«yr 9 ‘ ‘ .- < . .. - .« . A ’ . H.,.,,9_-_ku' > 1.3.. v . n 4... .. .., .,. .... a”, .... , w. . , \. . - - x. .. ., , , _ _. .. . ‘ .. u... 4.._,,_-‘__J‘_‘ .H‘V'nt...” 4. in. -.. ... v A. .u. C I Id 0 § 4 . ~ ~ . . .‘2.’.':.;‘.‘;u:_-_H_uu u . n MATHEMATICAL ANALYSIS OF STATE AND DYNAMiC LATERAL PRESSURES EN THE INFINITE UPRIGHT FLAT-BOTTOMED DEEP GRAIN BIN Thai: for the Segree of 91:; I} MICHIGAN STATE UNIVERSITY Joe! 35v lsaacson 1.963 ill-isms C . 9, This is to certify that the thesis entitled Mathematical Analysis of Static and Dynamic Lateral Pressures in the Infinite Upright Flat- ‘ Bottomed Deep Grain Bin .t a ‘ presented by Joel Dov Isaacson has 'been accepted towards fulfillment of the requirements for ' ‘ M—degree in Agricultural Engineering I. ‘4 "I / .ilfl‘ ’/ Mx—t/‘nd A Major professor ’ Date I “,1 [I ’ ,5 0-169 LIBRAR Y Michigan Sm: ‘ OVERDUE FINES: 25¢ per day per item RETURRING LIBRARY MATERIALS: Place in book return to remove charge from circulation records ABSTRACT MATHEMATICAL ANALYSIS OF STATIC AND DYNAMIC LATERAL PRESSURES IN THE INFINITE UPRIGHT FLAT-BOTTOMED DEEP GRAIN BIN by Joel Dov Isaacson The determination of static and dynamic pressures exerted by grain on the containing deep bin structure continues to be a hazardous task for the designing engineer. A mathematical analysis of this problem is pre- sented in this study. Two parallel mathematical models were constructed, corresponding to the mechanism of grain pressures in deep bins with symmetrical cross- sections. The first (analytic solution) showed that in the one-dimensional problem, the fundamental behavior of pressures is fully characterized by an ordinary linear differential equation of the first order with (possibly) variable coefficients. The second model (algebraic solution) was put in a compact form involving series of determinants and matrices that made it readily applicable to numerical calculations. The models were com- pared for various cases. For a case of constant coefficients they were found to be mutually identical as well as identical with Janssen's solution. For a case of variable coefficients (a linearly increasing density-function, a hyperbolically decreasing ratio-function, and a constant wall-friction parameter) the solutions stayed identical. A special choice of the varia- ble coefficients showed complete identity with the Reimberts' semi- empirical solution. An intermediate conclusion suggested a strict identity Joel Dov Isaacson between the models and showed that the analytic solution can serve suc- cessively as a prototypical solution-model for a two-dimensional dynamic, one-dimensional, the Reimberts', Janssen's, Rankine's and hydrostatic solutions, by the proper choice of the characteristic functions. The characteristic functions--density, ratio and friction functions-- defined, respectively, the unit weight, the ratio between the lateral to the vertical pressures, and the ratio between the frictional stress at the wall and the lateral pressure, as functions of the depth coordinate. Typical characteristic functions associated with static and dynamic conditions were derived and their nature was analyzed. The final chapter introduced a new approach based on topological considerations. A brief representation of the principles led to intermediate results which showed that the maximum dynamic lateral pressure in typical circular cylindrical grain bins should occur at about one-third of the total height H above the bottom and decrease sharply toward an elevation of about 0.12H above the bottom. These results agreed closely with experi- mental findings reported from various sources. It was also suggested that the dynamic ratio-function is monotonically decreasing versus the depth. Recommendations for further studies were made, emphasizing the need for the determination of the mechanical-rheologica1-biologica1 prop- erties of grains and the development of the mathematical theory of grain- pile transformations. A brief outline of the two-dimensional problem was presented in the appendix. MATHEMATICAL ANALYSIS OF STATIC AND DYNAMIC LATERAL PRESSURES IN THE INFINITE UPRIGHT FLAT-BOTTOMED DEEP GRAIN BIN BY Joel Dov Isaac son A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Agricultural Engineering 1963 ACKNOWLEDGEMENTS To Dr. James Sterling Boyd, my major professor, for his valuable contribution, guidance and encouraging friendliness. To Dr. E. A. Nordhaus (Mathematics) who supervised my mathematical studies and whose unfailing guidance and interest are much appreciated. To other members of the guidance committee: Dr. F. H. Buelow (Agricultural Engineering) and Dr. L. E. Malvern (Applied Mechanics) for their helpful advice and suggestions. To Dr. A. W. Farrall, chairman of the Department of Agri- cultural Engineering, for making available the graduate assistantship that made the undertaking of this study possible. To Leora, my wife, who-~besides other things--not only drew some of the drawings, but patiently yielded herself to become an expert on the subject. And above all, to the people of Michigan to whom we owe so much. ii LIST or FIGURES ....... .. ............................. LIST or TABLES ......... . ............................... LIST OFAPPENDICES ................ NOMENCLATURE ....................................... II. III. TABLE OF CONTENTS PART ONE: PRELIMINARIES INTRODUCTION ................................... A. The Nature of the Problem B. Objectives 1. General objectives 2. Specific objectives HISTORICAL AND BIBLIOGRAPHICAL NOTES ........ FACTORS AFFECTING ‘PRESSURES IN GRAIN BINS . . . A. General Classification 1. Grain characteristics 2. Bin characteristics a. Bin geometry b. The structural system 3. Operations a. Charging b. Discharging c. Simultaneous charging and discharging 4. Surroundings B. Discussion (DO‘O‘O‘ 19 19 20 22 22 23 25 25 27 29 30 3O Page PART TWO: ANALYSIS IV. DEFINITIONS .......... . ............................ 34 A. General Definitions 34 B. Definition of a Deep Bin 36 C. Classification of Materials 38 V. THE ONE-DIMENSIONAL PROBLEM .................. 41 A. Analytic Solution 41 1. Derivation of the governing differential equation 41 2. Boundary conditions and solutions 43 a. Without surcharge 43 b. With surcharge 43 B. Algebraic (Numerical) Solution 44 1. Construction of the fundamental model 44 2. A search for a compact notation 48 C. Investigation and Comparison of Solutions 51 l. Constant coefficients 51 a. Analytic solution 51 b. Algebraic solution 51 c. Comparison to Janssen's solution 52 2. Variable coefficients 53 a. Analytic solution 53 b. Algebraic solution 54 c. Comparison 55 3. Relationship of analytic solution to the Reimberts' semi-empirical solution 57 4. Discussion 58 D. The Characteristic Functions 61 l. The density-function 62 2. The ratio-function 63 iv inve 3 ti gati on 3. The friction-function a. Static conditions b. Dynamic conditions c. Practical considerations VI. RECOMMENDATIONS FOR FURTHER STUDY ......... A. Grain-Pile Transformations 1. The associated characteristic pile 2. Example of a set-theoretic attack on the problem a. The initial set b. The image set c. Definitions 3. Grain-pile transformations 4. Some intuitive intermediate results a. The decreasing character of the dynamic ratio-function b. The location of the center of singularity B. Recommendations for FurtherfiStudy 1. Theoretical a. General b. Specific . 2. Experimental 3.. General b. Specific APPENDIX .............................................. REFERENCES .......................................... a. Static c onditions b. Dynamic conditions, preliminary Page 64 68 71 72 72 74 77 77 77 78 78 79 8O 81 82 82 86 87 87 87 87 88 88 88 89 95 Figure 10. LIST OF FIGURES Geometrical representation of the nature of the problem ........... . ............................. The experimental findings of Platonow and Kowtun (1959) ............. . ..................... . ....... Diagramatic representation of Russian specifications fordynaminaCtors OOOOOOOOOOOOOOO ......... 00.... Three typical cases of bin walls affecting deflection and redistribution of external pressures . . . . . ....... Recommended areas to be used for dynamic factors arising from eccentricity of discharging orifice ...... Comparison between the analytic and algebraic solutions with variable coefficients . . . . . ............ Geometrical representation of a two-dimensional grain-pile transformation of the infinite bin ......... Two-dimensional density-function over a vertical section of the infinite bin .......................... Two-dimensional hyperbolic ratio-function over a vertical section of the infinite bin ...... . . ......... Two-dimensional dynamic ratio-function over a vertical section of the infinite bin ............ 7. . . . vi Page 13 14 25 28 56 85 94 94 94 LIST OF TABLES Table Page 1 Russian specifications for dynamic factors ......... l4 2 Classification of materials with respect to gravity pressures in storage containers ................... 39 3 Examples of material-types of materials stored in bins and containers . . . . .......................... 40 4 Analytic solution as a prototype of other solutions . . . . 60 vii LIST OF APPENDICES Appendix Page A-l General solution of the general ordinary linear differential equation of the first order ............. 89 A-2 An outline of the two-dimensional problem ......... 