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Some aspects of load duration behaviour in wood Clark, Kenneth Bruce 1975

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SOME ASPECTS OF LOAD DURATION BEHAVIOUR IN WOOD by KENNETH BRUCE CLARK B .A .Sc . ( 1 9 7 2 ) The Univer s i ty of B r i t i s h Columbia A THESIS SUBMITTED IN PARTIAL FULFILMENT THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n the Department of CIVIL ENGINEERING We accept th i s thes i s as conforming to the required standard The Univer s i ty of B r i t i s h Columbia A p r i l , 1975 In presenting t h i s thes i s i n p a r t i a l fu l f i lment of the requirements for an advanced degree at the Univer s i ty of B r i t i s h Columbia, I agree that the L ibrary s h a l l make i t f r e e l y ava i l ab le for reference and study. I further agree that permission for extensive copying of t h i s thes i s for s cho la r ly purposes may be granted by the Head of my Department or by h i s representat ives . It i s understood that copying or p u b l i c a t i o n of t h i s thes i s for f i n a n c i a l gain s h a l l not be allowed without my wri t ten permiss ion. K. B. Clark Department of C i v i l Engineering The Univer s i ty of B r i t i s h Columbia Vancouver 8 , Canada A p r i l 1975 ABSTRACT Present design stresses for s t r u c t u r a l timber include a reduct ion factor to account for the durat ion of loading. Recent studies have shown however that t h i s factor does not apply to commercial grade timber contain-ing knots and other i r r e g u l a r i t i e s of g r a i n . It has a l so been shown that the apparent Young's modulus perpendicular to gra in i n c lear mater ia l decreases grea t ly with longer durations of loading. It was therefore hypothesized that i n commercial grade wood, ;,'stress concentrations are made less severe i n long term loadings because of s tress redis tr ibut ion,made poss ib le by s t r a i n i n g perpendicular to g r a i n . This thes i s found experimentally that the amount of t e n s i l e s t r a i n i n g perpendicular to gra in around a knot i n timber beams subjected to bending increased s u b s t a n t i a l l y with long durat ion loadings . In conjunction with t h i s , a computer s imulat ion of the mater ia l around a knot showed that a decreasing apparent Young's modulus perpendicular to g ra in reduces the stresses perpendicular to g ra in . TABLE OF CONTENTS Page ABSTRACT i i TABLE OF CONTENTS i i i LIST OF TABLES v i LIST OF FIGURES v i i ACKNOWLEDGEMENTS i x CHAPTER 1. INTRODUCTION 1 1.1 Background 1 1 .2 Recent Invest igat ions . 2 1 .3 Purpose and Scope . . . . . . . . . 8 2 . EXPERIMENTAL OBSERVATIONS 12 2.1 Introduction 12 2 . 2 Se lec t ion of Method 12 2 . 3 Specimen Preparat ion. . . . . . . . . 15 2.k Loadings 16 2 . 5 Instrumentation 19 2 . 6 Test, Set-Up 20 2 .7 Analys i s 20 2.8 Results 31 2 . 9 F a i l u r e Modes 37 2 .10 Summary . ^0 3 . INFLUENCE OF STIFFNESS PERPENDICULAR TO GRAIN *H 3 . 1 Introduction ^1 i i i CHAPTER • Page 3 . 2 The F i n i t e Element . ^2 3 . 3 The Problem ^3 3 . ^ Results of. the Parametric Study . . 50 3 . 5 'The E /E Ratio 5^ y 3 . 6 Shear Modulus Effect, . . . . .':. . 60 3 . 7 Summary . . . . . . 60 4. CIRCULAR HOLE IN A FINITE PLATE HAVING GRAIN TYPE 0RTH0TR0PY ^.1 Introduction 6k k.2 The Problem 65 k.J> Preliminary Tests . 65 k.k Tension Zone Sizes and Positions. . 68 k,S Stress D i s t r i b u t i o n on Axes of Symmetry. . . . . . . 78 k.6 Summary .- 89 5 . CONCLUSIONS 91 REFERENCES . - 9 3 APPENDICES . 9^ A. THE FINITE ELEMENT 9^ A.L The Po t e n t i a l Energy Theorem. . . . 94 A.2 Derivation of the Six Node Plane Linearly Varying S t r a i n Orthotropic Triangle. . . . . . . 96 A.3 Derivation of Strains and Stresses . I 0 9 B. BEHAVIOUR AND TESTING OF THE FINITE ELEMENT. . . H I i v CHAPTER Page B . l Convergence • I l l B.2 Testing.'- . 112 C. THE COMPUTER PROGRAM.' . . . . . . . . . 118 v TABLES Table Page I Test ing Schedule 32 II Tension Zone Areas 35 III Parametric Analys i s Data ^6 IV Input Data: Hole i n Plate 70 FIGURES Figure Page 1.1 Madison Test ." . . . . . 3 1 .2 Duration of Maximum Load . . . . . . . . . . 5 1.3 F a i l u r e Loads: Clear .' , : . . . . . . . . . 6 1 .4 F a i l u r e Loads: No. 2 Construction . . . . . 7 1 .5 Tension Perp . -Fa i lu re S t res s : Commercial . 9 1 .6 Tension P e r p . - E l a s t i c Modulus: Clear . . . . 10 2.1 Specimen Preparation . . . . .' . . . . 17 2 . 2 Specimen Control Point Layout .' . . . . . . 18 2 . 3 Reference Frame , . .- . .' .' . .• 21 2 A Test Set-Up . . . .' . . . . . . . . . ! . : . 22 2 . 5 Va l ida tor Machine . . 24 2 . 6 Photograph Coordinates . . . . . . . . . . . 25 2 . 7 Specimen P .< . .' .' .' . .' . . ; 26 2 . 8 Specimen 2 . . . . . . . . . . . . . . . . . 27 2 . 9 Specimen 3 . .* . . . . . . . . . . . . . . . 28 2.10 Rapid F a i l u r e Mode . . . . . . . . . . .' . 38 2.11 Slow F a i l u r e Mode - 39 3 . 1 T y p i c a l Grain Pat tern . . ; 4 8 3 . 2 35 Element Model .' ." ." . , ! / . . . . . ^9 3 . 3 Bending . . . . . . .- . . . . .' 51 3 A Bending X Strains , . . . . . . .' . .• 52 3 . 5 Bending Y Strains . .' . 53 3 . 6 Tension Loading X Strains 57 3 . ? Tension Loading Y Strains . . 58 v i i 3 . 8 Bending 'Slow' Test Shear Modulus Ef fect . . 61 3.9 Tension Shear Modulus Ef fect . . y-v . .< 62 4 . 1 188 Element Model . .• .• .* . .< . 66 4 . 2 Edge Stresses : X D i r e c t i o n .' .• / , ! , . . 67 4 . 3 Edge Stresses : Y D i r e c t i o n .< . . . „• . . , 69 4 . 4 Stress D i s t r i b u t i o n X D i r e c t i o n Isotropic .• 72 4 .5 Stress D i s t r i b u t i o n X D i r e c t i o n E / E = 20 73 x y ^ . 6 Stress D i s t r i b u t i o n X D i r e c t i o n E /E= 40 74 x y 4 . 7 Stress D i s t r i b u t i o n Y D i r e c t i o n I sotropic 75 4 . 8 Stress D i s t r i b u t i o n Y D i r e c t i o n E/E, = 20 . 76 4 . 9 Stress D i s t r i b u t i o n Y D i r e c t i o n E / E = 40 . 77 x y 4 . 1 0 S t r a i n D i s t r i b u t i o n X D i r e c t i o n Isotropic .< 79 4 . 1 1 S t r a i n D i s t r i b u t i o n X D i r e c t i o n E / E = 20 . 80 x y 4 . 1 2 S t r a i n D i s t r i b u t i o n X D i r e c t i o n E„/E = 40 . 81 x y 4 . 1 3 S t r a i n D i s t r i b u t i o n Y D i r e c t i o n I sotropic .'• 82 4.14 S t r a i n D i s t r i b u t i o n Y D i r e c t i o n E / E = 20 . : 83 x y 4 . 1 5 S t r a i n D i s t r i b u t i o n Y D i r e c t i o n E/E = 40 .• 84 x y 4.16 Stresses i n the X D i r e c t i o n Side DE . . . . 85 4.17 Stresses i n the Y D i r e c t i o n Side DE . . J 87 4.18 Stresses i n the X D i r e c t i o n Side BC / .« . 88 4.19 Stresses i n the Y D i r e c t i o n Side BC . .• 90 A. l Element Configuration .' / . . .' .• . 97 B. l Uniformly Loaded Membrane .' , : .= .' .• , s H 3 B .2 32 Element Cant i lever . . . . . . < . .' .' 114 B . 3 Edge Stresses X D i r e c t i o n Isotropic . 116 B .4 Edge Stresses Y D i r e c t i o n Isotropic . .< . .• H 7 v i i i ACKNOWLEDGEMENTS I wish to thank my supervisors , Professor B. Madsen and Dr . M.D. Olson, for t h e i r advice and support throughout the preparat ion of t h i s t h e s i s . In a d d i t i o n , I would l i k e i n p a r t i c u l a r to thank Mr. Ron Ungless, the U . B . C . C i v i l Engineering Department program l i b r a r i a n , for advice and l o g i s t i c assistance far beyond the dut ies of h i s o f f i c e . Two scholarships provided by the Nat ional Research Counci l of Canada enabled me to undertake these s tudies . A p r i l 1975 Vancouver, B r i t i s h Columbia ix 1 CHAPTER 1  INTRODUCTION 1.1 Background Timber i s employed as the main s t r u c t u r a l y mater ia l i n many houses, apartments, shopping centres and i n d u s t r i a l bu i ld ings i n B r i t i s h Columbia, and i t s production i s a v i t a l part of the province^ economy. U n t i l r e c e n t l y , allowable design stresses had been derived by extrapolat ion from laboratory data , and l i t t l e e f for t had been made to understand the micro-scopic behavior of construct ion grade timber under load. Examination of th i s behavior may make poss ib le more accurate and r a t i o n a l u t i l i z a t i o n of information derived from experimental studies of small wood specimens. Because al lowable stresses for s t r u c t u r a l timber include reductions to account for long term loadings , the mechanisms that produce t h i s effect deserve deta i l ed study. The f i r s t major work on t h i s subject was carr ied out at the Forest Products laboratory i n Madison Wisconsin and was reported i n 1951• One hundred and twenty-six one inch by one inch matched c lear specimens 2 of Douglas F i r were tested i n bending. One of each p a i r was loaded to f a i l u r e i n a s t a t i c test taking about f i v e minutes. The matched specimen was then given a constant bending moment to produce a stress equal to from..60 to 95 per cent of the f a i l u r e stress of i t s partner. Based on these r e s u l t s , some of which were obtained from 6 per cent and some from 12 per cent moisture content specimens, and some from heat treated specimens, F i g . 1.1 was obtained. 1 It predicts that the long term strength of specimens ( f i f t y years of applied load) i s over 40 per cent lower than the short term strength (from a f i v e minute t e s t ) . This reduction, derived from bending tests on small c l e a r specimens has been empirically applied to the bending, shear and tension perpendicular to grain strengths of f u l l size timber containing knots, checks and adverse slopes of. ..grain.' 1.2 Recent Investigations Investigations from 1970 to the present of the load duration effects on commercial grade wood have been carried out at the University of B r i t i s h Columbia, and have shown disagreement with the Madison 2 3 Wisconsin r e s u l t s . In Madsen's (U.B.C.) experiments,'-" several rates of stepwise ramp loading were used to study the strengths of clear and commercial grade f u l l size lumber. One hundred and eighty-nine clear 2 x 6 's and two hundred and eighty-five number two grade 2 x 6 ' s were tested to f a i l u r e i n bending at s i x STRESS IN RATIO TO THE ULTIMATE STRENGTH IN A STANDARD TEST - % _ cn cn cn -si 0 0 C D to to O O c n O cn O cn O cn O CJ> O 4 rates of loading such that f a i l u r e occurred about 1; 10, 100, 1000, 10000 and 100000 minutes a f t e r i n i t i a l load applic a t i o n . The number two grade specimens contained knots and other grain i r r e g u l a r i t i e s . F i g . 1.2, adapted from the Madison Wisconsin report^ shows that approximately 80 per cent of the t o t a l strength reduction would occur within the longest of these time spans. From Fi g . 1.3 and F i g . 1.4 i t i s apparent that while the load duration strength reduction effects predicted by F i g . 1.2 might represent the behaviour of clear material, they cannot be b l i n d l y applied to materials containing i r r e g u l a r i t i e s of grain. This i s p a r t i c u l a r l y true i n the neighbourhood of the 5th percentile of strength, from which design stresses are derived. This should have been obvious since commercial material, containing knots and adverse slopes of grain, has d i f f e r e n t modes of f a i l u r e that clear material. The l a t t e r often develops wrinkles i n the compression zone as a prelude to f i n a l f a i l u r e , while the former usually f a i l s near, one of the i r r e g u l a r i t i e s 2 3 because of stress concentrations i n the tension zone. ' J Analagous r e s u l t s were obtained for dry lumber subjected 4 to shear.- Dry lumber subjected to pure tension perpend-i c u l a r to grain however exhibited considerable reductions 5 i n strength and apparent s t i f f n e s s with time. From a one minute test to a two month t e s t , the average reduct-ion i n f a i l u r e stress on the gross section f o r number two grade lumber was 33 per cent, of the same order as RATIO OF WORKING STRESS TO RECOMMENDED STRESS FOR LONG-TIME LOADING - % 31 I ro o c JO > o o >< r~ O > a I MONTH I05MIN. I YEAR I06MIN . 10 YEARS I0 7MIN. 5 0 YEARS 9 I 000 TO FAILURE - MINUTES I YEAR I ll II I 'I I 0 0 0 000 Figure 1-3 ON TIME TO F A I L U R E - MINUTES Figure 1-4 8 the 35 per cent reduct ion predicted by the Madison tes t r e su l t s of F i g . 1 . 2 . The time dependent s t i f fnes s was however an average of approximately eight times smaller for a 1 0 0 , 0 0 0 minute test than for a 1 minute t e s t . St i f fnesses were not considered i n the Madison work. The strength phenomenon was shown by F i g . 1 .5 fo r commercial m a t e r i a l , and the time dependent s t i f fne s s reduct ion by F i g . 1 .6 for c lear mater i a l . In t h i s paper, time dependent s t i f fnes s w i l l r e f e r to t o t a l s t ra ins ( e l a s t i c and creep) while under load. 1 .3 Purpose and Scope From-; these r e s u l t s , the hypothesis was drawn that the effects of s tress concentrators l i k e knots decrease with time because of r e d i s t r i b u t i o n s of stress i n mater ia l adjacent to i r r e g u l a r i t i e s of g r a i n . This ef fect would increase the f a i l u r e strength of commercial mater ia l to at leas t p a r t i a l l y compensate for the strength reductions with time found i n c lear lumber by the Madison te s t s . It was further hypothesized that the very large reductions of the time dependent s t i f fnes s perpendicular to gra in could , i n a long term loading , permit extensive t e n s i l e s t r a i n i n g perpendicular to g ra in and lead to some kind of flow i n the mater ia l surrounding stress concentratorsv Stresses perpendicular to g ra in to cause t h i s s t r a i n i n g would be set up by the more marked curv-ature of gra in i n commercial than i n c lear mater i a l . 9 3 0 0 TIME TO F A I L U R E - minutes T E N S I O N PER?. - F A I L U R E S T R E S S C O M M E R C I A L FIG. 1-5 10 150 100 co ID _ J 3 O O O I— 00 < _ J UJ 10 100 1000 10 000 TIME TO FAILURE - minutes 100 000 T E N S I O N P E R P . - E L A S T I C M O D U L U S C L E A R FIG. 1-6 11 The purpose of the present inve s t i ga t ion was then twofold. The f i r s t purpose was to experimentally determine whether or not the amount of t e n s i l e s t r a i n i n g perpendicular to gra in i n wood containing i r r e g u l a r i t i e s increases s i g n i f i c a n t l y with the durat ion of load . The second purpose was to discover through a computer s imulat ion whether or not a time-dependent s t i f fnes s perpendicular to g ra in could produce a s i g n i f i c a n t ef fect that d i l not exis t i n c lear mater i a l . 12 iCHAPTER 2 EXPERIMENTAL OBSERVATIONS 2.1 Introduction The load durat ion behaviour of wood containing stress r a i s e r s has been observed to be d i f f e r e n t from that of s t ra ight grained mater i a l . A necessary part of the proving of the hypothesis involves an experimental inves t i ga t ion of the s t ra ins under load i n i r r e g u l a r mate r i a l . Strains p a r a l l e l and perpendicular to the l o n g i t u d i n a l axis (approximately p a r a l l e l and perpend-i c u l a r to the assumed gra in pattern) of a beam having a s ing le edge knot i n the middle of an otherwise c lear segment were s tudied. The measured s t r a in s included creep, i f occurr ing , as i n e l a s t i c mechanisms contribute to r e d i s t r i b u t i o n of s t res s . 2.2 Se l ec t ion of Method Although any method of t e s t i n g would at best show only the behaviour of the surface plane of m a t e r i a l , a t the leas t an i n d i c a t i o n of the t o t a l s t r a i n d i s t r i b u t i o n ins ide the board could be obtained. One constra int on the t e s t i n g procedure was that the character of the wood surface 13 should not be a l tered by at taching any mater ia l more s t i f f than the specimen i t s e l f . Anything coupled to the surface must be guaranteed to deform exact ly as the mater ia l beneath i t . As w e l l , the gra in pat tern should be v i s i b l e so that the point of f a i l u r e i n i t i a t i o n could be seen and so that s t ra ins could be re la ted during analys i s to the board conf igurat ion . Because the behaviour was not p red i c t ab le , i t was a l so des i rable to have the c a p a b i l i t y of measuring s t ra ins at severa l stress l e v e l s . F i n a l l y , the s t ra ins had to be emenable to recording while the tes t was i n progress without d i s turb ing the specimen. Stra ins could have been measured d i r e c t l y using s t r a i n gauges or b r i t t l e coatings . Displacements could have been measured by using moire g r id s . There were how-ever objections and l o g i s t i c d i f f i c u l t i e s to these methods. Because the number of s t r a i n gauges required would have ob l i t e ra ted the face of the specimen, and the g lu ing process could have changed the surface character , t h i s method was discarded. B r i t t l e coatings had several drawbacks. They crack only once at a prescr ibed s t r a i n , and further cracking occurs elsewhere only as t h i s same s t r a i n i s reached. A complete s t r a i n d i s t r i b u t i o n cannot therefore be obtained at any time; the ent i re approach i s i t e r a t i v e and the behaviour at any p o s i t i o n a f te r a crack has formed i s unknown. As w e l l , coating spec i f i c a t ions indicated that viscous flow would occur during a slow 14 t e s t to such an extent that s t r a i n i n g due to creep would not be v i s i b l e . The cracking process i s i r r e v e r s i b l e ; shock loadings r e s u l t i n g from settlement of the apparatus or tearing of a few f i b r e s near the knot would cause permanent cracking although the board might rebound e l a s t -i c a l l y . F i n a l l y , the coating i s opaque and the cracks r e s u l t i n g from even extreme loadings were found not to be r e a d i l y v i s i b l e . Using moire grids, contours of displacement can be obtained d i r e c t l y and strains,can be calculated. Some problems were that i t would have been l o g i s t i c a l l y very d i f f i c u l t to maintain a reference g r i d and that r i g i d body rotations r e s u l t i n g from the unsymmetric curvature of a beam would appear as displacement fringes and would have to be subtracted from the r e s u l t s . In one attempt f o r t h i s study, a lithographer duplicated two hundred l i n e per inch screens onto transparent s t r i p p i n g f i l m . Ideally the grid would have separated from the f i l m a f t e r gluing so that only a matrix of l i n e s would have remained on the board. Two problems arose however. F i r s t , the f i l m did not s t r i p cleanly, and second, the s p e c i a l s t r i p p i n g f i l m cement caused the wood to expand i n ridges. Undercoatings were used to prevent the glue moisture from penetrating the wood, but a l l waterproof varnishes crack-ed on loading thereby a l t e r i n g the character of the specimen face. The method f i n a l l y used was the most d i r e c t . 