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Soil tillage studies with model plane chisels. Strong, Chester Ray 1971

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SOIL TILLAGE STUDIES WITH MODEL PLANE CHISELS CHESTER B.S.A., University of BY RAY STRONG B r i t i s h Columbia, 1966 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in the Department of A g r i c u l t u r a l Mechanics We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH May, 1971 COLUMBIA In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced degree at the U n i v e r s i t y o f B r i t i s h Co lumb ia , 1 ag ree tha t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s tudy . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y purposes may be g r a n t e d by the Head o f my Department o r by h i s r e p r e s e n t a t i v e s . It i s u n d e r s t o o d tha t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department o f AGRICULTURAL MECHANICS The U n i v e r s i t y o f B r i t i s h Co lumbia Vancouver 8, Canada Date A p r i l 14, 1971. ABSTRACT The physical c h a r a c t e r i s t i c s of p a r t i c l e size d i s t r i -bution, compactability and p l a s t i c i t y of Ottawa sand and Haney clay were determined. Direct shear tests were used to relate dry bulk density, s o i l water content and normal pressure to the shear strength of Ottawa sand and Haney clay. The s t a t i c and k i n e t i c values of soil-metal f r i c t i o n were determined for each of three c h i s e l shaped t i l l a g e machines with Ottawa sand and Haney clay. The f r i c t i o n values were then related to normal pressure, area of contact and s o i l water content for each s o i l . T i l l a g e studies were conducted and the forces r e s u l t i n g from soil-machine i n t e r a c t i o n were measured. For each s o i l , these forces were related to s o i l water content, dry bulk density, machine width and machine v e l o c i t y . The s o i l and c h i s e l variables were combined in accordance with the Buckingham IT theorem to form dimensionless r a t i o s . These dimensionless r a t i o s were combined to form equations f o r use i n model-prototype predictions. The accuracy of these predictions was found to vary with s o i l water content, -dry bulk density and machine v e l o c i t y . Since a l l measurements recorded during the course of t h i s study were analyzed by s t a t i s t i c a l procedures, the r e s u l t i n g equations do not represent basic physical r e l a t i o n s h i p s . Caution should therefore be used i f these equations are to be applied to values beyond the range of values analyzed i n t h i s report. - 1 1 TABLE OF CONTENTS PAGE INTRODUCTION 1 Statement of the problem 1 Study objectives 2 Project outline 3 REVIEW OF THE LITERATURE 5 S o i l t e s t i n g procedures 5 S o i l - t o o l interactions 7 T i l l a g e t o o l s i m i l i t u d e 8 T i l l a g e t o o l c h a r a c t e r i s t i c s 9 EXPERIMENTAL METHODS 1 1 S o i l t e s t i n g 1 1 P a r t i c l e size analysis 1 1 P l a s t i c i t y t e sts 1 1 Shear t e s t i n g 1 1 S o i l - c h i s e l interface 1 2 Soil-metal f r i c t i o n 1 2 T i l l a g e t e s t i n g and equipment used 1 3 Equipment 1 3 Instrumentation 1 3 T i l l a g e tools 1 4 S o i l variables 1 6 ANALYTICAL PROCEDURES 1 7 Shear strength 1 7 Soil-metal f r i c t i o n 1 7 T i l l a g e analysis 1 8 - i i i -PAGE RESULTS AND DISCUSSION 24 S o i l physical properties 24 S o i l - c h i s e l i n t e r a c t i o n 26 T i l l a g e forces 27 Direct relationships 27 Dimensionless equations 30 SUMMARY AND CONCLUSIONS , 5 3 SUGGESTIONS FOR FURTHER WORK 54 LIST OF REFERENCES 55 APPENDICES A. Tranducer plan 58 B. Amplifier construction 61 C. Correlation matrices f o r s o i l shear 64 strength test D. Correlation matrices for soil-metal 66 f r i c t i o n tests E. Correlation matrices for t i l l a g e tests 68 - i v -LIST OF TABLES Variables and corresponding dimensions' Comparison of s o i l physical c h a r a c t e r i s t i c s Internal angles of f r i c t i o n f o r Ottawa sand Reaction forces f o r Ottawa sand when v e l o c i t y i s varied and B = 0.057 l b / i n 3 and W = 1.99% Reaction forces for Ottawa sand when water content i s varied and B = 0.054 l b / i n 3 and V = 10.72 in/sec Reaction forces for Ottawa sand when dry bulk density i s varied and W = 1.99% and V = 10.72 in/sec Reaction forces f o r Haney clay when v e l o c i t y i s varied and B = 0.04 7 lb/-in" 3 and W = 8.6 8% Reaction forces f o r Haney clay when water content i s varied and B = 0.045 l b / i n 3 and V = 10.72 in/sec Reaction forces f o r Haney clay when dry bulk density i s varied and W = 8.68% and V = 10.72 in/sec Correlation matrix f o r d i r e c t shear tests f o r Ottawa sand Correlation matrix f o r d i r e c t shear tests f o r Haney clay Correlation matrix for soil-metal f r i c t i o n tests f o r Ottawa sand Correlation matrix for soil-metal f r i c t i o n tests f o r Haney clay Correlation matrix for ti l l a g e - tests for Ottawa sand Correlation matrix for t i l l a g e tests f o r Haney clay. - V -LIST OF FIGURES FIGURE TITLE 1 P a r t i c l e size d i s t r i b u t i o n for Ottawa sand 2 P a r t i c l e size d i s t r i b u t i o n f o r Haney clay 3 Results of standard Proctor test f o r Ottawa sand 4 Results of standard Proctor test f o r Haney clay 5 Ottawa sand. Actual draft force vs. value computed from Equation [11] 6 Haney clay. Actual draft force vs. value computed from Equation [14] 7 Comparison of predicted draft forces f o r 2.25 inch wide c h i s e l in Ottawa sand when ve l o c i t y i s varied and B = 0.057 l b / i n 3 and W = 1.99% 8 Comparison of predicted draft forces f o r 2.25 inch wide c h i s e l in Ottawa sand when water content i s varied and B = 0.054 l b / i n 3 and V = 10.7 2 in/sec 9 Comparison of predicted draft forces for 2.25 inch wide c h i s e l i n Ottawa sand when dry bulk density i s varied and W = 1.99% and V = 10.72 in/sec 10 Comparison of predicted draft forces f o r 2.2 5 inch wide c h i s e l i n Haney clay when ve l o c i t y i s varied and B = 0.048 l b / i n 3 and W = 8.68% 11 Comparison of predicted draft forces f o r 2.25 inch wide c h i s e l i n Haney clay when water content i s varied and B = 0.045 l b / i n and V = 10.72 in/sec 12 Comparison of predicted draft forces f o r 2.2 5 inch wide c h i s e l in Haney clay when dry bulk density i s varied and W = 8.6 8% and V = 10.72 in/sec 13 Computed vs actual draft force f o r 0.75 inch wide c h i s e l i n Ottawa sand - v i -LIST OF FIGURES CONTINUED FIGURE TITLE PAGE m Computed vs actual draft force f o r 1.50 48 inch wide c h i s e l i n Ottawa sand 15 Computed vs actual draft force f o r 2.2 5 49 inch wide c h i s e l in Ottawa sand 16 Actual draft force vs value computed by 49 General (G) Dimensionless Equation f o r Ottawa sand 17 Computed vs actual draft force for 0.75 50 inch wide c h i s e l i n Ottawa sand 18 Computed vs actual draft force f o r 1.50 50 inch wide c h i s e l in Ottawa sand 19 ' Computed vs actual draft force f o r 2.2 5 51 inch wide c h i s e l i n Haney clay 20 Actual draft force vs value computed by 51 General (G) Dimensionless Equation f o r Haney clay A l Transducer plan with s t r a i n gauge locations 59 A2 Wheatstone bridge configurations f o r forces 50 and moments to be measured Bl Schematic diagram of s t r a i n gauge amplifiers 62 ST OF NOMENCLATURE ABBREVIATIONS AND DEFINITIONS Angle of i n t e r n a l f r a c t i o n -- angle between Mohr f a i l u r e envelope and horizontal axis. s o i l dry bulk density •-- weight of oven dried s o i l per unit volume ( l b / i n 3 ) . maximum dry bulk density obtained with standard Proctor test ( l b / f t 3 ) . cohesive strength — s o i l normal stress ( l b / i n ^ ) . draft force -- horizontal of c h i s e l movement. v e r t i c a l force -c h i s e l action. shear strength at zero o reaction force along axis - v e r t i c a l reaction force caused by c o e f f i c i e n t of uniformity — indicates slope of p a r t i c l e size d i s t r i b u t i o n curve, determined by D60 / D1CT diameter at which 10% of the sample i s composed of smaller diameter p a r t i c l e s . diameter at which 60% of the sample i s composed of smaller diameter p a r t i c l e s . s o i l void r a t i o -- r a t i o of volume of void space t volume of s o i l s o l i d s in a given s o i l sample. k i n e t i c f r i c t i o n -- resistance to motion over s o i l -metal interface while motion i s occurring at a uniform rate. Lower Atterberg l i m i t -- minimum water content i n percent at which s o i l exhibits p l a s t i c i t y . Moment about the horizontal axis which passes at r i g h t angle to d i r e c t i o n of c h i s e l movement. machine scale factor -- r a t i o between prototype machine size and model machine s i z e . normal pressure -- the normal stress acting at right angles to f a i l u r e surface i n s o i l shear test or soil-metal f r i c t i o n . p l a s t i c i t y index -- the water content difference between upper and lower p l a s t i c l i m i t s . - v i i i -peak shear stress -- the maximum shear stress value as determined from shear stress-deformation curve. resultant force -- maximum force reaction to s o i l -c h i s e l i n t e r a c t i o n . s t a t i c f r i c t i o n -- resistance to motion over s o i l -metal interface when motion i s imminent. steady shear stress — that value of s o i l shear strength where shear strength remains r e l a t i v e l y constant i n spite of increasing deformation. Chisel area — c h i s e l width times length below s o i l surface. upper Atterberg l i m i t -- the maximum s o i l water content i n percent at which a s o i l sample exhibits p l a s t i c i t y . machine v e l o c i t y -- v e l o c i t y i n in/sec at which the c h i s e l being studied passes through a s o i l mass. s o i l water content -- the weight of water per unit weight of dry s o i l expressed on a percentage basis. the water content at which maximum dry bulk density occurs f o r standard Praetor t e s t . -••ix -ACKNOWLEDGEMENTS The writer wishes to express his sincere gratitude to Professor L.M. Staley f o r assistance and guidance i n con-ducting t h i s study. The encouragement and help rendered by Dr. J . deVries, Dr. R. Campanella and Dr. E, Nyborg i s greatly appreciated. The writer also wishes to thank Chief Technician Mr. W. Gleave and assistant Mr. H. Pehlke for help i n developing the equipment and instrumentation used i n t h i s study. Fin a n c i a l support for t h i s i n v e s t i g a t i o n was provided by the National Research Council of Canada, through Grant number A-1915. INTRODUCTION Statement of the problem While s o i l c u l t i v a t i o n machines helped form a basis f o r a g r i c u l t u r e , man has been unable to determine a complete mathematical r e l a t i o n s h i p involved between these machines and the s o i l . Consequently he has been unable to do quantitative design e i t h e r for minimizing the forces and energies involved or f o r creating a s p e c i f i c s o i l condition. In f a c t , t r i a l and error methods have been merely expanded i n order to develop increasingly complex t i l l a g e tools without knowing eithe r t h e i r reaction forces or t h e i r effects i n advance. In most instances where engineers or other s c i e n t i s t s have attempted to develop a quantitative soil-machine r e l a t i o n s h i p , they have been prompti by a need to develop an immediate, single complex t i l l a g e t o o l (such as an advanced mouldboard plow) or the study has been r e s t r i c t e d to an extremely small part of the o v e r a l l picture. One must note that while early workers did not have access to modern, high speed e l e c t r o n i c computers, th i s type of equipment has been used only to a li m i t e d extent for data analysis i n many recent projects. Following an observation of the almost complete lack of progress i n attempts to understand soil-machine interactions t h i s project was designed so that the i n d i v i d u a l effects of s o i l physical c h a r a c t e r i s t i c s , s o i l strength properties, s o i l machine size and operating variables might be studied and analyzed i n an independent, orderly fashion. Study Objectives 1) To select two basic s o i l s , one being cohesionless and the other exhi b i t i n g cohesive properties and to determine t h e i r physical c h a r a c t e r i s t i c s of p a r t i c l e size d i s t r i -bution, compactability and upper and lower Atterberg l i m i t s . 2) To determine the effects of dry bulk density, s o i l water content and applied normal pressure on the shear strength of each s o i l . 3) To determine the magnitude and c h a r a c t e r i s t i c s of s t a t i c and k i n e t i c f r i c t i o n when movement occurs between the s o i l - c h i s e l interface and to determine the eff e c t s of interf a c e area, applied normal pressure, s o i l water content and dry bulk density on the f r i c t i o n forces f o r each c h i s e l and s o i l to be studied. 4) To determine f o r each s o i l the effects of dry bulk density. water content, c h i s e l width and v e l o c i t y on the reaction forces f o r f l a t , c h i s e l shaped machines i n c l i n e d to enter the s o i l at 45 degrees to the d i r e c t i o n of motion. 5) To use the Buckingham TT theorem for developing a series of dimensionless r a t i o s involving a l l measured and calculated s o i l , s o i l - c h i s e l and c h i s e l variables and then by regression analysis, to develop prediction equations capable of c o r r e l a t i n g these dimensionless r a t i o s so that s o i l - c h i s e l reaction forces are indicated. 3. Project outline In accordance with the previously stated objectives, Ottawa sand and Haney clay were selected as s o i l s with the q u a l i t i e s desired for the scope of t h i s project. Both s o i l s were subjected to mechanical a n a l y t i c a l procedures i n order to determine t h e i r p a r t i c l e size d i s t r i b u t i o n and Atterberg l i m i t s . Both s o i l s were then subjected to standard Proctor tests i n order to develop a sound basis for understanding the effects of s o i l water content and dry bulk density on input energy relationships f o r these s o i l s . Direct shear tests were then carri e d out on each s o i l and the s o i l shear strength was related to the following variables; normal load, s o i l water content and dry bulk density. Three c h i s e l widths; 0.75, 1.50 and 2.25 inches were studied i n f r i c t i o n tests by moving each over prepared s o i l surfaces. The reaction forces were measured to determine the soil-metal f r i c t i o n involved. For each c h i s e l and f o r each s o i l , the normal load, s o i l water content and dry bulk density were varied so that t h e i r effects on the soil-machine f r i c t i o n forces might be developed on a quantitative basis. The f i n a l portion of the study was then carried out -by moving each c h i s e l at various v e l o c i t i e s through a large sample of each s o i l . The consequent reaction forces were measured as s o i l water content and dry bulk density were varied under measured conditions. These forces were then related to d i r e c t l y measurable s o i l and c h i s e l variables as w e l l as to ' h. the c o m p o s i t e v a r i a b l e s .of s o i l s h e a r s t r e n g t h and s o i l - m e t a l f r i c t i o n . 5. REVIEW OF LITERATURE S o i l t e s t i n g procedures Most of the accepted test procedures for determining s o i l strength parameters have evolved from testing and pre d i c t i n g for s t a t i c conditions. These test procedures have been d i r e c t l y applied to the dynamic reactions of s o i l t i l l a g e . With regard to the s o i l s considered for t h i s project, Ottawa sand i s cohesionless and i s a r e l a t i v e l y simple physical medium for study and prediction when compared with cohesive Haney clay which has been described as a v i s c o p l a s t i c material (13). Lambe (15) provides an excellent basic description of many"of the standard s o i l t e s t i n g procedures as well as depicting the methods of presentation and the usefulness of the test r e s u l t s . Each t e s t outline also includes a b r i e f description of the s o i l mechanics theories involved and the interactions and e f f e c t s involved when s o i l parameters and/or test procedures are varied. He also states that while the shear strength of a cohesive s o i l generally increases as the rate of shear i s increased, the shear strength of a cohesionless s o i l varies less than 2% f o r shear rates between 0.1 and 0.0006 inches per minute. Panwar and Siemens (19) were able to rel a t e s o i l f a i l u r e energy relationships and shear strength to water