9O 1. Analytic and algebraic solutions 90 a. Analytic solution 90 b. Algebraic solution 91 2. The characteristic functions 92 a. The density-function 92 b. The ratio-function 93 c. The friction-function 93 viii NOMENCLATURE cross - sectional area; initial set set element of set A element of set A' constant; image set set element of B element of B' constant; curve curve parameter; cohesion set of curves set of curves diameter diameter of kernel lateral force functional symbol weight functional symbol total height of bin height of deep bin height of surcharge, height identity matrix; set integer; subscript integer; subscript ratio between lateral and vertical forces ix 0 max k( y) KZL" 233 n n'- p. pn. py. p(y) Q q R RI ratio between lateral and vertical pressures coefficient of earth pressure at rest ko maximum ratio-function perimeter of bin cross-section mapping characteristic mapping subscript set of mappings constant inte ge r; subsc ript n l 2 lateral pressure vertical force vertical pressure hydraulic radius; re sidual force; rectangle center line of R length of base of T frictional force; set frictional stress; "center of singularity” load; triangle median of T integer; subscript intege r; subscript ‘< 9 ”GD -<-<-< max fly) owl> VOGJ M 3') Name $-&'o function set element of set X; width coordinate depth coordinate angle function function; unit weight initial unit weight maximum unit weight density-function increment parameter constant natural angle of repose imaginary angle 'of repose friction parameter friction-function coefficient of friction of grain on wall parameter functional symbol 3. l4 summation symbol hydraulic radius; length angle of internal friction angle Other Symbols (I there exist V for every 3 such that E belonging to exp e, base of Napierian logarithms tg tangent det Subi A see page 49 Subscripts Used O ht i :3 SOT! max PART ONE: PRELIMINARIES I. INTRODUCTION A. The Nature of the Problem Deep bins and silos are very commonly used for the storage of grain and other granular materials throughout the world. The first condition of structural design--knowledge of the forces. acting upon the structure--is not well defined in the field of silo design. There is lack of information concerning the values of lateral and vertical pressures exerted on the walls and bottoms of grain bins by stored materials. There is evidence that the use of the present design formulas has resulted in many partial or complete failures. On the other hand, in order to avoid structural failures, the safety factors allowed in silo design unnecessarily increase construction costs. Therefore, there is a vital need for more reliable methods of estimating silo pressures, under both static and dynamic conditions, for design purposes. Grain pressures in deep bins were first calculated as for a semi-liquid of the same density as the grain. Thus, for instance, lateral pressure was calculated from the hydrostatic formula = y (I. A. 1) Py Y where lateral pressure exerted by the grain on the wall at Py " the depth y Y - density of grain y — depth of cross-section calculated from the top of the grain This method had serious deficiencies since many structures buckled under the vertical load arising from the friction of the grain on the walls--a phenomenon unknown at the time. On the other hand, it was realized later that the calculations based on hydrostatic pres- sure distribution were exaggerated in comparison to the actual lateral pressure. The developments in the field of soil mechanics for calculating earth pressures gave rise to the definition of a new factor, ken-the "coefficient of earth pressure at rest, " which was borrowed for use in the silo problem. Formula (I. A. 1) thus became py = koyy (I. A. 2) where R0 was usually determined according to Rankine as _l-sin¢ — .A. o l +sin¢ (I 3) k where 4) is the angle of internal friction of the grain. This improvement did prevent the former exaggerated estimations of the lateral pressure, but still did not take into account the vertical load on the walls, resulting from friction effects; thereby, the whole weight of the grain mass was erroneously assumed to be transmitted directly to the bottom of the bin. The next development was proposed by Janssen (1895), and took into account the phenomenon of grain-wall friction py = 133- [l - exp {- (P%)y>] (I. A. 4) where R - "hydraulic radius" of the bin cross-section u - coefficient of friction of the grain on the wall k - Rankine coefficient of earth pressure at rest, proposed for this case by Koenen (1896). Since that time a considerable number of formulas have been proposed by various investigators. Most of them give similar results to that of the Janssen-Koenen method, and accordingly the latter has remained in widest use. Over a long period it appeared that the problem had been solved; however, during the last three decades it has transpired that the Janssen formula provides the solution of one particular case out of a whole set of quite complex phenomena. The basic deficiency is in the fact that Janssen's formula is static in nature; whereas it is becoming ever more widely recognized that the main problem of grain pressures in deep bins is of a dmamic nature. Accordingly, three basic states are defined in a deep grain-bin system: Rest--a state where no grain is added or removed 1 - Hydrostatic