611 2 52 2 0.17772 4 4.7 2497 0.24174 1 6 53 0.1 72 62 -/ 3 6 m m 2509 ( 3000 PSI) 2238 ( 3000 PSI) 2 5 2 2 ( 3 0 0 0 PSI) 2497 (4500 PSI) 1653 ( 3000 PSI) a < I 1 I 1 "'I I I 1 I • • *"I 1 I I I M i l | 1 I I I M l l | 1 l l l l l l l | 1 I I 1 I l l l [ 10-' 3 5 10-' 3 5 10° 3 5 10' 3 5 10' 3 5 10* 3 5 10* TIME (HRS) ( L O G SCflLEJ FIGURE 29.1. Creep curves for boards with same value of e l a s t i c deformation (36 mm). U l o a. 0 3 3 C o Board no. M.O.E. (*107 PS!) SR. (7o) 2 363 0.1643 6 3 5. 9 2 S84 017134 59. 5 2 82 6 0.23949 -ID L U Q 3 8 m m 2363 ( 3000 PSI) 2884 (3000 PSI) 282 6 (4500 PSI ) o o — i — | H 1 " r l 1—I H 11"| 1—| r| int| 1—| i 11ni] 1—| i 11 inr 10-* 3 5 10-» 3 5 10° 3 5 10' 3 5 10* 3 5 10' TIME (HRS) (LOG SCALE) ' i i i "ni 3 5 10* FIGURE 29.2. Creep curves for boards with same value of e l a s t i c deformation (38 mm). o — a. cc c_ ) a L U -U _ L U a o a a ' 10 Board no. MOE. (*107 PSI) SR. (%) 1973 0.14893 32.8 555 0.1 5825 50. 2 13 32 0.1 551 2 2489 0. 23 473 — 4 1 0 0 22932 -4 1 mm 1973 (3000 PSI ) 555 (3000 PSI) 1332 (3000 PSI) 24 89 ( 4500 PSI) 41 0 ( 4500 PS I ) 1 | i | mi | 1—| i | T f i • | 1—| i | in»| 1—| n; '"H 1—| H ""1 1 — I 1 "1' M11 * 3 5 10-> 3 5 10° 3 5 10' 3 5 10* 3 5 10* 3 5 104 TIME (HRS) (LOG SCALE) FIGURE 29.3. Creep curves for boards with same value of e l a s t i c deformation (41 mm). o—• o . LU -L U O o . ro a a Board no. MOE. (*107 PSI) S.R. (%) 22 1 0 2 5 4 9 0.1 4 580 0.22 66 4 74.7-42 mm 2210(3000 PSI) 254 9 (4 500 PSI) — I — | i-| Tnq 1—| H iur| 1—| i 11 ni| 1—| n " , T1 1—I ' \ ""I 1—| H 1 "H 10-* 3 5 10-> 3 5 10° 3 5 10' 3 5 10* 3 5 10» 3 5 10» TIME (HRS) (LOG SCALE! FIGURE 2 9.4. Creep curves for boards with same value of e l a s t i c deformation (42 B o a r d no. M.O.E. (*10 7 PSI) S. R. (%) 19 A 7 O.U 5 2 9 29.7 o a _ 03 1602 0. 21 890 — H o '— ID LU -LL. LU a a a . CM 43 mm 194 7 (3000 PSI) 1602 (4500 PS I ) io-* - i — | ' i TTiiq 1—i i~| i n i | 1—| i | 1—| l | MIT[ 1—| i 11 n i | 1—i H in-q 3 5 10-> 3 5 10° 3 5 10" 3 5 10* 3 5 10» 3 5 10d ' TIME (HRS) (LOG SCALE) FIGURE 29.5. Creep curves for boards with same values of e l a s t i c deformation (43 o a . a a . OD 2 E o L U I L i_ L U a 9 -a a ' Board no. M O E . (K1 0 7 PS I ) S. R. (V.) 1969 0.21 544 22 07 0 1314 4 58. 3 21 4 7 0. 2 1 3 6 4 2 6 0 0 0.20568 2067 0.21 1 57 2 838 0.21513 932 0.216 65 -45 mm 10-* FIGURE 29.6 1969 (4500 PSI ) 2207 (3000 PSI) 214 7 ( 4 500 PSI ) 2600 (4500 PSI) 2 06 7 ( 4500 PSI) 2 838 (4500 PS I) 9 3 2 (45 00 PSI) | i | r T111 1—| i | nn| 1—| i | nif| 1—| r\ nnj 1—| i T mi] 1—| i 11 nq 3 5 10-' 3 5 10° 3 5 10' 3 5 10* 3 5 10» 3 5 10* TIME (HRS) (LOG SCALE) . C reep c u r v e s f o r boa rd 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 (45 mm) O t-100 8 hrs I 6 w e e k s ) 151 P-8T R-8) O . * 6 o » • » » T e x « * - i — 30.0 t= 504 hrs ( 3 weeks ) oe»" «x * eo JB"^ 1 o x ^ " O O » „ X X O 0 * 8 O —1 20.0 30.0 40.0 SO.O 60.0 -I 10.0 t = 1 5 1 2 hrs ( 9 weeks ) a 0 " 1 8 • -o0 ~I— :o.o - i — 40.0 50.0 60.0 t = 2016 hrs ( 3 mont hs) R-D_ LU LU or o 0 — R - i I — < :-or 0 U- _ A — i — 10.0 — I — 10.0 - I — 13.0 t = 96hr s X* X 10.0 20.0 so.o a.o t= 48 hrs o o «f»o0 » , &'ff»Vfc xx «x X X X X x 13.0 30.0 0.0 40.0 t = 10hrs —r 20.0 90.0 . 40.0 t = 1 hr ) SO.O 20.0 X.O 40.0 SO.O O.O Time t = 5 mn 20.0 SO.O 40.0 SO.O 60.0 8 -60.0 0.0 0 o S x "„ 8 „ x * * —I 20,0 —\ SO.O 50.0 60.1 Q t MOD K l „ t 4500 PSI t = 2184 hrs ( 13 w eeks ) oP X I x ,«6"<5 f S ^ B X ^ x — 1 — 10.0 1 30.0 40.0 50.0 ^ . E L A S T I C DEFORMATION (mm) FIGURE 30. Fractional creep expressed at 10 d i f f e r e n t times versus e l a s t i c deformation, for 2 stress l e v e l s . Raw data. < z o CL Ixl LU CH * <_> < cr a" 9 X ° M « t = 50 A hrs ( 3 weeks ) e « _ K •> o a e a» a a a — I — 0.13 —> 3.16 — I 0.16 — I — 1.2 — I 0.22 I O.M — i — 0.26 —I 0.21 X * « as » i " x t = 1008 hrs ( 6 weeks) « X o « •*> * « ! • % & 4 | # ^ a" a" 0 •„ a ~ a"a«61a-ea* a" « a e 1 1 1 1 1 P 1 1 r-o.i o.i2 o.i4 o.ie a.it 0.2 0.22 0.24 0.26 2.2 t: 96 hrs o'.io . M (Lo o>t Zn Zn OJ a t = 1512 hrs x ( 9 weeks ) t= 48 hrs •a a — 1 1 1 1 1 p 1 1 1 1 0.1 0.12 0.14 J.l» 0.10 0.2 0.22 0.24 0.» 0.21 0.1 t= 1 0 hrs T 1 -— i i — ^ 1 1 r~ 0.1 0.12 0.14 0.16 0.(1 0.2 0.22 0.24 0.26 0.1 * a * " • 1 _ a , % /•/ * aa 0 8 M B a a • _ a o i oT»a I u 4 oT» iui o^ a 0T22 Zu iTa 1.1 t= 1 hr a *a •»•« 1 1 | M I H — l a ••» aw "na-) a • , , 0.1 0.12 0.14 0.11 0.11 0.2 0.22 0.24 0.21 0.21 0.1 lime t: 5mn 9 , " ~,«mr*»^v.w .u*. •, , B 0.1 0 12 0.14 i II 0.11 0.2 0.22 0.24 0.20 0.20 3.1 M.0. E. ( psi * 10 7 ) t = 2184 hrs ( 13 weeks ) a - i — 1 1 1 1 1 1 r 1 1 01 0.12 . il.)4 0.16 0.1» 0.2 O.a 0.24 0.21 0.21 1.1 FIGURE 31. Fr a c t i o n a l creep, expressed at 10 d i f f e r e n t times versus modulus of e l a s t i c i t y , for 2 stress l e v e l s . Raw data. Ul to 153 6 0 %<S.R.< 100 % c E IT) 5 0 %<S.R.< 6 0 % D_ LU U J rr -i r O ©DO - i B r -0.1 0.12 0.14 0.16 0.18 0.? 0.72 0.24 0.76 0.76 0.3 4 0°/o<S.R.< 5 0 % 0.1 0.12 . 0.14 ~1 1 1 0.16 0.2 0.72 0.24 0.26 0.78 0.3 I — O < rr LL 30%<S.R.< A 0 % 0 O O QD -i 1 r -e*a q r-0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 0.26 0 %<S.R.< 3 0 % — i — 0.12 M.O. E . ( ps i 1 0 FIGURE 32.1. Fractional creep plotted against the modulus of e l a s t i c i t y for s p e c i f i c ranges of stress r a t i o s , at time= 5 minutes. 154 60 %<S.R.< 100 % 1 1 1 1 — i 1 1 1 1 1 0.1 0.12 0.14 0.16 0.10 0.2 0.22 0.24 0.2S 0.26 0.3 O B-Q_ LU LU CC 5 0 %<S.R.< 6 0 % e a °o - I 1 1 ° i °—<«i " i 1 1 1 1 0.1 0.12 0.14 0.16 0.1B 0.7 0.72 0.74 0.76 0.76 0.3 4 0 °/.<S.R. < 50 7o O O o 1 1 1 1 » I " " — I — 1 1 1 1 0.1 0.12 0.14 0.16 0.16 0.2 0.22 0.24 0.26 - 0.26 0.3 I— O < * CC LL. A 3 0 % <S.R. < 40 V. o 0 a 1 1 r « — " — i = * i — " 1° i 1 1 1 0.1 0.12 0.14 0.16 0.10 0.2 0.72 0.24 0.26 0.20 0.3 0 %<S.R. < 3 0 7o »1 * — » I 1 0.1 0.12 0.14 0.16 0.16 0.2 0.2} 0.24 0.2S 0.20 0.3 B a ! — » rn » , i i — a _ ^ M . O . E. ( p s i x 1 0 7 ) FIGURE 32.2. Fractional creep plotted against the modulus of e l a s t i c i t y for s p e c i f i c ranges of stress r a t i o s , at time= 1 hour. 155 6 0 %<S.R.< 1 00 % o o ° LU LU CC O I < 5 0 %<S.R.< 6 0 % o 0 0 1 1 1 1 * l 1 1 1 1 1 0.1 0.12 0.14 0.16 0.18 0.? 0.72 0.24 0.26 0.28 0.3 4 0 %<S.R.< 50 % © © o e feo Q B 1 1 1 1 1 1 1 1 1 I 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 0.26 0.28 0.3 I— O e < * Ct L L _fl a_ 3 0 % <S.R< 4 0 % o - i 1 r — — " — i 1 I s 1 1 1 1 0.1 0.12 0.14 0.16 0.18 .0.2 0.22 0.24 0.26 0.20 0.3 0 % <S.R.