content and dry bulk density f o r a Drummer s i l t y clay loam s o i l . These relationships were developed from the results of a series of di r e c t shear tests and unconfined compression t e s t s . G i l l (8) was able to develop a rel a t i o n s h i p between progressive losses of s o i l water by a s o i l sample with corres-ponding dry bulk density increases and was consequently able to v e r i f y the existence of a shrinkage l i m i t on the basis of quantitative t e s t s . By applying X-ray techniques to s o i l studies , K i t a n i and Persson (14) developed procedures capable of d i r e c t measure-ment of a x i a l displacement within a s o i l sample. The displace-ment which they measured and were able to describe quantita-t i v e l y was caused by the compression of a s o i l sample by t r i a x i a l shear test apparatus. Using t h i s technique they were also able to relate normal stresses to measured variable l a t e r a l stresses. Kim (13) was also able to d i r e c t l y measure s o i l deformation induced by applied stresses by using Moire fringe techniques which he developed f o r cohesive s o i l s . Vomocil and Chancellor (28) related the compressive and t e n s i l e strength of remoulded samples of Yolo s i l t loam, Yolo s i l t y clay and Columbia s i l t loam to both volumetric water content and moisture retention pressure. • Nichols (17) was a pioneer i n the f i e l d of s o i l t i l l a g e studies and his series of a r t i c l e s e n t i t l e d "The Dynamic Properties of S o i l s " outlined a series of test results and theories capable of r e l a t i n g some of the s o i l strength properties to ph y s i c a l l y measurable s o i l variables. Fox, et a l . (7) determined the energy required to pulverize a s o i l sample to a desired state and related t h i s energy to the moisture content and p a r t i c l e sizes of the s o i l sample. They also related s o i l shear strength to s o i l moisture content. S o i l - t o o l interactions The i n t e r a c t i o n between a s o i l and a machine operated so as to rearrange t h i s s o i l i s an extremely complex area of study. Development of any r e l a t i o n s h i p attempting to explain such interactions must involve understanding the i n d i v i d u a l and/or cumulative effects of a l l s o i l and machine variables included i n the r e l a t i o n s h i p and the manner i n which they a f f e c t the i n t e r a c t i o n . Nichols et a l . (18) were able to determine the effects of plow share shape, amount of wear and angle of approach and the i n i t i a l s o i l condition to the types and extent of reaction forces imposed by a s o i l sample". They measured the physical forces involved and the modes of s o i l reaction as a t i l l a g e t o o l passed through a s o i l mass. The l a t t e r were determined v i s u a l l y through a glass walled t i l l a g e bin. Chisholm et a_l. (5) studied the relationships among the s o i l conditions and the forces acting on an i n d i v i d u a l t i l l a g e t o o l while i t s operation i s being affected by other tools operating in conjunction with i t . They determined that for a s p e c i f i c t o o l operating in an a r t i f i c i a l s o i l , d raft forces could be varied by over 25% depending on the degree and type of interference caused by the associated t o o l s . Wismer and Luth (30) were able to develop prediction 8. equations f o r c h i s e l s operating in saturated clay s o i l s . Their studies indicated a relationship between the apparent cohesive strength of a s o i l as determined by undrained t r i a x i a l shear tests and the resistance of the s o i l to the i n t r u s i o n of a cone shaped penetrometer. Nichols (17) was able to relate the force reactions involved in soil-metal f r i c t i o n to s o i l water content, t i l l a g e t o o l area, the surface condition of the t i l l a g e t o o l , the normal pressure applied to the s o i l - t o o l i nterface and, i n cases of extremely loose s o i l conditions, to the dry bulk density of the s o i l . He was able to observe four d i s t i n c t phases of soil-metal f r i c t i o n ; compression, f r i c t i o n , adhesion and l u b r i c a t i o n . The main distinguishing factor was s o i l water content. T i l l a g e t o ol s i m i l i t u d e A number of projects designed to evolve an under-standing of the interactions between s o i l s and t i l l a g e tools have been based on the theories of similitude and dimensionless r a t i o s . The dimensionless r a t i o s involve measurable parameters and are calculated by the Buckingham IT theorem. This procedure has been successfully used i n f l u i d mechanics and has been applied to the f i e l d of s o i l mechanics. Consequently, the s o i l and machine variables have been treated i n manners which may or may not indicate t h e i r precise e f f e c t on s p e c i f i c s o i l -t o o l i n t e r a c t i o n s . Some investigators have successfully used t h i s procedure to develop s a t i s f a c t o r y prediction equations f o r the s p e c i f i c conditions they were studying. Others, however, have not been so fortunate. A l l have been unable to provide 9. explanations f o r eit h e r success ov f a i l u r e i n terms of s o i l and/ or t o o l parameters and t h e i r e f f e c t on s o i l mechanics. Reaves, e_t a l . (21) were able to develop si m i l i t u d e based prediction equations f o r a variety of chisels operating i n an assortment of s o i l types. However, they have not included water content i n any of t h e i r dimensionless r a t i o s and did not mention the water contents at which the s o i l s were tested. . Wang, et a l . (29) state that they have developed equations capable of predicting draft forces with acceptable accuracy l i m i t s under any given range of s o i l conditions by conducting experiments in a d i f f e r e n t s o i l . They also claim the a b i l i t y to estimate draft force within a model-prototype scale factor of 2 to 1 without having to resort to d i s t o r t e d models. These conclusions were stated following tests con-ducted on a single s o i l at an unstated water content. Unfortunately, they have deemed as unimportant and therefore have not indicated the extent of the experiments to be conducted i n the d i f f e r e n t s o i l s under consideration. T i l l a g e t o o l c h a r a c t e r i s t i c s The lack of available quantitative design parameters has resulted i n most t i l l a g e t o o l designs being based on t r i a l and error methods and q u a l i t a t i v e observations. Very few t i l l a g e studies are designed to y i e l d d i r e c t quantitative i n f o r -mation regarding the interactions of various t o o l parameters or the e f f e c t s of these interactions on t i l l a g e forces. Kaufman and Totten (12) have outlined a q u a l i t a t i v e process f o r developing a s p e c i f i c mouldb.oard plow while 10. Soehne (24) has outlined the development of t i l l a g e tools i n r e l a t i o n to t i l l a g e requirements and indicates that improved quantitative knowledge might r e s u l t in modifications and improvements to t i l l a g e tools.-Carlson (3) has outlined the development of mould-board plows from the stages of q u a l i t a t i v e analysis to the development of a mouldboard plow from t h e o r e t i c a l quantitative knowledge. This quantitative knowledge i s analyzed and converted to design c r i t e r i a by use of a s p e c i a l computer program. 