pressure curve, Eq. (I. A. 1) 2 - Earth pressure curve, Eq. (I. A. 2) 3 - Conventional pressure curve (Janssen's), Eq. (I. A. 4) 4 - Typical empirical dynamic pressure curve, Fig. 2 Lateral Pressure p / ,/ / ANN Depth ‘K Figure 1. Geometrical representation of the nature of - the problem. Charging--an operation in which grain is added Discharging--an operation in which grain is removed from the bin (under the gravity forces alone). ie_s_t_ is considered a static state, and Janssen's formula is still regarded as applicable to this state. (For the sake of accuracy it should be emphasized that even "at rest, ” the system--both grain and bin--is actually in some degree of motion.) Charging is usually carried out by mechanical devices which permit the grain to fall freely through charging orifices at the top into the interior of the bin. This process is sometimes accompanied by prominent dynamic effects. Discharging is usually accomplished by gravitational forces, through discharging orifices in the bottom. It has been established that the discharging process is the most dangerous, since it is accom- panied by prominent and frequent dynamic effects, more so than the charging process. The lateral pressures which arise during this process are liable to be_tlv_o or t_hr_e<_e times as large as those calculated according to static formulas. Whereas the conventional formulas are more or less applicable to a state of rest, there do not exist satisfactory methods for estimating grain pressures in charging or discharging. In view of the critical pressure increase during the so-called "dynamic effects, ” analysis of 1 these pressures is of major importance. B. Objectives At the time of the initiation of the study the objectives were: 1. General objective 5 a. Study and investigate ideas which might lead to a better understanding of pressure phenomena in granular materials under silo (deep bin) conditions. b. Develop approaches and methods for calculating pressures exerted by granular materials on the containing silo structure. 2. Specific objective 5 a. Determine the relation of factors such as: (i) angle of repose (ii) grain density (iii) friction of grain on walls (iv) hydraulic radius of bin cross-section (v) bin total height (vi) cross-section shape to the normal pressures and vertical loads on the walls and bottom of grain bins, under static and dynamic conditions. b. Produce mathematical models that can be used to predict qualitative relationships of factors such as: (i) density-function (ii) friction-function (iii) ratio-function as well as mathematically verify the frustrating qualitative behavior of lateral pressure, as reported from experimental studies of dynamic effects in deep grain bins. The development of the study was kept roughly along these lines. II. HISTORICAL AND BIBLIOGRAPHICAL NOTES Research into pressures in grain silos commenced with the early experiments of Roberts (England, 1883), which showed that (i) the sum of the vertical forces acting on the bottom of a grain bin is much smaller than the weight of grain stored in the bin; (ii) pressure on the bottom does not increase when the height of grain exceeds two diameters. The experiments of the sequence of investigators who followed Roberts endorsed his observations and generalized the second conclu- sion with respect to lateral pressures as well. Ketchum (U. S. A. , 1919) summed up the experiments of his con- temporaries (Roberts, England, 1883; Janssen, Germany, 1895; Prante, Germany, 1896; Tolz, Canada, 1897-1903; Airy, England, 1897; Ketchum, 1902-1903, 1919; Bovey, Canada, 1903; Lufft, Argentine, 1904; Jamieson, Canada, 1905; Pleisner, Germany, 1906). as follows: "1. The pressure of grain on bin walls and bottoms follows a law which is entirely different from the law of the pressure of fluids. 2. The lateral pressure of grain on bin walls is less than the vertical pressure (0. 3 to O. 6 of the vertical pressure, depending on the grain, etc. ), and increases very little after a depth of 2 1/2 to 3 times the width or diameter of the bin is reached. 3. The ratio of lateral to vertical pressures, k, is not a constant, but varies with different grains and bins. The value of k can only be determined by experiments. 4. The pressure of moving grain is very slightly greater than the pressure of grain at rest (maximum variation for ordinary conditions is probably 10 per cent). 8 5. Discharge gates in bins should be located at or near the center of the bin. 6. If the discharge gates are located in the sides of the bins, the lateral pressure due to moving grain is decreased near the discharge gate and is materially increased on the side opposite the gate (for common conditions this in- creased pressure may be two to four times the lateral pressure of grain at rest). 7. Tie rods decrease the flow but do not materially affect the pressure. 