< 3 0 % o „ ° ° e „ 8 o, „ . . 1 i 1 1 i 1 1 1 1 1 0-1 0.12 3.14 0.16 0.16 0.2 0.23 0.24 0.26 0.21 0.3 _^ M. 0. E. ( ps i x 1 0 7 ) FIGURE 32.3. Fract i o n a l creep plotted against the modulus of e l a s t i c i t y for s p e c i f i c ranges of stress r a t i o s , at time = 10 hours. 156 6 0 ° / ,<S .R .< 100 % Ul » 1 1 1 I 1 1 1 1 1 1 1 0.1 0.12 0.14 0.16 0.16 0.2 0.22 0.24 0.26 0.26 0.3 CO 5 0 % < S . R . < 60 % 0_ LU UJ cr o < o —i 1 i 1 1 1 1 1 r— -i 0.12 0.14 0.16 0.16 0.2 0.22 0.24 0.26 0.26 0.3 4 0 % <S.R. < 50 % 1 1 1 1 1 1 1 1 1 1 0.1 0.12 0.14 0.16 0.16 0.2 0.22 0.24 0.26 0.2B 0.3 3 0 %<S .R .< 4 0 % < cr. L L A — i — 3.12 — r — 0.14 - | 0.16 O 0 ° ' —I 0.24 —I 0.26 0 ° /o<S .R .< 30 % a e 8 • a • a p. a a a a —i 1 1 — " 1 1 i 1 1 1 1 0.12 0.14 0.16 0.16 0.2 0.22 0.24 0.26 0.20 0.1 _^ M . O . E . ( p s i x 1 0 7 ) FIGURE 32.4. Fractional creep plotted against the modulus of e l a s t i c i t y for s p e c i f i c ranges of stress r a t i o s , at time = 2 days. 157 6 0 7.<S.R. < 10 0 % 1/1 — i 1 1 1 1 1 1 1 1 1 0.12 0.14 0.16 0.1B 0.2 0.22 0.24 0.26 0.28 0.3 O l 5 0%<S.R. < 60 % © © a • i 1 1 1 1 1 1 1Z. ? > 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 0.26 0.28 0.3 Q . LU LU CC O < z o „ — i-i — o a CC 4 0%<S.R. < 5 0 % T3 o oo 3 0 %<S.R. < 40 % 0 °/.< S.R. < 3 0 % » ©„ 0 ° „ 8 © o © — i 1 1 1 1 1 1 1 1 0.14 0.16 0.16 0.2 0.23 0.24 0.26 3.26 0.3 ^ M.O.E. (ps i x l 0 7 ) FIGURE 32.5. Frac t i o n a l creep plotted against the modulus of e l a s t i c i t y for s p e c i f i c ranges of stress r a t i o s , at time = 4 days. 158 © o B-6 0 7o<S.R. < 1 0 0 % —i 1 1 1 1 1 1 1 1 1 0.12 0.14 0.16 0.16 0.2 0.?? 0.24 0.76 0.26 0.3 5 0 ° / . < S . R . < 6 0 % R-0.1 0.12 0.14 —I 1 1 — 0.16 0.16 0.2 I I I I 1 0.22 0.24 0.76 0.2S 0.3 in 2 4 0 % < S . R . < 5 0 % e 8 ° "S. 0.1 _ _ 0 . 1 2 — 1 0.14 ~I 0.16 —I 0.2 l> X 0.26 0.3 Q_ UJ LU cr 30 %<S .R .< 4 0 % 0 oo „ i— < cr (JL A — I — 0.12 —1 1 1 1 1 0.14 0.16 2.16 0.2 0.22 "I 1 r 0.24 0.26 0.; 0 % < S . R . < 3 0 7 . R-°° e o o e IT i 1 1 1 1 1 1 i 1 1 0.1 0.12 0.14 0.16 0.16 . 0.2 0.22 0.24 0.26 0.26 0.3 ^ M. 0. E. ( p s i x 1 0 7 ) FIGURE 32.6. Fractional creep plotted against the modulus of e l a s t i c i t y for s p e c i f i c ranges of stress r a t i o s , at time = 3 weeks. l/J o o CL 8i LU UJ CC 9 i o < o I — o < CC LL A ^ 3 0 %<S.R.< /,0 V. an - 1 — 0.12 I 1 1 1 1 I 1 1 014 0.16 0.16 0.2 0.22 0.24 0.26 0.20 0.1 °. R 0 % <S. R.< 3 0 % — i 1 1 1 1 1 1 1 1 0.12 0.14 0.16 0.10 0.2 0.22 0.24 0.26 0.26 4 0%<S.R.< 50% o e e a o'.i2 TAA oTii TAB 0T2 0T22 0T24 ' o'.ie 0T20 0.1 50%<S.R.< 60% a •> _o'.i 0TT3 77* Zit 7A» oTn OTM oTJs tTm 0.1 60%<S.R.< 1 00% —1 1 1 1 i 1 1 1 r 1 0.12 0.14 0.1B 0.18 0.? 0.22 0.24 0.28 0.28 0.3 J> M. 0. E. (psi x 1 0 ) FIGURE 32.7. Fractional creep plotted against the modulus of e l a s t i c i t y for s p e c i f i c ranges of stress r a t i o s , at time = 6 weeks. — S i D_ LU LU 4-LT. O * i 40 %< SR. < 50 7, — i 1 1 1 1 1 1 1 1 0.14 0.16 0.16 0.2 0.22 0.24 0.26 0.26 0.3 R i 30 7. < S.R. < 40 7. 0.1 0.12 —I 1 1 1 1 1 1 1 I 0.14 0.16 0.16 0.2 0.22 0.24 0.26 0.20 0.3 5 0 7,<SR < 607. a ° a a —i 1 1 1 1 1 1 1 1 n 0.12 0.14 0.16 0.18 0.2 0.22 0.24 0.26 0.28 0. < LL. A «i R i 0 7. <S.R. < 307. R i 6 0 7.<S.R. < 100 7. o.i o^ ia o^  Z* Zit Zi Zn Z» Zx Zn 0.3 "Zi Zii Zi* oae ZTa Zi Zn Zu Zx Zx 0. > M . O E . ( ps1 v 10 7) FIGURE 32.8. Frac t i o n a l creep plotted against the modulus of e l a s t i c i t y for s p e c i f i c ranges of stress r a t i o s , at time = 9 v/eeks. • • o» AOVo<S.R.< 5 0 V , 30 % < S.R. < AO % 7!i2 77A O'.IB o':u 77i 7.a q'.M o'.» o.a 0.3 "i 1 GO O ~- 1 r ~& oTia oTn 0T24 IKS oTa 0.3 5 0 % < S . R . < 60 % 1 1 1 1 1 1 1 1 1 1 I 0.1 0.12 0.14 0.16 0.1B 0.2 . 0.22 0.24 0.28 0.28 0.3 0 V. <S.R. < 30 V. si 60 V. <S.R. < 1 0 0 % Zi 77, 77* 77> 77t °t 772 77A 77* 7\T Ii 7^ 77* 77,7, \> M O E . (psi x 1 0 ) 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 s p e c i f i c r anges o f s t r e s s r a t i o s , a t t ime = 3 months. i—• H 1 8 a •• 4 0 % < S . R . < 5 0 % « « 0 si — i — 0.13 L » 1^16 TAB o l o!?? ir?4 o1^ O'.TB . o.s 3 0 °/.< S.R.<40% si o o 8i — i — 0.12 —1 1 1 1 1 1 1 0.1B 0.3 3.22 0.24 0.25 0.28 0.3 0 %< S.R.< 3 0 % i 1 1 1 1 1 1 1 1 1 i 0.1 0.12 0.14 0.16 0.10 0.2 0.22 0.24 0.26 0.28 0.3 _>M .0.E. ( psi * 10 7 ) 50 % < SR.< 60 % —i 1 1 1 1 1 1 1 1 ~ i 0.12 0.14 0.16 0.18 0.2 0.22 0.24 0.26 0.20 0.3 6 0 %<S.R.<100 % o'.i2 0T14 0T16 OTIB 0T2 0T22 0T24 O~K 0T20 O.S FIGURE 32.10. Fractional creep plotted against the modulus of e l a s t i c i t y for s p e c i f i c ranges of stress r a t i o s , at time = 13 weeks. 163 A P P L I E D S T R E S S = 3 0 0 0 p s i I M E 3 m o n t h s 9 w e e k s 6 w e e k s 3 w e e k s 4 d a y s 2 d ay s 10 h o u r s 1 hour 5 m inu te s 10.0 20.0 E L A S 30.0 40.0 50.0 60.0 C D E F O R M A TI 0 N ( mm ) FIGURE 33. 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 A P P L I E D S T R E S S = 4 5 0 0 p s i IM E 3 m o n t h s 9 w e e k s 6 week s 3 weeks 10.0 2 0.0 E L A S T I C 30.0 40.0 D E F O R M A T I O N 50.0 ( mm ) FIGURE 34. 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 for 4500 p s i . A P P L I E D S T R E S S = 3 000 p s i 0.0 10.0 20.0 E LAST I C T IME 3 m o n t h s 9 week s 6 w e e k s 3 w eek s 1 5 30.0 40,0 50.0 D E F O R M A T ION ( m m 60.0 days days 10 h o u r s h o u r m i n u t e s FIGURE 35. Fractional 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 for 3000 p s i . FIGURE 36. Fractional 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 for 4500 p s i . A P P L I E D S T R E S S = 3 0 0 0 p s i T IME 3 m o n t h s 9 w ee k s 6 w ee k s 3 w e e k s 4 d a y s 2 day s .10 h o u r s 1 h o u r 5 mn 0 .10 0.12 0 . U 0.16 0.18 M O D U L U S O F E L A S T I C 0.2 0 0.2 2 0.2 4 I T Y ( p s i * 1 0 7 ) 0.2 6 0.28 0.30 FIGURE 37. Frac 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 for 3000 p s i . A P P L I E D S T R E S S = 4 5 0 0 p s i "IME 3 m o n t h s 9 w e e k s 6 wee ks 3 week s 4 day s 2 day s 10 h o u r s 1 h o u r 5 mn 0.10 0.12 0.14 M O D U L U S 0.1 6 O F 0.18 E L A ST I C 0.20 TY 0.22 p s i * 0.24 1 0 7 ) 0 2 6 0 .28 0.30 FIGURE.38. Fr 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 for 4500 p s i . 