11-EXPERIMENTAL METHODS S o i l t e s t i n g P a r t i c l e s i z e analysis Samples of Ottawa sand and Haney clay were subjected to a dry sieve analysis. Since no p a r t i c l e s of Ottawa sand passed through 0.149 mm sieve openings, the p a r t i c l e size analysis was deemed completed. Haney clay was, however, subjected to a Bouyoucos hydrometer analysis as outlined by Lambe (15). The r e s u l t i n g data were then plotted on semi logarithmic graph paper and the values of D ^ Q , Dg^ and the c o e f f i c i e n t of uniformity were determined from these graphs. P l a s t i c i t y tests Ottawa sand, being cohesionless, was not subjected to p l a s t i c i t y t e s t i n g . However, the cohesive Haney clay s o i l was subjected to Atterberg l i m i t tests as described by Lambe (15). The upper and lower Atterberg l i m i t s and p l a s t i c i t y index were thus determined. Shear t e s t i n g A major problem i n studying the relationships between the shear strength and dynamic strength properties of a s o i l i s the s e l e c t i o n of a suitable s o i l shear test procedure. Other investigators have noted differences r e s u l t i n g from varying the test procedures. Very l i t t l e information i s available to rel a t e the actions and r e s u l t s of these te s t procedures to the actions and results imposed during t i l l a g e . Thus, various test procedures were studied f o r t h e i r shear actions and t h e i r corresponding usefulness. The factors most considered i n se l e c t i n g the te s t procedure were the freedom of s o i l pore water movement and the r e l a t i v e degree to which the shear f a i l u r e planes would be predetermined during t i l l a g e . Consequently, the st r a i n - c o n t r o l l e d d i r e c t shear test described by Lambe (15) was selected as thi s procedure most clos e l y indicated the shear f a i l u r e behaviour during t i l l a g e t e s t i n g with plane c h i s e l s . Each s o i l was shear tested i n both compact and loose conditions at each of three water contents. Each of these combinations was also subjected to normal pressure of 3.75, 9.28 and 17.53 pounds per square inch. For Ottawa sand, the water contents were 0, 10.8. and 19.9 % while for Haney clay, they were 0, 16.8 and • 27.7 %. These water contents were obtained by c a r e f u l l y hand mixing water with the s o i l samples to obtain uniformity. The s o i l samples were then subjected to shear t e s t i n g . This mixing procedure was selected to maximize the s i m i l a r i t y between shear tes t i n g and t i l l a g e t e s t i n g where the volume of s o i l involved dictates this procedure be used. S o i l - c h i s e l interface  Soil-metal f r i c t i o n Soil-metal f r i c t i o n was the force required to move each c h i s e l across the surface of a s o i l sample. An Instron tester was used to provide a constant rate of movement and a continuous record of f r i c t i o n force on an associated chart recorder. Each c h i s e l was tested with normal loads of 0, 0.22 2.2 and 4.4 pounds plus the weight of the c h i s e l and associated brackets. The s o i l s were tested in both loose and compact conditions f o r water contents of 0, 10.0 and 19.2 % for Ottawa sand and 0, 10.0 and 2 6.3 % for Haney clay. 13 . T i l l a g e t e s t i n g and equipment used  Equipment A t i l l a g e test bed was designed and constructed to propel a t i l l a g e t o o l through an eight foot long s o i l sample at controlled v e l o c i t i e s between zero and f i v e miles per hour. The unit was powered by a one h a l f horsepower speed controlled Servo-Tek e l e c t r i c motor. A t i l l a g e t o o l was c a r r i e d on an aluminum carriage r i d i n g on Thomson b a l l bushings and case hardened, polished s t e e l shafts to minimize f r i c t i o n drag and vib r a t i o n . Instrumentation A transducer f o r measuring the forces along each of three orthogonal axes and the moments about each of these axes with each measurement being independent was developed for t h i s study. (See Appendix A). The basic measuring units were e l e c t r i c a l resistance s t r a i n gauges. These gauges, each having a resistance of 500 ohms and a gauge factor of 2.12, were connected i n wheat-stone bridge configurations. Attempts were made to construct amplifiers suitable for amplifying the r e s u l t i n g signals using Motorola MC 14 39 G operational amplifiers as a base. (See Appendix B). Serious and time consuming problems, including crosstalk between amplifier units and d i f f i c u l t y i n i s o l a t i n g them from e l e c t r i c a l noise i n the surrounding area were encountered. These problems were solved before discovering that at the high rates of amplification required, these units lacked long term s t a b i l i t y . 14. Consequently, the output from the transducers f o r F ( v e r t i c a l force) and M (moment about Y-axis) were fed into y Brush model RD561200 amplifiers and the signal recorded on the associated Brush model BL-202 two channel chart recorder. The transducer output for draft force (E^) was fed into an E l l i s model BAM - 1' amplifier and then recorded on a model 7100-A Mosely chart recorder. The chart speed for the Mosely recorder i s prec i s e l y c o n t r o l l e d . Therefore, the distance between 2 marks which are produced on the chart by the carriage passing over microswitches provides an accurate i n d i c a t i o n of machine v e l o c i t y . / The v a r i a t i o n i n angle of approach of a t i l l a g e t o o l attached to the transducer i s 0.16 degrees at the maximum design draft force of 150 pounds. This factor i s important as the angle of approach f o r d i f f e r e n t sized t i l l a g e tools must be constant to maintain geometric s i m i l a r i t y . T i l l a g e tools Three widths of plane c h i s e l s were used to produce scale factors suitable for use i n si m i l i t u d e with the smallest acting as a model for the other two, and the intermediate size acting as a model f o r the largest. Thus, scale factors of 1.5, 2.0 and 3.0 were studied. Observations of other experiments (10) indicate that depth of operation and c h i s e l width have diverse e f f e c t s on t i l l a g e draft forces. A l l ch i s e l s were therefore operated at the same depth of three inches and c h i s e l area was related to the scale factor. 15 . This factor creates dimensionally distorted models and adds to the d i s t o r t i o n caused by the s o i l s which are used f o r a l l machines. Also width, as a design variable, i s controlled by machine designers while depth, as an operating v a r i a b l e , i s controlled by any i n d i v i d u a l machine operator. In order to maintain uniformity, a l l c h i s e l s were constructed from one piece of 1/8 inch thick hot r o l l e d s t e e l . Each was milled to within 0.002 inches of the desired width and then hand polished, with crocus c l o t h , to a mirror f i n i s h . The f i n a l lapping was p a r a l l e l to the di r e c t i o n of s o i l movement over the c h i s e l face. The leading edge of each t o o l was sharpened to an angle of 30°. Thus, a clearance of 15° was formed between the c h i s e l under surface and the s o i l . The 0.75 inch wide c h i s e l was operated at 10% of the po t e n t i a l speed of the Servo-Tek motor while the 1.50 inch c h i s e l was operated at 14.14% and the 2.25 inch c h i s e l at 17.32%. These values were chosen to maintain model-prototype s i m i l i t u d e as the v e l o c i t i e s of each prototype c h i s e l are determined by the r a t i o : V = V / F T [1] p m when Vp = v e l o c i t y of prototype c h i s e l V = v e l o c i t y of model c h i s e l m J n = model-prototype scale factor. Although not a requirement for simi l i t u d e based pred i c t i o n equations, each c h i s e l was operated at a l l three v e l o c i t i e s i n order to develop a more complete understanding of v e l o c i t y as a factor a f f e c t i n g soil-machine reaction forces. S o i l variables Each c h i s e l variable was tested i n both the loose and compacted states for each s o i l at each of three d i f f e r e n t water contents. For Ottawa sand, the water contents were 0, 2 and 4 percent with the dry bulk density varying between 0.0 51 and 0.0 57 pounds per cubic inch. Haney clay was tested at water contents of 0, 8.7 and 13.9 percent while the dry bulk density varied between 0.043 and 0.047 pounds per cubic inch. P r i o r to each t r i a l , samples were taken for water content determination and fixe d volumes of s o i l were removed by a sampling core and weighed for bulk density determination. ANALYTICAL PROCEDURES Shear strength During each t e s t , stress and deformation were read from d i a l gauges and recorded manually and were then related graphically. Both the peak shear stress value and the steady shear stress value were derived from these graphs. For both s o i l s , each of these values was related to s o i l water content, dry bulk density and normal pressure for each t r i a l . This step was completed by analyzing these variables with the multiple l i n e a r regression and stepwise l i n e a r regression package available on an IBM 360/67 e l e c t r o n i c computer at the University of B r i t i s h Columbia as was a l l regression analysis f o r t h i s study. The sign i f i c a n c e of each factor's contribu-t i o n to the regression equation was provided i n the computer printout during t h i s analysis. Soil-metal f r i c t i o n As f o r most f r i c t i o n studies, both s t a t i c and k i n e t i c f r i c t i o n forces were determined. S t a t i c f r i c t i o n i s the peak resistance to s l i d i n g which occurs when motion i s imminent. Kine t i c f r i c t i o n i s the resistance which occurs during movement at a r e l a t i v e l y uniform rate. Both values were recorded by a chart recorder and measured by manually measuring the r e s u l t i n g deflections on the chart and comparing them to previous c a l i b r a t i o n s . Each f r i c t i o n force was then related to s o i l water content, dry bulk density, normal pressure and c h i s e l width by multiple l i n e a r regression and stepwise l i n e a r regression. 