15 A two-dimensional gr id of .013 inch diameter shallow holes spaced one-half inch centre to centre was punched into the surface of a board, and the distances between holes were measured before and a f ter loading. During s t r a i n i n g of course the holes were deformed,but i f the deformation was assumed to be symmetric (a reasonable assumption over .013 inch) then the centre of the hole was a su i tab le point for measurement. A greater problem was that s ince the diameters were much larger that the displacements occurring between them, shear s t ra ins c a l -culated from distances on the board face lacked s i gn i f i cance although the normal s t ra ins were meaningful. Advantages of t h i s method were that r i g i d body motions could be ignored, displacements at each stress l e v e l could be e a s i l y recorede on f i l m , and the measurement process d id not a f fec t the sample once the holes had been punched. An experimental study of c lear grained mater ia l was not under-taken because only the surface mater ia l was v i s i b l e , and while the presence of a knot can be guaranteed through the thickness of a board, the absence of gra in i r r e g u l a r i t i e s cannot. 2.3 Specimen Preparat ion To minimize the number of unknowns i n r e l a t i n g t h i s work to that of Madsen, the aspect r a t i o of the 3 4 specimens was made s i m i l a r to that of a 2x6. ' Pa ir s of specimens as near ly i d e n t i c a l as poss ib le were examined, one at a fast and one at a slow loading r a t e . Specimens 16 . 6 2 5 inch by 2 . 5 0 0 inches having an aspect r a t i o of . 2 5 0 (as. compared to . 2 7 3 for a 2 x 6 ) , made by s p l i t t i n g 2 x 6 ' s as i n F i g . 2 . 1 , were used. They were 84 inches long with a s ing le approximately semic i rcu lar edge knot of . 5 t o 1 . 5 inch radius i n the middle. The knot had to extend through the board with minimal change i n diameter from one side to the other and have i t s l o n g i t u d i n a l axis orthogonal to the faces of the specimen. To s impl i fy the l a t e r modell ing of the d i s c o n t i n u i t y on the computer, the knots chosen were as simple as poss ib le with apparently smooth flowing gra in around them. The 2x6*s were cut lengthwise to produce a sec t ion approximately 2 . 7 5 inches by 1 .50 inches so as to create a h a l f knot near the middle of one edge. This board was then jointed before being run lengthwise through a saw to become two sections 84; inches by 2 . 7 5 inches by . 6 2 5 i n c h , which were mechanically planed to the f i n a l s i z e . An eight inch long sect ion i n which the knot was centred was then f i n e l y sanded. A brass g r i d containing one hundred and e ighty- f ive holes was clamped over t h i s sec t ion and the . 0 1 3 inch diameter holes were punched into the specimen using a s p e c i a l t o o l designed to ensure that penetrat ion was perpendicular to the face of the board and of approximately . 0 2 0 inch depth. The pat tern of holes i s i l l u s t r a t e d i n F i g . 2 . 2 . 2 .4 Loadings To be consistent with previous work and because S p e c i m e n P r e p a r a t i o n . F i g . 2 - I S p e c i m e n C o n t r o l Po in t L a y o u t . F i g . 2 - 2 19 i t was the simplest type of arrangements, the specimens 2 , 3 were tested i n uniform bending. F a i l u r e stresses at the extreme f ibres of approximately 4 0 0 0 p s i were expected and at leas t f i v e increments to f a i l u r e were des i rab le i f intermediate s tress leve l s were to be examined. Stress increments of about 5 0 0 0 p s i on the gross sec t ion were therefore used. Rates of loading were selected so that f a i l u r e could be expected i n e i ther approximately eight minutes or eight days. These r a te s , i f uniform would have been equivalent to 4 8 0 psi /minute and 480 ps i /day (or . 3 3 p s i /minute ) , producing a r a t i o of loadings of 1440 to 1. This r a t i o of loadings w i l l be ca l l ed the rate factor throughout t h i s study. A pre l iminary set of specimens" was tested to f a i l u r e i n order to see the type of r e su l t s that would occur. When they were analyzed i t became apparent t h a t ' f u r t h e r tests would be required i n order to e s tab l i sh s i g n i f i c a n t results*:,. The analys i s however required monopolization of laboratory equipment. A t o t a l of three sets of specimens was loaded to f a i l u r e . 2 . 5 Instrumentation A reference frame, cons i s t ing of a t h i n aluminum border onto whose perimeter^were glued s t e e l ru les marked i n hundredths of an i n c h , rested on top of the specimen on a rocker and a r o l l e r so that the bending 20 was not res t ra ined and the te s t region was surrounded as i n F i g . 2 . 3 . The specimens were photographed onto black and white f i l m before and a f ter every load increment by an Hasselblad E.L.M. s ingle lens r e f l ex camera with a 50 mil l imetre wide angle lens onto 2 . 2 5 inch square negatives. The camera was set on a t r ipod with three dimensional contro l so that i t s platform could be maintained h o r i z o n t a l while i t was lowered to match the v e r t i c a l def lexions of the specimens. This kept the negative face of the camera p a r a l l e l to the face of the board. 2 . 6 Test Set-Up The specimen, reference frame and loading pan were set up as i n F i g . 2 . 4 . The specimen board was supported on metal semic irc les at each end and was r e s t -rained from l a t e r a l motion by plywood forks attached to a f ixed support mechanism. The forks were spaced at s ixteen inches centre to centre and were s u f f i c i e n t l y deep to prevent i n s t a b i l i t y type f a i l u r e s of the specimen. Loads were applied by twenty-five pound lead ingots l a i d on the loading pan shown i n F i g . 2 . 4 so that a uniform bending moment resul ted over the th ir ty- two inch long sec t ion i n which the knot was centred. 2 . 7 Analys i s The 2 . 2 5 inch square negatives of the test region were enlarged so that the specimen i n the photograph was approximately twice l i f e - s i z e . The enlargements were - 3 REFERENCE FRAME 22 F i g . 2-4 TEST SET-UP 2 3 then placed on a Browne and Sharpe Va l ida tor measuring machine equipped with a microscope (F ig . 2 .5) so that the X- and Y- coordinates of the centre of each hole could be measured with an accuracy of . 0 0 0 2 inch and a p r e c i s i o n of . 0 0 0 1 inch . The hole centre was located by an averaging of two readings as i n F i g . 2 . 6 . Because of the small displacements occurr ing between any hole and those adjacent, only the pre l iminary photo and the f i n a l photo before v i s i b l e cracking were usefu l for numerical a n a l y s i s . The d i f f e rent stress l eve l s t h i s procedure produced for each p a i r of photographs was j u s t i f i e d because the knots var ied i n s ize and the stresses ca lculated from the gross sect ion only spproximated the true stresses near the knot. The stress l eve l s used are shown i n F ig s . 2 . 7 » 2 . 8 and 2 . 9 . C a l i b r a t i o n was performed on the Va l ida tor machine by comparing the distances on the reference frame scales with the true d i s tances . A comp-uter program was wr i t ten which, given the photographic coordinate data , ca lculated s t ra ins i n both the X and Y d i r e c t i o n s by d i v i d i n g the ca l ibra ted displacements by the true distances between the centres of holes . The average s t r a i n from the end of each row or column to each point was a lso ca lculated but was not found to be a useful parameter. Because of f i l m over-exposure, the pre l iminary tes t s did not produce r e su l t s as complete as those pro-duced by l a t e r t e s t s . The former did however show i n VALIDATOR MACHINE Y ( Y „ r.) x C X Z > \ ) X F i g . 2-6 ' PHOTOGRAPH COORDINATES X STRAINS- FAST TEST 4800 PSI X STRAINS - SLOW TEST 1920 PSI Film Overexposed \ Y STRAINS - FAST TEST 4800 PSI Y STRAINS - SLOW TEST 1920 PSI SPECIMEN P TENSION SHOWN SHADED Fig. 2-7 ON Y STRAINS - FAST TEST Y STRAINS - SLOW T E S T 2880 PSI J 2880 PSI SPECIMEN 2 TENSION SHOWN SHADED Fig. 2 - 8 29 F i g . 2.7 that the hypothesis of an expanding tension region perpendicular to gra in adjacent to a knot might have v a l i d i t y . The pre l iminary tes t s were useful as we l l i n demonstrating how the experimental r e su l t s could best be organized for reduct ion on the computer. The remaining four tests v/ere performed and analyzed according to the method explained above. The s t ra ins i n the X and Y d i rec t ions were p lo t ted independently for each t e s t , and the fast tes t re su l t s were compared with those for the slow te s t s . Attempts were i n i t i a l l y made to p l o t contours of s t r a i n magnitude, but fo l lowing an error ana lys i s i t was found that the s t r a i n magnitude at descrete points had less s i gn i f i cance than the s izes of the regions i n tension and compression, which were consistent through the error study. Because of errors inherent i n the method, the ca lculated s t ra ins represented a set of numbers which was useful and consistent although i n d i v i d u a l s t r a i n magnitudes might not be s t r i c t l y accurate. Stra ins between two points were p lo t ted at the midpoint of the s t ra ight l i n e j o in ing them, and l i n e a r i n t e r p o l a t i o n was used i n separating the tension from the compression regions . Some d i s c r e t i o n was used i n smoothing the curves. F ig s . 2 . 7 » 2.8 and 2.9 represent i d e n t i c a l faces of the specimens for both the fast and the slow tes t of each set . It can be noted from F i g . 2.1 that i f the photographs had been used d i r e c t l y , mirror images of the s t r a i n f i e l d s would be shown. This cor rec t ion was 30 made i n t e r n a l l y by the computer program. . Before d i scuss ing the r e s u l t s , some of the uncer ta int ie s i n the analys i s should be mentioned. The dots on the photographs had poor ly defined edges so that they could not be trapped as p r e c i s e l y as i n F i g . 2 . 6 . After f i n i s h i n g the measurements on each photograph how-ever, several points were remeasured and the pos i t ions of t h e i r centres were found to be i n agreement to within .001 to .003 inches of the org ina l readings. Any incremental error as readings were taken from one s ide of the photo-graph to the other was minimal s ince a given point was compared only with those immediately adjacent to i t , and the pos i t ions of a l l points were independently re l a ted to the reference frame sca le . Another poss ib le error arose from the measurement of displacements i n the X d i r e c t i o n along s t ra ight l ine s rather than along curved l ine s p a r a l l e l to an assumed neutra l ax i s . This problem was examined and the di f ferences between the method used and a more accurate one were found to be n e g l i g i b l e i n even the most severe cases. The camera was set up only twenty inches h o r i z o n t a l l y form the specimen s c that wide var i a t ions i n c a l i b r a t i o n were observed over the area of the photo-graph and the l i n e a r in terpo la t ions used i n c a l i b r a t i o n s may not have been s t r i c t l y accurate. As w e l l , the photo-graphic enlargement process was carr ied out by a commercial establishment which, although doing everything poss ib le to 31 maintain accuracy, could not guarantee the p r e c i s i o n . A l l of these uncerta int ies were minor hut could have had some effect on i n d i v i d u a l s t r a i n magnitudes. Any changes measured between the fast and slow tests were s i g n i f i c a n t however s ince a l l of these considerations were consistent for a l l t e s t s . 2.8 Results A summary of the specimen f a i l u r e stresses and of the stresses on the gross sec t ion at which the s t ra ins were measured from the photographs i s presented i n Table 2.1. The photographs measured were those showing the board having only the weight of the loading frame appl ied and those at the maximum load before v i s i b l e cracking . The l a t t e r were chosen because the largest poss ib le d i sp lace-ments were des irable i n order to give the greatest s ign i f i cance of r e s u l t s . Measuring v i s i b l y cracked mater ia l involved discontinuous displacements and would have precluded comparison with simple computer simulations as we l l as with s t r u c t u r a l material , before f a i l u r e . In des ign, the important stress l e v e l i n a mater ia l i s that which can be accepted p r i o r to f a i l u r e . Specimen P had d i f f e rent loading ra te s , a d i f f e rent loading rate factor (as defined above), and a much smaller knot than d id the specimens of ser ies 2 and J. The P ser ies r e su l t s indicated that the l o g i s t i c s of ad just ing the camera required a slower fast t e s t , and that a l a rger knot 32 TESTING SCHEDULE Specimen Loading Rate F a i l u r e Stress on Gross Section p s i Stress Increment on Gross Sect ion Analyzed p s i P - l 480 p s i 15 sec 4800 4800 P-2 480 p s i 24 hrs 2880 1920 2-1 480 p s i min 3360 2880 2-2 480 p s i 24 hrs 2880 2880 3-1 480 p s i mm 2880 1920 3 - 2 480 p s i 24 hrs 3360 1920 Table 2.1 33 was required i f s i g n i f i c a n t displacements were to be e a s i l y measured. The s t ra ins i n the X d i r e c t i o n ( i . e . perpendicular to g ra in immediately to the l e f t and r i g h t of the knot and approximately p a r a l l e l to gra in elsewhere) ranged from approximately - . 0 1 5 0 to + . 0 2 0 0 . The average X- s t ra ins i n a l l regions were of s i m i l a r magnitude although pockets of high t e n s i l e and compressive s t ra ins appeared near the knots. The main di f ference i n X- s t r a i n magnitudes from fas t to slow tests was a strong tendency toward increased tension (or reduced compression) i n the l a t t e r . The s ize of t h i s s t r a i n change var ied from zero to about +.0020 although most of the regions enjoyed r e l a t i v e l y small a l t e r a t i o n s of the order of + .0050. The area around the knot d id not show any more of a d i f ference i n s t r a i n magnitudes i n the X d i r e c t i o n than d i d the re s t of the board when the fast and slow ser ies were compared. The s t r a i n magnitudes i n the Y d i r e c t i o n ( i . e . i n general perpendicular to g r a i n above the knot and some ^distance to the sides of i t , and approaching p a r a l l e l to gra in immediately beside the knot) were of the order of - . 0 1 5 0 to +.0150 i n fast t e s t s , - . 0 2 0 0 to +.0200 i n one slow tes t (specimen 2) and - . 0 1 0 0 to +.0070 i n the other slow tes t (specimen 3). The di f ference i n s t r a i n magni-tudes was about three times greater for specimens 2 than for specimens 3» i n d i c a t i n g perhaps that the creep ef fect perpendicular to gra in was grea t ly accented by a higher 34 stress l e v e l . As w e l l , i n specimen 2 shown i n F i g . 2.8, a l l of the di f ferences were towards t e n s i l e s t ra ins while specimen 3, shown i n F i g . 2.9, had some of i t s s t ra ins approach compression for the slow tes t as compared to the fast one. The general tendency for specimen 3 was how-ever toward increased tension i n the Y d i r e c t i o n . For both ser ies 2 and 3 the s t ra ins i n the Y d i r e c t i o n i n the middle t h i r d of the tes t r eg ion , where the knot was loca ted , were 75 per cent higher than were those far ther away. This indicated that the presence of a knot g rea t ly accents the s t r a i n i n g perpendicular to g r a i n . A l l ser ies showed a marked increase for slow tests over fast tests i n the magnitudes of the t e n s i l e s t ra ins and i n the s i ze of the t e n s i l e region i n the Y d i r e c t i o n . The areas of the t e n s i l e regions i n both the X and Y d i rec t ions i n the te s t area were measured using a planimeter on F ig s . 