8. The maximum lateral pressures occur immediately after filling, and are slightly greater in a bin filled rapidly than a bin filled slowly. Maximum lateral pressures occur in deep bins during filling. 9. The calculated pressures by either Janssen's or Airy's formulas agree very closely with actual pressures. 10. The unit pressures determined on small surfaces agree very closely with unit pressures on large surfaces. 11. Grain bins designed by the fluid theory are in many cases unsafe as no provision is made for the side walls to carry the weight of the grain, and the walls are crippled. 12. Calculation of the strength of wooden bins that have been in successful operation shows that the fluid theory is un- tenable, while steel bins designed according to the fluid theory have failed by crippling the side plates. " Most of the experimental findings of the early investigators have not been invalidated to this day. Conclusions #4 and #8, however, are in striking contradiction to the conclusions drawn from the later ex- periments of Reimbert (France, 1943) and others. (Conclusion #10, related to measuring techniques, seems too dogmatic in view of the argument that is still being waged over this question today. ) Whereas Frohlich (Germany, 1934) and Dorr (Germany, 1938) endeavoured to obtain theoretical solutions, others--Fordham (England, 1937) and McCalmont (USA, l938)--continued to develop techniques for the measurement of pressures exerted by granular materials on the containing structures. 10 At the beginning of the forties, with the expansion of grain ele- vator construction, a strong incentive was given to grain pressures research in two main centers--the USSR and France. Modern experi- ments in the USSR began in Baku (1938-1939), on a larger scale and with more up-to-date techniques and instrumentations than those of their predecessors. The experiments revealed some unexpected problems, none of which was completely solved. Accordingly, it was recommended that further experiments be carried out. The most important results of these experiments were: 1. Endorsement of the necessity to strengthen the region of the mid-height of the walls, in comparison to the sections corresponding to calculations according to Janssen, hitherto accepted in the USSR. 2. Ratification that the increase of pressures during dis- charging--which are far from pressure limits calculated according to Janssen--can be one of the major causes of damage to the walls. 3. It became clearly evident that lateral pressures are not necessarily uniform along the perimeter of a. bin cross- section. Additional investigations carried out at the USSR Institute of Building Research were aimed at clarifying what factors determine the nature of the effect of grain on the containing structure, and in what way it is possible to prevent the increase of pressure during discharg- ing. These experiments showed that the principal factors are-~the density of grain, the dimensions of grain, and the proportions of the bin, i. e. the ratio of the height to the width. 11 Kim (Moscow, 1959) summed up the Russian research on grain elevator s as follow 5: "1. Charging by the "rain" method increases exploitation of the capacity of the bin by 6 to 10 percent, compared to stream charging. 2. The sum of the vertical forces acting on the bottom of the bin is greater during charging than during discharging. 3. The sum of the vertical forces which stem from grain friction on the walls is greater during discharging than during charging. 4. The lateral pressures during discharging depend upon the flow pattern. If the grain flows in a stream of uniform cross-section coinciding with the bin cross-section, then high dynamic pressures are created (two to three times as high as Janssen's theory values); whereas in funnel flow dynamic pressures do not occur. 5. The flow-pattern during discharging depends upon the coefficient of friction of the grain on the wall, the density of the grain, the geometric proportions of the bin (height to diameter), etc. However, none of the above factors, separately or jointly, can always ensure, with reliable certainty, the desirable funnel flow-pattern. 6. Horizontal rings or perforated central pipe ensure funnel flow and negate the possibility of the development of high dynamic pressures during discharging. 7. Grain elevators consisting of a cluster of bins which ex- ploit a central bin as a perforated discharging pipe, are particularly advanced structures. 