169 D_ LU LU CC O *1 t= 1008hrs ( 6 w e e k s ) #a a e B • * a a I I I 1 49.0 00.0 00.0 100.0 t = 50 4 hrs ( 3 week s ) * * a 9 A* • OJ a S e a e _ . r i 1 1 1 3.0 30.0 40.0 00.0 80.0 !"» ' t = 96hrs . • o a at? % l a | 1 1 1 0.0 30.0 40.0 80.0 00.0 100.0 t= 48 hrs 8 o.o ao.o 4D.o ao.o m.o IOO.O 8 . • t= 1512 hrs (9 weeks) a * . e ' V . »* a e if I 1 1 I 1 3.0 30.0 40.0 00.0 00.0 100.0 t= 201 6 hrs ( 3 months) aa _ e a ee »oi « * Oo* e e oo a a a a i. i i 1 1 1 30.0 40.0 oo.o ao.o IOO.O < Z o < cc t = 10 hrs - r 30.0 4D.o ao.o ao.o 10.0 t = 1 hr o.o 30.0 40.0 ao.o ao.o ioo.o Time t= 5 mn i . o 40.0 n . o ao.o ioo.o S i STRESS RATIO (V.) t = 21 84 hrs ( 1 3 weeks ) o a a 1 1 1 1 : 3.0 30.0 40.0 ro.o oo.o ioo.o (I) FIGURE 39. 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 , f o r 1 s t r e s s l e v e l : 3000 p s i . Raw data. 170 A P P L I E D S T R E S S E S . 3000 ps i , 4 500 ps i M E 0.0 1 0.0 2 0.0 3 0.0 E L A S T I C D E F O R M A ' 4 0.0 5 0.0 ION ( m m ) 3 m o n t h s 9 w e e k s 6 w e e k s 3 week s 4 days 2 day s 10 hour s 1 hour 5 m inu te s 60.0 FIGURE 40. 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 points for 3000 and 4500 p s i . A P P L I E D 3 0 0 0 ps i , A 5 00 p s i 10.0 2 0.0 E L A S T I C T I M E 13 w e e k s 3 mon th s 9 w e e k s 6 w e e k s 3 w e e k s A d a y s 2 d ay s 10 hours 1 hour 5 m i n u t e s 30.0 AO.O 50.0 60.0 D E F O R M A T I O N ( mm ) 41. Frac 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 points for 3000 and 4500 p s i . A P P L I E D S T R E S S E S . 3 0 0 0 p s i , A 5 00 p s i T I M E 0.10 0.12 O U 0 1 6 M O D U L U S 0.1 8 0.20 0 OF E L A S T I C I T Y 0.28 3 months 9 week s 6 w e e k s 3 w e e k s 4 days 2 d a y s 10 hour s 1 h o u r 5 mn ( p s i » 1 0 ' ] FIGURE 42. Fract 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 points for 3000 and 4500 p s i . ^ to APPENDIX A - l MATHEMATICAL TREATMENT OF CREEP by BMD:02R COMPUTER PROGRAM. CREEP DEFORMATION INITIAL VARIABLES: XI = 6 c X2 = SR X3 = MOE X4 = <S e ADDED VARIABLES: X5 = X2 X3 X6 = X2 X4 X7 = X3 X4 X8 = X 4 2 X9 = X2 X 4 2 X l O = X3 X 4 2 1 BMD02R - STEPWISE REGRESS I ON - REVISED MARCH 27, 1973 HEALTH SCIENCES COMPUTING FACILITY, UCLA 4 5 6 ~T~ 8 9 T O 11 12 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 13 14 15 16 ' 17 VARIABLE MEAN STANDARD DEVIATION 18 1 14.40954 5. 72 505 19 2 41.34435 14.25918 20 3 19.73979 2.90453 21 4 29.54172 4.4892 f 22 S 797.24268 242.95103 23 6 1247.42139 543 .12256 24 7 570.56250 6.60479 25 8 892.55151 280.28857 26 9 38453.046 88 21553.41797 27 13 16849.933 59 2527.26147 28 1 CORRELATION MATRIX 29 0 VARIABLE 1 2 3 4 5 6 7 8 9 13 30 NUMBER 31 32 1 1 .000 0.4019 -0.7358 0.7785 0.9591E-01 0.5923 -0.22 66 0.7917 0. 6856 0.7693 33 2 1. 000 -0.4635 0.4133 0.9041 0.9431 -0.3542 0.3944 0.8377 0.3909 34 3 1.000 -0.9810 -0.7057E-01 -0.6913 0.2557 -0.9597 -0.7880 -0.9732 35 4 1.000 0.4453E-02 0.6713 -0.1897 0.9954 0.7978 0.9972 36 5 1.000 0 .7135 -0 .2 7 75 -U.1590E--01 0.5338 -0.1651E-C1 37 6 l.OOu -0.34 79 0.6615 0.9708 0.6521 38 7 1.000 - 0 . 1897 - 0 . 3204 -0.1162 39 8 i.ooo 0.7994 0.9925 40 9 1.000 0.7819 41 10 1.000 42 1 SUB-PROBLM 1 43 DEPENDENT VARIABLE 1 44 MAXIMUM NUMBER OF STWS 20 45 F-LEVEL FOR INCLUSION 3.0C0000 46 F-LEVEL FOR DELETION 3.000000 47 TOLERANCE LEVEL 0.001000 48 49 50 51 52"" ~STEP NUiBER 1 53 VARIABLE ENTERED 8 54 55 MULTIPLE R 0.7917 56 STD. ERIOR (JF EST. 3.5264 57 ~ 58 ANALYSIS OF VARIANCE 59 DF SUM JF SQUAKES MfcM SQUARE F RAT 1 0 60 REGRESSION 1 1273.578 12 73.57a 1C2.417 f 6 1 6 2 6 3 R E S I D U A L 6 1 7 5 8 . 5 4 8 1 2 . 4 3 5 6 * 6 5 6 6 VARIABLE V A R I A B L E S IN E Q U A T I O N COEFFICIENT STO. ERkOR F TO REMOVE VARlAbcE VAKIAuLEi NUT PARTIAL CORK. IN E Q U A T I O N TOLERANCE f TO ENTER , 6 7 6 8 6 9 ( C O N S T A N T - 0 . 0 2 3 0 6 I • "•< 7 0 ~ 7 1 7 2 8 0 . 0 1 6 1 7 0 . 0 3 1 6 0 1 0 2 . 4 1 6 9 ( 2 ) . 2 3 4 0 . 1 5 9 6 8 0 . 1 3 9 6 0 - 0 . 1 6 3 2 2 0. 8 4 4 5 0 . 0 7 8 9 0 . 0 0 9 1 1 . 5 6 9 9 ( 2 ) 1 . 1 9 2 5 ( 2 1 1 . 6 4 2 3 ( 2 ) 7 3 7 4 7 5 * 5 6 7 0 . 1 7 7 O 0 0 . 1 4 * 7 7 - 0 . 1 2 7 3 4 0 . 9 9 9 7 0 . 5 6 2 5 0 . 9 6 4 0 1 . 9 5 4 3 ( 2 1 1 . 3 7 6 7 1 2 ) U . 9 B 9 0 ( 2 1 7 6 7 7 7 8 * 9 1 0 0 . 1 4 3 5 5 - 0 . 2 2 0 0 4 0 . 3 6 0 9 0 . 0 1 4 9 1 . 2 6 2 3 ( 2 ) 3 . 0 5 2 9 ( 2 1 7 9 8 0 8 1 8 2 8 3 8 4 STEP N U M B E R 2 V A R I A B L E E N T E R E D 1 0 8 5 8 6 8 7 M U L T I P L E R S T O . E R I O R O F 0 . 8 0 3 0 E S T . 3 . 4 6 8 5 8 8 8 9 9 0 A N A L Y S I S O F V A R I A N C E O F S U N OF S Q U A R E S M E A N S Q U A R E R E G R E S S I O N 2 1 3 1 0 . 3 0 5 6 5 5 . 1 5 3 F R A T I O 5 4 . 4 5 8 9 1 9 2 9 3 R E S 1 0 U A L 6 0 7 2 1 . 8 2 1 1 2 . 0 3 0 9 4 9 5 9 6 V A R I A B L E V A R I A B L E S I N E Q U A T I O N C O E F F I C I E N T S T O . ERROR F TC REMOVE V A R I A B L E V A R I A B L E S NOT PART I A L C O R K . I N E Q U A T I O N T O L E R A N C E F TO E N T E R 9 7 9 8 9 9 ( C O N S T A N T 2 2 . 1 0 4 7 1 1 \ 1 0 0 1 0 1 1 0 2 8 1 0 0 . 0 3 8 5 2 0 . 0 1 2 8 9 8 . 9 3 5 1 ( 2 1 . - 0 . 0 0 2 5 0 0 . 0 0 1 4 3 3 . 0 5 2 9 12 I . 2 . 3 4 0 . 1 6 2 7 1 0 . 0 0 9 3 4 0 . 0 1 7 8 0 0 . 8 4 4 5 0 . 0 5 0 4 0 . 0 0 3 4 1 . 6 0 4 4 ( 2 1 0 . 0 0 5 2 ( 2 1 0 . 0 1 8 7 ( 2 ) 1 0 3 1 0 4 1 0 5 • 5 6 7 0 . 1 8 0 7 3 0 . 1 4 2 8 1 0 . 0 0 6 6 1 0 . 9 9 9 7 0 . 5 6 1 2 0 . 6 1 4 5 1 . 9 9 2 1 ( 2 1 1 . 2 2 8 4 ( 2 1 0 . 0 0 2 6 ( 2 1 1 0 6 1 0 7 1 0 8 • 9 0 . 1 1 3 1 0 0 . 3 5 2 0 0 . 7 6 4 5 ( 2 1 1 0 9 1 1 0 1 1 1 F - L E V E L OA T O L E R A N C E ( N S U F ^ I C I E N T FOR F U R T H E R C O M M U T A T I O N 1 SUMMARY T A B L E 1 1 2 1 1 3 1 1 4 S T E P NUMBER V A R I A B L E M U L T I P L E E N T E R E D R E M O V E O R RSQ INCREASE F IN RSC CM VALUE TO ER OK R E M O V E NUHbEK OF I N O E V A R I A B L E S Ih 1 1 5 1 1 6 1 1 7 1 2 8 0 . 7 9 1 7 0 . 6 2 6 7 1 0 0 . 8 0 3 0 0 . 6 4 4 8 0 . 6 2 6 7 1 0 2 . 4 1 6 9 O.Oldl 3 . 0 5 2 9 1 2 1 1 8 1 1 9 1 2 0 1 L I S T O F R E S I D U A L S " ' " ' " " " C A S E Y Y f 121 lot N U M B E R XI 11 C O M P U T E D R E S I D U A L X( 81 X( 10 1 lc£ 123 1 32. 0000 25.5395 6.4605 1514.5242 22295.1C16 124 2 ' 21. 0000 18.8129 2.1871 1169.2295 19354.2461 125 3 12. 5790 12.8225 -0.2435 846.5190 16775.4688 <• 126 4 14.0000 1C.8858 3.1142 755.9746 161*4.4219 -? 127 5 7. 5800 9.3148 -1.7348 521.4375 13165.7695 128 6 18.5820 15. 2861 3.2954 964.7859 1 7613.1250 129 7 16.0000 17.7308 -1.7308 110 7.62 52 1HH37. 3628 130 ""8 12.5830 11.0126 6 47.2441 " 14426.4219 131 9 14. 5840 13.9572 0.626(1 682.5059 16876. 1602 132 10 19.5860 19.1C98 0.4762 1185.7690 19490.4805 153 1 1 11.0000 10.0880 0.9120 663.6802 15050.2734 134 1 2 11.0000 10.6657 0.3343 671.7947 14944. C664 135 13 14.0000 13.4317 0.5683 946.8540 18079.2227 136 14 " 30. 