2.7? 2.8 and 2.9. The r e s u l t s are presented i n Table 2.2. The rate fac tor i s the r a t i o of the loading rate i n the fas t case to that of the slow ease. Tests P were not included i n the averages because of the d i f f e r e n t loading r a t e s , the smaller knot and the smaller number of points ava i l ab le for the c a l c u l a t i o n of s t r a i n s . Although the t e n s i l e s t r a i n area i n the X d i r e c t i o n experienced subs tant ia l magni f icat ion, i t appeared that the majority of r e l axa t ion took place i n the Y d i r e c t i o n . Relaxation i s defined here as a tendency 35 TENSION ZONE AREAS Specimen Rate Factor Fast Test Per cent Tension Slow Test Per cent Tension Magnif icat ion % Slow Tension % Fast Tension P 5760 54 8 .15 2 1440 39 82 2 .10 3 1440 57 65 1.14 Average magnif icat ion of 2 and 3= 1 . 6 2 Table 2 . 2 A X D i r e c t i o n Specimen Rate Factor Fast Test Per cent Tension Slow Test Per cent Tension Magni f icat ion % Slow Tension % Fast Tension P 5760 14 66 4 . 7 1 2 1440 23 86 3-74 3 1440 34 62 1.82 Average magnif icat ion of 2 and 3- 2 .78 Table 2 . 2 B Y D i r e c t i o n 36 toward i n e l a s t i c s t r a i n i n g under constant load. It does not imply that l o c a l stresses are constant. Examining F ig s . 2 . 7 , 2.8 and 2 . 9 for s t ra ins i n the X d i r e c t i o n , an increase i n the s ize of the t e n s i l e s t r a i n zone adjacent to the knot can be seen for the slow tes t s . To the sides of each knot t h i s s t r a i n i n the X d i r e c t i o n has a sub-s t a n t i a l component perpendicular to g r a i n . The t e n s i l e s t r a i n f i e l d areas i n the Y d i r e c t i o n were s u b s t a n t i a l l y larger for a l l specimens i n the slow tests than i n the fast tests as can be seen i n Table 2 . 2 . Further , the f i e l d s became more uniform i n tension on the knot per iphery. That i s , the t e n s i l e zone tended to surround the knot rather than just abut onto i t i n p laces . The majori ty of the tension perpendicular to g ra in zone increase occurred above the knot and along the edges of the board. By combining the tension f i e l d s i n the X d i r e c t i o n beside the knot and i n the Y d i r e c t i o n above i t a tremendous growth i n the s i ze of the region i n tension perpendicular to gra in can be seen for the slow tests over the fast t e s t s . It was unfortunate that because of the small displacements observed i t was impossible to examine intermediate stress l eve l s to determine the stresses at^which the majori ty of the s t r a i n i n g perpendicular to gra in took p lace . The s t r a i n diagrams seem to show that the laws of equi l ibr ium i n bending have been v i o l a t e d . It was not c lear exact ly what caused t h i s ef fect but i t was seen i n 3 7 a l l cases. Most l i k e l y , the s t r a i n pat tern through the wood var ie s somewhat from that v i s i b l e on the surface so that o v e r a l l a t y p i c a l bending s t r a i n d i s t r i b u t i o n i s present. This does not a f fect the s i gn i f i cance of the measured increases i n s t r a i n perpendicular to the surface g r a i n . 2 . 9 F a i l u r e Modes The r a p i d l y loaded specimen of ser ies P f a i l e d away from the knot, but the f i ve others broke adjacent to i t . I n i t i a t e d usua l ly by perpendicular to g ra in cracking at the adverse gra in slope adjacent to the knot, f a i l u r e occurred i n one of two ways. The f i r s t , which happened i n fast t e s t s , involved a rap id f r ac tur ing of mater ia l so that the board was broken explos ive ly as i n F i g . 2.10. The second, occurr ing i n the slow t e s t s , was preceded by a large amount of t e n s i l e s t r a i n i n g perpend-i c u l a r to g ra in above the knot. This caused a crack or cracks to open up above the knot, and t h i s crack gradual ly spread along the gra in with further app l ica t ions of load u n t i l e i ther a shear type f a i l u r e occurred or the cracked gra in reached the edge of the board. There was v i s i b l e cracking for some time before f i n a l f a i l u r e . An example of t h i s second mode i s shown i n F i g . 2.11. These two types of f a i l u r e re in force the suspic ion that s ince d i f f e rent modes of f a i l u r e are l i k e l y to occur for d i f f e r e n t rates of load ing , the effect of a knot or other d i s -Fig. 2-10 RAPID FAILURE MODE 39 CRACKS PROPPED OPEN BY SPACERS F i g . 2-11 SLOW FAILURE MODE 4 0 cont inu i ty on the strength w i l l a lso vary with the rate of loading . 2 . 1 0 Summary The f i r s t part of the two part problem defined i n sec t ion 1.3 has now been invest igated within the scope of t h i s present research. The t e n s i l e s t r a i n i n g per-pendicular to gra in i s s i g n i f i c a n t l y greater for slow loadings than for fast loadings i n both magnitude and the s ize of the area which i s affected. . The majority of re lax-a t ion occurred perpendicular to gra in above the knot, i n d i c a t i n g that the presence of a gra in i r r e g u l a r i t y i n -creases the amount of s t r a i n i n g and therefore the c a p a b i l i t y for stress r e d i s t r i b u t i o n . That some smoothing of stress r a i s i n g i r r e g u l a r i t i e s occurs with long durat ion loadings i s indicated by the greater uniformity of the shape of the t e n s i l e s t r a i n zones than with fast loading . The evidence of t h i s smoothing effect i s further re inforced by the • so f te r ' or more gradual type of f a i l u r e experienced by slowly loaded specimens. 41 CHAPTER 3 INFLUENCE OF STIFFNESS PERPENDICULAR  TO GRAIN 3.1 Introduction The work of the previous chapter showed that cons iderably more t e n s i l e s t r a i n i n g perpendicular to gra in occurred i n slow tests than i n rapid te s t s . The next stage i n inves t iga t ing the hypothesis of sec t ion 1.3 was to invest igate whether or not a time dependent s t i f fnes s perpendicular to g r a i n , such as found i n pure specimens , would produce s t r a i n f i e l d s s i m i l a r to those found experimently around a knot. The procedure chosen for the inves t iga t ion of the effect of s t i f fness perpendicular to gra in was the f i n i t e element method. Plane stress behaviour i n bodies of i r r e g u l a r shape can be thus programmed for so lu t ion on a computer. In t h i s way as w e l l , g ra in s tructure can be modelled and then assembled to form the f i n a l s tructure of the mater ia l . 42 3.2 The F i n i t e Element The basic f i n i t e element theory i s we l l known^ so that only a b r i e f d e s c r i p t i o n need be presented here. Consider a body subjected to stresses and displacements along parts of i t s boundary, and then consider d i v i d i n g t h i s domain into a number of subdomains with nodes along t h e i r edges. By deducing how each subdomain or element behaves i n terms of displacements and forces at these i nodes, and then combining a l l of these elements by matching degrees of freedom and summing corresponding forces at the nodes a so lu t ion can be obtained. This produces the standard s t i f fnes s problem and can be solved by t r a d i t i o n a l methods. In order to model the in-plane behaviour of a beam, plane stress f i n i t e elements were used. The con-s t r a i n t s on the p a r t i c u l a r element for t h i s problem were that s t i f fnesses must be amenable to a l t e r a t i o n p a r a l l e l and perpendicular to the gra in of the wood and not just to g loba l axes, s t r ings of elements must be able to change d i r e c t i o n to model the curvature of the g r a i n , the elements must be able to represent tens ion , comp-res s ion and shear s tresses , and there must be s u f f i c i e n t accuracy to ensure that the changing of a s ing le e l a s t i c modulus w i l l produce s i g n i f i c a n t and consistent r e s u l t s . Constant s t r a i n t r i ang le s were i n i t i a l l y used but were found to be gross ly inaccurate when tested on a can t i l ever beam. A l i n e a r s tress orthotropic s i x node t r i a n g l e with ^3 di f f e rent e l a s t i c modulae p a r a l l e l and perpendicular to one edge was therefore se lected . The der iva t ion of th i s element was carr ied out and the elements were tested i n a few simple cases. Deta i l s are presented i n Appendices A and B, and a copy of the computer program i s enclosed as Appendix C. 3.3 The Problem To obtain s a t i s f ac tory re su l t s and to model the gra in curvature as accurate ly as po s s ib l e , i t was des i rable to examine a f ine mesh of f i n i t e elements. Because of the expense involved however, a parametric analys i s was performed on a f a i r l y coarse mesh i n order to se lect the optimal modulae for input to a l arger problem, to invest igate how some of the e l a s t i c parameters a f fect the s t r a i n d i s t r i b u t i o n around a knot, and to obtain some pre l iminary re su l t s for comparison with the experimentally obtained s t r a i n d i s t r i b u t i o n s . I t was beyond the scope and the purpose of th i s inves t i ga t ion to t r y to re f ine a f i n i t e element that would accurate ly r e f l e c t the behaviour of wood. One stage modell ing only was used. S impl i f i ca t ions l i k e assuming the same e l a s t i c modulae for wood i n tension and compression and ignoring the effects of l o c a l f i b r e tear ing were made. The sole purpose here was to examine the effect of one set of orthotropic s t r e s s - s t r a i n modulae on the o v e r a l l s t r a i n pat tern of the model described below. 44 Madsen's recent work has shown that the time dependent s t i f fnes s (combined e l a s t i c and i n e l a s t i c s t ra in ing) decreases with the rate of loading. Tests on mater ia l containing knots, adverse gra in s lope, and other d i s c o n t i n u i t i e s showed l i t t l e v a r i a t i o n i n e i ther the modulus p a r a l l e l to gra in or the bending s trength, with the durat ion of load a p p l i c a t i o n for those boards i n a sample which f a i l at the lower per cen t i l e s of s trength. - The time dependent s t i f fnes s perpendicular to gra in d id however exhibi t a very marked decrease as the durat ion of loading was lengthened. To repeat the o r i g i n a l hypothesis then, large magnitude t e n s i l e s t r a i n i n g perpendicular to gra in might provide a mechanism for stress r e d i s t r i b u t i o n around d i s c o n t i n u i t i e s so as to minimize t h e i r ef fect on the strength of the mater i a l . Decreasing the e l a s t i c s t i f fnes s perpendicular to gra in i n the f i n i t e elements was used to model the effects of long durat ion loadings on the stresses and s t ra ins around a simulated knot. Since i n the parametric analys i s of an e l a s t i c body a l l changes are r e l a t i v e and l i n e a r , i t was found usefu l to a r b i t r a r i l y f i x the ap-parent modulus p a r a l l e l to gra in at 1.8x10^ p s i and the Poisson's r a t i o for s t r a i n i n g perpendicular to gra in r e s u l t i n g from a p p l i c a t i o n of load p a r a l l e l to gra in at .30. The former i s bel ieved to decrease by only 10 to 20 per cent during the time periods examined here, and r e l i a b l e information could not be found about the l a t t e r . 4 5 Table 3*1 shows the input data used for the main part of the inve s t i ga t ion . The subscripts x and y on E and E x y r e fe r to the element x and y axes, not to the g loba l X and Y axes. A ser ies of test ca lcu la t ions was run i n which the r a t i o of Poisson's r a t i o s 2J /U was var ied yx xy and i n which the shear modulus was increased by a factor of 10. In the 'slow-G reduced' test c a l c u l a t i o n s , the shear modulus was reduced by 30 per cent to approximately model to f indings of a previous study. It was recognized that by a l t e r i n g the apparent Young's modulae without changing the Poisson's r a t i o s , the laws of conservation of energy and i n p a r t i c u l a r the r e c i p r o c a l theorem were v i o l a t e d . This problem was ignored for two reasons. F i r s t , i n the r e a l mater ia l i t was unclear how much each Poisson's r a t i o would change and second, the reduced apparent Young's modulus perpend^-i c u l a r to gra in was i n the r e a l case l i k e l y caused by creep, an i n e l a s t i c e f fec t , so that conservation of energy d id not apply. As w e l l , a ser ies of tests showed a l t e r -ations i n the Poisson's r a t i o to have had n e g l i g i b l e ef fect on re su l t s i n the program. Using a high power microscope, the gra in around f i v e knots (that i s , twenty quarters of knots) was t raced . Using these t r a c i n g s , the distance i n r a d i i from the knot centre to the beginning of gra in curvature, and the angle of g ra in at s p e c i f i c normalized coordinate locat ions were measured for each case. After averaging 46 PARAMETRIC ANALYSIS DATA Case Name Ex p s i %. p s i ^ x y A* G p s i Ex/Ey Isotropic 1 . 8 x l 0 6 1 . 8 x l 0 6 . 3 0 • 30 6 . 9 0 x 1 0 ^ 1 Fast 1 . 8 x l 0 6 , 9 x l 0 5 . 0 5 .30 1 . 1 5 x l 0 5 20 Median 1 . 8 x l 0 6 . 3 x l 0 5 . 0 5 .30 l . 1 5 x l 0 5 60 Slow 1 . 8 x l 0 6 . I x l O 5 . 0 5 . 3 0 1 . i 5 x i o 5 180 Slow G red 1 . 8 x l 0 6 . l x l O 5 . 0 5 . 3 0 . 8 5 x l 0 5 180 Young's modulus p a r a l l e l to gra in Young's modulus perpendicular to gra in Poisson's r a t i o for s t r a i n i n g p a r a l l e l to gra in caused by stress appl ied perpendicular to gra in Poisson's. r a t i o for s t r a i n i n g perpendicular • to gra in caused by stress appl ied p a r a l l e l to gra in v V ^ x y ~ Table 3-1 47 these parameters, the s ing le quarter knot which most c l o s e l y resembled the average was used for model l ing. Unfortunately the gra in densi ty could not be d i r e c t l y measured. When l ine s were made continuous to fol low the known d i rec t ions of gra in at every point however, a gra in densi ty re su l t ed . F i n a l l y , an element mesh was drawn into the diagram and the r e s u l t was a problem cons i s t ing of t h i r t y - f i v e elements and eighty-eight nodes. The r e s u l t i n g model i s shown i n F i g . 3.1. This was small enough to be solved i n the core of the computer and was s u f f i c i e n t l y inexpensive to run that i t was su i tab le for a parametric study. F i g . 3.2 shows t h i s mesh. The f i n i t e elements were placed into the modelled gra in by forc ing a ce r t a in edge of each element (the 1-2 edge i n Appendix A) to be oriented p a r a l l e l to the gra in boundary. The e l a s t i c modulae of each element were oriented p a r a l l e l and perpendicular to th i s edge. The edges p a r a l l e l to gra in have double l ine s i n F i g . 3.2. The loadings were of a r b i t r a r y magnitude and were applied i n both bending and pure tens ion . The examination i n bending was for comparison with experimental r e su l t s and that i n tension was to show a l i t t l e more c l e a r l y the effects of parametric manipulations s ince the appl ied s t r e s s / s t r a i n gradient would be absent. Since the problem was programmed as l i n e a r e l a s t i c , a pure compression loading would have produced exactly the same re su l t s as for pure tension but with the signs reversed. >Fig. 3--1 TYPICAL GRAIN PATTERN 50 Symmetry was employed by forc ing the nodes above the r i g h t hand edge of the knot to have zero h o r i z o n t a l d i s -placement. V e r t i c a l support was supplied by r e s t r a i n i n g one of the nodes at the l e f t hand side of the mesh. The boundary condit ions did not provide any r e s t r a i n t along the bottom edge of the model. 3 . 4 Results of the Parametric Study The ca lculated re su l t s obtained d i r e c t l y were s t ra ins i n the g loba l X and Y d i rec t ions at each node. By r o t a t i n g a l l s t ra ins into these axes an average s t r a i n was obrained that could be presented i n the same form as the experimental r e s u l t s . Using the s t ra ins i n the g loba l coordinate d i r e c t i o n s , p lo t s showing the tension and compression regions of the model were drawn. These were important i n showing those regions i n which tension per-pendicular to gra in was produced and i n which therefore large scale stress r e d i s t r i b u t i o n would be encouraged. The r e l a t i v e magnitudes of the s t ra ins were a lso examined. Figures 3.3> 3«4 and 3 . 5 show the pos i t ions and s izes of the tension and compression: regions for bending moment loading . F i g . 3 . 3 was produced for comparison with F ig s . 2 . 7 . 2 . 8 and 2 . 9 . It may be observed that the t e n s i l e s t r a i n f i e l d s i n the g loba l Y d i r e c t i o n grew l a t e r a l l y as the r a t i o of E / E was increased. S i m i l a r l y , x y the s t ra ins i n the g loba l X d i r e c t i o n tended to become more t e n s i l e beside the knot as E / E increased. 