8. The installation of horizontal rings or a perforated pipe is an effective measure for the rehabilitation of damaged grain bins. " In conclusion, the author emphasized that "in spite of the large number of experimental and theoretical investigations during a long period of time, the problem of grain pressures in bins has not been finally solved to this day. Without clarification of this question it will not be possible to design and construct grain bins correctly. ” Accord- ingly, grain pressures research has continued to proceed in the USSR. 12 Platonow and Kowtun (19 59) of the Stalin Technological Institute in Odessa, carried out extensive investigations and measured pressures under static and dynamic conditions in three full-scaled structures. Their findings showed that lateral pressures under dynamic condi- tions are twice as high as Janssen's, but decrease sharply at the lower region of the walls as shown in Fig. 2. No theoretical explana- tion for this unpredictable phenomenon was suggested. Previous Russian design specification (recommended by C. N. I. P. S.) require specific allowances for lateral pressures in- crease during discharge. These specifications are given in Table l and are illustrated in Fig. 3. With the construction of many larger grain elevators in France and Algeria a number of studies, which have become widely known, were made by Marcel and Andre’ Reimbert (1943, 1954, 1959), Caquot (1957), Despeyroux (1958), Pamelard (1959), Kellner (1938, 1960) and others. The essence of the findings of the French experiments was that lateral pressures during charging were in good correspondence with the Janssen-Koenen equation, or the extended formula of Caquot. On the other hand, the increase of pressures during discharging was, according to Despeyroux's terms, ”quite capricious. " The most striking experiments were carried out by the Reimberts. Inter alia, they carried out a series of experiments in steel grain bins, employing electrical resistance strain gages. They found that the l3 Lateral Pressure (1b. /sq. ft.) 500 - 1000 1500 I 1 - Janssen's curve 2 - Recommended curve 0. 35H 3 - Dynamic curve :I: O Ln 1 o' ’7 \ a \ .r: 60 \ a \ o \ Q \ l | l ‘. l / I 80 ‘/‘H < “7"“ - In I I—4 ' o' 'I ,0, h I Figure 2. The experimental findings of Platonow and Kowtun (1959). l4 Table 1. Russian specifications for dynamic factors. Dynamic factor in: Zone from bottom External bins Internal bins I - 0. 12h 1. 0 l. 0 II - 0.18h 2. 0 1. 5 III - 0. 40h 1. 5 1. 5 IV - 0. 30h 1.0 1.0 Bottom hopper 1. 5 l. 5 4 Figure 3. Lateral Pressure —Janssen '-'-Inte rnal bin IV .30h ---'Exte rnal bin anssen ax. ans sen l. 5 x Janssen DEEP BIN h .0 x Janssen . 5 x Janssen Bottom Diagramatic representation of Russian specifications for dynamic factors. 15 increase in lateral pressures during discharging was up to 2. 4 times as great as the corresponding pressures in charging. The results varied somewhat for the same bin and same grain. The ratio of the lateral pressures during discharging to that in charging was called the "dynamic coefficient, " and accordingly, it was recommended that a dynamic factor equal to 2 be introduced into the calculations of grain bins. These experiments, like those of the Russians, showed that the maximum lateral pressure during discharge arises in the region of the mid-height, and possibly even above it. This phenomenon is in contradiction to all of the accepted theories, and demands a theoretical explanation that has not yet been found. Later observations of damaged silos proved beyond a doubt that the damage is indeed endangered in the upper sections of the walls. Despeyroux surmised that the uncertainty in the calculations of pressures during discharge stems from a number of factors, for example--discharge rate, flow-pattern and the properties of grain. He recommended the institution of further investigations into these questions. Pamelard suggested certain modifications for the method of cal- culation proposed by the Reimberts. Caquot, K’erisel, Buisson (1960) and others dealt with the subject from the point of view of soil mechanics and the mechanics of granular materials. The two former and Kellner proposed calculation methods of their own. The publication of the French reports aroused a wave of interest 16 among various investigators in Europe. Under the auspices of the Royal Swedish Geotechnical Institute, Bergau (1959) initiated in 1951 experimental studies on actual structures with the aim of clarifying the processes of charging and discharging. The conclusion drawn from these experiments emphasized: ". . . during the emptying process, the silo walls can be ex- posed to pressure increases. These can be in excess of the 10-20 per cent referred to in earlier literature. . . . It should therefore appear important that more detailed studies be made of the emptying process. These studies should be combined with investigations about the shape and the location of the outflow and their effect on the stress distribution in the mass so that suitable steps can be taken to prevent the build-up of dangerous and indeterminable excess pressure. " Publications in England--Gray (1944), Pasfield (1950-1956), and the proceedings of the British Institute of Agricultural Engineers (1955), indicated considerable interest in the problem of grain and silage pres- sures as well. Moss (1955) conducted an important and advanced experimental investigation in a raw sugar bin. In Germany, Theimer (1949-1959) published a number of papers on problems of calculations of pressures of grain and flour. An inves- tigation on an actual grain bin is being carried out by the German Com- mittee of Norms. Investigators from Ireland (Mallagh, 1958), Austria (Friedrich, 1962; Torre, 1963), Hungary (Jaky, 1948), Poland (Nowacki and Dabrowski, 1955), Italy (Fumagalli, 1960), Sweden (Jacobson, 1958), South Africa (Zakrzewski, 1959) and elsewhere have devoted considerable attention 17 to the treatment of various aspects of the silo problem. However, none has suggested a complete analysis. Concluded Zakrzewski: ”. . . The conventionally accepted method of calculating the required reinforcement for bin wall . . . without allowance for dynamic effect during the emptying of the bins, is considered dangerous practice, often leading to a defective structure . . . . The analysis of stresses is very complex and the usual simplified assumptions are not safe. ” ‘ Isaac son (1960) carried out experimental and theoretical studies at the Israel Institute of Technology. The experimental findings showed that critical lateral pressures occur at mid-height of bin walls. Edel- man and de Leeuw (1961) introduced the ”biological - mechanical - rheological complex" of grain stored in deep bins and suggested a comprehensive study along these lines. Parallel to the research into grain bins, there are being carried out in Europe at present a number of researches into silo-structures for storage of cement. The research techniques and the findings are often similar and have importance for the understanding of the problem of grain bins. An important investigation of a real cement silo was made in England by Rowe (1959). Similar investigations were made in Russia by Petrow (1958), Holland (Hass flfl° , 1958) and Germany (Bohm, 1956 and Leonhart _e_t_a_1. , 1960). The contribution of USA research in this area was relatively limited since Ketchum and his contemporaries. Although several studies (Amundson, 1945; Barre, 1958; Caugheysta_l., 1951; Collins, 1962; 18 Dale and Robinson, 1954; Stahl, 1950;. Weiland, 1962; and others) were made during the years, no organized effort is recorded. Wrote Brandes (19 61) : ". . . There is lack of definite information concerning the values of horizontal and vertical pressures exerted on the walls and bottoms of bins by most fillings. Considerable effort has been and is presently being spent in an empirical approach to the solution of this problem based on tests conducted on small- sized containers. But a few attempts have been made to deter- mine the actual pressures existing in full-scale bin structures, either to corroborate those found late in the nineteenth century for grain or to extend those findings to other fillings. Nor has any continuing effort been made toward an understanding of the fundamental theory of bin pressures. That this condition con- tinues to exist in a day when the means of measuring these values are available makes not too happy commentary on the thorough- ness of this branch of the profession. " Boyd and Yu (1960, 1963) were among the few investigators that conducted experimental studies on full-scale (silage) bins. Jenike (1961) made an important contribution to the related field of grain mechanics. The meetings of the American Society of Agricultural Engineers fre- quenlty discuss some related problems. Claimed Collins (1962) re- cently: ". . . Until better data is obtained and results correlated with type of flow and characteristics of stored material and structure, it will not be possible to predict when and to what degree overpressures will occur.” III. FACTORS AFFECTING PRESSURES IN GRAIN BINS A. General Clas sification The following is a general non-mathematical discussion aimed at introducing various factors recognized as being involved with grain pressures. This discussion is primarily based on extensive literature survey, and--to a certain degree-~on experience as well as on plain engineering reasoning. The approach is, in general, qualitative rather than quantitative. Factors affecting grain pressures in storage structures can be classified in four main groups as follows: 1. Grain characteristics 2. Bin characteristics 3. Operations 4. Surroundings Critical pressures are likely to occur under the worst combina- tion of factors belonging to above four groups. It is, of course, the task of the design engineer to minimize the probability of such occur- rence. A first step in this direction is to gain a general understanding of these factors by means of a systematic classification. 