18. T i l l a g e analysis Draft force (F ), v e r t i c a l force (F ) and the moment about the horizontal axis running p a r a l l e l to the c h i s e l face (My) were continuously monitored on chart recorders during the entire length of each t r i a l . F and M were then read d i r e c t l y z y J from the Brush chart at 5 mm. i n t e r v a l s on the chart, while F was determined by measuring the deflections on the Mosely chart to the nearest 1/100 inch at 1/10 inch l o n g i t u d i n a l i n t e r v a l s on the chart. The data for each test was then averaged f o r the duration of the s p e c i f i c t r i a l and the forces converted to pounds and the moments to foot-pounds by comparison with previous c a l i b r a t i o n s . The resultant force (R) and the normal pressure (N) exerted on the c h i s e l were calculated for each t r i a l . Multiple l i n e a r regression and stepwise l i n e a r regression were then used to relate each force to the water content and dry bulk density of each s o i l as well as to c h i s e l width and v e l o c i t y . The forces were then related to the calculated weight of s o i l disturbed, v e l o c i t y , shear strength and soil-metal f i r c t i o n by the same process. Using the Buckingham IT theorem, dimensionless r a t i o s , which included the variables and t h e i r corresponding dimensions as shown i n Table 1, were developed. TABLE 1 Variables and Corresponding Dimensions Variable Symbol Dimensions Dry bulk density Ch i s e l v e l o c i t y Water content Chisel area T i l l a g e forces Shear strength Soil-metal f r i c t i o n Gravity B V W TL R Using B, V and TL as the repeating variables, the following TT terms were developed. TT 1 = W TT„ = F F D 2 x z R BV 2TL * BV 2TL ' BV 2TL ir q = S (S may be ei t h e r the peak or the steady value) j 2 BV TT^ = F (F may be ei t h e r the s t a t i c or the k i n e t i c B V 2 value) TT c = ( T L ) 1 / 2 G 5 V 2 The ^2 terms were then related to the remaining TT terms by multiple l i n e a r regression and stepwise l i n e a r regression. The TT terms for each c h i s e l were f i r s t analyzed separately so that regression equations were developed f o r each c h i s e l i n each s o i l . The next procedure involved determining the regression equation r e l a t i n g the TT terms formed f o r a l l chisels operating at t h e i r respective v e l o c i t i e s as determined by equation [ 1 ] . This step determined the effectiveness of si m i l i t u d e i n model-prototype predictions for t i l l a g e studies. The r e s u l t s , f o r each equation, were then displayed i n both tabular and graphical form i n order to allow optimum comparisons. The r e s u l t s f o r each equation were compared graphically with the predicted r e s u l t s . 20. D IAMETER (mm.) FIGURE 1. PARTICLE SIZE DISTRIBUTION FOR OTTAWA SAND 100 o i 1 . I . . 0-1 001 0-001 , 0 0001 D IAMETER (mm.) r-o FIGURE 2. PARTICLE SIZE DISTRIBUTION FOR HANEY CLAY FIGURE 3. RESULTS OF STANDARD PROCTOR TEST FOR OTTAWA SAND. 23. 104 i 10 15 20 25 30 WATER CONTENT (%) FIGURE 4. RESULTS OF STANDARD PROCTOR TEST FOR HANEY CLAY. RESULTS AND DISCUSSION S o i l physical properties The r e s u l t s of tests involving the basic physical c h a r a c t e r i s t i c s of Ottawa sand and Haney clay are depicted i n Table 2 below. TABLE 2. Comparison of S o i l Physical Characteristics SOIL Test Factor Ottawa sand Haney clay P a r t i c l e size D ^ Q (mm.) analysis n , m m > J D F I ~ (mm.) 0.48 0.65 0.000038 0.005 C u 1. 353 131.58 Compaction Wopt ( % ) o B max(lb/ft ) 7.7 104.72 19.6 103.17 P l a s t i c i t y UL (%) LL (%) PI -47.9 19.85 28.05 The detailed results of the p a r t i c l e size : analysis may be observed i n Figures 1 and 2 while the res u l t s of the compaction tests are depicted i n Figures 3 and 4. A l l p a r t i c l e sizes for Ottawa sand were within the range f o r sand whereas the Haney clay contained 18% sand, 37% s i l t and 45% clay. For Ottawa sand, the shear strength values were found to be related to the s o i l variables and normal pressure as i s depicted i n equations 2 and 3. PS = 6.153 - 14.2267 E + 0.0133W + 7.9962E2 + 0.5 84 8N [2] SS = 1.9113 - 4.8335E + 2.8998E2 + 0.0004203W2 + 0.5449N [3] For Haney clay the corresponding relationships were as depicted i n equations 4 and 5. PS = 0.9066 + 0.1701W - 0.3006E2 - 0.007139W2 + 0.6575N [4] SS = 0.6213 + 0.1720W - 0.2521E2 - 0.006943W2 + 0.6541N [5] 2 when PS = peak shear strength ( l b / i n ) 2 SS = steady shear strength ( l b / i n ) W = s o i l water content (%) E = void r a t i o 2 N = normal pressure ( l b / i n ) Equations 2 to 5 i n c l u s i v e were a l l s i g n i f i c a n t at F-$ 0. 0002 and by comparison with the Mohr f a i l u r e envelope equation, may be used to indicate the cohesive strength and the angle of in t e r n a l f r i c t i o n of the s o i l by S = C + o tan <J> [6] 2 when S = s o i l shear strength ( l b / i n ) 2 C = cohesive strength ( l b / i n ) 2 a = normal stress ( l b / i n ) <fj = angle of i n t e r n a l f r i c t i o n (°) As may be noted from equations 2 and 3, the cohesive strength of Ottawa sand i s very low (C -*• 0) while f o r Haney clay, equations 4 and 5 indicate that the cohesive strength i s , as expected, a much larger value. Also, equations 2 to 5 i n c l u s i v e indicate that the angles of i n t e r n a l f r i c t i o n depicted i n Table 3 are r e l a t i v e l y constant values f o r each s o i l . TABLE 3 Internal F r i c t i o n Angles f o r Ottawa Sand and Haney Clay S o i l Tan F r i c t i o n Angle F r i c t i o n Angle ((J)) Peak Steady Peak Steady Ottawa sand 0.5848 0.5449 30° 18' 28° 36' Haney clay 0.6578 0.6541 33° 20' 33° 12' Soil-machine i n t e r a c t i o n For both Ottawa sand and Haney clay, s t a t i c and k i n e t i c values of soil-metal f r i c t i o n were found to be related to c h i s e l width, normal pressure and s o i l water content. These relationships are described i n equations 7 and 8 for Ottawa sand SF = 0.009176 - 0.01T + 0.00281T2 + 0. 2457N [7] KF = -0 . 001471 + 0 .003388W <• 0 .2433N [8] Equations 9 and 10 describe the corresponding relationships f o r Haney clay SF = 0.001801 - 0.006722W + 0.00005255W2 . + 0.3689N [9] KF = 0.002441 + 0.3151N [10] when SF = 2 s t a t i c f r i c t i o n ( l b / i n ) KF = 2 k i n e t i c f r i c t i o n ( l b / i n ) W = s o i l water content (%) 27. T = c h i s e l width (in) 2 N = normal pressure ( l b / i n ) Equations 7 to 10 in c l u s i v e were a l l found to be s i g n i f i c a n t at F 0.0. The s o i l bulk densities at which the soil-metal f r i c t i o n tests were conducted included no values i n the compression phase described by Nichols (17). Consequently, s o i l bulk density was not a s i g n i f i c a n t factor i n the r e l a t i o n -ships described by equations 7 to 10 i n c l u s i v e . T i l l a g e forces Direct relationships During each t i l l a g e t e s t , the measured forces resulted from the interactions between the c h i s e l involved, i t s v e l o c i t y and the s o i l conditions at the time of t e s t i n g . For Ottawa sand, these relationships are indicated by equations 11, 12 and 13. F = -175.7H"41 + 3 . 