1 1 (a ) 'Fast ' X Strains E = 9 » IO5 ( b) 'Slow' X S t r a i n s E =. I » I 0 5 ( c ) ' Fast ' Y St ra ins E,=.9 , IO5 (d I 'S low ' Y St ra ins E,= . I . IO 5 (a) , (b) , (c) , (d) E,= 1.8 < 10 ' „ = 0 5 »,.= .30 G =1.15 - 10 s TENSION SHOWN SHADED B E N D I N G Fig. 3 - 3 52 (a) Isotropic (b) ' F a s t ' E,= 9 . IO 5 (c )' M e d i u m ' E , = .3 , 10 (d) ' S l o w ' E , =.l . IO5 ( e ) ' S l o w - G R e d u c e d ' E y = . I « 1 0 * (a) (b) , (c) , (d) (e) E,= l . 8 . 1 0 s 1.8 . IO 6 1.8. IO 6 E , = l . 8 . 10 s ' . , = .30 .05 .05 ' , . = .30 . 30 . 30 G =6.9.10* 1.15, 10* . 8 5 . 1 0 * TENSION SHOWN SHADED BENDING X S T R A I N S F i g . 3-4 (a) I s o t r o p i c jps»>> (b) ' F a s t ' E,= . 9 , 1 0 * ( c) ' M e d i u m ' E y = .3 « 10 (d) ' S l o w ' E , =. I • 10 ( e ) ' S l o w - G R e d u c e d ' E = . I « I 05 (a) (b),(c),(d) (e) E «= 1.8 . 10 s 1.8, 10s 1.8 . IO 6 E y = l.8 » IO6 ',,= 3 0 .05 .05 ',.= .30 .30 . 30 G = 6 .9 ,10* 1.15.10* . 8 5 . 1 0 ° TENSION SHOWN SHADED BENDING Y S T R A I N S F i g . 3 - 5 54 Adjacent to the knot, the s t ra ins i n both the X and Y d i rec t ions were t e n s i l e . These trends are the same as the experiment i f one takes higher E x / E y r a t i o s to r e -present slower test loadings . However the amount of t e n s i l e s t r a i n region growth here was not as much as i n the experiments. The d i s p a r i t i e s i n the s izes of tension regions between the experiment and the f i n i t e element so lu t ion probably resul ted from the crude modelling of the problem. In r e a l m a t e r i a l , the e l a s t i c modulae need not be i d e n t i c a l throughout the specimen and may vary with changes i n the gra in dens i ty as i t curves around the knot. As w e l l , of course the modulae are not p e r f e c t l y e l a s t i c i n r e a l mater i a l . Further the di f ference i n the e l a s t i c modulae for tension and compression was not included. However the modell ing d id reproduce the experiment trends and therefore further work using t h i s model seemed j u s t i f i e d . 3.5 The E / E Ratio x y  Under bending s imulat ion , the s t ra ins i n the X d i r e c t i o n along the bottom of the board became smaller as the knot was approached, and the distance from the knot at which t h i s decrease became s i g n i f i c a n t increased with the E / E r a t i o . As the simulated knot was approached from x y the l e f t the maximum t e n s i l e s t ra ins moved away from the base of the board toward the midheight u n t i l the knot was reached at which point the maximum s t ra ins i n the X d i r e c t i o n 55 occurred again at the bottom of the sec t ion (that i s , d i r e c t l y above the knot) . This effect was accentuated as E / E was increased i n d i c a t i n g that the mater ia l x y immediately beside the knot was carry ing less load as the s t i f fnes s perpendicular to gra in was reduced, and that therefore the semic i rcular notch was becoming r e -l a t i v e l y more shallow and less of a s tress or s t r a i n concentrator. Stra ins i n the X d i r e c t i o n near the knot were much larger than those away from i t i n the same gra in i n the tension reg ion , and the s t r a i n concentration factor (comparing average s t ra ins i n the bottom gra in away from the knot with those i n the same gra in above the knot i n an ad hoc manner) increased from I . 5 6 for the i s o t r o p i c case to 2.88 for E / E = 20 to 4 . 9 5 for x y E / E = 180. That i s , reductions i n E , the apparent x y y s t i f fness perpendicular to g r a i n , produced much greater changes i n the X- s t r a i n i n the curved area near the knot than i n the s t ra ight gra in farther away. The s t ra ins perpendicular to gra in were a lso much greater near the knot than away from i t , but s t r a i n concentration factors as ca lculated for the X d i r e c t i o n could not be obtained here because at some r a t i o s of E / E many of the nodal x y s t ra ins i n the affected region were i n compression. F i g s . 3 . 4 and 3 . 5 show as wel l that the t e n s i l e strain.: f i e l d s both p a r a l l e l and perpendicular to gra in increased i n s ize as the E / E r a t i o was increased. x y With pure tension loading there was of course 5 6 no neutra l ax i s . F ig s . 3.6 and 3-7 show the tension regions loaded for t h i s case. The uniform t e n s i l e s t ra ins i n the X d i r e c t i o n at the l e f t hand edge of the model became concentrated toward the centre of the beam as the knot was approached. As i n bending however, the maxima occurred immediately above the knot. With E / E = 20 a x y compression zone existed to the l e f t and s l i g h t l y above the knot. Both the s i ze of t h i s compression zone and the magnitude of i t s s t ra ins decreased as the degree of orthotropy was increased. At the same time, the bottom edge of the board became less h i g h l y s tra ined i n the X d i r e c t i o n . The s t r a i n concentration factors as defined above were for t h i s case 2.77 for i s o t r o p i c , . 4 . 1 3 for E / E = 20 and 7.18 for E / E = 180 so that again increas ing x y x y orthotropy had greater effect on s t ra ins near the knot than on those away from i t . The s t ra ins i n the Y d i r e c t i o n were, as expected, i n uniform compression away from the knot. As the knot was approached from the l e f t , t h i s compression became l o c a l i z e d toward the centre of the board u n t i l near the knot the ent i re sec t ion went in to tens ion. Maximum s t r a i n magnitudes were found d i r e c t l y above the knot and, as for bending, the s izes of the tension f i e l d s i n both the X and Y d i rec t ions increased with E / E . x y The d i rec t ions of p r i n c i p a l s tress and s t r a i n were produced by the computer program, but the coarseness of the mesh and the small number of points at which the stresses and s t ra ins were ca lculated made i t impossible (a) Isotropic ( c ) ' Med ium' E , = 3 , 10* (a) (b),(c),(d) E«= 1 . 8 « 1 0 s 1.8 , IO6 E,= 1.8.10 s ',,= .30 .05 ',,= .30 .30 G = 6.9 . 10° 1.15x10* (b) ' F a s t ' E y=.9» IO* ( d ) ' S l o w ' E , = 1 .10* T E N S I O N S H O W N S H A D E D T E N S I O N LOADING X S T R A I N S F ig . 3 - 6 (a) I s o t r o p i c ( c ) ' M e d i u m ' E , =.3 « 10* (a) (b),(c),(d) E x= 1.8 . IO 6 1.8.10 6 E,= 18 . IO 6 ',,= .30 .05 V = 3 0 .30 G =6.9,10* 1.15, I0 5 (b) ' F a s t ' E y = 9 , IO5 ^ ^ ^ ^ ^ ^ (d) ' S l o w ' E, = 1 , I 0 5 TENSION SHOWN SHADED T E N S I O N L O A D I N G Y S T R A I N S F i g . 3 - 7 CO 59 with t h i s problem to produce a coherent p lo t of the r e s u l t s . This was done however for a l arger problem which w i l l be discussed i n the next chapter. 1 Under pure ly t ens i l e : model l ing, the p r i n c i p a l strains-were p a r a l l e l to gra in throughout the mesh except for a region to the l e f t of the knot where they seemed to tend towards being perpendicular to g ra in . This tendency and the s ize of the region increased with increas ing l eve l s of or th-otropy. The same was true for the bending moment model except that because of the appl ied s t r a i n gradient the effects were a l i t t l e more d i f f i c u l t to see d i r e c t l y . The amount of tension s t r a i n .perpendicular to g ra in increased subs t an t i a l ly as the apparent Young's modulus r a t i o was increased. The p r i n c i p a l s tress d i r e c t i o n s i n pure tens ion were p a r a l l e l to gra in throughout except near the knot and along the bottom of the beam as the knot was approached. The increase i n orthotropy tended to take stress away from the bottom of the beam and r e d i s t r i b u t e i t so that the p r i n c i p a l l ine s of tension flowed smoothly around the knot and apparently made the sec t ion approach that of a beam v/ith a gradual ly decreasing cros s - sec t ion instead of that of a beam of constant cross- sect ion having a semi-c i r c u l a r notch. This same behaviour was v i s i b l e i n the moment loading case as w e l l . 6o 3 .6 Shear Modulus Ef fect A r b i t r a r i l y increas ing the shear modulus G by a factor of 10 also produced v i r t u a l l y the same r e s u l t as with G l e f t unchanged. ' It was found i n general that the s t i f f e r the model was i n shear, the larger was the t e n s i l e s t r a i n f i e l d i n the g loba l X d i r e c t i o n . Since the region of compressive s t ra ins i n the X d i r e c t i o n more c l o s e l y approached the knot ( in the region where X- s t ra ins approach being perpendicular to grain) when G was smal l , a more marked reduct ion i n load with time might be expect-ed for materials weak i n shear. F i g . 3.8 shows these effects and indicates as wel l the danger involved i n draw-ing conclusions from such a pe r iphera l e f fec t . F i n a l l y , a larger shear modulus produced a more uniform d i s t r i b u t i o n of s t r a i n s , and. s l i g h t l y smaller s t ra ins i n the Y d i r e c t i o n . F i g . 3«9 for t e n s i l e loading shows that the tension f i e l d s i n the Y d i r e c t i o n expand for ' s low' tests with the shear modulus magnified by 10, i n the same manner as with the o r i g i n a l shear modulus. 3.7 Summary From the preceding work a few conclusions were drawn. F i r s t , as the orthotropy of the model was increased, the mater ia l on the tension edge of a bending problem adjacent to the knot became r e l a t i v e l y s t ress les s and be-came thereby a les s important part of the load carry ing mechanism. Second, as th i s was happening, the t e n s i l e 61 J (a) X S t r a i n s (b) Y S t r a i n s (c) X S t r a i n s (d) Y S t r a i n s (a ) , (b) ( c ) , (d ) E , = l.8 « IO 6 1.8 « IO 6 E y = . 1 , IO 5 .1 X IO 5 "„ =. 0 5 .05 »„= 3 0 . 3 0 G =1.15x10 5 1.15, IO 6 TENSION SHOWN SHADED B E N D I N G ' S L O W ' T E S T S H E A R M O D U L U S E F F E C T F ig . 3 -8 62 i ( a ) ' F a s t ' X S t r a i n s ( b ) ' F a s t ' Y S t r a i n s ( c ) ' S l o w ' X S t r a i n s ( d ) ' S l o w ' Y S t r a i n s (a) , (b) (c ) , (d ) E«= 1.8 . IO 6 1.8 , 10 s E,= 9 , IO* . 1 , IO 5 »»,= . 05 . 0 5 » „ = . 3 0 . 3 0 G =1. 15,10 s 1. 1 5 , 1 0 s TENSION SHOWN SHADED T E N S I O N S H E A R M O D U L U S E F F E C T F i g . 3 - 9 63 s t r a ins i n the Y d i r e c t i o n (roughly perpendicular to grain) were increas ing i n t h i s reg ion and could have been a factor i n a l lowing s tress r e d i s t r i b u t i o n . Therefore t h i r d , the ef fect of the knot as a s tress r a i s e r was being decreased. This bears out the r e su l t s of previous studies which have shown construct ion grade mater ia l not to experience s i g n i f i c a n t decreases i n strength with time.^»3 A r a p i d l y loaded beam f a i l s at a knot as i f i t had been notched, whereas slowly loaded beams have been found to f a i l i n shear or at some adverse slope of gra in away from a knot. These d i f f e rent f a i l u r e mechanisms imply that mater ia l with defects cannot be considered to be the same as c lear mater ia l into which a notch has been cut . Slope of gra in apparently magnifies the t e n s i l e s t r a i n perpend-i c u l a r to g ra in and contributes to a smoothing of d i s -c o n t i n u i t i e s . The r e su l t s indicated that the, semic i rcu lar notch i n the model tended to behave as a broader defect , avoiding the very high stress concentrations which occur at sharp corners . 64 CHAPTER 4 CIRCULAR HOLE IN A FINITE PLATE  HAVING GRAIN TYPE ORTHOTROPY 4.1 Introduction In Chapter 3» a f i n i t e element inves t iga t ion was made of the effects of orthotropy on the stresses and s t r a ins around a semic ircular notch on one edge of a beam. The model was subjected to uniform t e n s i l e stresses i n one case and to a l i n e a r l y varying s tress gradient i n another. These simulations were re la ted to experimental work with an edge knot. In r e a l mater ia l however, the knot or other d i s c o n t i n u i t y may occur away from an.edge, As an approach to a more general case, a c i r c u l a r hole i n a f i n i t e p la te was modelled. Curving gra in surrounded the hole so that the problem was d i f f e rent from that of a hole i n a mater ia l having orth-otropy r e l a t i v e to g loba l X and Y axes. The elements were t r i angu la r and had t h e i r e l a s t i c modulae p a r a l l e l and perpendicular to one edge. This edge (the 1T2 edge of F i g . A . l ) was oriented along the l o c a l gra in l i n e . The model of th i s inves t i ga t ion was more deta i led than 65 the 35 element s imulat ion of F i g . 3 - 2 . 4.2 The Problem The problem conf igurat ion was s i m i l a r to that of F i g . 3 . 1 . F i g . 4.1 shows the model used. It has 419 nodes and 188 elements. Double symmetry was employed by fo rc ing the nodes along the bottom of the f i n i t e element mesh to have zero Y displacement and the nodes along the r i g h t hand side to have zero X displacement. A consistent uniform t e n s i l e load vector of a r b i t r a r y magnitude was appl ied to the l e f t hand side of the model'. The elements were those described i n Sect ion 3*2 and Appendix A . 4.3 Pre l iminary Tests The model was f i r s t examined i n the i s o t r o p i c and E / E = 100 cases. The re su l t s were recorded i n F ig s . x y 4.2 and 4 . 3 for stresses i n the X and Y d i rec t ions r e spec t ive ly . F i g . 4.2 for stresses i n the X d i r e c t i o n shows that the f i n i t e element so lu t ion for F i g . 4.1 had good agreement with the a n a l y t i c s o l u t i o n for an i n f i n i t e p la te when e l a s t i c modulae are i s o t r o p i c . When a large orthotropy (E / E = 100 with subscripts x and y r e f e r r i n g x y to element axes) was app l ied , the region adjacent and to the l e f t of the hole went into compression i n the X d i r e c t i o n .1 This would tend to decrease the l i k e l i h o o d of t e n s i l e cracking as a prelude to f a i l u r e . In the region above the ho le , the peak stress i n the X d i r e c t i o n was great ly magnified by the a p p l i c a t i o n of orthotropy. F i g . 4-1 <. 188 , ELEMENT MODEL TENSION LOADING tSOTROPlO-ANALS-Tl^- PLATE. ISOTROPIC- FINITE ElEMEHT Bx/Ey/OO -F/NITZ ELEMENT F i g . 4-2 . EDGE STRESSES X DIRECTION 6 8 The loca t ion of the peak stress (immediately above the hole) was unchanged. F i g . 4.3 for stresses i n the Y d i r e c t i o n shows subs tant ia l di f ferences between the f i n i t e element so lu t ion of F i g . 4 .1 , and the a n a l y t i c so lu t ion for a hole i n an i n f i n i t e p l a t e . This was because the f i n i t e element model was f i n i t e i n the Y d i r e c t i o n , although i t could be considered i n f i n i t e i n the X d i r e c t i o n . The d i s t r i b u t i o n of stress was however s i m i l a r i n both cases, with peak stresses occurring i n the same loca t ions . With E / E = 100, the peak s t resses 'a long both edges of the x y model were reduced and the stresses i n the Y d i r e c t i o n above the hole o s c i l l a t e d before reaching zero at the top edge of the model. The Y stress ' immediately above the hole was non-zero, thereby apparently v i o l a t i n g equ i l ibr ium. For th i s reason, further analys i s was per-formed at E / E r a t i o s of 1,20 and 4 0 . At an orthotropic x y r a t i o of 40 i t was found that the normal stress above the hole was approximately zero. At E / E = 20, the modulae x y i n timber subjected to short durat ion loadings are approx-imately represented. 4.4 Tension Zone Sizes and Pos i t ions The e l a s t i c modulae input for the ser ies of tes t ca lcu la t ions are given i n Table 4 .1 . The G modulus for orthotropic cases was taken to represent the values found by Madsen. It was found in Madsen's tests that the shear modulus var ies very l i t t l e with the durat ion 70 E / E •xf y \ p s i E y p s i xy yx G p s i 1 100 100 . 3 0 . 3 0 38 20 100 5 .015 . 3 0 6 4o 100 2 . 5 . 0 0 7 5 . 3 0 6 E x= apparent Young's modulus p a r a l l e l to gra in E = apparent Young's modulus perpendicular to gra in ^Xy~ Poisson's r a t i o for s t r a i n i n g p a r a l l e l to gra in caused by stress applied perpendicular to gra in cJ.= Poisson's r a t i o for s t r a i n i n g perpendicular yx to gra in caused by stress appl ied p a r a l l e l to gra in Table 4 . 1 Input Data 71 of load. The s izes and locat ions of the t e n s i l e stress zones i n the X and Y d i rec t ions are shown i n F ig s . 4.4 to 4.9 i n c l u s i v e . A heavy l i n e separates regions of tension and compression, and the tension zones are ind ica ted . The numbers wr i t ten at the element corners are r e l a t i v e values of s t res s . The a r b i t r a r y scale on which the values are measured i s d i f f e rent for stresses i n the Y d i r e c t i o n than i n the X d i r e c t i o n . One uni t of stress i n the X d i r e c t i o n i s equivalent to ten uni t s of stress i n the Y d i r e c t i o n . Analagous to the re su l t s of Chapter 3 i the stresses i n the X d i r e c t i o n as shown i n F ig s . 4.4, 4 . 5 and 4.6 ind-ica te that the region immediately to the l e f t of the hole became less h igh ly stressed i n tension as the degree of orthotropy was increased. This bore out the hypothesis that i n general the mater ia l to the l e f t of the hole cont-r ibutes less to the load carrying capacity of the sec t ion as the degree of orthotropy i s increased. The stresses i n the Y d i r e c t i o n as presented i n F ig s . 