6400 24.0110 6.62 90 1433.8337 21 354.0820 137 1 5 12.0000 1C.5651 1.4349 715.0278 15651 .2461 138 1 6 14.0000 13.0258 0.9742 878.1741 17182.34 77 139 1 7 11.0000 13.9868 -2.9868 889.8879 16978.1641 140 1 8 11.0000 11.4386 -0.4386 697.4883 15030.8711 141 19 12. 0000 11.8870 0.1130 695.0603 14813.8203 142 2 0 13.6460 12.0334 1.6126 735.4402 15378.0547 143 2 1 12.0000 13.5486 -1.5486 838.3337 16358.4063 144 22 12.0000 10.8398 1.1602 706.9751 15417.0078 145 23 14. 0000 12.1641 1.8359 717.9187 15055.4648 146 24 13.0000 9.9072 3.0928 573.2668 13728.0234 147 25 11.0000 14.4913 -3.4913 928.9082 17378.0039 148 26 6.6540 10.9442 -4.2902 665.6912 14738.3984 149 27 13.6540 16.9747 -3.3207 U04.2993 19088.9102 150 28 16.6550 19.1341 -2.4791 1177.3130 19350.3203 151 29 25.0000 21.0990 3.9010 1288.7383 20282. 1602 152 30 26.0000 20.7952 5.2048 1312.6855 20 773.2 461 153 31 9.6590 9.5106 0.1484 523.3115 13116.2773 154 32 8.6600 18.2693 -9.6093 1194.8774 19967.5938 155 33 13.0000 13.0310 - C.0310 806.3896 16072.9609 156 34 9.0000 9.8922 -0.8922 622.5525 14494.2656 157 3 5 27.7lflo 32.8641 -5.1541 1888.7720 24826.0156 158 36 12.0000 11.0992 0.9008 636.9060 142 32.2969 159 37 21.7110 17.9612 3.7498 1161.6501 19578.4531 160 38 12.0000 13.3194 -1 .3194 816.6445 16115.6563 161 39 11.0000 10.5408 0.4592 627.2517 14306.9844 162 40 19. 0000 18.3726 0.6274 1134.4775 18994.5508 163 41 23.0000 25.2173 -2.2173 1474.4832 21497.9609 164 42 11.7180 10.2235 1.4945 559.3220 13366.2500 165 43 16.0000 19. 7644 -3.7644 1267.4309 20*88.0117 166 44 8.0000 12.8505 -4.8505 793.0415 159 39.3398 167 45 16. 7230 16.2267 0.4963 1003.2419 17d29.6172 168 46 23. 7240 14.2800 9.4440 920.2124 17328.5117 169 47 14.0010 17.4663 -3.4653 1087.2844 18o29.5J13 170 48 12.0000 11.9866 0.0134 754.5454 I5t.9l.6273 171 49 7.0000 10.9323 -3.9323 721, 4055 I5o02.!3i>25 172 50 9.0000 13.0595 -4.0595 824 ?236 16344.3672 173 51 9. ooor, 9. 5440 -0.5440 47^.4575 12364.'461 174 52 9.0000 14.3754 -5.3754 905 .1)281 17063. 7<S56 ITS 53 15.0000 12.7002 2.2998 770.9509 15658.Y8S2 176 54 12.0000 9.9469 2.0531 533.0554 13091 ,b359 177 55 10. 5100 13.6323 -3.1223 876.2190 16909.2 773 "178" 1 179 180 C A S E Y Y C T l ( 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) XI 10) 182 183 56 13.5110 13.5821 -0.0711 851.2974 1 0 5 4 4 . 9 6 0 9 " 1 8 4 " 5 7 6.0000 10.1448 -4.1448 578.7395 1 1717.^013 " ~ 185 58 16.0000 13. 7865 2.2135 914.C942 17431.7813 . 186 59 10. 5650 15.9597 -5.J94/ 1017.1636 18151.2852 ( 60 10.0000 12. 1646 -2.1646 729.0542 15227.0234 * 188 61 14.CC00 11. 1142 2.8854 687.2788 1 5 0 0 3 . 2 9 o 9 189 62 16.5680 11.7989 4.7691 780.7554 161 71.007b 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 I d l 4 9 . 3 i . 9 4 ' " " " 191 192 T93 " 194 FINISH CARD ENCOUNTEREO 195 PROGRAM TERMINATED END O f F I L E 1 7 8 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 = X4 2 X9 = X2 X4 2 XlO = X3 X4 2 I 2 3 1 BHD02R - STEPWISE REGRESSION - REVISED MARCH 27, HEALTH SCIENCES COMPUTING FACILITY. UCLA 1973 > -• 5 6 PROBLEM CQOE NUMBER OF CASES NUMBER OF ORIGINAL VARIABLES CRPDEF 91 4 7 8 9 NUMBER OF VAR IABLES ADDED TOTAL NUM1ER JF VARIABLES NJMBER OF SUB-PROBLEMS 6 10 1 — < 10 11 12 THE VAR 1 A3 LE FORMAT IS < 10X ,4F8.31-13 14 15 rt> 17 18 VARIABLE MEAN 1 17.15468 ST ANOARO DEV IAMON 7.52491 19 20 21 2 34.61537 1 19.90491 4 33.61629 6.96142 2.68173 7.57275 22 23 24 5 690.73291 6 1205.94775 7 659.26001 173.17403 506.50879 134.05516 25 26 27 8 1186.76831 9 44135.88672 13 22970.50391 541.76416 28639.96875 9687.78906 28 29 30 1 0 CORRELATION MATRIX VARIABLE 1 2 NUMBER 3 4 5 6 7 a 9 13 31 32 33 1 1 .000 0.5502 2 1.000 -0.5258 0.8102 0.9290E-01 0.8116 0.1673 0.85B2 0.7143 0.9520 0.54 91 0 .99 80 0. 8102 0.B191 0. 7481 0.9142 0.7147 0.9529 34 3 5 36 3 4 5 1.000 -0.4916 1.000 0.5a 29 0.4022 1.000 -0.1957 0.9480 0.6643 0.10 57 0.80 54 0.db34 -0.4539 0.9948 0.419a -0.2611 0.96 32 0.5924 -0.1880 0.9458 0.6691 37 38 3 9 6 7 8 1.000 0.9479 1.0 00 0.9547 0.8133 1.000 0.9924 0. 9098 0.9766 0.9994 0.9509 0.9527 40 41 42 1 9 10 SU B-PR0BL1 1 1 .000 0.9915 1.000 43 44 45 DEPENDENT VARIABLE 1 MAXIMUM HUMBER OF STEPS 20 F-LEVEL FOR INCLUSION 3.000000 46 47 48 F-LEVEL FOR DELETION 3.000000 TOLERANCE LEVEL 0.001000 4 9 50 51 52 5 3 54 STEP NU1BER 1 VARIABLE ENTERED 4 55 56 57 MULTIPLE R 0.8102 STO. ER<OR OF EST. 4.4355 58 5 9 60 ANALYSIS OF VARIANCE DF REGRESSION 1 SUM OF SQUARES 3345.24 8 MEAN SQUARE 3345.248 F RATIO 170.039 c 6 1 62 63 R E S I D U A L 8 9 1 7 5 0 . 9 3 9 1 9 . 6 7 3 > 65 66 V A R I A B L E V A R I A B L E S IN E Q U A T I O N C O E F F I C I E N T STD. ERROR F TO REMOVE VARIABLE VARlAULtS NO I P A R T I A L CORK. IN E Q U A T I O N T O L E R A N C E F TO ENTER . ? 67 68 69 ( C O N S T A N T - 9 . 9 0 9 1 2 1 * \ . . . . 7 Q 71 7 2 4 0 . 8 0 S 0 8 0 . 3 5 1 7 4 1 7 0 . 0 3 8 7 ( 2 ) . 2 3 5 -6 .3 l325~ - 0 . 2 4 9 8 5 - 0 . 2 9 5 6 1 0 . 3 4 1 4 0 . 7 5 8 4 0 . 8 3 8 2 9.57<rB 5 . 8 5 9 3 6 . 4 2 6 3 ( 2 1 (2) ( 2 1 73 74 7 5 6 7 8 - 0 . 2 8 8 5 1 - 0 . 2 9 7 9 2 0 . C 7 0 8 3 0 . 1 0 1 3 0 . 3 5 1 3 0 . 0 1 0 5 7 . 9 9 0 0 8 . 5 7 1 0 0 . 4 4 3 7 (21 (21 (2) 76 77 78 • 9 1 0 - 0 . 2 0 4 3 9 - 0 . 2 7 1 3 8 0 . 0 7 2 3 0 . 1 0 5 4 3 . 8 3 6 4 6 . 9 9 6 3 (21 (21 79 80 81 82 83 84 S T E P NUiBER 2 V A R I A B L E E N T E R E D 2 85 86 8 7 M U L T I P L E R 0 . 8 3 0 7 S T O . ER3.0R OF E S T . 4.2361 88 89 90 A N A L Y S I S O F V A R I A N C E D F SUM OF S Q U A R E S R E G R E S S I O N 2 3 5 1 7 . C 6 4 MEAN S Q U A R E 1 7 5 6 . 5 3 2 F RATIO 9 7 . 9 9 8 91 92 93 R E S I D U A L 88 1 5 7 9 . 1 2 3 1 7 . 9 4 5 94 95 96 V A R I A B L E V A R I A B L E S I N E Q U A T I O N C O E F F I C I E N T S T D . ERROR F TO REMOVE V A R I A B L E V A R I A B L E S NOT PART 1 A L C O R R . I N E Q U A T I O N T O L E R A N C E F TO E N T E R 97 98 99 (CONST A N T -6 . 66 9 6 6 t • 100 101 1 0 2 2 4 - 0 . 3 3 9 7 1 0 . 1 0 9 7 8 1.05 8 5 2 0 . 1 0 0 9 2 9 . 5 7 4 8 ( 2 ) . 1 1 0 . 0 0 9 8 ( 2 1 . 3 5 6 0 . 2 1 7 5 9 0 . 1 5 4 6 7 0 . 1 0 8 2 9 0 . 0 4 9 8 0 . 0 0 9 7 0 . 0 0 3 5 4 . 3 2 3 8 2 . 1 3 6 1 1 . 0 3 2 4 ( 2 1 ( 2 1 W\ 103 104 105 * 7 8 9 0 . 1 3 4 1 8 0 . 1 4 2 7 7 0 . 1 1 7 5 7 0 . 0 0 4 0 0 . 0 1 0 1 0 . 0 2 0 8 1 . 5 9 5 0 1 . 8 1 0 2 1 . 2 1 9 5 ( 2 1 (21 121 106 107 1 0 8 * 1 0 0 . 1 7 1 9 0 0 . 0 0 4 8 2 . 6 4 9 0 (21 109 110 111 S T E P N U I B E R 3 112 113 114 V A R I A B L E E N T E R E D 3 M U L T I P L E R 0 . 