5582T + 3.6683W + 4983.5552B x - 0.0819V - 0.5359W2 - 42580.OB2 [11] F = -233.9427 + 4.8502T + 4.0905W + 8116.0604B z + 0.0465V - 0.5715W2 - 70650.OB2 [12] R = -275.9427 + 6.0246T + 5.4603W + 9536.4463B - 0.011V - 0.7783W2 - 82590.OB2 [13] The corresponding relationships f o r Haney clay are described by equations 14, 15 and 16. F = 53.549 + 2.9051T + 0.1368W - 3731.8906B x + 0.2295V + 0.0212W2 + 56420.OB2 [14] F = 158.3130 + 5.0347T + 0.2797W - 9414.6966B z + 0.3920V + 0.0337W2 + 128800.OB2 [15] 28. R = 158.8711 + 5.7662T + 0.3062W - 9774.0242B + 0.4536V + 0.0396W2 + 136900.OB2 [16] when F = draft force (lb) x F = v e r t i c a l force (lb) z R = resultant force (lb) T = c h i s e l width (in) W = s o i l water content (%) 3 B = s o i l dry bulk density ( l b / i n ) V = c h i s e l v e l o c i t y (in/sec). The relationships described by equations 11 to 16 in c l u s i v e were a l l s i g n i f i c a n t at F 0.0. Comparison of equations 11 to 13 with equations 14 to 16 indicates that each s o i l type presents unique t i l l a g e r e l a t i o n s h i p c h a r a c t e r i s t i c s which must be recognized and understood on a quantitative basis before complete s o i l t i l l a g e r e l a t i o n s h i p s can be developed. For the two s o i l s studied, the e f f e c t s of c h i s e l v e l o c i t y , s o i l water content and dry bulk density were almost completely opposite. However, the negative v e l o c i t y e f f e c t attributed to Ottawa sand by these equations must be considered to be exaggerated. A possible explanation f o r t h i s e f f e c t might be that a s l i g h t v i b r a t i o n -and corresponding draft reduction, may have been imparted to the ch i s e l s operating at higher v e l o c i t i e s . However, the ov e r a l l e f f e c t of c h i s e l v e l o c i t y indicates that the shear strength of cohesionless s o i l s tends to be n e g l i g i b l e while the shear strength of cohesive s o i l s i s d e f i n i t e l y rate dependent. 3 0 xi 0 10 2 0 30 COMPUTED DRAFT F O R C E (lb.) FIGURE 5. OTTAWA SAND - ACTUAL DRAFT FORCE VS. VALUE COMPUTED FROM EQUATION 11. 3 0 0 10 2 0 3 0 C O M P U T E D DRAFT F O R C E (lb.) FIGURE 6. HANEY CLAY - ACTUAL DRAFT FORCE VS. VALUE COMPUTED FROM EQUATION lU. 30. Attempts tp develop s a t i s f a c t o r y equations r e l a t i n g t i l l a g e forces to c h i s e l v e l o c i t y , soil-metal f r i c t i o n , s o i l shear strength and weight of s o i l disturbed were unsuccessful due to the low l e v e l of s i g n i f i c a n c e of each contributing factor. Consequently, no comparisons with t h e o r e t i c a l force-reaction equations as proposed by G i l l and Vanden Berg (10) were possible. Dimensionless equations Attempts have been made to develop dimensionless t i l l a g e r elationships using the cohesive strength plus the angle of i n t e r n a l f r i c t i o n of the s o i l to describe the s o i l shear strength value. However, cohesion and f r i c t i o n angle were shown i n equations 2 to 5 i n c l u s i v e (by comparison with equation 6) to be determined by the s o i l and i t s condition at the time of t e s t i n g , and bear no rela t i o n s h i p to t i l l a g e v a riables. Consequently, the normal pressure applied to the f a i l u r e surface must be known i n order f o r s o i l shear strength to make a meaningful contribution to a s o i l t i l l a g e r e l a t i o n -ship equation. S i m i l a r l y , the normal pressure value i s required f o r studying soil-metal f r i c t i o n i n r e l a t i o n to s o i l t i l l a g e . Since the chisels were i n c l i n e d at an angle of "45° ._ to the s o i l surface and t h i s value i s very s i m i l a r in magnitude to the angle of the shear f a i l u r e planes formed during t i l l a g e , the same equations were used to indicate normal pressure for c a l c u l a t i n g both s o i l shear strength and soil-metal f r i c t i o n . For Ottawa sand, the normal pressure i s indicated i n equation 17. N = 3.4248 - 1.1366T + 0.9807W + 0.2106T2 - 0.1379W2 - 2.0957E2 [17] For Haney clay, normal pressure i s indicated by equation 18. N = 32.1408 - 2.8869T - 36.4914E + 0.00122V + 0.5836T2 + 0.0115W2 + 10.9012E2 [18] 2 when N = normal pressure ( l b / i n ) T = c h i s e l width (in) W = water content (%) E = void r a t i o . The normal pressures derived from equations 17 and 18 were included i n the appropriate soil-metal f r i c t i o n and s o i l shear strength equations to y i e l d the numerical values of soil-metal f r i c t i o n and s o i l shear strength values f o r each t e s t . These values were then included i n the previously developed dimensionless r a t i o s and regression equations developed for t i l l a g e reactions. For Ottawa sand, the' 0.75 inch wide c h i s e l ' s reactions are described by equations 19, 20 and 21; the 1.50 inch wide c h i s e l ' s reactions by equations 22, 23 and 24; and the 2.25 inch wide c h i s e l ' s reaction by equations 25, 26 and 27. When each c h i s e l i s operated i n Ottawa sand at i t s proper sim i l i t u d e based v e l o c i t y (as determined by equation 1, the interactions were as depicted by equations 28, 29 and 30. 32. F — £ — = -0 . 0553 + 0.0127W + 0. 6634 + 1. 0308 [19] BV*TL BV* BV F — | — = -0.0276 + 0.0118W + 0.9189 + 1.0169 - ~ [20] BV TL BV* BV* - \ = -0.0514 + 0.0161W + 1. 1065 - ~ + 1.4871 ™ - [21] BV TL BV BV F — £ — = -0. 007588 - 0. 001506W - 0. 2736 ^ + 3. 0657 [22] BV TL BV BV F — \ — = 0. 004433 + 0.00152W - 0. 1676 — + 3.4876 [23] BV TL BV BV -5-— = -0. 0004752 + 0. 0001727W - 0. 2994 — 9 + 4. 6332 — [24] BV TL BV BV F — ^ = 0.0111 - 0.0101W - 0. 3835 + 3. 2361 [25] BV TL BV* BV F — | — = 0,043 - 0.011W - 0.3719 + 3.65 [26] BV TL BV* BV -^5 = 0.0419 - 0. 0153W - 0. 5427 + 4.9038 §L- [27] BV TL BV* BV* F — £ — = -0. 0393 + 0. 005241W + 0.4985 + 1.4391 [28] BV TL BV BV — ^ — = -0.0309 + 0.008054W + 0. 7783 + 1. 3964 [29] BV TL BV* BV* — = -0.046 + 0.008841W + 0.9012 + 2.0249 ^ ~ [30] BV TL BV BV The corresponding relationships f o r Haney clay are described by equations 31, 32 and 33 f o r the 0.75 inch wide c h i s e l ; equations 34, 35 and 36 f o r the 1.50 inch wide c h i s e l ; and equations 37, 38 and 39 f o r the 2.25 inch wide c h i s e l . When 33. each c h i s e l was operated i n Haney clay at i t s proper si m i l i t u d e based v e l o c i t y (as determined by equation 1) the interactions were as depicted by equations 40, 41 and 42 p — £ — = 0.0135 - 0.00626W + 0.5038 ^ ~ + 0.5091 [31] BV TL BV^ BV F v ^ F — | — = -0.0241 - 0.001027W + 0 . 0694 ^ ~ + 1.9370 [32] BV^TL BV BV = -0.008109 - 0.004942W + 0. 3872 — 9 + 1. 7879 ££y [33] BV^TL BV^ BV^ F — — = 0.0602 - 0.005017W + 0.2557 + 0.7783 [34] BV^TL B\' BVZ F — 5 = 0.0549 - 0.005742W + 0. 3453 + 1.0868 [35] BV TL BV BV 34 . TABLE 4 Calculated Reaction Forces f o r Ottawa Sand when Velocity i s varied and B = 0.05 7 k b / i n 3 and W = 1.99%. Chise l Width Equation (in.) Numbers F X (lb) F z (lb) R (lb) V (in/sec) 0.75 19,20,21 6 .80 8.59 10 .94 6.820 6 .44 8.55 10.73 10.725 - - - 6.09 8.51 10 .51 13.435 28 ,29 ,30 6 .91 8.68 11. 08 6 .820 1.50 22,23,24 12.11 15.25 19.25 6. 820 11.46 15.42 19 .27 10 .725 11.22 15.63 19.30 13.435 28 ,29 ,30 11.22 14.64 18.41 10 .725 19,20,21 11.70 14.83 18.89 6.820 10 .97 14. 72 18.42 10.725 10.21 14 .60 17.80 13.435 2.25 25,26,27 15.15 19.03 24 .45 6 .820 14.93 19 . 84 24 .85 10.725 14.58 20.50 24 .25 ' 13.435 28,29,30 15.51 21.50 26.45 13.435 22,23,24 15.90 21.05 26 .40 6.820 15 .60 21.40 26 .45 10 .725 15 .24 21.65 26.40 13.435 19 ,20,21 17.65 22.60 28.90 16.55 22.65 27.90 15.48 22.35 27.30 6.820 10 .725 13.435 35. o 1 — — 6 8 10 12 14 VELOCITY (in./sec.) FIGURE 7. COMPARISON OF PREDICTED DRAFT FORCES FOR 2.2 5 INCH WIDE CHISEL IN OTTAWA SAND WHEN VELOCITY IS VARIED AND B = 0.057 l b / i n 3 AND W = 1.99%. TABLE 5 Calculated Reaction Forces f o r Ottawa Sand when Water Content i s varied and B = 0.0541 l b / i n 3 and V = 10.725 in./sec. C h i s e l Width Equation (in.) Numbers 0.75 19,20,21 F F R W X z (lb) (lb) (lb) (%) 1.64 2.76 3.28 0 5.00 6.75 8.44 1.994 6.17 8.05 10.13 3.967 1.50 22,23,24 3.82 5.27 6.54 0 8.88 12.00 14.94 1.994 9.94 13.65 16.49 3.967 19,20,21 1,25 2.98 3.31 0 8.03 11.03 13.69 1.994 10.42 13.69 17.20 3.967 2.25 25,26,27 4.98 7.72 9.24 0 12.78 16.60 18.97 1.994 16.29 20.95 24.50 3.967 22,23,24 4.11 6.16 7.41 0 11.65 16.25 19.98 1.994 16.20 21.35 27.30 3.967 19,20,21 2.45 5.32 5.96 12.13 16.85 20.80 16.63 21.70 27.40 0 1.994 3.967 37. 25 20 0 1 : — 0 1 2 3 4 WATER CONTENT (%) FIGURE 8. COMPARISON OF PREDICTED DRAFT FORCES FOR 2.2 5 INCH WIDE CHISEL IN OTTAWA SAND WHEN WATER CONTENT IS VARIED AND B = 0.05*4 l b / i n 3 and V - 10.72 in/sec. 3 8 . T A B L E 6 Calculated Reaction Forces f o r Ottawa Sand when Dry Bulk Density i s Varied and W = 1 . 