4.7i ^.8 and 4.9 show that the s ize of the tension zones adjacent to the hole increase with increas ing ortho-tropy. In the region above the hole the stresses i n the Y d i r e c t i o n (approximately perpendicular to grain) are s i g n i f i c a n t l y smaller for E / E = 40 than for E / E = 2 0 . As a crude measure of t h i s , the average of a l l t e n s i l e Y s t ra ins above the hole i n the former i s 2 5 , and i n the l a t t e r i s 30 ( the s t ra ins being r e l a t i v e only ). This C N T R CLRR PLOT« 00839048. DRTfl CHECK—NEWKNOT SCRLE FACTOR-20.000 Oo C N T R •CLRR PLOT« 00839048. DflTR CHECK—NEWKNOT CNT-R CLRR PLOTS 00839048. DATA CHECK—NEWKNOT SCALE FACTOR- 20 .000 LL 78 r e d u c t i o n of s t r e s s p e r p e n d i c u l a r to g r a i n with i n c r e a s i n g o r t h o t r o p y could tend t o reduce the d e l e t e r i o u s e f f e c t of c u r v i n g g r a i n on the s t r e n g t h p e r p e n d i c u l a r to g r a i n i n wood. The s t r a i n s i n the X d i r e c t i o n as. shown i n F i g s . 4 . 1 0 , 4 . 1 1 and 4 . 1 2 i n d i c a t e t h a t as the degree of o r t h o t r o p y was i n c r e a s e d , the r e g i o n to the l e f t of the h o l e was l e s s h i g h l y s t r a i n e d i n t e n s i o n . F i g s . 4 . 1 3 , 4.14 and 4 . 1 5 f o r s t r a i n s i n the Y d i r e c t i o n show an expanding zone of t e n s i l e s t r a i n s p e r -p e n d i c u l a r to g r a i n as the degree of o r t h o t r o p y was i n -creased. As d i s c u s s e d i n p r e v i o u s c h a p t e r s , the a d d i t i o n a l s t r a i n i n g p e r p e n d i c u l a r to g r a i n could a s s i s t i n the r e d i s t -r i b u t i o n of s t r e s s e s . The magnitudes of the s t r a i n s p e r -p e n d i c u l a r to g r a i n i n c r e a s e d s u b s t a n t i a l l y as the apparent Young's modulus p e r p e n d i c u l a r t o g r a i n was reduced. 4 . 5 S t r e s s D i s t r i b u t i o n on Axes of Symmetry Graphs were produced showing the s t r e s s d i s t r i -b u t i o n s a l o n g edges BC and CD (see F i g . 4 . 1 ) of the model of t h i s chapter. D i s t r i b u t i o n s were shown f o r the f i n i t e element s o l u t i o n s of E /E equal t o 1 ,20 and 4 0 . In x y a d d i t i o n , the a n a l y t i c s o l u t i o n f o r an i n f i n i t e p l a t e with a c e n t r a l h o l e was shown i n F i g s . 4.16 to 4 . 1 9 i n c l u s - . i v e . Along edge DE (above the h ole) the s t r e s s e s i n the X d i r e c t i o n were shown by F i g . 4.16 to have a fund-a m e n t a l l y d i f f e r e n t d i s t r i b u t i o n f o r o r t h o t r o p i c than C N T R CLHR PLOT« 00839043. DflTR CHECK—NEWKNOT 6Z 08 CNTR CLIR PLOT« 00839043. DflTR CHECK—NEWKNOT SCfiLE FACTOR- 20.000 y> -< T8 CNTR CLRR PLOT« 00839048. DATA CHECK—NEWKNOT 28 CN TR CIRR PLOTH Q0839048. DATA CHECK—NEWKNOT SCALE FACTOR-20.000 tr8 86 for i so t rop i c e l a s t i c modulae. The i so t rop i c case had a r e l a t i v e l y smooth decrease i n stress from D towards E. The orthotropic cases experienced a large o s c i l l a t i o n i n stress p a r a l l e l to the Y ax i s . The i s o t r o p i c ana ly t i c so lu t ion was provided only for reference since i t was ca lcula ted for an i n f i n i t e p la te and did not apply d i r -e c t l y to the model of F i g . 4 .1 . From E / E = 20 to E v / E = x y x y 40, the change i n behaviour was minimal although the peak s tress d i d increase s l i g h t l y with increas ing ortho-tropy. In F i g . 4.17 for stresses i n the Y d i r e c t i o n on side DE, a 60 per cent decrease i n peak stress perpendicular to gra in was experienced from the i s o t r o p i c case to E / E _ = x y 20. A further 15 per cent decrease was experienced from E / E = 20 to E / E = 40. The stress d i s t r i b u t i o n was sim-x y x y i l a r for a l l cases. The reduct ion i n stress perpendicular to gra in would diminish the l i k e l i h o o d of the cracking per-pendicular to gra in that was described i n Chapter 2 as being the usual mode of f a i l u r e i n i t i a t i o n for s lowly loaded timber containing knots. Along edge BC to the l e f t of the h o l e , the mat-e r i a l was shown by F i g . 4.18 to be less h i g h l y stressed i n the X d i r e c t i o n as the degree of orthotropy was increased. In the i s o t r o p i c case, a l l of the mater ia l along the edge was i n tens ion. In going to E / E = 20 and E / E = 40, the x y J magnitudes of t e n s i l e stresses decreased and the magnitudes of t e n s i l e stresses decreased and the magnitudes of comp-89 res s ive stresses adjacent to the hole increased. F i g . 4 .19 for stresses i n the Y d i r e c t i o n along edge BC showed a s i g n i f i c a n t decrease i n peak t e n s i l e s tress perpendicular to gra in near the hole i n going from i so t rop i c modulae to increas ing l eve l s of orthotropy. This was p a r t i c u l a r l y apparent at sect ion 1-1. The smaller stresses perpendicular to gra in would i n h i b i t the opening of cracks adjacent to the hole . 4 . 6 Summary The study of th i s chapter produced re su l t s analagous to those of Chapter 3- The mater ia l to the l e f t of the hole became less h igh ly stressed i n the d i r -ec t ion of load a p p l i c a t i o n as orthotropy was increased. The s izes of the regions i n tens ion perpendicular to gra in increased with increas ing orthotropy. The most important r e s u l t of t h i s study was however the decrease i n peak stress perpendicular to gra in above and to the l e f t of the hole as the orthotropy was increased. Since the degree of orthotropy i s bel ieved to increase with the durat ion of a loading , the opening of cracks perpendicular to gra in would be i n h i b i t e d i n slow loadings . The stress reduct ion perpendicular to gra in would therefore p a r t i a l l y compen-sate i n slow tests for the reduct ion i n strength found i n c lear mater ia l by the Madison Tests . 91 CONCLUSIONS The region surrounding an edge knot i n a timber beam experienced greater t e n s i l e s t r a i n i n g perpendicular to g ra in with long durat ion loadings than with short durat ion loadings . Both the s ize of the t e n s i l e s t r a i n zone perpendicular to gra in and the magnitudes of i t s consituent s t ra ins increased for slower te s t s . In a d d i t i o n , the s t r a i n f i e l d became more uniform. It was found, through a computer s imulat ion of the p h y s i c a l t e s t s , that a reduct ion of s t i f fnes s perpendicular to the l ine s of gra in would reproduce the trends found experimental ly. The area of the tens ion f i e l d perpendicular to gra in increased with increas ing orthotropy. A de ta i l ed modell ing of a hole i n the centre of a p la te having gra in type orthotropy showed an expanding tens ion f i e l d and increas ing t e n s i l e s t r a i n perpendicular to g r a i n . It a l so showed that i f the mater ia l d id behave e l a s t i c a l l y , the stresses perpendicular to g ra in above and beside the hole would be reduced with increas ing l eve l s of orthotropy. These r e s u l t s provided some basis for the 2 , 3 , 4 f indings of previous s tudies . Madsen's work at U . B . C . showed specimens subjected to t e n s i l e loadings perpendicular to g ra in to be accompanied by very 9 2 subs tant i a l decreases i n s t i f fnes s perpendicular to g ra in as the durat ion of the loadings was increased. ' It was shown i n t h i s thes is that the increased s t r a i n i n g perpendicular to gra in a lso took place i n beams subjected to long term bending a p p l i c a t i o n s . It was shown by modell ing that th i s s t r a i n i n g could be caused by decreas-ing s t i f fnes s perpendicular to g r a i n . Large sca le s t r a i n i n g perpendicular to g ra in could be a mechanism to promote stress r e d i s t r i b u t i o n and reduce peak stresses perpendicular to g r a i n . This would be a reason for the d i f f e r e n t / behaviours found for c lear mater ia l and mater ia l containing knots or other gra in i r r e g u l a r i t i e s when load durat ion i s considered. \ 93 REFERENCES 1. " Lyman Wood.' R e l a t i o n of Strength of Wood, t o D u r a t i o n of Load. Report 1916 F o r e s t Products Laboratory, Madison Wisconsin, F o r e s t S e r v i c e , U.S. Department of A g r i c u l t u r e , 1951*' 2. Borg. Madsen. D u r a t i o n of Load T e s t s on Dry-Lumber i n Bending. S t r u c t u r a l Research S e r i e s Report No. J., Department of C i v i l E n g i n e e r i n g , U n i v e r s i t y of B r i t i s h Columbia, 1971'. 3. Borg Madsen. D u r a t i o n of Load Tests on Wet Lumber i n Bending. S t r u c t u r a l Research S e r i e s Report No. 4 , Department of C i v i l E n g i n e e r i n g , U n i v e r s i t y of B r i t i s h Columbia, 1972. 4. ' Borg Madsen.1 D u r a t i on of Load T e s t s on Dry Lumber Subjected t o Shear. S t r u c t u r a l Research S e r i e s Report No. 6, Department of C i v i l E n g i n e e r i n g , U n i v e r s i t y of B r i t i s h Columbia, 1972. 5. ' Borg Madsen.1 D u r a t i o n of Load Tests f o r Wood i n Tension P e r p e n d i c u l a r t o G r a i n . S t r u c t u r a l Research S e r i e s Report No. 7 i Department of C i v i l E n g i n e e r i n g , U n i v e r s i t y of B r i t i s h Columbia, 1972. 6. O.C. Zienkie,wicz. The F i n i t e Element Method i n En g i n e e r i n g S c i e n c e . McGraw-Hill, London, 1971. 7. H.K. Ha and R. Sen. "HASENSXNDS". Computer program, U n i v e r s i t y of B r i t i s h Columbia C i v i l E n g i n e e r i n g Department Program L i b r a r y , 1970. 8. S.P. Timoshenko and J.N. Goodier. Theory of E l a s t i c i t y , pp. 90 - 97, 3rd Ed., McGraw-H i l l , Toronto, 1970. • Appendix A  The F i n i t e Element A . l The P o t e n t i a l Energy Theorem The f i n i t e elements used i n t h i s paper were derived from s t r a i n energy considerat ions so tha t , as an i n t r o d u c t i o n , the p o t e n t i a l energy theorem should be examined. Let TTe = p o t e n t i a l energy of an element "TT = t o t a l p o t e n t i a l energy U = t o t a l s t r a i n energy W = t o t a l p o t e n t i a l energy of the load Lie = s t r a i n energy of an element We = p o t e n t i a l energy of the load for one element {?e} - matrix of loads a c t i n g on an element [<£j = vector of displacements for an element elemental s t i f fne s s matrix { X J = vector of displacements for the ent i re problem = master s t i f fne s s matrix [P] = master load vector The p o t e n t i a l energy theorem states that of a l l the displacement f i e l d s which s a t i s f y c o m p a t i b i l i t y and kinematic boundary cond i t ions , the true displacement 95 . f i e l d which s a t i s f i e s 'equi l ibr ium and s tress boundary condi t ions provides a minimum for the p o t e n t i a l energy. The t o t a l p o t e n t i a l energy of an element i s a funct ion of both the s t r a i n energy and the p o t e n t i a l energy of the load such that a>-we It has been shown that given c e r t a i n cont inu i ty between elements.the elemental energies can be summed to produce the t o t a l p o t e n t i a l energy of the problem. w - i i x S ' L K K x l - f p r f x ] ( A > 1 ) Applying the ca lcu lus of v a r i a t i o n s ( A . l ) to get a minimum p o t e n t i a l energy gives C K J / X j - {P} -0 (A.2) This i s the standard form of the s t i f fne s s problem where {Pj i s the vector of external loads and [ KJ i s a s t i f f n e s s matrix which comes from the s t r a i n energy c a l c u l a t i o n s . 96 A.2 D e r i v a t i o n of the S i x - Node Plane L i n e a r l y Varying  S t r a i n Orthotropic Tr iangle Since i t was necessary to be able to a l t e r e l a s t i c modulae r e l a t i v e to the d i r e c t i o n of the gra in at every p o i n t , the element was derived i n terms of i t s l o c a l coordinate axes. Fig". A . l shows a t y p i c a l element rotated at some angle to the g l o b a l (X,Y) coordinate system. Let U = displacement of a node i n the g loba l X d i r e c t i o n ll = displacement of a node i n the g loba l Y d i r e c t i o n •f = l o c a l coordinate axis p a r a l l e l to the 1 - 2 s ide of the t r i a n g l e Y = l o c a l coordinate axis per-pendicular to t h e / axis and .pass ing through node 3 U••= displacement of a node i n the l o c a l f d i r e c t i o n -V = displacement of a node i n the l o c a l ?? d i r e c t i o n a , b , c = length of s ide The p r o v i s i o n of a l i n e a r l y vary ing s t r a i n d i s t r i b u t i o n i n an element requires that the displacement f i e l d be quadratic i n both d i r e c t i o n s . 97 Fig. A - l ELEMENT CONFIGURATION 98 (A.3) The twelve degrees of freedom necessary to be associated with the twelve constants of (A.3) were provided by u-and 2/ degrees of freedom at each of s i x nodes. Nodes 4,5 and 6 were located at the midpoints of the sides of the t r i a n g l e . Given the g loba l coordinates of the corner nodes, simple geometry was employed to ca l cu la te the length parameters a,b and c and the angle S . cos.© ^f^- sikQ' a » [tx2-x^cxz-x,) -Cys- ya)Cy2-\j,)l/r b^CZj-xjcxt-zj+tyj-y^tyryJl/r (A .4) r- Jcxi-x,)2 *'.y*-*/.)* The polynomial c o e f f i c i e n t s Q-i of equations (A.3) were expressed as functions of the nodal d i s -placements through the c rea t ion of a transformation matrix. 99 U, - U (-bjO) - a, - o2b + q5]d (A.5) 5 r if(-bjO) &7 - a6b *rQ„ hZ and s i m i l a r l y f o r other displacements so that the form •I'd] " [T] fAl w a s obtained where (utJ 1), s uti0t J ... ifb ) (A.6) f T ] -/ -b 0 0 bZ 0 0 0 0 O o 0 0 o o 0 0 0 I -b o O b* O / a. 0 0 a* 0 o 0 0 0 o a 0 O o 0 O 0 1 a o 0 a\ o 1 0 c 0 6 cz 0 0 0 0 0 0 0 o o o 0 0 i 0 c 0 O c2 / a-b Z o 0 ( ¥ J 0 0 0 0 0 0 0 •0 0 0 o O o 1 a-b o 0 m 0-( Q I c 2 ac 4 a* 4 c2 4 D 0 0 0 0 D • 0 0 o O O o 1 a Z c 1 0.C 4 a1 4 o 4 1 -Jl 2 c T k " 4 T T o 0 0 O O O 0 0 0 O o o 1 -i c. T .Ac 4 12 4 cl T (A.7) The transformation matrix was inverted to give 100 The s t r a i n energy equation for plane stress i s U•-• ?///cv,/* rVjfj < rtf) dy ' (A.9) where (j; , / - ^ ^ tf, ^ (7j -• hifytly* (£j +dj*&) (A. 10) and for t h i s problem, CJ^  = normal s tress i n the l o c a l f d i r e c t i o n (7^  = normal s t ress i n the l o c a l ^ d i r e c t i o n T* = shear s tress = normal s t r a i n i n the l o c a l f d i r e c t i o n cf^  = normal s t r a i n i n the l o c a l ^ d i r e c t i o n K - shear s t r a i n £ y = Young's modulus i n the l o c a l f d i r e c t i o n £y = Young's modulus i n the l o c a l ^ d i r e c t i o n ^ = shear modulus lAy = Poisson ' s r a t i o de f in ing the s t r a i n i n the $ d i r e c t i o n r e s u l t i n g from stress i n the 101 ^ d i r e c t i o n = Pois son ' s r a t i o d e f i n i n g the s t r a i n i n the d i r e c t i o n r e s u l t i n g from s tress i n the f d i r e c t i o n Subs t i tu t ing (A.10) into ( A . 9 ) and assuming constant thickness of the element y ie lded U -• 2 ff [ /- tysjj* fa  2 * tJy ix £j ) (A.11) '^pit^ (t/'^&Sj)* GX"] dA The s t r a i n s required i n (A.11) were obtained i n terms of the polynomial c o e f f i c i e n t s <2; from the assumed d i s -placement f i e l d s (A.3) " a$ h<*l0$+ 2Q,z>l . (A. 12) Subs t i tu t ing (A.12) in to (A.10) gave 1 0 2 + (a/ *asz *• £Q3CX6) $]dA i 2 Jj' f^[2jSvaza4 ^4fiyac,alZ*(^l^yi-^y^)a4a9 ^lljltf* aza5 + (firtfiy^)azal0 + 2(fi*Uy*fi^x) a*a? +2j8y a9ala + 26?(as a4 * Za^ a„+a4ae+ 2<*e a»V d A + 4-L al0 an+2(i (2a4 + a4 al0 + 4a6 Q„ +?alo a„ )]dA jJl fufito,9 • n ^ y *fj4,)% al0*4j <*,i ( A . 1 4 ) * 6 (a/ + 4 a/ + 4a4 Q„ ) ]d A ij'j \ Ztj8> a/ +2 (fix iJy <-fiy £JJX) a4 alZ *$(4a(azta/ + 4akaID)] dA 103 Equation (A.14) was integrated not ing that JjjCI^)^-JJ fcf,n) dfdn (A.15) to produce an elemental s t i f fne s s matrix / kl] i n terms of the polynomial c o e f f i c i e n t s a i . . . a i2 f U Ik] Ik] Ik] (A.16) [ K] -0 0 0 0 0 (9 0 1* b A J 0 ^ (J-t>')c 6 <o 0 & 3 P* il „ <a 3 ^)6 G / A* J 6 j (A.17) 105 0 0 0 0 0 0 0 0 H J M ) c P*Hj 4 fl"4y IZ fiy^X H 0 A , (attic1 0 G z 0 0 b G 3 0 0 Ac^*y IZ A (Ol+^C1 Pyty* iZ fixl)^ 48 h4» 4b c o 2 4 fixJ/y H 0 , Ca+h)c3 0 0 D caUo*)c 0 M*y Io fWy it fatty It fb^y tt 0 6 0 „Catl>)c3 6 6 & 0 urn) (A.19) 0 0 0 0 0 0 " z 0 0 3 0 h z 0 Pj 3 r Ca.+b)c3 6 ,z h it it h iz / 6 3 0 Q (A+t>)c3 h 3 (A.-20) 1 0 7 The transformation matrix was employed to produce an elemental s t i f fne s s matrix i n terms of the general ized displacements i n the l o c a l system. - Hir-jirn'ikiur-'HU (A.21) whore [WJ - [T -7 T [kl][T' 'J In order to assemble the elements by matching d i s -placements at the nodes i t was then necessary to b r i n g each elemental matrix into the g loba l system. [ZViKUhl (A.22) Then U'l&yikAU] where [ k j • [ RV[TTl U] [ T-][ R j U.23) ' v e c t o r of elemental displacements iiv-the g loba l system 108 [«,] 0 0 0 0 0 0 M 0 o o 0 0 0 [R.l 0 0 0 0 0 0 [R,l 0 0 0 0 0 0 m 0 o 0 0 0 0 (A.24) and [R,-]-. cot, 6 Si>i 9 - sin 6 CO, Q and 0 was defined i n (A.4) The matrix;./ ke1 i s the elemental s t i f fnes s matrix used i n the s o l u t i o n of the s t i f fnes s problem. 