8 3 9 5 115 116 1 1 7 S T D . E R R O R O F E S T . 4.1563 A N A L Y S I S O F V A R I A N C E 118 119 120 D F SUM UF S Q U A R E S R E G R E S S I O N 3 3 5 9 1 . 8 3 0 R E S I D U A L 8 7 1 5 0 4 . 3 5 8 MEAN S Q U A R E 1 1 9 7 . 2 7 6 1 7 . 2 9 1 F RATIO 6 9 . 2 4 1 r 121 122 123 VARIABLES IN EQUATION VARIABLES NUT IN EQUATION 124 s. 125 126 VARIA3t.fi COEFFICIEM STD. ERROR F TO REMOVE • VARIABLE PARTIAL CURR. TOLERANCE F TO ENTER r 127 < 128 (CONSTANT -37.84108 1 129 2 -1.18513 0.42061 7.9389 (2 1 . 5 0.01700 0.0056 0. 0249 (21 130 131 3 4 1.52324 0.73254 1.95439 0.44208 4.3238 19.5442 (21 . (21 . 6 7 -0.01062 0.0025 0.05456 0.0034 0.0097 (21 0.2568 (21 132 8 -0.05699 0.0035 0.2603 121 133 9 -0.03677 0.0117 0.1164 12) 134 135 • 10 0.04362 0.0028 0. 1640 121 136 137 F-LEVEL OR TOLERANCE INSUFFICIENT FOR FURTHER COMPUTATION 138 1 SUMMARY TABLE 139 140 STEP VARIABLE MULTIPLE INC REASE F VALUE TO NUMBER OF IKOfc 141 NUMBER ENTERED REMOVED R RSQ IN RS C ENTER OR REMOVE VARIABLES IN 142 143 144 1 4 0.8102 0.6564 0.6564 170.0387 1 U 5 2 2 0.8307 0.6901 0.0337 9.5748 2 146 147 3 1 LIST OF «. ESiOUALS 3 0.8395 0.7048 0.0147 4. 3238 3 148 149 CA SE Y Y 150 NU1 BER XI It COMPUTED RESIDUAL XI 4) XI 2 ) XI 31 151 152 1 32.0000 25.2957 6.7043 39.1730 30.0000 14.5290 153 2 21. 0000 18.6478 2.3522 34.1940 30.0000 16.5530 15* 3 12.5790 13.6542 -1.0752 29.0950 30.0000 19.8 170 155 4 14. 0000 12.8912 1.1088 2 7.4950 30.0000 21.3690 156 5 7. 5800 9.6939 -2.1134 22.8350 30.000C 25.2490 157 6 18.5820 15.1188 3.4632 31.0610 30.0000 16.2560 158 7 16. 0000 17.5550 -1.5550 33.2810 30.0000 17.0070 159 8 12.5830 10.2783 2.3047 25.4410 30.0000 22.2890 160 9 14.5840 13.7932 0.7908 29.7070 30 . 0 000 19.1230 161 10 19.5860 18.9421 0.6439 34.4350 30.0000 16.4370 162 1 1 11.0000 11.4966 -0.4966 25.7620 30.0000 22.6 776 163 12 11.0000 11.1454 -0.1454 25.9190 30.0000 22.2450 164 13 14. 0000 15.8284 -1.8284 30.7710 30.0000 19.0 94C 165 1 4 30.6400 23.2957 7.3443 37.8660 30.0000 14.8930 166 15 12.0000 12.2078 -0.2078 26.7400 30.0000 21.8890 167 I 6 14.0000 14.3253 -0.3253 29.6340 30.0000 19.5 660 168 17 11.0000 13.9684 -2.9684 29.8310 30.C000 19.0790 169 18 11. 0000 11.0464 -0.0464 26.4100 30.0000 21.5 500 170 19 12.0000 1C.5955 1.4045 26.3640 30. OCOO 21.3130 171 20 13.6460 11.4572 2.1888 27.1190 30.0000 <!0.9100 172 ~ 2 1 ~ 12.0000 ~ " 1 2 . 9 1 5 5 -0.9155 23.9540 30.0000 19.5130 173 22 12.0000 11.7877 0.2123 26.5890 30.0000 tl.-.OJO 174 23 14. OCOO 10.9149 3.0851 26.7940 30.000U 2 0. v 710 175 24 13.0000 9.8761 3.1239 23.9430 30.0000 23.9't70 176 2 5 ll.OCOC 14.6678 -3.6678 30.4780 30. it000 18.7 0S0 177 26 6. 6540 10.7549 -4.1009 25.8010 30.0000 22.1 iGO 178 " 27 " 13.6540 17.8822 ' -4.2282 33.2310 30.0000 17.2U6C 179 28 16. 6550 18.7002 -2.0452 34.3120 30 . 0000 l c * . )60 • 180 29 25. 0000 20.7386 4.2614 35. 3990 30. J JOO 15. .' -80 f 181 182 183 30 3 1 32 26.OCOO 9. 65 90 8.6600 21.5200 9.4922 19.6174 4.4800 0.1668 -10.9574 36.2310 22.8760 34.5670 30.0000 30.0000 30.0000 15.8250 25.0640 16.7 110 s 184 185 186 3 3 34 3 5 13. ccob 9. OOUO 27. 7100 12.4652 10.8332 31. 5645 0.5348 -1.8332 -3.8545 28.39/0 24.9510 43.4600 30.0000 30.0000 30.0000 IV.932o 23.2 820 13.1440 r 187 188 189 36 37 3 8 12.0000 21.7110 12.0000 9.9664 18.8893 12.5154 2.0336 2.821r -0.5154 25.2370 34.0830 28.5770 30.000C 30.0000 30.0C0C 22.3460 lb.8540 19.7 340 190 191 192 39 ~ ~ 40 41 11.0000 19. 0000 23.0000 10.2964 17.9366 23.8607 0.7036 1.0634 -0.8607 25.0450 33.6820 38.3990 30.00uu 30.0000 30.OCOO 22.8090 16.7430 14.5 800 193 194 195 42 43 44 11. 7180 16.0000 8.0000 9.2821 20.8066 12.2583 2.4359 -4.8066 -4.2583 23.6500 35.6010 28.1610 30.0000 30.0000 30.0000 23.9330 16.1650 20.0 990 196 197 198 ~~ «>s" 46 47 16. 72 30 23. 7240 14.0010 15.5795 14.5757 17.1484 1.1435 9.1483 -3.1474 31.6740 30.3350 32.9740 30.0000 30.0000 30.0000 17.7 720 18.8310 17.1340 199 200 201 48 49 50 12.0000 7.0000 9.0000 11.9676 12.0427 12.9189 0.0324 -5.0427 -3.9189 27.4690 26.8590 28.7180 30.0000 30.0000 3o.00oG 20.7960 21.6280 19.8180 202 20 3 204 "ST 52 53 9. 0000 9.0000 15.0000 8.8339 14.1206 11.8093 0.1661 -5.1206 3.1907 21 .8050 30.0920 27.7660 30.0000 30.0000 30.0000 26.0060 18.8440 20.3110 20$ 206 207 54 5 5 1 12.0000 10. 5100 9.1388 13.8526 2.8612 -3.3426 23.0880 29.6010 30.0000 30.0000 24.5600 19.2980 208 209 210 CASE NIH BER Y X( l l Y COMPUTED RESIDUAL XI 41 X( 2) X( 3 1 211 212 213 56 57 13.51 10 6. 0000 13.2326 9.7257 0.2784 -3. 7257 29.1770 24.0570 30.0000 30.0000 19.4350 23.7020 214 21S 216 58 59 60 16.0000 10. 5650 10.0000 14.7424 16.1188 11.1900 1.2576 -5.553d -1.1900 30.2340 31.8930 27.0010 30.0000 30.0000 30.OCOO 19.0700 17.8450 20.8 860 217 218 219 bl 62 63 14.0000 16. 56 80 22.0000 11.0938 12.7641 16.1950 2.9062 3.8039 5.8050 26.2160 27.9420 32.1830 30.0000 30.0000 30.OCOO 21.B300 20.7120 17.5230 220 221 222 64 65 66 21.1490 18. 1420 29. 1410 26.8012 20.9279 31.6370 -5.6522 -2.7859 -2.4960 46.0440 41.4060 49.5580 45.0CC0 45.0000 45.COO0 18.3720 20.4670 17.0380 223 224 225 67 68 69 25.9410 18. 1470 34.1550 28.3133 18.2525 34.2159 -2.3723 -0.1055 -0.0609 46.8200 38.9280 50.6110 45.0000 45.0000 45.0000 18.3690 21.8 900 17.3800 226 227 228 " 70 71 72 32.2050 16. 0000 19.0000 25.5782 14.9264 19.3405 6.6268 1.0736 -0.3405 44.5500 35.4460 40.0560 45.0000 45.0000 45.000C 19.4 860 24.1740 21.1570 229 230 231 73 74 75 38.2080 29. 0000 27.0000 27. 7644 22.2050 23.0485 10.4436 6.7950 3.9515 46.2960 41.2200 42.8620 45.0000 45.0000 45.0000 18.6810 21.5 440 19.9910 " 212 233 234 7 6 " 77 78 27.9470 14.0000 18.0000 24.9570 14.4664 18.6321 2.9900 -0.4664 -0.6321 44.2820 35.3860 38.5190 45.00CO 45.CC0C 45.00CO lv.4220 23.9490 22.66*0 235 236 237 79 80 81 17.0000 11.OCOO 24.8830 17.4078 16.5173 20.8505 -0.4073 -5.5173 4.0325 37.2620 37.2280 40.8980 45.0000 45.0000 45.0000 23.4730 22.9320 21.0b80 238 239 240 8 2 "" 83 84 21.9400 21.9390 15. 0000 21.4277 28.6495 23.0321 0.5123 -6.7105 -8.0321 41.5830 47.3630 42.9830 45.0000 45.0000 45.00GJ 20.5b80 17.8930 I9.b250 241 2*2 243 244 24S 246 85 86 87 88 89 90 12.00C0 23.0000 30.4980 38.4960 28. 9940 23.4920 19.70 20 34.0300 25. 