9 9 % and V = 1 0 . 7 2 in./sec. Chisel Width Equation (in.) Numbers F x (lb) F z (lb) R (lb) B ( l b / i n 3 ) 0 . 7 5 1 . 5 0 1 9 , 2 0 , 2 1 2 2 , 2 3 , 2 i 4 4 . 1 7 5 . 6 5 7 . 0 6 0 . 0 5 1 1 4 5 . 0 0 6 . 7 5 8 . 4 4 0 . 0 5 4 1 6 . 4 4 8 . 5 5 1 0 . 7 3 0 . 0 5 7 3 7 . 8 5 1 0 . 5 7 1 3 . 2 0 0 . 0 5 1 4 8 . 8 8 1 2 . 0 0 1 4 . 9 4 0 . 0 5 4 1 1 1 . 4 6 1 5 . 4 2 1 9 . 2 7 0 . 0 5 7 3 1 9 , 2 0 , 2 1 6 . 4 4 8 . 9 5 1 1 . 1 2 0 . 0 5 1 4 8 . 0 3 1 1 . 0 3 1 3 . 6 9 0 . 0 5 4 1 1 0 . 9 7 1 4 . 7 2 1 8 . 4 2 0 . 0 5 7 3 2 . 2 5 2 5 , 2 6 , 2 7 2 2 , 2 3 , 2 4 9 . 6 1 1 3 . 4 5 1 6 . 5 8 0 . 0 5 1 4 1 2 . 7 8 1 6 . 6 0 1 8 . 9 7 0 . 0 5 4 1 1 4 . 9 3 1 9 . 8 4 2 4 . 8 5 0 . 0 5 7 3 1 0 . 0 9 1 4 . 1 5 1 7 . 3 0 - . 0 . 0 5 1 4 1 1 . 6 5 1 6 . 2 5 1 9 . 9 8 0 . 0 5 4 1 1 5 . 6 0 2 1 . 4 0 2 6 . 4 5 0 . 0 5 7 3 1 9 , 2 0 , 2 1 9 . 7 9 1 3 . 8 2 1 7 . 0 0 1 2 . 1 3 1 6 . 8 5 2 0 . 8 0 1 6 . 5 5 2 2 . 6 5 2 7 . 9 0 0 . 0 5 1 4 0 . 0 5 4 1 0 . 0 5 7 3 39. 25 O I : 0-050 0-052 0-054 0-056 0-058 DRY BULK DENSITY ( lb./ in? ) FIGURE 9. COMPARISON OF PREDICTED DRAFT FORCES FOR • 2.25 INCH WIDE CHISEL IN OTTAWA SAND WHEN DRY BULK DENSITY IS VARIED AND W = 1.994% AND V = 10.725 in/sec. M O . R = 0. 0812 - 0. 007683W + ).4302 + 1. 3355 [36] BV TL BV BV F X = 0.0949 - 0.008653W + 0.4779 - 0.0691 ^L- [ 3 7 ] BV TL BV BV* F ~ Z ~ 0.1157 - 0 . 0088T5W + 0 . 6063 £ ~ + 0. 0536 [38] BV TL BV* BV* R = 0. 1497 - 0 . 0122W + 0 . 7722 + 0. 0008708 [39] BV TL BV* BV* F — ~ — = -0. 0327 - 0. 005457W + 0. 5968 + 0.319 [40] BV TL BV BV F 7 P ^ ^ F — 5 — = -0. 007707 - 0. 001265W + 0. 2309 —A - + 1.4872 [41] BV*TL BV* BV P PS 9 F -A* = -0 . 0259 - 0. 004571W + 0. 5689 • — + 1.3260 ™ - [42] BV TL BV* BV when F = draft force (lb) x F = v e r t i c a l force (lb) z R = resultant force (lb) 3 B = s o i l dry bulk density ( l b / i n ) V = c h i s e l v e l o c i t y (in/sec) T •= c h i s e l width (in.) L = c h i s e l depth (in.) These relationships are a l l s i g n i f i c a n t at F 0.0 The dimensionless r a t i o s involving g r a v i t y , steady shear s t r e s s , and k i n e t i c soil-metal f r i c t i o n were not included i n equations 19 to 42 in c l u s i v e as they made no s i g n i f i c a n t contribution to the dimensionless relat i o n s h i p s . The effectiveness of prediction from the equations f or Ottawa sand are displayed by Tables 4, 5 and 6 and by Figures •41. 7, 8 and 9, while the predictions f o r Haney clay are displayed in Tables 7, 8 and 9 and Figures 10, 11 and 12. As these graphs and tables i n d i c a t e , prediction equations based on si m i l i t u d e may be developed successfully at some s o i l water content values and s o i l dry bulk density values. Thus, the values of dry bulk density and water content at which t h e i r tests were conducted may provide an explanation for the success of some of the studies referred to i n the l i s t of references included i n thi s report. At the same time the f a i l u r e of others i s explained. Comparisons of actual draft forces with those predicted by the dimensionless equations f o r Ottawa sand are presented i n Figures 13 to 16 i n c l u s i v e while the corresponding comparisons for Haney clay are presented i n Figures 17 to 2 0 i n c l u s i v e . Comparison of these graphs with Figures 5 and 6 yi e l d s an i n d i c a t i o n that the treatment of the d i r e c t l y measur-able c h i s e l and s o i l variables in the dimensionless equations does not indicate t h e i r true e f f e c t on the s o i l t i l l a g e r e l a t i o n s h i p . Indeed by comparing the variable treatment i n the dimensionless r a t i o s with the treatment of these variables i n equations 11 to 16 i n c l u s i v e , the conclusion i s reached that the eff e c t s of c h i s e l v e l o c i t y , c h i s e l width, s o i l water content and dry bulk density are treated i n a completely distorted manner during i n c l u s i o n i n dimensionless r a t i o s . Also explained by t h i s f a c t i s that while the si m i l i t u d e based predictions are s a t i s f a c t o r y at certa i n s o i l variable values, they are not f o r others. 42 TABLE 7 Calculated Reaction Forces for Haney clay when Velocity i s Varied and B = 0.04 7 l b / i n 3 and W = 8.6 8% Chisel Width Equation F F R V (in.) Numbers / i ^ i t- i \ (lb) (lb) (lb) (m/sec) 0.75 31,32,33 9.09 10.55 13.95 6.927 9.13 10.90 14.28 10.725 9.57 11.05 14.35 13.572 40,41,42 9.09 10.53 13.47 6.927 1.50 34,35,36 11.68 15.79 19.64 6.927 12.79 16.96 21.25 10.725 13.62 17.88 22.55 13.572 40,41,42 13.72 16.75 21.70 10.725 31,32,33 14.23 15.50 21.15 6.927 14.45 16.18 21.75 10.725 14.27 16.54 21.90 13.572 2.25 37,38,39 . 14.05 19.43 24.00 6.927 15.38 21.65 26.70 10.725 16.55 23.70 28.95 13.572 40,41,42 15.68 22.45 27.95 13.572 34,35,36 15.23 20.55 25.60 6.927 16.95 22.35 . 28.10 10.725 18.30 23.70 30.00 13.572 31,32 ,33 18.94 19.46 27.15 19.05 20,50 28.10 18.79 21.00 28.30 6.927 10.725 13.572 UJ U cr O Li-< cr 10 0 -e- - e - a c tua l — f r o m 1 - 5 0 " ch i se l - e - f r o m 0 - 7 5 " ch i s e l 8 10 12 VELOC ITY (in./sec.) 14 FIGURE 10. COMPARISON OF PREDICTED DRAFT FORCES FOR 2.2 5 INCH WIDE CHISEL IN HANEY CLAY WHEN VELOCITY IS VARIED AND B = 0.0<4 7 l b / i n 3 AND W = 8.6 8%. 44. TABLE 8 Calculated Reaction Forces f o r Haney clay when Water Content i s Varied and B = 0.045 l g / i n 3 and V = 10.72 i n / s e c . Chisel Width Equation F F R W (in.) Numbers ( * b ) ( * b ) Q b ) 0.75 31,32,33 5.71 6.66 8.80 0 7.77 8.99 11.93 8.679 9.65 12.75 15.82 13.926 1.50 34,35,36 7.68 9.64 12.34 0 10.40 13.68 17.20 8.679 12.80 18.60 23.20 13.926 31,32,33 7.57 7.69 10.80 0 11.71 12.36 17.07 8.679 15.42 19.45 24.85 13.926 2.25 37,38,39 10.40 13.10 16.50 0 12.95 18.17 22.35 8.679 15.10 21.55 26.10 13.926 34,35,36 9.27 11.37 14.67 . 0 13.34 17.40 21.95 8.679 18.45 24.80 30.90 13.926 31,32,33 8.71 7.79 11.68 14.95 14.77 21.05 20.70 25.45 32.70 0 8.679 13.926 45. 25 2 0 15 LU Ot O U L L L 10 0 0 actual — * - f r o m 1 -50 " chisel f r o m 0 - 7 5 " chisel 4 8 12 WATER C O N T E N T (%) 16 FIGURE 11. COMPARISON OF PREDICTED DRAFT FORCES FOR 2.25 INCH WIDE CHISEL IN HANEY CLAY WHEN WATER CONTENT IS VARIED AND B = 0.04 5 lb/in3 AND V = 10.72 in/sec. 4 6 • TABLE 9 Calculated Reaction Forces f o r Haney clay when Dry Bulk Density i s Varied and W = 8.68% and V = 10.72 in/sec. Chisel Width Equation (in.) Numbers F F R X z (lb) (lb) (lb) B 3 ( l b / i n ) 0 .75 31,32,33 5.87 6 . 44 8. 75 0. 0430 7.77 8.99 11.93 0.0451 9.13 10.90 14.28 0.0469 1.50 34,35,36 7.14 9.24 11.67 0.0430 10.40 13.68 17.20 0.0451 12.79 16.96 21.25 0.0469 31,32,33 7.97 7.22 10.71 0.0430 11.71 12.36 17.07 0.0451 14.45 16.18 21.75 0.0469 2.25 37,38,39 9.56 13.28 16.40 0.0430 12.95 18.17 22.35 0.0451 15.38 21.65 26.70 0.0469 34 , 35 ,36 8 .47 10 . 74 13 .70 ... 0 .0430 13.34 17.40 21.95 0.0451 16.95 22.35 28.10 0.0469 31,32,33 9.37 7.08 11.54 14.95 14.77 21.05 19.05 20.50 28.10 0.04 30 0 .0451 0.0469 25 0 i 0-040 0-042 0-044 0-046 0-048 DRY BULK DENSITY (lb./in. 3) FIGURE 12. COMPARISON OF PREDICTED DRAFT FORCES FOR 2.25 INCH WIDE CHISEL IN HANEY CLAY WHEN DRY BULK DENSITY IS VARIED AND W = 8.6 8% AND V = 10.72 in/sec. 30 UJ o § 2 0 L L LL < cn Q 10 < 3 < 0 H8. actual = 0 -249 «• 0 • 9 5 5 4 computed 0 10 20 3 0 COMPUTED DRAFT FORCE (lb.) FIGURE 13. COMPUTED VS. ACTUAL DRAFT FORCE FOR 0.75 INCH WIDE CHISEL IN OTTAWA SAND. 30 LU O § 2 0 b_ f-< cn O 10 < ZD U < 0 actual = - 0 -154 + 1 -0157 computed 0 1 10 20 3 0 COMPUTED DRAFT FORCE (lb.) FIGURE IH. COMPUTED VS. ACTUAL DRAFT FORCE FOR 1.50 INCH WIDE CHISEL IN OTTAWA SAND. 49. 