109 A.3 D e r i v a t i o n of S tra ins and Stresses Stra ins at any po int (f ,^) i n an element can be ca lcula ted from equations (A.12) once the p o l y -nomial c o e f f i c i e n t s ( A^ have "been ca l cu l a t ed . In t h i s program, s t r a ins were evaluated at the nodes so that they could be averaged between elements. To solve for [ A ] , the deformations i n the g loba l system for a given element were r e t r i eved and rotated back into the l o c a l coordinate system. {A}' IT-KRUSI (A.26) Using the s t r a i n s , s tresses were ca lcu la ted using the or thotropic e l a s t i c i t y matrix for plane s t re s s . [<r]--lD1Ul where f (T j ( <TX t CT J ; t) 1*1 - <s*>t,J) (A.2?) 110 0 (A.28) O O O 6 -A Mohr's c i r c l e approach was used t o r o t a t e these s t r e s s e s i n t o the g l o b a l system f o r a v e r a g i n g a t the nodes t o produce a more a c c u r a t e r e s u l t . The averaged s t r a i n s and s t r e s s e s were then s u b j e c t e d t o Mohr's c i r c l e i n order t o y i e l d the d i r e c t i o n s and magnitudes of p r i n c i p a l s t r e s s e s and s t r a i n s a t the nodes. I l l Appendix B Behaviour and Test ing of the Element B . l Convergence In order to converge onto a s o l u t i o n , the f i n i t e element displacement f i e l d assumed must provide s t r a i n -free r i g i d body motion, as we l l as constant s t r a i n modes. Further , plane s tress f i n i t e elements require that the displacements i n both the f and d i rec t ions be cont in-uous along element edges. %, y The f i r s t two c r i t e r i a were c l e a r l y s a t i s f i e d by the formulations of equations ( A .3). To check the l a s t c r i t e r i o n , the displacement along each edge of the t r i a n g l e was found to be quadratic both p a r a l l e l and perpendicular to the edge. Three constants were therefore required to define the displacement along the edge, and these were provided by the displacement i n the appropriate d i r e c t i o n along that edge at each of the three nodes. By forc ing the displacements at each node along the edge of an element to match those of adjacent elements the c r i t e r i o n was automatical ly s a t i s f i e d . The rate of convergence was e a s i l y determined. Since the element displacement f i e l d was quadrat ic , the error from a T a y l o r ' s ser ies t runcat ion was of the order of some length parameter^ cubed. D i f f e r e n t i a t i n g once gave an terror i n s t r a i n of the order of ^ / . S t r a in i s 112 ra i sed to the second power i n the s t r a i n energy expression so that an error of the order of J to the fourth power r e s u l t s . I f t h i s length parameter i s taken as the r e c i p r o c a l of the number of elements along an edge, the s t r a i n energy should converge as the order of 1/N\ B.2 Test ing The elements were tested i n three cases.- The f i r s t was the load case i l l u s t r a t e d i n F i g . B . l where a uniformly d i s t r ibu ted load was applied to the top edge of a membrane. This problem gave uniform v e r t i c a l d i s -placements, s t ra ins and stresses as requ i red , of exact ly the correct magnitudes. The second case was the t h i r t y -two element cant i l ever shown i n F i g . B.2. Here the re su l t s did not agree exact ly with theory but were wi th in reason-able bounds. The stress i n the X d i r e c t i o n was accurate to within a maximum of 6 per cent and the shear s tress was accurate to within a maximum of 17 per cent e r r o r . This problem was run i n order to compare the element with an i so t rop i c f i n i t e element which had been programmed 7' prev ious ly by a d i f f e rent method.' The re su l t s agreed exact ly . A few other tests were r u n . i n which the modulae were reversed i n some elements, and the r e s u l t s were as predicted by theory. A t h i r d test of the program was made by modelling a c i r c u l a r hole i n the centre of a p late that could be considered as i n f i n i t e l y long i n the d i r e c t i o n of loading , 113 F i g . ' B-l UNIFORMLY LOADED MEMBRANE F i g . B - 2 . 32 ELEMENT CANTILEVER 115 and of f i n i t e length i n the transverse d i r e c t i o n . The g r id used was that created for the analyses of chapter 4 and i s i l l u s t r a t e d i n F i g . 4 . 1 . F i g . B.3 and F i g . B .4 show stress comparisons of the f i n i t e element so lu t ion (for the f i n i t e plate) with the a n a l y t i c solut ions for a Q . pla te i n f i n i t e i n both d i r e c t i o n s . This comparison was made for a uniform t e n s i l e loading along the l e f t hand s ide of the model, the upper l e f t hand corner of which i s shown i n F ig s . B .3 and B . 4 . The re su l t s were very s i m i l a r i n pat tern and magnitude for stresses i n the X d i r e c t i o n where the f i n i t e element model might be considered i n f i n i t e . In the Y d i r e c t i o n , the stresses showed the same type of v a r i a t i o n as for an i n f i n i t e p l a t e , but were of d i f f e rent magnitudes because of the equi l ibr ium requirement that the stresses i n the Y d i r e c t i o n be zero at the free boundaries. In the program as wr i t t en , a l l elements have the same e l a s t i c modulae r e l a t i v e to t h e i r l o c a l coordinate axes, although the d i r e c t i o n of orthotropy can be rotated through ninety degrees when required for ease of assembling the elements. The system of equations was solved using a Choleski decomposition type l i b r a r y rout ine . TENS ION LOADING F i g . B-3 EDGE STRESSES X DIRECTION TENSIOKJ FINITE ELEMENT ANALYTIC-/NF/NITE PLATE ISOTROPIC 118 Appendix C The Computer Program 1 1 9 fr ^  x / w - >^ w / ^ „ ? 3 4 6 7 01 I'LN S 1 . >i ( 43 C ) » Y (4 ; C ) , I f. C ( 4 J:G . 6 ) , 1 X ( fif - Q > t JX ( Hdi) ) . T i r - r s i r r ? n : , 1 2 ) , s (! 2, i : ) , i ( i ? , ? : ) ,P < 1 2 . 1 2 ) . . 1 IN!>f I O / (S Jv.<0) . I.J ( 12 ) r i K r K S l O r D E L { ] 2 ) , / P C . 2) r s ? o F»- ir.f .•) . 8 . 5 9 r i T ? SIGN' S I C X ( 6 > , 5 i r . Y ( f - ) , T / - u ( t l , ? P ' ; > ( < 1 , ! : P S Y ( o ) , G / M 6 ) C I K L f S I Q r . r x x ( 4 3 C ) , h Y Y < 4 2 ' ) , F X Y ( 4 ? 0 ) C 1 (' '-"NJTO SXX (43C ) , SYY ( 42 • > , SXY ( 43C ) , f - C n L N T ( 4 3 • ) ] X 12 P I P F N M O " ' i I n (2 j J ) , r.S (j 2 ) , c v r r. S I j 2) CIN:. h'SIOr AGP : . 'SS( FcO) [IN '-.>5!0N- S I C l X ( 6 ) . S I G L Y I 6 ) , TAUPL ( i ) 13 14 • ' 15 : C C. n iK ; - r .5tcN x t u t Y . r , : . MJTB&R CF >JCCES r ;p. [ = ; i . r o P I M - C ^ I C N i c n . G E . N U M R E " C F C E G R E F S C F FRF'SOS-JM 16 :• V 17 " •' 18 C C C C - I M - C S M N IX JX . 6 i . MIKBr 8". CF C F G F ' i E S CF F R E E COM C I ^ S i O N £ .GE-o C L ' F * L E A r ' 0 BUT R? C O S I S T S N T TH -S OL C H S LP. F, OUT I NF. S • C T M ' K S I G N Ff .C.""". M.'MPfF np CEGRFF.c CF F c C - C 0 M •. . 19 . . . . . . . 20 • ' '.. 21 ' C , C X ICC IS TF-r 0 NODE M.'Mi!ERS F (. P FfiCH C L E f - E ^ LJ IS T h f I 1ST OF C E G P - E S C F F R : i : C O , " FCk AN ELEMENT TEL IS A MATRIX DF GLCESAL C F C r E S CF FFFFCOM FLIP, E L E V E N 1 " . ?2 • Zi 24 ''• C '• • '•C . C " LP IS /• STC.FAGF MATRIX LSFD IN THE S I " E S S S LB RCILTI h = WHICH MUST . ('LST RE CTKf:NSICI-iC SAKE AS NUM PEP O F DrG'^sES CF F R 2 E- 0 CI' Pz~- E L C h E M ' • , •25 26 " ' " 27 •-" c N5IZ f t ' S T HAVE TFT SAME MAGNITUOR AS TFF F I R S T D I ' C C N S I C N OF ICC IN M/PPtlG F F V ' ^ C 1 F - v ^ n 2 •; 29 2 3 . 2 5 29 r s ' i z = 4 ; 0 U E P l - G = 3 P r v. I'. n i - ' 0 . • ' ' . 2 1 22 ZCQ re- 2-J,i i = i,^oco-.t A l l l ^ - . C C CCr-TT,VL'F •. '.33 • ; ; ; ••• 35 ' • •; CC 235 1=1,430 S X X ( i ) =• •? • 10 SY> ( I ) = C . r ,0 37 , 3 7 . 2 ; S X Y ( ! ) = ? „ C 0 E X X d ) = o.r-r- • F Y Y l ' l )=--,.no '•..•'• 3 7 . 6 ... v.." 2a ... '33 22^ rXY ( I ) =loF1> N C C U - : T ( T )=C.HC -• -C r K T ' M l e . • ; 4 C , .;. 4 i " " 4 1 . 2 5 256 CC 236 !=1,P6C* . . . F '• ( i ) = c . r c '•. 42 •' ' 43 ; • " 44 ' • F K ( D = - i . r o P V ( 2 ) = - 4 . n O F f-' (4 1 = - 2 o r Ci . .• 4 5 • • •'; 4 6 4 7 F n ( e ) = - 4 . c p P M £ ) = - 2 . r O Ff(i" i)=-4„r''-4e - • • \ 4 2 . 2 5 . 4 8 . 5 ; Ff ( 1 2 ) = - 2 . 5 C 0 P:-I 14 )=-6.oo F ' ' ( I M = - ? ' . C : .' i . 4 E . t ' 4 r t .7 4 P . P -Ff-Jio) = - f . n c r f ' ( 2 ^ ) = - : - . r r Ff-(?^i=-f.n-'> • . 4 a . s i . =.'"!•' 4 8 . 8 2 " " 4 8 . €3 R - ( : 4 ) - - 2 . r c Fl' ( ?P ) =-! . 5T2 * 3 * UCI-/ (6,221-.) • • » , i ' ' 323 •. FHPPAT ( ' 0 ' , ' PASTF'a LCiC 'VECTOR ' ) ; 1.7' (f., 2Z-2 > t r.K L) , L = 1 ,200) . . : ? 2 2 r-rf:y/:T(/, i' :C13of») . . . . . CALL "L « Y P L T (X ,Y , I \ S 1Z , ICC , IX ,: JX , N E , NN.NV; " , 54 '" C/LL' TLAStTL ,G,-X,rY,I.XY,UYXl " •• CALL E/.NTWK ICC , JX, N ~ , N V A"., NV3 , NT-1 Z , LP A NT-, K'HOO) -nr . 4 7 0 1 M I ,f.E"„-:- • ' * ' " -57 r r 4 7 1 L = i,r.r>nr: . • • -58 . J - = I . C C ( I N , L ) • : u 5 C : NC.fLHT ( JT) =KCLNT < JT) + 1 • 60 471 COM TN'I.T. ' • • • - • * . 61 470 C C M ! M ? 62 • PC. 206 L L = 1 , N F -62.25 IF ( I r F PUCo K .5 ) GO T Q z n v 63 •V .PITP (5 ,212) LL • £4 2 - 2 F C P ' K A K . ' l ' . ' F L E M E M NG. ' , I5) - ; ' . - '. .-, - ' — "• — - ' ,64.25 2 1 3 c r N T i N u E • • ' • 65 c 1 = 3 c n ( L L , 1 ) - . -66 C2= l fP ( LL , 2 ) •: :• 67 C 3 = ! f C ( L L , 3) 68 >1=X(C1) 65 > 2 = > i C .? ) - ; 7 ) X3 = X ( C3 ) • • •' . 71 Y1=Y (C1) . . 7: . Y2=Y(C2) 73 Y3=Y(C3) . -.7 3. 25 IF(!DEPUG.FC.O) GC TC 25'J 74 : V.F ITE< 5 , 209) LL •••••i5''- 209 FTP'-iAT ( '0 ' , ' FLFMENT ST IFFMSSS M A T p. I X FCF. EL EMcNT1, 15) " • • ."15.25 250 CCNTTfcl.E 76 C A L L L S T ( X l , X Z , X 3 , Y l , Y 2 , Y ? , t X , E Y , L X Y , L Y X , R ,G,TL ,LL , NE L R »I DEBUG) — - 76.25 IF(ICEFUG.EC.-T) GC T'J 251 7 7 - V.F I T ( 5, 2IC ) 78 ... 220 FORMAT(• S'ELEMENTAL STIFF'- ESS C: A T F 1 X ') • - - . ... — — — 70 K I T ? (5 ,2 J8 ) ( J R < I ,K) , ^ = 1 ,12 > , L = 1 ,3.2 ) e j 2 c a F{!FM;. 1 ( • ' , 121)13.? ) .-. e;„ 25 251 COTiraE , . 6 1 PC 55 N = ) , N V A R ' . . . 82 Vt: 55 J=i,NI>CD -< C3 J l = (J- l ) <-N'VAR ' • • . • - ' 8 4 L J ( N+ Jl) = JX ( N V A * . * ICC (LL , J) -NVAR + N ) 85 5 5 Cf NT !MJf. ' • • € 5 . 2 5 I F < I TE EUC. ECO > G C TC 252 ' 66 V F I T P ( 5,3GC ) ( L J ( K L ) , K L=1 , 1 2 ) t,7 .300 FCF-'-'AT ( >C ' , ' L J -IS'. 1215) F / . 2 5 25 2 CCN7!M. r -S<3 C A L L SETUPfA.NV3,LJ,<,LBAND,N'SIZ> 89 206 CCNTINL^ 8S.25 IFIIOFPUG.FC.J) G C TC 253 ' 90 ' WF iir:.{ 5, - i r . ) 91 320 FCPI-'AT 1 ' 1 • , • NA5TEF- STIFFNESS NATFIX1) 92 C A L L t.:PFKAT < A , 3 6£, j fcf , I6f!,3 68 ,1 ,1 ,65 ,2 ) ,. -93.25 2 5?. fCNTIMJ? 94 <:ATir = l . C - l t 95 C M'.TF THAT CFPAh.f) ic MOPE EFFICIENT "HAN CP AND CNLY Wh£N WATFIV . "• "' 96 • r. IS NOT USED 5 7 CALL CFPANE { A,r;-i, NCtiGjLf/lMD, l , P . A T i c , . C E T , J E X P, ./) 53 c THE "H'lr.O TRY IN C A L L CB ANC I S THF CRCGF CF THC ^/S 'TS's STIFFNESS • C O r M A T R I X — f . ' U S T BF F X * CTLY THE SAKE THE NUKHc'P. c p NOM-zr. pp' 121 >' •:„.-,> • '." '_• "J- — .." — — » J—— -'— . . . . — _ . — . . . . C <.-.Vf. HcE-RLES: CF. . F CCf. \t\ T h . " p c c F L ' f , v . - . . . . ••. • -c 10_* * 1 H . 25 c FM. fjCV f.C» Tfl IMS T F E 'SDLLTION U I SPLAC i f ' - " > V • v F i (e, i f c ) rsT .;. ••. ; •'• .- -\'-.\. "•. •'. • ' - " • . 1 0 i . 5 150 .•'•FCF.f A 7 { • • , '-PETrfPMI N.aM .-Cf' V A STER ST I F F tilS S K A * P. 1X - IS' , G2C .f; ) '-' " • • ' " . ' ' .' • " ^ 1-52 ** ' • CALL . EXPAriD ( AGF .HS S > Mf'AT ,MV AF- , P" t IX) • -•: ' ' ' • 1 )*. "5 i r i i r i p u c . r - . - - " ) c.C'Tc 2^ - - ..' • i' : M. IT:. I C , i!„.l ) N M T . • . . ' 1,4 I ' l ' .' FCFMT 1 • • , ' M><AT= ' , ! 4 ) ' 1^5 ' « " i T r ( S , i 0 2 ) ( J X U ) , T = I ,hwn ) 1 J 6 .. lt-2 . F C F N 1 T ( «f: ' , 2 0 1 f) U f . 25 . 254 • C C J \ T I f a c - . . ' .-1C7 v.Fjr-r u , i c ? ) N V A R , P . C : - G TOO r-:- P C ' F . M A T ( <C < , 'NVAR= • ; .1 b , 'NCEC= 1 , 1 5 ) 109 * H 1 T 1 7 (fc , 201 ) LFANC . . . ' . • -. . " 11 J 201 FC R*-'A T ( * C ' i ' THF HALF G ANO WIDTH I S ' , ' 4 , / ) ~ ' ' • " 111 WFIT.E16. 202) N'V3 - ~ ; 112 113 2 02. FCF.M-M <•->••• V T H E R E A3,"' , 1 3 , 1 X , ' V AR I / EL ' S PER SLFM=NT-« )'T'" r^ *. ~~." • V»F I " E ( t , 7 7 9 ) * AT!C ' '. ^ : ~ • - :t 114 779 F C Ff-' AT ( • C * > ' F A T I C = ' » C 2 0 . 1 0 ) 11S > P I ~ F ( £ i 7 ' ' E ) . ' , 1 1 6 " 79" F C" F M / . T ( • ] . • , • S T F " S S E S' RELATIVE T O FL " M " t\ T AX^S') " 117 P H I M . 1 , 110 F! V. I •• P 2 119 FfWIKP " 12- V> FIT £ ( 6 , 2 3 2 ) 1*1 2 2 2 FfPKATf »0«, 'cLEMFJN'7 ' , 6 X NGDE N L:fD E F • , 6 X , • S 1 GX • , 1 ? X , ' S IGY • , 1 2 X , ' T A 12~ * - U ' , 1 2 X , • " P S X 1 2 X , • EPSY 1 2 X , 'GAMMA • , / / / ) 1 123 L C 1 7 0 ht• = ! ,Nf: 124 u n r ( t , 2 ; . - ) 125 233 F C F N A T ( • - . , J 5 ) 126 126.25 C/ L L L S T F L S ( N A , P N , E X , r : Y , t . l X Y , U Y X , C - , S X X , S Y Y , S X Y . , ! j e , I C G , N £ L P , E X X , r * r X Y , ICEIU.G ) Y Y , ' • . ' 127 - 1 7 0 " C C t- Tit IF -12" DC 4 7 5 I =) »r>.fi 129 " / S X X ( 7 ) = S X X ( I ) / (•' C C L N T ( I ) . . . 1 3 " SYY ( I )=SYY( I ) / fCCLf>T ( ' l ) _. ... . . . . i _ : SXYI I )=SXYII) /^C0LKT(I ) 1*1.2 EXX ( I )= EXX ( I ) / N C.0L'h;T ( I ) -r-1*1.4 H Y Y ( I ) = E Y Y ( I ) / N C C L f T ( I ) ' ' - i : : . t E X Y ( I ) = E X . Y ( I ) / NO C L N T ( I ) 132 4 75 c. c N T i N L E . 133 ' VF I T 1 6 ,505) ' •• • ' '' " . „ 1J4 135 5C5 - F C P f A T ( M • , i n x , ' P R I K C i F A L STRtSSES A N C CIBECTION RELATIVE TO Tti GLCPAL SYSTEM' ) • -F ... ' • •' i - t V F I T " ( f , f r o .37 C . f.'CW CILCLL/TE TFE PRINCIPAL STRESSES . . -13" . C C . 501; I = 1 »N: f» . ' . - . • ?39 F(>p.f-:*T('«C«. • NODE ' , 1 6 X , ' PSIGX • , 1 5 X , ' F<; 1GY' ,1 4 X . ' Af\ C-L " -CFC-CL C C • ) I4 j SF= ( S > X ( I l-SYY ( :••) ) * » - 2 + 4 . D e * S X Y ( I ) " " T 2 1*1 P .A = . 5 P C * C S C F T ( S F ) 14* C= (SXXI I ) + S Y Y (I> ) / * o r C 14 3 C PSIGX t FSIGY ARE TF-C F P if.jc 1P *L STPFSSFS 144 145 C : r Af C L S = P.nTAT]CN CLOCKWISE OF PRINCIPAL i"TFcSS c.S F R O T H C GLCPAl ' ^YSTf-P 146 F-'!CX = C + FA 147 P S IGY = C-F A . . . - - -14* T r = ?„f-**sxY( I )/(S.xx( I ) - S Y Y ( i ) ) •; -.: - • 149 A N C L F = r A T A K ( T F ) / 2 . C O / 3 . 1 4 1 5 9 ^ 1 P C . P O 3 49.1 IF ( SXX I J.L T.SYY( I ) ) A G LE = 9 0 . Di\' +A f*G L E '' 1 50 VF-lTf ( t . E l i : 1 I .F'SICX.PSIGY,ANGLE ' 122 , i S l ~ ; - J ^ -'0 •iZ.ri-W ( • t i A»F 2A. 2.,F-20.3 , F21.3 ) ^ 151.25'1" KF.ITE:<A,'fc(!0) X m , V ( H , ANGLEr, . / X V " i£i.'s" *<.: ' ft Ft / a {2F K ; . M . , F I V).5)V - ! . V ' / \ . ? , ; ^ • ., • " ^ V . - ' : - : - . V - . ' ^ • ' . .• 152.25 ' C C n C l ' L i T f T F E PRINCIPAL STRAINS -. •.-• f *~15?. 5 ' j.Fl- LF(6 , 550) - • ' C • V v y / . - ' ' "('1', 10X , ,'PF INC! FAL STRAINS -UX C I F. CC TI >: • F 1.1TI V" TC SYSTEM) ' ' •. r .- •• . - X - : ^ - • .52.6.,, 55C FPFM i •  ', 10X » ,'PF IMC! FAL STRAINS -UX C I F. C TI >: • F L.1TI V" TC Th 152.7 ^CLCFAL fcF.ITr<6,56C) •152.«1 .560 rr-FMAT (•(;•,' riDDE ••» 16X , 'PEP5.X • , 15 X, 'REFSY • , 14X , ' AN G LE -DFG-CLCC • ) 15 2.6 2 Cr 570 1=1,NN 152.63 EF=(CXM T ) — t>V< T ) ) '•' *2 + 4 »C0-;' X Y (I ) * * 2 -152.84 F/F = .5C0*CSCKT(SF) U2.E5'\" C F = c r x x < I ) + f Y Y 11 ) ) 12. D C 152.86 -.C - PFFSX f. PrFSY A P.E TH': FPI NC I FAL STPAIN5 1521,87**0 . ANCLE=RCTA-!C.N CLTCCKWISE- OF'PF INC'IP'AL STRAINS FROM THE GLOPAL AXES: 152. 68 F(-FSX = CE-+F AF?W— . - — . ; ••..'•.'>' • 152.89 •'• • ' PFFSY = CF.-'-'AF' 152.9 TFr = 2.na* L'XYl ! )/<EXX< I )-FYY ( I ) ) 152.51 AFGLF = CATAN(TFE)/2.nO/2.14155-;aCC.CC 152.911 !F (EXX( I ) olT 0FYY<I ) ) AN'CLE = 50.OC+ANGLE 152. c2 , WRITE, (fc ,580) I ,.PEFSX, FE FSY , ANGLE 1 52. 53-;.,' 58C FC:Ff«A.T( 1 , 14 ,F23. 5 ,1715. 5, F 22. ? ) 152.531 VP IT! (7.600) X < I ) . y ( I ) . ANGLE 152.5i 57 r CCNTJ^L 3 5;; '« K I T M 6,476 ) • . ' -JL14 476 .FrrrAT('l',15X,'AVERAGE STRESSES AT THE MOOES ' ) 155 WFITc ( 6 ,47 7) 156 _ 4 77 PCF'.MA.T ( • - • , TJOC E ' , 1 5X , • SI G X ' , 1 5 X , ' S I G Y * , 1 5X , ' TAL" ) 1 57 • t.T 479 1 = 1 , N N 158 WRITE! 6 , 476) I , SX X ( I ) , S Y Y ( I") , S X Y {I") ' 155 478 F( F M t T(' • , I 4 , F?2.6,2F 15„6) 160 ' 479 f CNTIM.; 160. 1 . V>F ITF ( 6 , 46C ) T . . . . 16 J . 2 A$0 F C F v. A T ( i l < , .15 x , • A V E R A G E^- S T F. A IN S .AT .-3 Ft-, MODES') - '•-16r-3 M Hf 16 ,461) - • ------ • • .. ' 16C.4 ...;v4Pl FfPr-'AT ( '-• , "NOTE • , 1 5X, 'EPSX' , 15> , ' tPE Y • , 15X , "GAMMA ' ) : ., , 163.5 r r 4C2 1 = 1 , NN • '-: ' 16C.6 VFTTE(6,46?) I , E X X ( I) , E Y Y (I ) , E X Y ( I ) • : .:," -V. - --: r , • • • .160.7 .A. 483 Ff Fr--AT( • • , I 4 , F 20. 6 , F 1 5. 6, F 1 6. 6 ) . , " . 160.8 402 CCNTINLF • • .': j - : . - \ '•161 >••"•*••••• STCP 1 6 2 . . . E N C • •- " ... - • ' 163 SLPRCLTIN'C LAYCUT (X , Y, NSIZ , I C C IX, JX.NF, NN , NV AR. j fJM AT , N CFG, NELP ) JJJ! - l r P L l C l T FrAL»F(A-H,0-Z) •165..v;,. . -C X = X-COC-o.C 1N'AT£ OF EACH NODE 1/6' C Y= Y-C CCP.C I NATE. CF E ACK N CDF' 167 - If.O = L 1ST -CF- NODE NUMBERS FOR EACH ElE^F.r'T 168 C • IX = v IF N'CCAL. CEGrEE.CF FREECCM IS -RE-STRAINED 165 -C : =1 IF NCT P?ST'RMNcC 170 •• c j>=rr n--? r r LIST OF THE C^GF.EES CF rFsr-nn^ 171 C M=T(1T/L IHlNEtP Ci F EL EN F N'T S 172 , C Nt:=.TCTAL NLMEER OF NOflE < 173 " C NVtF=MJVEr-g CF VAMAfHES PEP NODE 174 C NNAT^TCTAL NLt-'.PER C F UNKNOWNS 175 , C NTEC^CruKTEF. L S E D I N OETEFMMNG JX FN T Fo I E S 176 C N ' S J Z T C P F / T E R T F A N N U V H P R OF E L E M ' - M T S 177 C JC LPt:v = ELFK cNT TG HAVE ELASTIC fCCVitE FEVI-RSiC 17B ' t C N'ELF.FV IS.THE TOTAL WUfBER CF T H £ S E ELEMENTS • 175 C TF I S SUHFTL'T IN E R E t PS ELEMENT ANT NCCE r ATA - " 123 1 3 0 ui fr-t-ifuh' x< n ,'YU) ,fc cifCsiT,6j.,i xU') ,*JX (>7'. > . - ; -; . , . . , ., - : •j t"Yil) ,FCC(NSI7',6) ixm ."JX <>7'. . -P. ( 2 , 0 ) ••'•>:.