7033 -7.7020 •11.0300 4.7947 29.2513 24.7671 22.1458 9.2447 4.2269 1.3463 39.8450 5Q.89J0 44.9000 47.7980 44.4280 41.3300 45.000C 45.0000 45.00C0 4S.0GCC 45.00C0 45.OCOO 21.6650 16.9000 19.1190 17.7 300 19.1100 21.3640 247 248 249 "256" 251 252 9 1 17.0000 18.7278 -1.7278 39.4650 45. OCOO 21.5130 ENO bt H I E F I N I S H C A R D E N C O U N T E R E D PROGRAM T E R M I N A T E D 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 = X4 2 (forced variable) X9 = X2 X4 2 XlO = X3 X4 2 1 6 M D 0 2 R - 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 2 7 , 1 9 7 3 H E A L T H S C I E N C E S C O M P U T I N G F A C I L I T Y , U C L A P R O B L E M C 3 D E C R P D E F N U M B E R O F C A S E S 9 1 NUMBER OF O R I G I N A L V A R I A B L E S 4 NUMBER OF V A R I A B L E S A D D E D 6 T O T A L N U M 3 E R O F V A R I A B L E S 1 0 NUMBER O F S U B - P R C B L E H S 1 T H E V A R I A 3 L E F O R M A T I S I T 0 X . 4 F 8 . 3 r T O -l l 1 2 1 3 1 4 1 5 1 7 V A R I A B L E MEAN S T A N O A R D OEV I A T I O N 1 8 1 1 7 . 1 5 4 6 8 7 . 5 2 4 9 1 1 9 i 3 4 . 6 1 5 3 7 6 . 9 6 1 4 2 2 0 3 1 9 . 9 0 4 9 1 2 . 6 8 1 7 3 2 1 4 3 3 . 6 1 6 2 9 7 . 5 7 2 7 5 2 2 5 6 9 0 . 7 3 2 9 1 1 7 3 . 1 7 4 0 3 2 3 6 1 2 0 5 . 9 4 7 7 5 5 0 6 . 5 0 8 7 9 2 4 7 6 5 9 . 2 6 0 0 1 1 3 4 . 0 5 5 1 6 2 5 8 1 1 8 6 . 7 6 8 3 1 5 4 1 . 7 6 4 1 6 2 6 9 4 4 1 3 5 . 8 8 6 7 2 2 8 6 3 9 . 9 6 8 7 5 2 7 1 0 2 2 9 7 0 . 5 0 3 9 1 9 6 8 7 . 7 8 9 0 6 2 8 2 9 3 0 V A R I A B L E N U M B E R 1 0 3 1 3 2 1 1 . 0 0 0 0 . 5 5 0 2 - 0 . 5 2 5 8 C . 8 1 0 2 0 . 1 6 7 3 0 . 7 1 4 3 0 . 54 91 0 . 8 1 0 2 0 . 7 4 8 1 0 . 7 1 4 7 3 3 2 1 . 0 0 0 0 . 9 2 9 0 E - 0 1 0 . 8 1 1 6 0 . 8 5 8 2 0 . 9 5 2 0 0 . 9 9 80 0 . 8 1 9 1 0 . 9 1 4 2 0 . 9 5 2 9 3 4 3 1 . 0 0 0 v - 0 . 4 9 1 6 0 . 5 8 2 9 - 0 . 1 9 5 7 0 . 1 0 57 - 0 . 4 5 3 9 - 0 . 2 6 1 1 - 0 . 1 B 8 0 3 5 4 1 . 0 0 0 0 . 4 0 2 2 0 . S 4 8 0 0 . 8 0 54 0 . 9 9 4 8 0 . 9 6 3 2 0 . 9 4 5 8 3 6 5 1 . 0 0 0 0 . 6 6 4 3 0 . 8 6 3 4 0 . 4 1 9 8 0 . 5 9 2 4 0 . 6 6 9 1 3 7 6 1 . 0 0 0 0 . 9 4 7 9 0 . 9 5 4 7 0 . 9 9 2 4 0 . 9 9 9 4 3 8 7 1 . 0 0 0 0 . 8 1 3 3 0 . 9 0 9 8 0 . 9 5 0 9 3 9 8 1 . 0 0 0 0 . 9 7 6 6 0 . 9 5 2 7 4 0 9 1 . 0 0 0 0 . 9 9 1 5 4 1 1 0 1 . 0 0 0 4 2 1 S U B - P R O B L M 1 4 3 4 4 4 5 4 6 4 7 4 6 D E P E N D E N T V A R I A B L E 1 M A X I M U M N U M B E R OF S T E P S 2 0 F - L E V E L F O R I N C L U S I O N 3 . 0 0 0 0 0 0 F - L E V E L F O R D E L E T I O N T O L E R A N C E L E V E L 3 . 0 0 0 0 0 0 0 . 0 0 1 0 0 0 ^ 9 5 0 5 1 - 5 2 -5 3 5 4 - S T E P ~ ~ N U > 4 B E R I " V A R I A B L E E N T E R E D ~5T" 5 6 5 7 " 5 8 5 9 6 0 MULTIPLE R S T D . E R I O R OF E S T . A N A L Y S I S OF V A R I A NC E R E G R E S S I O N 0 . 8 1 0 2 4 . 4 3 5 5 D F 1 SUM OF SQUARES 3345.228 ME >N SQUAKE 33-.3.228 f RATIO 170.035 CO 61 62 63 RESIDUAL 89 1750.960 19.674 ... s 64 65 66 VAR I A8L E VARIABLES IN EQUATION COEFFICIENT STO. ERROR F TC REMOVE VARIABLE VARIABLES NUT PARTIAL COKk. IN EQUATION TOLERANCE F TO ENTER , 67 6.8 69 (CONSTANT 3 . 79 9 57 1 \ 7 0 71 72 8 0.01125 0.00086 170.0356 (3) . 2 3 4 -0.33735 -C.302o5 0.07091 0.3290 0. 7940 0. 0105 11.3012 121 8.6731 (21 0.4448 (2) 73 74 75 • 5 6 7 -0.32491 -0.33947 -0.32224 0.8238 0.0886 0.3385 10.36o5 (21 11.4621 (21 10.1968 (21 76 77 78 9 10 -0.3423B -0.32111 0.0462 0.0924 11.6852 (21 10.1167 (21 79 80 81 82 83 84 STEP NUMBER 2 VARIABLE ENTERED 9 85 86 87 MULTIPLE R STO. ERROR Of 0.8347 EST. 4.1910 88 89 90 ANALYSIS OF VARIANCE OF SUM OF SQUARES MEAN SQUARE REGRESSION 2 3550.477 1775.238 F RATIO 101.067 91 92 93 RESIDUAL 88 1545.711 17.565 94 95 96 VARIABLE VARIABLES IN EQUATION COEFFICIENT STO. ERROR F TO REMOVE VARIABLE VARIABLES NOT PARTIAL CORK. IN EQUATION TOLERANCE F TO ENTER 97 98 99 (CONSTANT -0.40476 1 100 101 102 8 9 0.02392' 0.03379 34.7443 "(31 . -0.00025 0.00007 11.6852 (2 1 . 2 3 4 -0.05685 0.07979 -0.06BU4 0. 0465 0.0753 0.0090 0.2821 (21 0.5574 (21 0.4047 (21 1 0 3 104 105 5 6 7 -0.01431 -0.05562 -0.01683 0.1035 0.0105 0.0498 0.0178 (2) 0.2700 (21 0.0246 (21 106 107 108 * 10 -0.00256 0.0115 0.0006 (2) 109 110 U l f-LEVEL OR TOLERANCE INSUFFICIENT FOR FURTHER CoHPuTAflON 1 SUMMARY TtBLE 112 113 114 STEP NUMBER V A R I A B L E M U L I I P L E ENTERED REMOVED R RSQ INCREASE F VALUE TO IN KS C tMIEK UK REMOVE NUMBER OF INOE VARIABLES lit 115 116 117 1 2 8 0.8102 0.6564 9 C.8347 0.6967 0.6564 170.0356 0.0403 11.6652 1 2 118 119 120 1 L IST OF R E S I D U A L S ' " ' " " " " ~ CASE Y Y ( TU NUMBER XI II CCPPUTED RESIDUAL XI b I XI V) 122 123 1 32.OCOO 25.0071 6 . 9 9 2 9 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 / I T 7 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 1 3 £ j A ' 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 1 9 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 2 5 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 7 523.3115 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 1 5 7 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 l l o l.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 . 180 C A S E V V f 181 182 183 NUMBER 56 XI 11 13. 5110 COMPUTED 13.6928 RESIOUAL -0.1813 XI 81 851.2974 XI 9) 25538. 9UC 184 185 186 57 58 59 6.0000 16.0000 10.5650 9.1792 14. 7327 16.4396 -3.1792 1.267J -5.8746 578.7395 914.0942 1017.1636 17362.1836 27 4 22 . 8 2 4 2 30 514.9063 -187 188 189 60 61 62 10. 0000 14.0000 16.5680 11.6685 10. 9767 12.5246 -1.6685 3.0233 4.0434 729.0542 687.2788 780.7554 21871.6250 20ul8.3633 23422.6602 —< ~ ""190 191 192 6 3 64 65 22.0000 21. 1490 18. 1420 16.7473 26.9011 21.6771 5.2527 -5.7521 -3.5351 1035.7454 2120.0503 17 14.4573 31072.3594 954C2.2500 77150.5625 19) 194 195 66 67 68 29.1410 25.9410 18.1470 31.2280 27.8292 19.1131 -2.0870 -1.8882 -0.9661 2455.9951 2192.1130 15 15.3687 110519.7500 98645.0625 68192.43 75 196 197 198 ' 6 9 70 71 ' " 3 4 .1550 32.2050 16.0000 " 3 2.5865 ~ 25. 1578 15.7776 1.5685 7.0472 0.2224 2561.4724 " 1984.7026 1256.4187 115266.2500 ~~ ' 89311.5625 56538.8398 199 200 201 72 73 74 19.0000 38.20 80 29.0000 20.2607 27.2008 21.4792 -1.2607 11.0072 7.5208 1604.4829 2143.3201 1699.0834 72201 .6875 96449.3750 76458.9375 202 203 204 75 76 77 27.0000 27.9470 14.0000 23.2574 24.8512 15.7229 3.7426 3.0958 -1.7229 1837.1509 1960.8953 1252.1689 82671 .7500 88240.2500 56347.6016 20 S 206 207 78 79 80 18.0000 17.0000 11.0000 18.7052 17.4783 17.4456 -0.7052 -0.4783 -6.4456 1483.7131 1388.4561 1385.9236 66767.0625 62400.5195 62366. 