3 0 LU O cr £ '20 "r— Lu < or a < ZD I— o < 10 0 actual = -1 -598 + 1 -1516 computed 0 10 20 30 COMPUTED DRAFT FORCE (lb.) FIGURE 15. COMPUTED VS. ACTUAL DRAFT FORCE FOR 2.25 INCH WIDE CHISEL IN OTTAWA SAND. 30 LU O § 2 0 li_ LL a < ZD I— 10 0 actual = 0 - 7 8 0 + 0 - 9 0 7 6 computed O 10 20 30 COMPUTED DRAFT FORCE (lb.) FIGURE 16. ACTUAL DRAFT FORCE VS. VALUE COMPUTED BY GENERAL (G) EQUATION FOR OTTAWA SAND. -J < ZD t-y 30 L U O rr O 20 L L I-u_ < cn a 10 0 50. actual = - 4 - 6 5 9 +1-6366 computed 0 10 COMPUTED 20 30 DRAFT F O R C E (lb.) FIGURE 17. COMPUTED VS. ACTUAL DRAFT FORCE FOR 0.75 INCH WIDE CHISEL IN HANEY CLAY. _ 30 J6 0 10 20 30 COMPUTED D R A F T F O R C E (lb.) FIGURE 18. COMPUTED VS. ACTUAL DRAFT FORCE FOR 1.50 INCH WIDE CHISEL IN HANEY CLAY. 51. LU U cr £ h-L L < cr a < o i — u < 3 0 2 0 10 0 actual = 2 -253 + 0 -7972 computed 0 10 2 0 3 0 COMPUTED DRAFT F O R C E (lb.) FIGURE 19. COMPUTED VS. ACTUAL DRAFT FORCE FOR 2.25 INCH WIDE CHISEL IN HANEY CLAY. 3 0 L U g 2 0 O L L I— L L C? 10 a < ZD § o actual = 0 - 6 3 + 0 • 9 6 4 4 computed 0 10 20 3 0 C O M P U T E D DRAFT F O R C E (lb.) FIGURE 20. ACTUAL DRAFT FORCE VS. VALUE COMPUTED BY GENERAL (G) DIMENSIONLESS EQUATION FOR HANEY CLAY. The d i f f e r e n t treatments of these variables w i l l give close numerical results at some values but not at others. A c a r e f u l re-evaluation of other studies available would lead one to conclude that the ojbective of most of these studies was to test the theories of similitude rather than the stated objective of developing a s o i l t i l l a g e mechanics. A further disadvantage of r e s t r i c k t i n g a s o i l t i l l a g e study to similitude based models i s that the e f f e c t s of i n d i v i d u a l t o o l variables are not distinguishable from one another. The main advantage to the use of dimensionless equations i n t i l l a g e studies i s the p o s s i b i l i t y of t e s t i n g a single model machine and then predicting the r e s u l t s f o r larger prototypes. The re s u l t s from t h i s study indicates that the procedure has d e f i n i t e p o t e n t i a l f o r use but requires further work so that the variables being studied are treated i n a manner which r e f l e c t s t h e i r actual e f f e c t on s o i l t i l l a g e i n t e r a c t i o n s . 53. SUMMARY AND CONCLUSIONS 1) Ottawa sand i s a cohesionless s o i l while Haney clay i s d e f i n i t e l y cohesive. 2) S o i l void r a t i o , s o i l water content and normal pressure may be combined to predict the s o i l shear strength of e i t h e r Ottawa sand or Haney clay. 3) S o i l water content, area of contact and normal pressure may be combined to predict the soil-metal f r i c t i o n between eithe r Ottawa sand or Haney clay and the s o i l machines studied. 4) S o i l water content, dry bulk density, c h i s e l width and c h i s e l v e l o c i t y may be combined to predict s o i l - c h i s e l reaction forces f o r the s o i l s and chi s e l s studied. 5) The dimensionless r a t i o s developed may be combined to predict s o i l - c h i s e l reaction forces f o r scaled implements. However, large discrepancies do exis t for certain s o i l conditions and the time required for preliminary t e s t i n g i s very extensive. 6) Since a l l measurements recorded during the course of th i s study were analyzed by s t a t i s t i c a l procedures, the re s u l t i n g equations do not represent basic physical r e l a t i o n s h i p s . Caution should therefore be used i f these equations are to be applied to values beyond the range of values analyzed i n thi s report. SUGGESTIONS FOR FURTHER WORK . The work i n i t i a t e d in th i s study should be expanded by adding an increased number of t i l l a g e t o o l variables and s o i l types and to determine t h e i r e f f e c t s on soil-machine in t e r a c t i o n s . Possible machine variables to study would be depth of operation, angle of approach and machine shape. The other s o i l types would add to the body of knowledge deve-loped to the possible extend that dimensionless r a t i o s and consequently prediction equations might be developed by combining the s o i l and machine variables i n such a manner as to indicate t h e i r e f f e c t on the force interactions involved. The next step would be to determine the ef f e c t s of t i l l a g e t o o l variables on the production of desired s o i l conditions. An i n t e r e s t i n g note of great merit i s that equations 11 to 16 i n c l u s i v e are i n such a form that v e l o c i t y , c h i s e l width and reaction forces are d i r e c t l y related in such a form as to indicate p o t e n t i a l development of t i l l a g e energy r e l a t i o n s h i p s . The consequence of th i s r e l a t i o n s h i p would be a very meaningful study on t i l l a g e cost minimization. LIST OF REFERENCES Bailey, A.C. and G.E. Vanden Berg, 1968. 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Siemens, 1970. S o i l Shear Strength and Energy f o r Failure Related to Density and Moisture. Annual Meeting, Amer. Soc. Agric. Eng., Paper No. 70 - 146. 20. Reaves, C.A. 1966. A r t i f i c i a l S o i l s Simulate Natural S o i l s i n T i l l a g e Studies. Trans, Amer. Soc. Agric. Eng. Vol. 9, No. 2, pp. 147-150. 21. Reaves, C.A., A.W. Cooper and F.A. Kummer, 1968. Similitude i n Performance Studies of S o i l - C h i s e l Systems. Trans. Amer. Soc. Agric. Eng., Vol. 11, No. 5, pp. 658-661. 22. Ross, I.J. and G.W. Isaacs, 1961. Forces Acting i n Stacks of Granular Materials. Trans. Amer. Soc. Agric. Eng. Vol. 4, No. 1, pp. 92-96. 23. Schafer, R.L., C.W. Bockhop and W.G. Lovely, 1968. Model-Prototype Studies of T i l l a g e Implements. Trans. Amer. Soc. Agric. Eng., Vol. 11, No. 5, pp. 661-665. 24. Soehne, W.H., 1966. Characterization of T i l l a g e Tools. Grundforbattring 19, pp. 31-48. 57. Timbers, G.E., L.M. Staley and E.L. Watson, 1965. 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Similitude of Soil-Machine Systems. Trans. Amer. Soc. Agric. Eng., Vol. 11, No. 5, pp.65 3-658. 58. A P P E N D I X A 60. FIGURE A2. WHEATSTONE BRIDGE CONFIGURATIONS FOR FORCES AND MOMENTS TO BE MEASURED. 61. A P P E N D I X B 62. FIGURE B l . SCHEMATIC DIAGRAM OF STRAIN GAUGE AMPLIFIERS 63. PARTS LIST FOR STRAIN GAUGE AMPLIFIERS Rl = 1 K ohms R2 = switch allows choice of IK, 1.5K, 2.2K, >4.7K, 10K, 22K, "47K, 100K, 2200K, 4700K, 10 ,000K ohms. R3 = I K ohms R4 = 10K ohms R5 = 10K ohms R6 = 1 K ohms R8 - 10K ohms, 10 turn potentiometer R9 = 500 ohms e l e c t r i c a l resistance s t r a i n gauges CI = 2200 pF C2 = 0.1 uF C3 = 800 yF T. - NPN Transistor ) 2N4920 1 ) T 2 = PNP Transistor ) 2N492 3 NOTES: 1. Amplification = R2/R^ 2. A l l grounds to be carr i e d separately to a common ground. 3. C2 capacitors are used to eliminate crosstalk as 6 s t r a i n gauge bridges and 6 amplifiers were connected in p a r a l l e l from the same power source. •4. Switch f o r R2 must be of make before break type. 64. A P P E N D I X C 65. TABLE CI CORRELATION MATRIX FOR DIRECT SHEAR TESTS FOR OTTAWA SAND. . w 2 w E 2 E Normal Steady Peak Peak 0.062 0. 061 -0.517 -0.527 0.964 0.970 1.000 Steady 0. 080 0. 062 -0.417 -0.423 0.982 1.000 Normal 0. 000 0. 000 -0.405 -0.40 8 1.000 E 0.133 0. 250 0.997 1.000 E 2 0.128 0. 245 1. 000 W 0.960 1. 000 w 2 1. 000 -TABLE C2 CORRELATION MATRIX FOR DIRECT SHEAR TESTS FOR HANEY CLAY W2 W E 2 E Normal Steady Peak Peak -0.2714 -0. 160 -0.124 -0.115 0. 767 0.994 1.000 Steady -0 .218 -0. 102 -0.122 -0.118 0. 769 1.000 Normal 0.000 0. 000 -0.259 -0.2 60 1.000 E -0.369 -0. 305 .0.994 1.000 E 2 -0.326 -0. 249 1.000 W 0.963 1. 000 w 2 1.000 • 66. A P P E N D I X D 67. TABLE DI -CORRELATION MATRIX FOR SOIL-METAL FRICTION TESTS FOR OTTAWA SAND. T 2 T W2 Water Normal Kin e t i c S t a t i c " S t a t i c -0. 396 -0. 426 0. 069 0. 065 0. 978 0. 985 Kine t i c -0. 370 -0. 393 0. 114 0. 112 0. 982 1. 000 Normal -0. 339 -0. 360 0. 033 0. 015 1. 000 Water -0. 108 -0. 086 0. 962 1. 000 w2 -0. 128 -0 . 105 1. 000 T 0. 988 1. 000 T 2 1.000 TABLE D2 CORRELATION MATRIX FOR SOIL-METAL FRICTION TESTS FOR HANEY CLAY T 2 T W2 Water Normal Kinetic S t a t i c S t a t i c -0. 352 -0. 378 0. 380 0. 332 0. 838 0. 936 Kine t i c -0. 358 -0. 385 0. 216 0. 187 0. 890 1. 000 Normal -0. 375 -0. 396 0. 000 0. 000 1. 000 Water 0. 000 0. 000 0. 952 1. 000 w2 0. 000 0. 000 1. 000 T 0. 990 1. 000 T 1.000 68. A P P E N D I X E 69. TABLE E l CORRELATION MATRIX FOR TILLAGE TESTS FOR OTTAWA SAND B 2 B w2 W V T F X -0.165 -0.160 0.495 0.588 -0.082 0.656 F z -0.141 -0.136 0.473 0. 555 0.012 0.708 R -0.151 -0.146 0.483 0.570 -0.023 0.695 T 0.029 0. 028 0.000 0.000 -0.002 1.000 V 0. 016 0.013 -•0.010 • -0.Oil 1.000 W -0.573 -0.569 0.961 1.000 w2 -0.486 -0.484 1.000 1.000 R B 0. 999 1.000 1.000 0.996 F z B 2 1.000 1.000 F X 0.969 F z 0.988 R F X TABLE E2 CORRELATION MATRIX FOR TILLAGE TESTS FOR HANEY CLAY B 2 B w2 W V T F X 0. 648 0. 644 -0. 281 -0.308 0.178 0. 504 F z 0.591 0.584 -0.219 -0.252 0.188 0 . 541 ' R 0. 613 0.607 -0.241 -0.272 0.187 0. 532 T ,0.000 0.000 0.000 -0.000 0.003 1.000 "V 0. 004 0.004 -0.010 -0.009 1.000 W 0. 805 -0.800 0. 964 1.000 w2 -0.803 -0. 802 1.000 1.000 R B 0.999 1.000 1.000 0.997 F z B 2 1.000 1. 000 0. 971 0.986 F X F F R x z 

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