-';•'•-: -.: > . : ' 1 , '22 " 1 r 5>.r. ( 5 ,'f0) ,r-N NELCCV ' / : • ' • : , , : ' * ' * l s - ^ AU r c i - i < > ] ' . : ) ' , : . " : ' • : • " : ' ; - v , > - ' : " " : : - ' " • ' ' : . ~- l o 4 ,V-.'.P I TP (6 , A l ) "NE' ,NN ,NJV.«P , N - L PCV '-• ' .V "-VS -S ' ..' . •' l l l i l J 1... Ai . . . j ^ L F N M ULi'.J. T C T f l L NT. CF " L E P c f'.'T 5. V..li.L.2.Xx.'.LifL.. .r.f N'CL'E S ' , I 5 , 5X , l f 6 ; ' V A f I *PLF'i ,PPF. , .Nnnt ', 1 5 , 5 X , ' N O . CP H t ^ M S R 7: V i - 5 fc D ' , 15 , / ) • ., v : 187 ^ — \\'y i T n it: * A ? ) ' • • . '-. ' • '" . • ' i ca 4 rcPNATt/, 1 NCTF 'WX, ' X-CCRC ,f..x,1 y-cr •• , c , x , 1 cx C.Y 1 ) - • ' 139 C CX AN C CY IWDIf ATP CONSTRAINT OF THE- F If. S T ANO SHCCNO C'GfF.E". CF PRE: OEM l y j C t~ ~\X NCOE • • - , • 191 C PC A NUN FR j C A L PF CC ECU R F • T C RFIC IN IX AS A * IMGL U "CW V EC TOP. ' " " -192 C FTAC IN NCPE ] N F G K M A 7 ] C N . . , 193,,. " C F C ' F S I D E NODES SE" X ANC Y ZERO AS THEY ARE N;V£P. L SED---J US T—~~~" ~ "T..T~~ ' 19A r •• F FAC IX BCUNCARY CCNOITICNS " ' • --• - - '-195 - Cf 10 1=1,NN . .. .... • - -1 = 6 « I2 = NVAF.M ' ' : ".• - r \ . ' ' ' • ' .-197 I1 = I2-NVAR-H'- : ; - • • - - • • ' . . 195 C _ K IS.TKE- NQCc ' NUK E E S — R E A D . I N TO KEFP CARDS DEC I P F E R A 3 L E -1^9 RE AD ( 5 , 4 2 ) K , X ( I ) , Y ( I ) , ( I X ( J ) ? J = 11 i 12) .' > ., , • ' r '. ,'• :-.. .'• : r 2< ) 4^ FGPVAT( I t , 2 F 1 0 . 0 , 21 5) : • ' ' - • • ; * ' - ' • 211 t TX FCP' EACH NODE F A S . T F E SAKE Nt'MP-ER OF .FNTP.I 'IS "AS MVAR : . . 2u. V » F I T E ( £ , 4 4 ) I , X ( I ) , Y ( I ) , < IX ( J ) , J = 11 ,1 2 ) ' . ' > ' ; ' 2L3 AA r ( F f . A T ( l x , T 5 , 5 X , F l C , A , 2 X , F 1 0 . , A , 5 X , 2 I 4 ) . „..: ' ' 2JA IC f CN71NLH ' , .' • --' : • • , • 2G5 V-F I Tr' ( 6 , 4 7 ) : ' . . ' •' -. 2C6- ' -47 - FC.RMAT ( / , 5 X . ' ELEMENT t , 5X, ' • NODE N 'Cr ' lTRS ' ) ' - • ' ' ' 2(7 C INPLT F L E K E N 7 C 4T * .. 203 C FOR' CPTHOTF.OPIC TRIANGLES TF.E—CFDEP. -OF THE NOOKS IN PUT MAY BE I '-'PORT AWT -2 09 •••'• '"-C - INPUT N0D2 H.'PEERS ANTICLOCKWISE STARTING WITH THE TWO PARALLEL '• 211 C TO T H f CASE 21 1 C KK IS T FE r L E M C H J NUMBER . : •' . 21 > CC 11 1=1 ,NE • • ' "  21 •-E*C(5»45) KK , I IC.C( I , J) ,J = 1 ,6 ) • - — . . . ' — - . . . . 214 45 F C F N AT ( 7 I 5 ) . . - - . . ' ; . . ' ~ : " . '.-:_zr^- • • :-\..~,-'..\. . ^ ; •.-•'"_••-"••. * 215 VFITF(6.46) I , ( ICC (I , J ) ,J = 1 , 6 ) ' ' " ^ ' • • - - ' • -'—: - " • - ' - -21 6 46 F O F N : A T ( 5 X , I 5 , 5 X , 6 I 5 ) . . . . . . ' 217 11 CC N7 1M , C •. • .> . • ... ' •'• '-'.,..' .- . :- '--.'• ' •• ' ' : • 213 CC 55 J = l ,NE ° - " '•• ' • ' ' ' ' ' ' ' ' ' • :  219 NFL?(J) = C - ' • , •-,-...,.,....,' -•;:' c ; -OV' . ' ; - . • • 22" 55 CONTINUE ' • "'•.'"'.' -.'•• -: rfi'^^': ' • •'-•'".-:''-''.."' • •" '-• ' • "• • 221 C .PF-VEF SE NCOLLI WFERF NECESSARY ' ; - - . . :  222 I F C K P l F.tV . C C . O ) GO JO. f 20 • •, '-' . . . ' •-"•;. ,>•'-.•. . 22i V> P I TE ( 6 , 6 2 0 ) • . . • r - ' . -*24 FPR^AT( '()' , • THESE c LENENTS HA \i F. PEVEPSED P C D J L l ' ) • - - '• '' • " ' '''•' 225 00 f-.l J=l,NELRtV ' ..' -:• .-. . .. : \ . -, ,;,•,'.'.-": 226 ^ "EACI5 . 65 ) J'LPEV '.'•'•-.':•,. • ' • ' ."V;.;'-.-227 "hF'Ml - ( 6 , £ 2 - ) jF LrEV [ ' 1_ _J " — 228 625 F C F * AT (• ' , 1 2 0 ) , ... . ' . • - - ; 229 65 FCFf- 'AT(I5) ' . ' ' • •} • ' •';' ..' • . ' - • • ' • 2"J NELR( J F L F r y ) = i " • -•• • ' ' • ' •  2J1 6 C C N ' I M'E .•.'.; 1 • . ; . 2.2 632 CCNTINLE . • -V.".'' ' -' - "'' • .- •". ' • ' ..' 2 - \y7,T=Nv:>p«NK . ' : : : : • ' - ' "• 234 tsC£(=C 235 C NC'V NLMRFF. CfGFFPS CF FFEEDCM 236 pp i ? T = ?.M'AT" • 1 237 iniX.ID) 1,2,3 236 3 NPEG = N'DEG+1 ' . 239 JX( I ) = Nr>EG -12k 240 . CC T," 12 241 2 JX(I)=C - . 242 CC TC 12 .. •-; 243 1 . WP1TF{6,17) J • ' 244 ' 17 FCPMTf./ / , < ll,PLT !> FCR DEGREE CF F PEE CON ' , 14,1X, 1 IS NEGATIVE' ) 245 12 Ft NT !NU E - *-246 C 'C !:G IS Ttr SIZE CF TF. F PFCU'LEf. -247 SFTUr.N -24B F N 0 24<; SLPPfUTUP F L AS (TI , G .EX ,:-Y , CXY, LYX I 250 IMPLICIT REAL* f U - F , 0 - Z ) 2 51 F£fr(5,6) TL,G,EX,EY,LXY,UYX 2 c o 6 FCF A7 ( F 1.' • 5 , 3 r i J . 5 ,2T 1C.5) -252 . kPI7E(6,7) 2 54 7 F'CFMT (/ / , IX, 'THICHN'SSi, 5X, • G ' , 10 X , ' F > ' ,8 X , ' 2 Y • , 5 > ' U X Y • , 8X , ' L'XY' 25 5. * ) 256. VF ITS ( 6, E) . . . 257 8 F P F" AT ( ' - « , 3 X , ' : N C F E S ' , 5 X , ' P S I ' , 8 X , ' P S ! ' , 7X) - - ~ ' - -256 VPITE<6,5) TL,G.EX,EY,L>Y,LYX 259 .9' • . PCFNAT l'0',FEo2>2C!lo2,C10o2,2011o2)- -26C PETUPN 2 61 • END": -262 . SLFt'OUTINE 6 AM CWH{ ICO , J X , MS , N V AP. , N VE ,M SI 2 , LfiAND , N'N CD ) 263 !MFL1CIT PF#L*fl(A-F,0-2) 264 C TF'I S * CL 7 IN C FKJCS THE HALF RAND VlICTH LPAi»C • ' 265 C JX ETC APE AS DEFINED IN LAYOUT 266 'C NV2=N0.CF VARIABLES PER ELEMENT 2 67 CIMENSION ICP(NS1Z,6 ) ,JM i ) ,L J (12 ) 263 CIwENS!CrNSTFF(12) 2 6 5 CCMNTN /ELT2/LJ 2 70 COL I VAL ENCE (NSTFF,LJ(1)) ?71 , c N N C D = N C o CF NCCES PER ELEMENT " " : 27? c riM 'NSICN LJ F C R T H c M-CRER CF CfGP E ES CF F REE POM FEP-ELEMFNT 2 72 NNOL = 6-274 L r A N 0=0 275 276 . hf ITT( 6, 202 ) 277 203 FCFMA7(«r« , 'ELEMENT NC . • ,20X, • CEGFEES CF FREEDOM ' ) 278 PC 2 1=1 .ME 279 C LCT EF IN E LJ= N Co CF PLCFEE OF FFEEFCM IN AN ELEMENT 28) C U + K l ) IS TFE NUMEER FFCM 1 TC NV3 CF THfiT DEGREE CF FREEDOM 281 PC 4 J = l ,*>V/ c 282 CC 4 K = l , NNCC 263 Kl= ( K - l )*NVAR 284 LJtJ+K - )=JX{NVAR~*ICC(I ,K)-NVAR+J) 2*5 4 CCNTir-LE 286 N\2-t;VAP«NNC(l 287 lr.c I 7<": ( 6 » 204 ) I , (LJ(L) ,L=1,NV2) 2 8b WPI"'(2) (LJ(L),L=1,12) 289 204 FCFM/T ( <r> , T c ,2"iX, 1215 ) 29^ C FIND riFFfP .rACFS BETWEEN DEGREE N:!JMRFP S UTHIN THE ELEMENT 291 C MN IS SCME NUNRER GREATER 7 h.AN THE EANDKIfTH 292 N7. > = C 29? M N = : - . - . C O -294 CP 8 J = l ,t.V3 255 C IF NO FP.ErOCP AT SCME CCOPDI NATE,THE Bltlt HCTH DCES NCT CKMGf* 296 IF(| J ( J ) . E OO O ) GO 70 f. 257 1F<LJ<J)-NAX) 6,6,5 29 8 ' 5 MAX = I.J(J) 259 6 IF<LJ<J)-NIN) 7,8,6 125 . „ . ^  * .. :< •) 7 . M1 r = 1. J < j) •: --\ . • , ' n • CCNTINUE ' ..' \'...*''.- . ". -: • 30? N P l = f M - M N •••• •.•y''^••. -" .' :-: >3 I F (MP 1 • G7« LEAN1;) LP A I i p = N[:l.. ' ;-' / " 3JA 3 C C M I N U ? '• \:' . . "•• • ' •' • .' • ."• 'C5 LFANr = L B A N C + l •' • '• -06 Rf TU- M 'C7 FME ' ' 3" 8 ' S L P F F t T 1 NT L5T ( X I ,X2 ,X 3 ,Y1 , Y2 , Y2 , EX , f Y ,1>X Y,UYX, P ,G , TL , LL , N E L K , IPCP C 8 . 25 * L ( ) *» *C5 I I ' F L K I T K f / L ^ £ ( . ' - - F , C - Z ) . 'ID D I M E N S I C r F (1 2 . 1 2 ) . S (1 2 ,3 2 ) , 7 (1 2 ,12 ) , P ( 1 2 ,1 2 ) _-n C - 1 P E N S I O N P ( 12 ) , Fl I 2 ) , 5 7 ( 1 2 , 1 2 ) , T P r . ^ K t 2 ^ i ) ^12 C I M N S I O N N F L ^ ( 2 0 : ) . . . . . . -J3 DIMENSION A P C i A ) , C-PD s 1 1 2 ) , E V E N s ( 1 2 i " ••• -' - - - • . ... 3 1 A CCMKCN / T L T 1 / S ... a 1 5 C f N N C N / F L T 3 / f L , EL ,CL, 7FET A - , v - _ . 1 _ — - . ' - i I f F Q L I V A L F f c C F ( A B C , A L ) " ' " "' ' ' ' " — _ " s 17 E Q U I V A L E N C E I S T F . S ( 1, 1 ) ) — 318 c ' S T I F F N E S S M A T R I X FCR 7 F E C P T H C T R O P I C T R I A N G U L A R P L A N E S T R E S S - • ' - . 319 - c ELEMENT WITH L I N E A R L Y V A R Y I N G STRESS = 2* CC 23.1 1 = 1,12 321 r •> P ( I )=G.DC • ... • . 222 F ( I ) = 0.r*rt • - -323 211 CC.NTINLE ' cA C C A L C U L A T E R E L E V A N T LENGTH PARAMETERS FCR T F E ELEMENT . •225. C S L = L F N G T H CF S I D E 3-2 ' • '• 326 S l = J X 2 - X l ) * - * 2 + ( Y 2 - Y l ) * * 2 . 27 5 1 = P S C R T ( S L ) ' '2* AL= (( X 2 - X 3 )=--( X 2 - X 1 )•( Y 2 - Y 3 )* ( Y 2 - Y 1 ) ) / S L 2 2 •» • P.L = ( ( X 3 - X i ) M X 2 - X i H ( Y 3 - Y l ) M Y 2 - Y l ) ) / S L 3 2.0 C l = ( ( X 2 - X I I ' ( Y 2 - Y 1 ) - ( X : - - X D * ( Y 2 - Y 1 ) ) / S L 331 Cf = ( X 2 - X 1 ) / S L - - _ -' 2 S i = I Y 2 - Y 1 ) / S L T F E T A = C A P C C . S ( C C ) - --- • - ' ' ' ' . " ~ T • 33A~ - • Vf I T - : (3 ) i P C " - — 235 A F. F /. = (A L + P L ) *C L / 2 • C 0 , - . 5. 25 I F ( I C E E U C o E C o O ) GO TO 2 5 5 , . •' '. 336 WF T T ? ( 9 ,215 ) L L , A R E a - 7 21 5 F C R F M C ','EL-NENT NC . • , IA , 5 X , • Ap f .' I S ' , F i 0 . 5 ) ' 7 . 25 255-, '• CCNT3MJE ;-• ; . . 3^P IF ( A P c A . L t .CCD ) CC TC 2-20 ~~ * GC T C 2 2 2 . • V " 4 j 2 20 ; V > P I T E ( 6 , 2 2 1 ) LL ' F • •»4 l 22 1 ' F f" F '•' A T ( ' - 1 .»'• s - A P. - A CF ELEMENT • , I A , 2X j • I S NEGAT I V F * * * " * * ' ) • A l . 25". 1F{ i r . E B U G . E C . C ) GO TO 256 3A2 222 W F 17 f - ( 9 , 21 C ) A L , B L , C L , C C , S T 342 2 1 0 • F f RKA 7 ( • f • • , ' A = ' , P 1 f' . 5 ,2 X , ' P = * , F 1 J . 5 i 2 X , 1 C = 1 i F10 . 5 1 2 X, 'COS= • , F S . 5 , 2 2.4A * X, «SIN'= 1 r F 8 . 5 ) . . 3 A i . 25 2 5 6 C C N 7 I N L E r H ' T T T S L I Z P P , S , T AND R NAT R I C E S T C 7 P R C A6 . r. c = r D T A T i r N M A T F IX 3A7 c T= TPANSFCFN'/ ,T1CN CAT.RIX :" 1 -: -A8 c S = P F i " C l C T n F • T T N U P T E T ANP P ' ' ' • A9 c F = T P A N S P C S E CF S 25" c. 7 = ! N V E R S : OF T P A N S F C R N AT 1 CN MATRIX AFTFF INV. STATEMENT . 5 i r( ; 5 1 = 1,32 J52 Cr 5 J = l ,12 : 3 e 3 R( I , J) =O.PC . • . . , . ' ?5A P t ( l,J)=cpn • i 3^5 w ^ . s c f . j j . o . r c 2 5 6 7 ( l , j } = n . n o .... 357 * . r * . r . F ( l , . J ) = - . C O 3 5 c - - T 5 f C N T ! M t • 4 ' " 5 9 . C " f V . ' S r U S T I C KCOLL*."- kti<:N N£C:S«. ' 56C IF (Kt I 0 (LL ) . C C . l ) CC TC 6 0 0 2 t i • : CC Tf. t C E 2 6 2 6 C 0 S ! * = ' X 3d r X = f Y  2 6 4 f V - S ' X * -2 6 5 S t X Y = L X Y 3 6 6 L X V = I Y X - 3 6 7 L Y X = S L X Y 2 6 8 6 0 5 Cn-'TI ' - 'UE 2 6 9 •" f . P l ' I L C . TEE Tp.AN.SEO*F'A~ICN MfiTPIX 3 7 C F 2 = P L * " 2 2 7 1 : C 2 = f L * * 2 3 7 2 f 2 = * l " ?  2 7 2 1 1 ! , : ) = ! . C O 2 7 4 T U , 2 > = - e i 3 ~ < 5 " T ( 1 . - ) = P 2 2 7 6 7 ( 2 , 7 1 = 3 . 1 C 2 7 7 T ( 2 , 3 ) = T ( ! , 2 ) 3 7 8 T ( ? . i x ) = P ? 2 7 9 " I I 2 , 2 1 = 3 . E T 2 8 0 T ( 2 , 2 ) = U L S I T ( ; , 5 ) = 2 2 8 2 T ( 4 , 7 ) = ? . . n o 3 8 3 T { ' , , P ) = A L 3 6 ^ 1 ( 4 , 1 1 ) = A ? : E 5 " ( 5 , 1 ) = ! . 0 0 3 8 6 T ( C , 3 I = C L 2 f 7 T ( 5 , 6 ) = 0 2 - f a " ( 6 , 7 1 = 1 . 0 0 3 6 9 - ( 6 , < 3 > = C L 1SI ' U . ' - ) = c •' 3 9 1 T ( 7 , 1 1 = 1 . C O 3 9 2 . T ( 7 , r j = ( 4 L - F L ) / 2 . C C 2 9 3 7 ( 7 , 5 1 = 7 ( 7 , 2 1 2 ' ( H , 7 ) = l . r 0 T ( E , £ ) = - < 7 , 2 > 7 ( f , 1 1 1 = 7 ( 7 . - ) 7 ( 9 , i ) = i . r c 7 ( 9 , 2 ) = A L / 2 . C 0 7 ( 9 , ; T ) = C L / : 7 ( 9 , 4 > = « L - C L / 4 . 0 0 7 ( 9 , 5 1 = 4 2 / 4 . C C T ( 9 , 6 ) = C 2 / 4 . C * T ( i f - , 7 ) = i . n c T ( 3 C , £ 1 = 7 ( 9 , 2 1 T ( l ^ , c ' ) = 7 ( 9 . 2 1 " ( K , 1 0 ) = 7 ( c , 4 ) 7 ( 1 0 , 3 3 > = T ( 9 , 5 ) T ( ? - , 3 2 ) = ~ ( 9 , 6 ) 7 ( 1 1 , 1 1 = 1 . r c T ( l i , 2 ) = 7 ( 1 , 2 1 / 2 . D C 7 ( 1 1 , 3 ) = T ( C , ? ) r ( l l , A ) = - P L * C L / 4 . C C T ( 1 3 , E ) = P r / 4 . 0 0 7 ( 1 1 . 6 1 = 7 ( 9 . 6 ) $?^£&'.:£fl • • • • : ^ V J : ^ ' ? ^ ^ - \ - ' : ' ^ . • V i " • .. _:. -• • • • < ; . ^ * r A i 7 ^ • . T( ? 7 >=1„1>C ; v: :r v'.\^-; = • •;.:,',".. V-..': V;-/.:: >* 'V.. ,: •,/ -. . ' , • = 1 T.C 1 2 , C »=T ( 1 1 . 2 )• ^ 0 .tc-. .'.-•.>:• .- .<:•:. &••-••} :•. • '. :• . - *•;•-• ;,- -:':c-. r;.T13?,5) = T c 1 1 « 2 i • - ' - , - : f . : . • •:.,:'-:-;vr- ,.- -.. .. . •• • > n 2 , : c ) ^ u n , < r v r - : : : . - , , . V ' • • • '.' .' • v ; - . T ( i 2 , i i ) = - ( i i , 5 ) . - v- '-v. '••-.- ,". -• ; T H 2 . ^ - : ) = T ( < : . f . ) . ' " " : • ;•• - ' 4 * 1 4 4 2 1 . 2 5 - 4 ^ " A I L P T F F P t M P . I E S ' AP . K ZE . - .O •.'•••• '.. - - . . . . • . ' . ' • •-. ' .- .-. I f ( I f E P U G . f C . 0 ) G C T C . 2 5 7 : • ; " -:v - ' . . ' ' W R I T E < 5 , 2 1 2 ) :- ' - • • ' " ' • 4 2 ' 424 4 < _ 5 2 1 2 214."-' F O R M A T I '0 ',. ' T C A N S F C - R M A 7 ION M A T R I X ' ) ... • „>...: .. WF ! " 2 ( 5 , 2 1 4 ) ( < T ( L , n ,N = 1 . 1 2 ) .L = l, 1 2 ) :' • \> - f > -V - :,. • ' . " - ' • • . ' . ' F('F."-'.T( • • , 1 2 F j Oo 5 ) ' • • ' ' " " "'•"" " " •'=" ' ' " ' 4 2 5 . ? 5 ^•26 •* 427 257 C ' C C N T 3 M U E . r .vrv.'. ; ' . . . , • •. . . ' , . ,' . •-•• ? : . .• N O V C A L C U L A T E .THE I N V S . F S F . C F T H E . T R ANS'FO R M A T ION "t'ii_T R I X - ' .' r^-' -C A L L P I N V R T ( T , j 2 , x 2 > D r i [ 1 , D C C N O ) ' '~ - - - -'"' 4 2 7 . 2 5 426 425 :co ' I F ( I C E F U C- o E C ) G O TO 2 5 8 . , . ----- . W R I T E ( 9 , 2 0 0 ) O C E T . C C C N C - ' * - ' - •"-'] '•' . '"• • s F O P M A T ( ' — ' , , D O E T = ' , 0 1 5 o f , ' D C C N P = ' i P 1 5 . 6 ) 4 3 0 4 2 1 <• 4 2 2 . c 212' T N'CW C O N T A I N S T F E I N V E R T E P T R A N S F O R M A T I O N M A T R I X ' ~ : ~ " . W R I T E ( 5 , 2 3 2 ) ••'''• " . - " ' ;" ' ;» ' " "'• ' : . •"• F O R M A T ! ' 0 « , ' I M V E R S E O T P ; , N S F C F . M A T ! O N Pf-Plf). ' 433 4 . 2 5 4 ' 4 <: 5 6 C WF ITF ( 9 , 2 1 4 ) . ( I T ( L , M ) ,N = 1 , 1 2 ) » L = 1 , 1 2 ) . - . .;. ... ; C E N T I M E . '• . - - • "'' -• ' NOW C A L C U L A T E T F E R O T A T I O N ' ' M A T R I X ' . '• 415 4 ' 6 s-437 R ( i , 1 ) =cc .... -. - :.'•••'. ' ••• '• R ( ! , 2 ) = S I ' • ' ' ' ••' -•'. " " . " • •'-P ( 2 , 1 ) = - S 1 -' ' ' •' ' ' " - • 436 4 " 9 4 4 2 P ( 2 , 2 ) = C C .'.•:• • - - • . P P 7 1 = 3 / 2 " . . ' .-' '- .' ' • . ' • - . C P 7 J = 1 .2 4 ' 1 4 4 2 4 4 " P 1 I + ? , J + 2 ) = F ( I , J ) " ': ' •- . F.C i+ 4 , J 4 4 ) = F < i , J ) ~ ' •..--.-.•._.•._.•••'•. . • '.; : ''.-;• R ( 1 4 6 , J + 6 ) = D ( I , J ) .'-'• , ' — 445 446 7 - • P ( I + E , J t - 6 ) =R ( I . J ) •- --~—y=- : . ...' P . ( T - » 1 0 , J + 1 0 ) = R ( I . J ) • . > — - — * r ' - - ^ — - . • . . -'-• . c r M u i ' - • -• ' ' •—• ':- • ' ' 1 •' ••' '.' ...' 4 t 7 4 4 7 . 2 5 4 4 " i TS l F I T f ( 5 , 2 7 6 ) -,-' .' . •- ' I F l l D E F l j G . t C . O ) G O TO 2 5 9 ' . " ' ?.\ '. • "•..'•.-' .-. '" "'-":'.' . F C F M AT ( ' • , ' PCT AT I C N f A T P I X • ) '' '' " "'' :.. ' ' 449 4 5 - 1 4 c . 2 5 2 0 0 " 5 C W F ' I T F ( 5 , 2 0 £ ) I I P ( L , M ) , M = l , 1 2 ) , L = i , I 2 ) : - : . . . - . -F O R M A T ( • ' , 1 2 C 1 3 ) • ' ' * •'' - ' • ; . • • . • , • - . ; ' . - ' ' . ' - , ' U - ' ; . / - . " J • . • - . ' C C f> TT N L C ' '''-'-' • : • • ' . 4 C 3 452 '•• : * . « ? ; P5 C .. M L L T T P L Y T F. Y. F T O -GET . . ? - . ' , . . ' ' . • ! C A L L P C - M U L T ( T , F , S , 1 2 , 1 2 , 1 2 , 1 2 , 1 2 , 1 2 ) .... , • .; -.' I F ( I r F P U C . F C . O )' G C T C 260 '•' - ' 4 5 . 4 5 i 4 ? 5 2 2 6 WF I T E ( 5 , 2 2 6 ) .. . , • - • , ' . .; • • F C F M A T (• • , <T • I N V E F S E * ' F = S • ) : -:;'' .: ' - • ' • • .- •-.'. ' , . V R t T F C 5 . ? 2 7 ) ( ( S ( I i J ) , J = 1 » i 2 ) , 1 = 1 , 1 2 ) . - ' . . ' ' : : . -456 41.6. 25 457 2 2 7 260 ; F O F ^ A T C ' •, 1 2 C 1 0 . 3.) _':-.— - . - . -C C N T I N L E V . F I T F C 1 ) ( C S ( T , J ) , 1 = 1 ,3.2 ) , J = l ,12 ) 458 4 55 461 • G ... t ' t I N I T I A L I Z E P • . cr 1 2 1 = 1 , 1 2 ".:•-'-".''.' ..'.-'-• P P 3 2 J-l, 3 2 " - : • '. • • ' - . .,..' ' 4 6 1 462 46 ? 1 2 0 R ( I , J ) = 0 . C C . . . •.' -' -- - .-, . -. ; ' ' C C N T I N L E " ' ' .'••- - ' - • " . ' . . • ' • - ' • ' . - " '••--' •-'••'-. NOW F I T I N T P - R . T H r . =2l.cNRNT S T I F F N E S S f/T F I X I N T . fcp KS C F P O L Y N O M I A L 464 -465 466 •C ccr r F i c I E N T S . ;-. P X = E X / ( 1 . C 0 - L Y X * L X Y j - t-:. • ' . ' ; P Y - E Y / I l o O O - U Y X - ^ U X Y ) " * ' • ' ' • ' ; **>e L F X = P X * U X Y L C Y = F Y * l \ X * 4 6 9 * : c I N T C C R / T I C N F A C T O R S ' 4 7 2 4 7 1 . C l = C L " ( A l + R L ) / 2 . C 0 C2 = C L 1 A L + F L ) / 1 2 . D 0 4 7 ? C 3 = C L * ( A l ' * 3 4 R L T * 3 ) / 1 2 . C 0 4 7 i C 4 = CI * * 2 * ( A U ' r 2 - ~ t . * * - 2 ) / 2 4 . C 0 4 7 4 C f = C L < < - ? , T ( / ' L * R l ) / 6 . C ? 4 7 5 C f =Ct* ( A l J=*-2 - U L * * 2 ) / 6 . TO 4 7 6 F ( 2,2 )=C 1-P.X 4 7 7 F ( 2 , 4 ) = C 5 - e x 4 7 6 P( 2 , 5 ) = C f < 2 . C C * 3 X 4 7 9 P ( 2 , 9 ) = C 1 / 2 . C ' J ' - C U E > * U ~ V ) 4 8 0 ."-»-• - P ( 2 , 1 0 ) = C 6 / 2 . 0 C * ( L F X + U E Y ) 4 61 P ( 2 , ? 2 ) =C.-"( t,~> + L B > ) 4 8 2 P ( 3 , 3 ) = C 1 * C 4 t 3 F (2,4 )= C 6 " G 4 6 4 P C - , 6 ) = 0 5 < 2 . 0 0 * G 4 6 5 P ( 3 , £ . ) = P ( 3 , 3 ) 4 6 6 P ( 3 , : o = C 5 * G ' 4 8 7 * i r , 1 3 ) = C f i r . r r * G 4 f 8 F ( 4 ,4 ) = C2* P.X+C2*G 4 6 9 P.( 4 , 5 ) 8X 4 9 0 P. ( 4 , f ) = C 4*2.r : 0*G 4 9 1 R U , f ) = P ( 3 , 4 ) 4 9 2 F ( 4 , 9 ) = R ( 2 , 1 2 ) / 2 . 0 C 4 9 3 P ( 4 , 1 0 ) = C 4 / 2 o D 0 * ( l . E X - H J E Y + 2 . C 0 + G ) 4 9 4 F < 4 , 1 1 ) = C 3 * 2 . C C * G 4 9 5 P . ( 4 , 1 2 ) = C 2 * ( U C X + L P . Y ) 4 9 6 P ( 5 , 5 ) = C 3 * 4 . C C * 3 X 4 9 7 R ( 5 , 9 ) = 2 . C 0 - F ( 2 , 1 C ) 4 9 8 P ( 5 , I C ) = C 2 * ( I E X + U E Y ) 4 9 9 D I 5 , 1 2 ) = C 4 * ? . n c ' ' ( L 8 X + L E Y ) 5 0 0 P ( 6 , 6 ) = C . 2 * 4 . C l ) * G 5 C 1 — ». - P ( 6 , 8 ) = 2 . C 0 * F ( 3 , I C ) - - - - -5 C 2 P ( 6 , : C ) = F ( 6 , f ) / 2 . C C 5 0 2 F ( 6 , 1 1 > = 2 . C ( T ? ( 4 , £ ) 5 0 4 ] P ( 8 , f ) = R < 3 , 2 ) 5 0 5 R ( F , 3 C ) = P ( 3 , 1 C ) 5 0 6 5 C 7 F ( 6 , 1 1 ) = - . C C * ~ ( 2 , 4 ) F ( 9 , o ) = C l = E Y " 5 C 8 R ( 9,3 C)=C 6 * P Y . 