55d6 208 209 210 81 82 83 24.88 30 21.9400 21.9390 21.1386 21.8663 28.4879 3.7444 0.073/ -6.5489 1672.6458 1729.1453 2243.2542 75269.0000 77811.5000 10094b.3750 211 212 213 fl4 85 86 15.0000 12.0000 23.0000 23.3912 20.0435 32.9512 -8.3912 -8.0435 -9.9512 1847.5383 1587.6240 2589.7920 83139.1875 71443.0625 1U540. O250 214 215 216 87 88 89 30.49 80 38.4960 28.9940 25.5610 29.0211 25.0180 4.9370 9.4749 3.9760 2016.0093 2284.6492 1973.8464 9072o.3750 102809.1875 88823.C625 217 218 219 90 91 23.4920 17.0000 21.5961 19.6554 1.8959 -2.6554 1708.1689 1557.4858 76867.5625 70086.8125 220 221 222 FINISH CARD ENCOUNTERED 223 END OF FILE PROGRAM TERMINATED • 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 X6 = 1/X4 = 1/X3 f 1 1 B M D 0 2 R • STFPMlSf RFGRfSblfiN - R E V I S E D MARCH 27# 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 ! P R 0 8 I . F M C O C F NUMBER OF C A S F S NUMBED OF OnlGlMAt VAHIABl.F.S FRACRr 6i 4 t • NUMBER OF V»RIABLFs ADOtD TOTAL NUMBER tlF V A R I A B L E S NUMBER OF S U B - P R O B L E M S 2 b 1 1 0 II L a THE VARtABLF FORMAT I S ( t OX,4F8 . I N -IS 1 4 Is 1 6 IT i s V A R I A B L E MEAN 1 47.7732? STANDARD D E V I A T I O N 1 3 , 0 7 6 6 4 1 9 2 0 2 1 2 4 1 , 3 4 4 J « 3 1 9 . 7 J 9 7 9 4 B9.54l7g 14,25918 2,90453 4.46927 22 2 S 24 5 0,0345* 6 0 , 0 5 l 8 » 1 C O R R E L A T I O N MATRIX 0 , 0 0 5 0 1 0,00802 • 2. 26 2T 0 VARIABLE I 3 N U M B E R 3 1 i 6 28 29 SO 1 1 ,1 )00 0.3549 2 I.000 3 •0.4794 0 , 5 1 1 1 • 0 . 4 6 3 5 0 , 4 1 1 3 1 . 0 0 0 •0,9810 •0.4720 • 0 . 4 4 3 3 0 . 9 9 6 9 0,5209 0,4127 • 0 . 9 8 3 1 3 1 3 2 3 S 4 5 6 1 . 0 0 0 • 0 , 9 8 3 1 1 . 0 0 0 0 , 9 9 7 « • 0 , 9 7 9 9 1 , 0 0 0 14 SS 3 6 1 8U8 - PR0RLM 1 DEPENDENT VARIABLE MAKIMIJM NUMBER OF S T E P S 1 2 0 ST 3 6 3 9 F»LEVEL FOB INCLUSION 3 , 0 0 0 0 0 0 F-LEVEL FOR DELETION J , 0 0 0 0 0 0 TOLERANCE i'EVEL 0 . 0 0 1 0 0 0 « 0 4 1 4B 4 3 4 4 «5 S T F P N U M O F R 1 VARIABLE FNTEdED 6 0 6 «T 48 M U L T I P L E 0 g STD. F R R O P OF E S T . 1 1 , 9 2 0 9 . 2 5 3 9 49 5 0 3 1 A N A L Y S I S nF V A R I A N C E OF SUM OF S Q U A R E S MEAN S Q U A R E F RATIO 5 2 5 1 5 4 ("FGRtSSION 1 RFSIDllAL kl 2 8 7 6 . 5 3 b 7 7 2 5 . 6 9 1 2 876 , 535 1 2 6 . 6 5 1 2 2 , 7 1 2 5 5 5 6 57 V A R T A H L E S I N E Q U A T I O N t • V A R I A B L E S NOI I N E Q U A T I O N 5 8 59 6 0 V A R I A H L F C O E F F I C I E N T STD. ERROR F To R E M O V E , • -1 . VAR IABLt P A R T I A L C O K R , T O L E N A N C L F TU fcNTEH 62 6J (CONSTANT 6 3.7«iS?0 ) 849.712J2 tTB.JObJO 22. 7121 (2) ! i 1 0,16810 0,21091 0.6128 O.Oiil 1,7491 (2) 2,7912 (2) < s. 64 69 66 • 4 5 -0.I05U9 0,22506 0,0052 0,0197 0,6700 (2) 1,2194 12) -67 68 69 70 71 72 S T E P N U H B F K 2 VARIABLE FNTERED S 7J 74 75 M U L T I P L E R STO. ERROR OF E S T . 0,5554 11.0546 76 7? 78 ANALYSIS nF V A R I A N C E RrORtSSION OF SUM OF SQUARES 2 3269.961 MEAN SQUARE 1634.980 F RATIO 179 79 80 81 RFSIDUAL 60 7312.266 122.204 82 81 84 VARIABLES IN EQUATION • VARIABLES NOT IN EQUATION V A R I A B L F C O E F F I C I E N T STB, ERROR F TO REMOVE . VARIABLE PARTIAL CORR. TOLERANCE F TO ENTER 85 86 81 (CONSTANT -16J .55901 ) 8* 89 90 ' k 2S2S 2J95 .05811 1406.18091 .14380 878.91479 3. 7, 2194 (2) , 4263 (2) , 2 3 4 0,19886 0,0094? •0.01702 0,6014 0.00S2 O.0044 2,4292 (2) 0,0053 (2) 0,0)7} (2) 91 92 9 ) F-LEVEL OR TnLERANCE INSUFFICIENT-FOR FURTHER COMPUTATION 9« 95 9* 1 SUMMARY T M l E S T E P V A R I A B L E MULTIPLE INCREASE F VALUE TO NUMBER OF INDfc 9» 98 99 NUMBER ENTERED REMOVED R RSQ IN RSQ ENIEH QN REMOVE VARIABLES IN too 101 103 1 2 1 LIST OF RESIDUALS 6 5 0,5209 0,5554 0,2711 0,3084 0 0 .2713 .0371 22,7121 1,2194 1 2 10J l o a to«j C A S E V NUMBER v( l l Y COMPUTED RESIDUAL X( 6) X( 51 106 107 10ft 1 Hi.6900 2 A t . 41/10 65.7016 54.9210 15.9884 6.4910 0,0688 0.0604 0 0 .0255 .0292 109 110 t i l J 43.2J50 4 50.9190 5 i*.lQ40 44.0220 40.2902 U\.7927 -0,7870 10.6288 -8,5987 0,0505 0,0468 0,0196 0 0 0 ,0144 ,0164 .0418 112 H I 114 » S9.8240 7 U8.07S0 6 ,j4.4590 as.8677 53.0845 ^3.0725 10,9561 -5.0095 6,3865 0,0548 0,0558 0,0449 0 0 0 .0122 .0300 .0141 n s 116 117 9 (19.0910 10 S6.8T70 11 H2.6990 46,6217 •45.4278 19.9981 2.4693 1.4492 2,7009 0,0521 0,0606 0,0441 0 0 0 ,0117 ,0290 .0188 i t s 119 120 12 fl2.4600 l i 45.4980 14 AO.9150 41,4561 43.8753 61.8956 6.98J9 1.6227 17.0194 0,0450 0,0524 0,0671 0 0 0 ,0186 .0125 .02o4 —< 121 122 121 IS 16 17 /I4.H770 /,7.2a .0 .6.8750 40.2184 41,9952 4t».5S7b 4.6566 3.2478 •9,6826 0,0457 0,0511 0.0524 0,0174 0,0317 0.0135 \ < .24 125 126 16 19 20 at .6510 /|5.5|70 .0.3190 41,1187 44.5213 44.0229 ... , j t l 4 6 7 7 - -0.9957 6.2961 0,0464 0,0469 0,0476 0,0179 0.0179 0,0169 12? 128 129 21 22 23 11.4450 45.1120 .2.2510 46.3273 41,1657 44,8182 •4 , 6 8 2 3 3.9661 7.4128 0,0512 0,0459 0,0477 0,0145 0,0176 0.0171 130 131 132 2(1 25 26 S4.2960 .6.0910 . 5 p 7«80 .... 4.43,2 47,2517 42,4119 ' 1 2 . 4 5 8 8 •11,1607 •16.6219 o.oate 0,0515 0.0452 0.0418 0,0128 0.0188 133 131 I3S 27 26 29 at.0840 aS.Saoo ..9.6400 50.9256 55.6992 58.9116 •9,8366 -7,1592 10,7264 0,0579 0,0608 0,0635 0,0101 0,0291 0.0279 "136 I3T 138 30 31 32 71.7*10 d2.2?40 . 5.0S20 57,4369 42,2949 9?.7567 1471101 •0.0709 •27.7067 0,0b32 0,0399 0,0598 0,0276 0,0417 0.0289 139 l«o M l 33 34 33 45.7790 16.0710 fcl.7600 45.4561 40,4368 76,7190 0 .3229 •4.3656 •12.9590 0,6502 0,0430 0,0761 0.0152 0.0401 0,0210 1MB 145 11 . 36 37 38 47.5490 63.7020 al.9910 41.6000 52.9791 46.1023 1,9490 • 11.1229 •4,1111 0,0448 0,0591 0,0507 0,0196 0.0291 0.0150 145 146 141 S4 40 41 43.9?20 .6.4090 .9.8*70 42,1907 54.4086 66,4212 1,7111 2.0064 •6.5262 0,0418 0,0597 0,0686 0,0199 0,0297 Oj0260 148 149 150 42 43 44 49.5470 ' q4.9420 >R.4o80 43.2012 55.4800 45.2023 6,1457 •10.5180 •16,7941 0,0418 0,0619 0,0496 0,0421 0,0281 0.0155 151 152 15J 45 46 47 .2,7970 .8.2060 42.4h20 $0.6687 46.6097 52.7465 1.9261 31,4003 -10.2845 0,0561 0,0531 0,0584 0,0116 0,0110 0,0101 154 159 156 48 49 50 43.6860 96.0620 .1.3.90 43.46B4 41.1206 49.1543 0,2206 •15.0569 •13.6153 0,0481 0,0462 0,0505 0,0164 0,0172 0.0148 15T 156 159 S i S2 53 41.2760 i»9.9o«0 S4.0740 44.2507 47.1696 49.2330 - 2 !«747 •17.4616 6,7910 0.0JB5 0.0511 0,0492 0,0459 0,0112 0.0160 160 161 u a • 4 55 1 5 t .«740 15.5070 43.2432 49.7901 6,7306 •10.2631 0,0407 0,0516 0,0411 0,0136 16* 164 «*5 CASE NUMBER V X( 1) r COMPUTED RESIDUAL X( 6) X( 5) 166 16? 168 56 57 46.3070 >4r9410 46.1539 42.3717 0.1511 •17.4107 0,0515 0.0422 0,0343 0.0416 169 170 >TJ 58 59 60 S2.92Q0 ^3.1270 17.01.0 45.4894 49,7704 44.3611 7.4106 •l6.o414 -7,5261 0,0524 0.0560 0,0479 0,0331 0,0114 0.0170 172 173 174 61 62 63 S3. 4021) S9.29J0 *.H.lS90 42.4001 4*.3776 51.5240 11 .0019 16.9154 16.6150 0.0456 0.0481 0.0571 0,0181 0,0158 0.0511 175 176 177 178 179 FINISH PROGRAM CARD PNCOllNTfREn TERMINATED END OF FILE 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 3 1 BMD02R - STEPWISE REGRESSION - REVISED MARCH 27, HEALTH SCIENCES COMPUTING FACILITY, UCLA 19/3 4 5 \ 6 PROBLEM C3 0E FRACRP NUMBER OF CASES 91 NUMBER OF ORIGINAL VARIABLES 4 r 7 8 9 NUMBER OF VARIABLES AOOEO 2 TOTAL NUM3ER OF VARIABLES 6 NUMBER OF SUB-PROBLEMS 1 10 11 12 THE VAR I A3 LE FORMAT IS 1 10X.4F8.3I 13 14 15 16 17 18 VARIABLE MEAN STANDARD DEVIATION 1 49.63556 13. 59948 19 20 21 2 34.61537 6.96142 3 19.90491 2.68173 4 33.61629 7.57275 22 23 24 1 5 0.03121 0.00671 6 0.05119 0.00728 CORRELATION MATRIX 25 26 27 0 VARIABLE 1 2 3 4 NUMBER 5 6 28 29 30 1 1 .000 0.2066 -0.4425 0.4604 2- 1.000 0.9290E-01 0.8116 3 1.000 -0.4916 -0.4498 -C.7546 0.5684 0.4558. -0.1256 -0.9840 31 32 33 4 1.000 5 6 -0.97/9 1.000 0.4673 -0.5345 1.000 34 35 36 1 SUB-PROBLM 1 DEPENDENT VARIABLE 1 MAXIMUM NUMBER OF STEPS 20 37 38 39 F-LEVEL FOR INCLUSION 3.000000 F-LEVEL FOR DELETION 3.000000 TOLERANCE LEVEL 0.001000 40 41 42 43 44 45 STEP NIN BER 1 VARIABLE ENTERED 4. 46 47 48 MULTIPLE R 0.4604 STO. ERROR OF EST. 12.1403 49 50 51 ANALYSIS OF VARIANCE OF SUM UF SQUARES MEAN SQUARE F R A T I O ' " " 52 53 54 REGRESSION 1 352 7.766 RESIDUAL 89 13117.367 3527.766 147.3b6 23.936 55 56 57 VARIABLES IN EQUATION V A K I A B L L i WOT IN EQUATIUN 5 8 " ' 59 60 VARIABLE COEFFICIENT STD.ERROR F T C REMOVE V A K I A B L E P A R T I A L CORK. TOLERANCE F TO ENTER I—1 c 61 62 63 (CONSTANT 4 21.84 328 I 0.82675 0.16899 23.9355 (2 ) . 2 -0.32206 3 -0.2797C 0. 3414 0.7584 10.18tI 7.4690 \ 121 12) 64 65 66 • 5 G.00248 6 0.30672 0. 0436 0.7817 O.O0U5 9.1384 121 (2) 6. 68 69 — — C TO 71 72 STEP N U . 8 E R 2 VARIABLE ENTERED 2 73 7 4 75 MULTIPLE R STD. ER-OR OF EST. 0.5419 11.5585 76 77 78 ANALYSIS OF VARIANCE DF SUM OF SQUARES MEAN SQUARE F RATIO REGRESSION 2 4888.359 24.4.180 18.295 79 80 81 RESIDUAL 88 11756.773 133.600 82 83 a. VARIABLES IN EQUATION • VARIABLES NOT IN EQUATION VARIABLE COEFFICIENT STD. ERROR F TC REMOVE VAKIAbLE PART IAL CURR. TOLERANCE F TO ENTER 85 86 87 (CONSTANT 30.95923 1 * 88 89 90 2 4 -0.95595 1.53 994 0.29955 0.27537 13.1841 (21 . 31.2729 (2) . 3 0.13031 5 0.10168 6 -0.03991 0.0498 0. 0402 0.0352 1.5028 0.9089 0.13B8 121 (21 U l 91 92 93 F-LEVEL OR TOLERANCE INSUFFICIENT FOR FURTHER COMPUTATION * 94 95 96 1 SUMMARY TABLE STEP VARIABLE MULTIPLE INCREASE F VALUE TO NUMBER OF INOE 97 98 99 NUMBER ENTERED REMOVED R RSQ IN RSC ENTER OR REMOVE VARIABLES IN 100 101 102 1 2 1 L IST OF .ESIDUALS 4 2 0.4604 0.2119 0.5419 0.2937 0.2119 . 0.0817 23.9355 10. 1841 1 2 103 104 105 CASE Y NUMBER XI 11 Y COMPUTED RESIDUAL X I 41 XI 2) 106 107 108 1 81.6900 2 61.4140 62.6046 54.9373 19.0854 39.1730 6.4767 34.1940 30.0000 30.0000 109 110 111 3 43.2350 4 50.9190 5 33.1940 47.0851 44.6212 37.4451 -3.8501 29.0950 6.2978 27.4950 -4.2511 22.8350 30.0000 30.0000 30.0000 112" 113 114 6 59.8240 7 48.0750 8 49.4590 50.1127 53.5313 41.4582 9.7113 31.0610 -5.4563 33.2810 B.JOOd 25.4410 30.CC0C 30.0000 3 0 . 0 0 0 0 U S 116 117 9 49.0910 10 56.8770 11 42.6990 48.0276 55. 3084 41.9525 1.0634 29.7070 1.5686 34.4350 0.7465 25.7620 3 0 . 0 0 0 0 3 0 . 0 0 0 0 30.CCO0 ' 118 119 120 12 42.4400 13 45.4980 14 80.9150 42.1943 49.6661 60.5919 0.2457 25.9190 -4.1681 30.7710 20.3231 37.8660 3 0 . 0 0 0 0 3 0 . 0 0 0 0 30.CC0C f 121 15 44.8770 43.4586 1.4184 26.7400 30.0000 122 16 47.2430 47.9152 -0.6722 29.6340 30.0000 123 1 7 36.8750 48.2185 -11.3435 24.8310 30. 0 0 0 0 " ~ " 1 2 V 18 41. 6510 42.9504 -1.2994 26.4100 3u.0OuG 125 19 45. 5170 42.8796 2.6374 26.3640 30.CC00 126 20 50.3190 44.0422 6.2768 27.1190 30.0000 ? 12* 21 41.4450 46.8680 -5.4230 28.9540 30.0000 128 22 45.1320 43.2261 1.9059 26.589C 3C.C000 129 23 52.2510 43. 5417 8.7C93 26.7940 3O.OOU0 130 24" 54.2960 " 3 9 . 1 5 1 4 15.1443 23.9430 30.0000 131 25 36.0910 49.2149 -13.1239 30.4780 30.0000 132 26 25.7880 42.0126 -16.2246 25.8010 30.0000 133 27 41.0890 53.4543 -12.3653 33.2310 30.0000 134 28 48.5400 55.1190 -6.5790 34.3120 30.0000 135 29 69.6400 57.5629 12.0771 35.8990 30.0000 136 3 0 " 71.7610 58.0741 13.6869 ^6.2310 30.0doc 137 31 42.2240 37.5083 4.7157 22.8760 30.0000 138 32 25.0520 55.5117 -30.4597 34.5670 30.0000 139 33 45.7790 46.0103 -0.2313 28.3970 30.OOOC 140 34 36.0710 40.7036 -4.6326 24.9510 30.0000 141 35 63. 7600 69.2063 -5.4463 43.4600 30.0000 142 36 47.5490 41.1441 6.4049 25.2370 30.0000 143 37 63.7020 54.7663 3.9357 34.0830 30.0000 144 38 41.9910 46.2874 -4.2964 28.5770 30.0030 145 39 43.9220 40.8484 3.0736 25.0450 30.0000 146 40 56.4090 54.1488 2.2602 33.6820 30.0000 147 41 59.8970 61.4127 -1.5157 38.3990 30.GCQG 148 42 49.5470 38.7002 10.8468 23.6500 3C.000C 149 43 44.9420 57. 1039 -12.1619 35.6010 30.0000 150 44 28.4080 45.6468 -17.2388 28.1610 30.0000 151 45 52.79 70 51.0566 1 .7404 31.6740 30.0000 152 46 78.2060 48.9947 29.2113 30.3350 30.0000 153 47 42.4620 53.0585 -10.5965 32.9740 30.0000 • 154 48 43.6860 44.5812 -0.8952 27.4690 30.0000 155 49 26. 06 20 43.6418 -17.5793 26.8590 30.0000 156 50 31.3390 46.5046 -15.1656 28.7180 30.0000 157 51 41.2760 35.8590 5.4170 21.8050 30 . 0 000 158 52 29.9080 48.6204 -18.7124 30.0920 30.0000 159 53 54.0240 45.0386 8.9854 27.7660 30.0000 160 54 51.9740 37.8347 14.1393 23.06 80 30.0000 161 55 35 .50 70 47.8643 -12.3573 29.6010 .30.0000 162 1 163 164 CASE 7 Y 165 NUMBER XI 11 CCMPUTEO RESIDUAL XI 41 XI 2) 166 167 56 46.3070 47.2114 -0.9044 29.1770 30.OOOC 168 57 24.9410 39.3273 -14.3860 24.0570 30.0000 169 58 52.9200 48.8391 4.0909 30.2340 30.0000 170 59 33.1270 51.3939 -18.2669 31.8930 30.0000 171 60 37.0350 43.8605 -6.8255 27.0010 30.0000 ~ 1 7 2 ' 6 1 " " " " 53.4020 ~42.6517 10.7503 ~ 2 6 . 2 1 6 0 . 30.0000 173 62 59.2930 45.3096 13.9834 27.9420 30.0000 174 63 68. 3590 51.8405 16.5185 32.1830 30.0000 175 6 4 45.9330 58.8462 -12.9132 46.0440 45.U000 176 65 43.8150 51.7040 -7.8890 41.4060 45.COCO 177 66 58. 8010 64.2576 -5.4566 49.5580 45.0000 178 -'-"-6-7' 55.4050 60.0412 -4.6362 46.8200 45.OCOO 179 66 46.6170 47.8880 -1.2710 3d.9280 45. CO C O 180 69 67.4850 65.8791 1.60 54 50.bl10 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 ' T B 7 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 ' " 7 9 " 4 5 . 6 2 3 0 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 r w _ g g 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 9 0 56 .84 00 51.5870 5.2530 41.3300 45.OOP. 202 91" 43.0760 4877150" -5.6390 39.4650 45700 CO 203 204 2o? 206 FINISH CARD ENCOUNTERED 207 PROGRAM TERMINATED  END OF FILE 

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