5 0 9 P ( 9 , 1 2 ) = C 5 * 2 . D 0 * e Y 5 1 0 P f l C I C ) = C 2 * G + C 3 * B Y 5 1 1 F ( 2 < - , 1 1 )='( 4, 6 J 5 1 2 P ( l ' - , 1 2 ) = C 4 * " . C J * " Y " 5 ! 3 P( 1 1 , 1 3 ) = C 2 * 4 . r o * G . 5 1 4 R ( 1 2 , 1 2 ) = C 2 * 4 0 F 5 1 5 CC E C O ! f i = l , 1 2 5 1 6 C O ' C O 1 F = 1 , 1 2 5 1 7 F ( I F ,3 / )=" ( ! / » , I P . ) £ 1 6 9cC C O T I N L " 5 1 9 CC 9 0 1 IA = 1 » 1 2 5 ? D CC 9 0 1 I P = 1 , 1 2 5 2 1 * ( I P , I M = T L < F ( I 3 , T / - ) 5 2 2 . 9 0 1 C O N T I M J * -5 2 2 . 2 5 l F ( ! r , - p u c . E C . * ' ) G O TC 2 6 1 5 2 2 IrFIT- ( 0 , 2 7 6 ) 5 2 4 2 7 6 " r r - . T l • ', 'IA'T.<»NSFIP.V'O S T I F F N E S S C A I F ix«) 5 2 5 V I I i J ( 9 , 2 * 6 ) ( ( ' ( ! , J ) , J = 1 , 1 2 ) , 1 = 1 , 1 ? ) -129 c 2 b . 25:.' '261 • CCNT U L u 5 26 *27 •' r V. C ' U L l C C NT A IN IF - / .T AN E POS E • CALL OGlpAMSiP , J 2 VI 2 . 1 2 , 1 2 ) CF '.S\'i'''\'\' c-.'\V.f,i'J • '-•''•«•".' "•""•v""'*'..-• : . v.*/* •••'=«";'•'•"•-..'•» • 52b C ! FF CPU CE.. THE rLI-MEN'T ' T I F F N - - - T> ' F CL i I Cf C, CI v l S 529 c 3 r • R E I N I T I A L I Z E . T F G R . U S E =AS /M CC I 1 M r f r c i A T C c T i ^ 0 f A T IX CO J = l , 12 32 ' , . . ' • - T.l 'I', J ) =0 . CO 0 CCNT IN'L='"'• ' ' 534 CALL D C M U L T I R , F , T , 1 2 , 1 2 , 1 2 , 1 2 , 1 2 , 1 2 ) 53" 00 21 1=1 ,12 :..':'••'•• * " .," • PQ 21 J = l , 12 ' -• -37 ' K ( I , J ) = 0 „ C 0 ... ..• • .... . C ~ E CCNTINLE .. '.:'' '•' •' '. : ' ; • . . ' '. . ' ' ' ' V v«'=M:>' it;,-..;';' 5 9 " C / L L O C W U L T ( T . S . R , ' 1 2 , - ' l ' 2 . 1 2 , , 1 2 , 1 2 , 1 2 ) "' '-- " ' - " ' • - • . — ^ " ^ ^ ' - 1 - > • 5 4 J L F NCW CONTAINS THE. ELEMENT .STIFFNiv-SS MATRIX" IN GLC B A'L" COOP D11 c 41 C NOV CHANCE'' NC CL LAE* 3 AC K T 0 . C R . fG I N AT -FC r T F E N^x-T- ELEMENT 54" IF (NE IF I L D ' . F C V l - ) C O T C 610 • ' ' ' •'•• ' 54 * . GC ~r 615 . • , . •.- • "44 610 SEX = - X . J ;V;i''' ;.V> V : - - ' .'"•'•:•'; " :'-- '•• . • - . ' - • : , ' . - v':-"^^C :-%f;V-v --^  •  545 Fx = '-Y • '-'! r'* '''" "•''- ' •'' • •" • ' -' " ."• - • .-5 H 6 F Y - S 1 X - - v '•• ; C ,'..:'•".-•>;'• ' . . . . . . . . . . . . . . . • ,. .. .,.'.-.•••; "4 7 • SLX> = LXY •'• '. ' '^r-;^-'- '"' ''-U-;-^ - - . i - ' : ' - ^ i - " - ^ - ^ l - . - " . ^ : ^ : ~: - - ,V'-> /" 54" • .•• • ' ' LXY = UYX • • - --" '.'•-' •'•'' •. . • ' l v . ' • ' . - ' - - .v.': vVV• - "?''; > • ''' •' ' ''" ;' 549 . L Y X - S U X Y : - .,',, \ - V-v ;•'•••.>'••• v • > - > V ' ' - - . • V - i . "5C 615 CCNTINLE ' • . - ' ^ -''":'--:V'-^ - '• ,'• V-" :5 • C CF C CK. TG SEE W F ETF F R I NC I VI C L A L ' COL L PN S' CF THE '-ELEV EN'T AL S T I F F N E S S ZJ ' ~5 . 5 C MATRIX -ARE IN ECU IL I EF. IG'M. ' S J . ' l IF ( I D E E U G . E C . O ) GC 'TC _ 262 •' ;-'::f'"*^V. '.- - ' ; ; 55.J.6 ' MF T~r ( 9 . 940) " " • . . . . . . ;,55C. 7 940 - FCPMAT ( ' 1 • t * CEGSEE "CF . F F EEDOf" ,"1C X-, ' CT'CS *7 10X , ' L VE NS ' ) —" r - - -"5 . 6 • oc. 9 6 0 v - 1 , 1 2 '•:.-• "J.-. .-•••'• " '" ' ' .• '>"• ' . , ' ,": . ' •' ' •' •• f . - • • - ^ . c - v . - ' " 55 ) . 61 CCCS ( M =0oC" •"' -' -•"'" - -• . ' • - . • ' ' • ' • - • ' • " - . :-'" " •• 55i . 6 2 ' E V ^N f ( M ) =0 «.C.O .• . . . v •. ' 5 0 . 6 3 960 5 5 0 . 6 4 Cr 9 K J - - I . I 2 . 6 5 C G 92C • I =1 ,11 ,2 •••-•.••-••'.•..•..- •-. , '., . - . > - . _ . . . . . ' , • c c . 6 6 . C O G S ! J ) = C D D S « J ) «-F { I , J ) 'V '(.-•' - ' -^r.'f-:Vv:'" ."• - - :V . { ;^ '^? . . ' ' . :> i ;V ; - ; " ' -5 5 : . 6 7 9?r. cc NT i *- L F " .  r- - r ' - -•' ••••.<•; '•. • ' ; • . '>-v ".ff'-Vi"' 55 . 68 • 00 930 1 =2 , 1 2 , 2 - . •. • :. - - . ••'•.• •"• ....y.:.. ••' •. ' - ........ •,, . -e 5 o . 69 F V E N S ( j ) = s V E N S t J ) + R ( i , J ) r- V '."^v. ^ ^ % ^ . T - . yy •,;' --i^;.- >;^ -.vv^  ,.-.>**-t,r:;/-" ' c 5 " n c 9 3 r C C N ' I N U E - - 1 -. . - : - W 'V'- •• • • *''''; ;" - v ' e e . ' 1 9 i c C CNTl .NL ' ., , ,. . .. . 5 5 0 . 9 2 . ' . . 00 960 J = 3 , 1 2 ••'.'••". ••'' :; .,' "= . ^'-55 . . 93 TT?'- (9 .950) J , CPCS ( J ) . FVEN'S ( J ) , R i 1 . J ) ' : ' .550. 94 ;> 950» . . FCFKAT ( ' ' , 1 9 , 0 2 4 . 6 ,01 5 . 8 , 0 4 0 . 8 ) , . : ' • . , 5 5 C . 9 5 96L CCNTINUS '•:. '. •,'••>- .: *: r v " - ' ; - : . . y'-"'-r. v - ' - • . ' • v ^ ' s ' ' • ' • 55C .96 ' ?6? ' CC NT IN L ~ ' ' ' : " ' ' ' " . ' • • • - - ' ' " ' ' 5 c i . . F.fTLF.N •-• . • • - ' " : . . . - • . ' . 5 5 2 "' rNT • '• '•-:•/•:•.::•••'•-'.' :;<?-?'•• — 1 - - - " ' • - L R R O L T I N E S E T l P l f l . NV3'.1. J . P . LF A NC . NS IZ ) ' ' c £ 4 IMPLICIT R E " ^ L * F ( A - H , 0 - Z ) . . . . . V.:. .•555 •,"-•'.- " .v - - . C ! f i :NS IC N L J 11 ) , A 190 COC ), P. 112 , 1 2 >;. y-^fz : 55-> C A = y l. S T F" R S T I F F N E S S "-AT F i x ' : ' • ' ; • v ' ' ••' 557 C NV:-'=MC. Cr V A? 1 Abl. F S -P EF £-1. EM E N Ttv -(••; sri,.<:y :'i-.v,.> c - . - r, 5 5 ° C L J = CCPE . NUM. FERS FCP T F F . EL ENENT, ;'-..;V; ^ : :. ' 559 C F = F i F N' c M T S T I F F N E S S MATRIX : - ' '- " ' " " ' - - v r ' ' ' " ' ' --- 6 ) N f l = L P A N C - l 6 ) NF I = LCANC.-1 , .- -• •• ••^ •>^ >,/-iv.: .... .'' v.' .-..'=,,•'•••-61 CC 12 I = 1 , N V 2 - ' -'•0.^s:: v-:^%^ 62 : " " " ' C T = m l.'KN N l . ' f H E F IN T H F F l E y P N T > A T P T X ' ' " ' ' 130 * V > J 5 6 3 >* ^  it „ c 5 6 6 -v 1 ' 7 l . C L J I I I ' F d J D 1^ ,1^,1 rr ' i j I,MV IIIB88K r - "J=F(.I-. :. M I M P r f ' -11> """EE LFr> M '".A •' ' . ' A ; - -, 5 6 7 K 6 3 " - f C C J \n T C S F F t N 1 MA T c T X S T P R F ' TG NV"-. S ' N C r E V E U A h T . - . C N L Y 1 r H t L G W E F H A L F O F T H : 5 71.w^ • I C R I F H J C - L J R ) l i , M , i 4 * * - A A A ,>572'«5i5Sp»,vC.;.-- T H I S F A F T O F T K S S U B R O U T I N E I S A C A P T E C F R O M H O O L FY • S P L A N E F P - ' M E N C T ~ S • "'' 5 7 : ; ^ 1 1 2 L = ( u c - l ) - N P . + l J R . • •-• 5 7 4 - " J f - ^ - ' - • C - f i Tp 1 1 5 • ... . ' - : . ... :. •.• 5 7 5 * 1 1 4 • L = ( U P - 1 J N F l + U C ~ , - „ 5 7 6 ^ t l l l t M L ) = M L ) + P < J , I ) 5 7 7 * 7 ^ 1 C C N T I N L E = _ u - 7 6 , ^ , 1 1 - C C N T I N L E - . "575f-OT- 1 2 C C N T I N L E , . : . * F ) ^  - F C T L EN -••'*• _ 5 6 1 A , . - E N D 6 8 - ' '-r ' S L ~ E F O U T I N t L S T F S 1 N A , P V ,r X , C Y , L X Y , I Y > , C , c > X , Z Y Y , S X Y , N E ,I C 0 , N F L P i F X 5 6 2 . 2 5 '- • " X . E Y Y . H X Y . i r . F p U G ) : 5 E S , v w * I F L I C 1 T R E A L * 6 < A - F , 0 - 2 ) 5 8 4 * C l j v r . - N S I C N L J ( 1 2 ) , PI- t 8 6 0 ) 5 E 5 • € ! N"FN S I G N f F L F ( ? i j ) J 5 E 6 = 6 7 ~% ' ' r £ f '.:>**••>-*• C I H E N S I O N P = L ( 1 2 ) , / P ( ' 2 ) C I N E N S I C N S ~ F ( i 4 4 ) , N S T F F ( 1 2 ) C l N P N S I C N / P C ( 4 ) 5 8 9 5 92! ' , ' £ 5 1 C I'M E N S J O N S I G X ( 6 ) , S I G Y l 6) , T A U ( 6 ) , EPSX< 6) , G P S Y ( 6 ) , G A M ( 6 ) , I C P ( 4 " (1 , 6 ) . , P i M E N S I C N ^XX ( 43 i 2 ) , S Y Y ( 4 3 ( ) , S X Y ( 4 ^ ' ) - — D I M E N S I O N ^ I C L X ( 6 ) , < : I G I Y < 6 ) , T i L C L l ( < ) — ~ 5 9 1 . 2 5 5 2 5 5 -. C . l . M E N S I G N L t « L X ( 6 ) , CD«:LY( 6 ) , G A C L < 6 ) , E > > ( 4 ^ 0 ) , E Y Y ( 4 2 0 ) , E X Y ( 4 3 u ) - - .-C C N f ' C N / E L T 1 / S ( 12, 1 2 ) ^ - " C C N N O N / E L T 2 / L J _ 5 9 4 -5 C 5 5 5 6 ^ * C O f ' M E N / E L 7 3 / / L , L'L , C L , T F C T A — - E C L I V A L E N C F ( STF ,<-( i , i ) ) ~ ^  -~ E C U I V A L E N C r ( N C T F F , L I I ' ) 1 - - - - _ £ 9 7 5 5 8 « 5 - , 5 " C~" r -E C L I V A L F N C E ( A P C , A L ) T H I S S L P T L T I N E P C C r i l C F S S T R E S S E S A T T F F f C C c S R c L A T I V t T O " _ — = ~ t l E K F N " ' - N r T G l r > 6 A L - A > r ' S ? 6 1 0 tai" . . &')2 ~ ' C C c S = - T I f i V F h S . E * ' P F C R E L E N E N T - N A -.. . • L. •• . •. • •. •. . :., i-.,. .. • N A = F L E M E N " W r P K r C C N 1~ F N = C I S F L A C f N f N T S i " L L T T C N V C T C ^ " * * Jt 6 0 ^ ^ t , 6 C 4 ' * 6 ' 1 5 c c C . " . E X , S T C . , A R E E L A S T I C C C N S i A N T S ^-'.^ •; S 1 G X = S T K F S S P A R A L L E L T C T H E L I N E - J P I N I N G T H E . F I R S T T W O N O D E S - 0 F ' • ' " ' -•• - T H E - T D I A N G L E -•• - • • : 6< - 6 t C B ^  ' • r. •. • •• C , c : S I G Y = N C F N A L S T F E S S P E R F t N C I C U L A R T C S I G X ^ • T A L = S H F A P S l r - ' S * : " / -F F S X = S T R A I N F A F A L L F L T C R I G X -; .• ' :. "... -rr: •• . •; :t • .-6 C 9 „ , 6 1 0 6 1 1 c c c E P ' E Y E S T R A I N F A R A L L E L " i C S I G Y •- : '- • • -, C - / r = S F E A R S T " A I N P'- L - N A T R I X F F C L O F A L C I S F L A C E M C \ T C f.CF A F L L C , K T _ -6 1 2 » 6 1 5 -6 1 4 .c r c • A F = K A T F I X O F P C L Y N C M * L C O E F F I C I F N T S F C R A N - L C " M E ^ T . . . . . . • M F 1 . F . A R R A Y K ^ ^ ^ ^ o r H K H E L i ^ F f T * F A V f F r V t P ' - c O C i L L I • I N I T I fil I 7 r 6 1 5 6 1 6 ? ° * 6 1 7 " P O 3 E C I - . . I . . • P E L ( I)=nonc A P ( I ) = P . D f 6 1 6 6 1 9 6 2 2 1 5 0 C C N T I N L E •.. -:~ . . : ........ -D C 4 8 2 1 = 1 , 6 E P S X ( I ) - C . r c i -waawicMfli _ „ - -* * , ? • * • . » » ^ * < i ""^ P S Y ( I ) - r „ T i , £ 2 2 ^ v ? " " c / M i ) - i . r - . ^ f z a * f " " ) r c i c x i i ) c i r - * - " £ 2 4 * ^ T C \ ( " ) f D T I ^ 9 2 S • ~ T fl ( J ) ". 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It'."- FORMAT ( ' - ' , 1 2 1 1 0 ) •- ••••'".-.•-•'• '• -£ 4 9 . 25 263 C C N T I N L E - . , 6 5 J C P F j c . r v F ELC"> N T A L D I c F L " C E N E N T " IN C L C B A L C " C D ' r A ^ E S c R C f P I — 6 C 1 r r l 1 " T 1 t c 2 JL L J ( I ) r 6 5 j ' TF (-d-L .EO.O ) GO T O T P O 6 5 4 r r l ( 7 ) P M J L )  £ 5 5 C C T ( i c C 6 5 6 1 8 C C C L I I ) ~ . r c £ 5 7 "-: ' 1 9 0 C C N T ? N U E £ 5 £ 11" C C N " T H L r 659 C C A L ' U L A T r MATF IX CT P C L V\n u I * L C O E F F I C I N l c AP 66C C "F T TNVF>-TFC F ^ L I I  •.ft v >.??»'«. CALL OGMLLT { S . C E L . A F , 1 2 , 1 2 , 1 , 1 2 , 1 ; 6 6 1 . 2 5 If ( I C E F U C . K . C ) GO.TO 264 : .662 •'-• V.P I T f | c , 1 i ' ) N t v" • -l- • •- • • •. • t t 3 ' $;;•;•'>;; ) 3 C F ( Rf.'AT ( ' C ' , ' PCL YN'CMiI AL C O E F F I C I E N T S . FC F E LEMr.NT ' , 1 4 ) 664 M r I T l ~ , 1 4 0 ) IAP(K ) , K = 1 , 1 2 ) • 6.fc5"->»" • '"•'140 F f R v f T ( ' - ' , 6 F 1 £ . 6 ) ' • • ' " -' • ' - - ' • • " • " "'• £ 6 5 . 2 5 264 O C N T I N L * -666 L f - 1 . C C - L X Y " L Y X 667 C C / I C t L A T I C N CF S T R A I N S ANC S T R E S S E S -AT T F E MODI 6 6 3 r r S X ( M f P ( c J - z - . T f - A P I c ) " B L 6 £ 9 " P C X ( " ) Al (2 ) + ^ . r i* A ° ( 5 )•• AL <:70'-^" 7 ~T 5 X C ) = A f > C ) + A P ( 4 ) * C . L 6 7 1 C P " X ( 4 ) A P (2) + A P I ' ) • » ( * L - O L ) £ 7 " P c X t 5 ) A P ( 2 ) - » A F ( 4 ) ' ' C L / ? . 0 a + A r t ' : ) * A L £ 7- ' ~ r f X ( c ) = A P ( 2 ) + A P ( 4 ) * C l / Z . n O - A P ( - ) * R L 6 / 4 ' P " Y ( l ) A P ( 9 ) - A P ( 1 C ) EL £ 7 5 r P c Y ( 2 ) A P ( c ) + / P ( 1 C ) ~ A L 676 ' E P S Y ( 2 ) = A r { c ) + 2 . 0 0 * A P < 12)^CL 132 rt T*V £ 7 7 * l ' V ( - ) AP(<-j + a F ( i C ) U L l ? 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C O 697 F 5 I <~ . 0 ^ 693 -, 701 F n . r i c x ( j ) c x f C F C X ( J ) + X Y E X E P < Y ( J . • • - I C Y ( J ) = Y X F Y * E P S X (J ) + £ Y M * E P S Y ( J ) : * • 7 0 1 V T /• L ( J ) C» C / M J ) v * I 702 c R ; =F ' i.niU5 OF--MCHR • S • CT FCLE ,-FCR.. THIS ...ST FES S,' CAS —..... —^—~rsr.r:'-—• " • ^ r-^ -7"3 S F = ( S 1 C X ( J ) - S I C Y ( J ) ) ~" 2 +4o CO*T AU { J ) v * 2 ' ' r>" -7 ~ J . £ < c r ( F P ' X t . ) c P ^ Y (J) ) * 2+4.CC«CAN ( J )^ -« . 7L-4 F A „5C C - C F T I ^ ) ' , 70- , .5 R/^N . C 0 C r ' C F 1 ( ' F ' M 7 - 5 0 . FFi'l =C'F'.T G I NA L ANGLE FROM PR I NC=I:P i L . S T Pf: SS • PLANE — _=-5C_3sr *^  ,., _ ~ 7 0 5 . 1 C FKIN = CP. IC INAL -ANCLF: F R ON - PR-I 'JC'I'P ALT«i.STPA I K. -P LAN ^  "~":=rZZL _ — *" 7 " 5 . 2 TFN r . P - * C i M J ) / I c F " X ( J ) - r P S Y ( J ) J -7 C 5 . PHIf C A T A M TFN ) 7 L 5 . 4 c F ' -TN / f CLE FPCN CLC E 1L "YSTEN T n P P I N C T FA L c 1R A I N PLANE - _ _ 7 r 5 . 5 1 F ( EFSX ( J ) . I T . F P S Y ( J ) ) F H I N r P I - . I M 3 . i 4 3 59 -~ ' EH - sr : 7 06 TF ^ . l v.^T / 1 ( J ) / ( S I C X ( J ) S I G Y t J ) ) — 7C7 TH1 C A T A N ( T F ) 7 ' f l r • P S I ='ANGLfc FROM GLCFAL S Y ST Ef* T C P F I N C 1 F A L STR:t S r PLANF 709 -IF ( S 10 X ( J J . L To STGY ( J ) ) P H I = P F I + 3 . 1 41 5 9 - ... 710 7 H . C F C I P H - 2 . L C - ' T F E T A F 5 N = ( E P S X ( J ) + c F S Y ( j ) ) / 2 o O O . •• • -7 1 i F ( " I C X ( J ) + SICY 1J ) ) / 2 „ r o 7 1 1 . 5 F-^If F F I N-_.C F r 7 1 712 c - • • - I C I X - K O F M A I STRFSS I M - ' H E G L C B A L' X - C I F E C T I C N 713 • r ; S IGLY = NOPNAL- ST » ES S ; : l N. .. GLOBAL A Y - C I • R F CT 3 OM 7 1 i C T A L'HL = ?HE A R "S TR. -SS REL A T I V E . T C GIOEAL . /X ES V 7 . ;:• 715 - i c i > ( J ) F + F ^ r c o s ^ s i ) 716 - ' C L Y ( J ) F F A - T - C S I F ^ I ) 717 T A L C l ( J ) » M r q \ I F " l 7 1 7 . 1 r F S L X ( J ) = F S N •» F. A S N * CCOS ( PS IN') " ' 1 • .• • • --, 7 1 7 . 2 f ^ L Y J J ) f " f ^ A S N * C C r S ( F S I f ) 7 1 7 . " 7ie Cl> I ( J ) K A ' f * N ' IN ( F N ) K F IT ' ( 6 , 231 ) IC 0( r;A ,'J ) , SIGX (J ) ? SI G Y ( J ) ,T ^  U ( J ) , E'T'SX ( J ) , EPS Y ( J ) , GAM ( "." 7 1 ' * J) 720 2' ' Fr Rf / T ( • • , 1 X , I 5 , F 6 . ' , F 6. £ ) 721 2 3'i C C N T T N l *~ 7*2 C .7 <L« THP S T P f S S F S . A N 0 . S T R AI N S A T • J F E.- N C C E S . F 0 R •.: AV FRAGINC 723 J i I C O I N ' . M tr 724 J2 I T I N * , ? ) 133 * - 7 2 7 - * \ . J r ICO(NA,5> -.72.5 --7 3-5 / I R 2 ^ ' • X X I I J . J ^ y x « J l ) * S I C L X ( 1) C X X ( J ) *XX<J H S K L M ) * < : Y Y t J l ) - c Y Y ( J l ) + S I f L Y ( i ) v * - 7 «• - c Y Y( I") c Y Y ( J > + c I 0 L Y ( 2 ) 7 -»» J. S \ Y ( J ) c YY ( J )+< ICLYI ) 7 -o c X Y ( J ' ) <XY< J l ) + T A L C l 1 1 ) 4 7 6 C X Y U ) r X Y t ) + 7ALGL( 2 ) 7-7 ^ X Y I J ^ J C X Y ( J 3 ) + T A L C L < _ ) * 7 3 E < X X ( J M <XX( 1 + S T C L X K ) 7 - ^ ^ x x u s i ^XXI J ^ J + ^ ' f L X J n £. 7 4 0 . 1 l X X < J 6 ) S X X ( J 6 ) + S i C L X { 6 ) ^ Y Y U ' ) c Y Y ( J 4 ) t S ' C L Y l 4 ) — 74? ^ r Y Y ( 1*) c Y Y ( J c ) + I C L Y ( c ) "*T_ W U. ^ — . 74J < : Y Y ( J 6 ) C Y Y U 6 H S I C L Y { 6 ) 744 J X Y ( J 4 ) = S X Y ( J 4 ) + TALGl:'(4) 74r> C X Y <JS) <:yY(J c ) + TAI f L ( c ) 7t6 ^ X Y U C ) c X Y ( 06 ) +T A l CL ( 6 ) — 7 4 6 . C 5 E X X ( J • ) = F X X ( J l ] + F P 5 C X ( 1 ) * 7 4 6 . 1 E>. X ( J 2 ) = l 'XX(J2 )+FPSLX( e. ) 7 4 6 . 1 5 p X X t J ° ) _ F X X ( J ) + «-F cLXe) -. 7 4 6 . 2 P X X U 4 ) C XX< J 4 ) + C P < : L X ( 4 ) 7 4 6 . 2 5 c > X ( J r ) E X X ( J c i ) + F F c L y ( c ) 1 ' 1 V-?f : ' ' " " ' .V 1 - * ' '> f ^ S ' . f ^ - -'•"» :'7 :.-'' ;' -:v '','v.r;:--W5f-:'7-~.vv' / 4 6 . 2 E X M J 6 ) FXXC J 6 ) t C P ^ L X (6 ) 7 4 6 1 . 3 5 C Y Y ( J I ) EYY< J 1 H _PSLY< 1) 7 4 6 . 4 —-EYY ( J2 ) =f YY ( J2 ) + ETSLY ( 2) * - — "w — 7 4 6 . 4 5 F Y Y ( J 2 ) = F Y Y ( J3 J + E P S L Y C ) 7 4 6 . 5 cYY( .J4) = E Y Y ( J 4 ) + 2 P S L Y { 4 ) 1 _ V-: • : ' 7 46 . 55 " ' E Y Y ( J 5 ) = F Y Y ( J 5 ) + t : F S L Y ( 5 ) *~ =- •+-->, ^_ _ 7 4 6 . 6 r \ Y ( J 6 ) = E Y Y ( J 6 ) +E P 5 L Y ( 6 ) 7 4 6 . 6 5 EX Y ( J 1 ) = EXY ( j 1 ) +GA-M. I i ) - — — ^ -— 7-*6.7 ~ "~ XV ( J ? ) = EXY< J2 ) + G A M (? ) * =—v_ ^ . _ — f 7 4 6 . 7 5 C X V ( J 3 ) = tXY( j : - ) + GANL( ) 746. E 1 yY (J4 ) = r XY ( J 4 )+GAM. (4 ) * 7 4 6 . £ 5 E X Y ( J 5 ) = F X Y ( J ^ ) + G A F L { r ) r -1 7 4 6 . 5 F > Y I J 6 ) = C X Y ( J 6 ) + G A K L ( 6 ) K 747 * . ' r • - .. CHANGE MCOL'L I P£CK FOR T F t 1* EXT ELF / FNT 74B IT ( K L r (NA) . F O . I } GC TC 6 C 749 » - .GC-vTO 615 7 5 : ^ - 610 C FX=EX 751 * F X = F Y 752 •* EY=SCX U 7 5 3 - EL>Y=LXY ;"'""...':;'-->- •v:>^^--l^,rvi-^4.-;." l;r^L''..''^ CCr/' *^ v-.4^ ."'V.>'v..~ 75M IXY=UYX LYX=SLXY 756 ' '/ 615 CCNTINLE 7 57 Ff TLF.-N 7 5 3 E( C * 7 5 9 SL PP. OUT in c j XP A\'D ( A GRO S S , MMAT , MVAP , PM , IX) %.*.X.< 762 c THIS E L H F C U T ] NF ' E X F ANOS TH - • S C L U T T C N V f C T CP' 0 F <: IZ F NN-T r / C K TC 761 C •. S I Z E ,NM/ .T,HY. I N S E P T H C Z w C " . VhC " 2 S.L F u l N C ^ P Y C O N P I T I C " : 762 c ... , ' U f P G >PPl IEC A O FF INTS IT PUT 76T T AC P P S ^ c X P A N r T S C i l . T l C N VFCTCP T r **LS I Z f NM /T 'v ( F r - j f u c p ) * " 764 f . N F 7 M = G R 0 S S S IZE C F M H . FP r PLFf 765 c NVAP = NL'MFEF CF V A R I A B L E S F EF f C C F CF E LPF F 1 1 T 766 c FN=N!ET S C L L T I C N ' VECTCR CF S I Z E N N C T J- 767 ' f C JX ICLNr-AKY CLN01TICN CCuE V( C~or 76 8 c THIS SLBFPLT11 f HCFK^ CNLY IF / L l C f f F f F " : CF F F r r C C AT A NOPE i 76<3 # c /•FE f I Tt FF t F S l R ^ I i \ c D OF Lf\ n r L" 7 ~ A-1 t 1 C ^ I* L « Pf Lf V ^ 7 7 J I L L I C I T FWL»-E IA-F,'-Z) < . ^ 771 77? J. r u ^ M c r F M 1 ) ,A( FCS< (1 ) , IX M ) -L L - -. 773 CC) * I-l.fr-Z'T 774 /- -J L L L L + I X P ) . - J r, < 9 ~775 K F r i ; t (I l - C . C " % 776 I F ( I X ( I ) . t f . C ) GC TC 5 777 ACPI" r>< ( T J - F F ( LL ) 778 C r M IK'E ^ 775 N T NNAT/NV/F 780 -.ii »v-i(Tf- • ' • 1 ViF 1 -( ( t , 4CJ / " 781 F f r N / T l « 'NCC c ,»l K X f • r" L X 1 » 1 c X , T i L Y ' ) — —1 — — <- - - — 7E? 782 rc IP i-i,Nr<; , - _ ^ P-NV/F* U - l ) * l A -* ~ -- 7E4 12 P+NVAP-I — 1-7E5 V r i T F ( c , ^ l ) I , ( * r " : C S S I J } , J = I l , 1 2 ) 7E6 41 FC P'A T( 15,2r20.5) 7E7 l r< CCNTIf\L c 7E8 • END CF F I L E 

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