19 20 1. Grain characteristics Grain characteristics certainly have prime importance in con- sidering the resulting pressures. It is recognized lately that grain is quite an unpredictable engineering material. This is partially due to its unique biological properties in comparison to other bulk solids such as coal, stone-aggregates, chemicals or cement. Since grain is a living substance, it is reasonable to assume that there is some degree of "biolOgical motion” in progress, even in a state of mechanical rest, where mechanical immobility is visually evident. Grains inspire, expire and continually absorb or generate heat, moisture and gases. These subsidiary processes--particular1y variations in moisture contentnaffect both the internal friction of grain and the friction of grain on the walls. They are also liable to affect the density and its distribution throughout the grain mass. Sometimes they may cause a conglomeration of clods of a number of kernels which grow to a considerable size. In such cases the grain must be treated as a material whose mechanical properties are definitely different from those that it acquired in its original state (see notes in section IV. C. ). An additional biological factor that directly affects the stresses within the walls of the storage structure itself is the biological source of heat. It may create a temperature gradient between the interior and the exterior of the bin and give rise to heat stresses in the walls. This phenomenon is particularly evident in reinforced concrete bins 21 without insulation, in cold geographical regions. Therefore, stored grain is justifiably looked upon as a biological- mechanical-rheological complex that demands very detailed and careful studies before any specific characteristics can be regarded as being determined. Accordingly, there exists a detectable trend in literature, as well as in various pertinent research activities, to establish a branch of Grain Mechanics which is somewhat similar in its approach to the study of soil properties in the field of Soil Mechanics. The scope of this study does not permit but to list some properties that were found to be of some significance in studying grain characteristics. The single kernel: Average weight, average volume, specific weight, average surface area, typical geometric form, surface roughness, hardness (friction resistance), breakage and mechanical damage, other irregularities, modulus of elasticity, compression strength, shear strength, impact strength. Mass of grain: Unit density, void ratio and porosity, angle of natural slope, internal friction, coefficients of friction on various materials, moisture content (under various temperatures and humidities), modulus of elasticity, compressive strength, static and dynamic shear strength, modulus of resilience, modulus of toughness, stress relaxation, flow- factor, effective yield locus, susceptibility to vibration, consolidation, specific heat, dialectric constant, thermal conductivity, coefficient of volume expansion, organic and biological variations in volume, other 22 biological properties (such as: adsorbtion of moisture, generation of heat, carbon dioxide etc. ), percentage of foreign matter and dust. 2. Bin characteristic 3 a. Bin geometry. Main factors related to bin geometry are: (i) Hydraulic radius--R, i. e. the ratio of the cross-sec- tional area to its circumference. (For more precise definition, see section IV. A. ). (ii) Cross-sectional geometric shape and dimensions. (iii) Total height of bin. Unless the walls are not strictly vertical (a case which was ex- cluded by the definition of a bin, see section IV. A. ), the hydraulic radius is a constant. The overall effect is that lateral pressure is directly proportional to R, whereas the friction effect is inversely proportional to R. These facts are satisfactorily introduced in most known calculation methods. However, the ratio of the hydraulic radius to the total height which is experimentally recognized as a fundamen- tally important factor, is ggt usually represented?” for theoretical difficulties in the derivation of most methods. Generally, this ratio determines whether a bin is shallow (a bunker), intermediate or deep. (See section IV. B. ). There is sufficient evidence--experimenta1, as (*) In fact, this ratio is completely deleted from important theories such as those of Janssen and the Reimberts. I. II ill. 11 ill-ill ‘ l I'd-Ills lullllllrl ‘- Illlll! E: .I {{{If 23 well as theoretica1--that grain pressures behave fundamentally different according to the degree of "deepness. " In chapter VI. A. an attempt is made to include this ratio in the overall formulation, and theoretically understand some of its influences. The geometric shape of the horizontal cross-section is of con- siderable importance too. In practice, most cross-sections are sym- metric and of the shape of either circles or regular polygons. However, a cluster of regularly-shaped bins may forcibly contain smaller bins of quite odd shapes; i. e. a cluster of circular bins that contains smaller "astroidic" binslfllt is apparent that an astroidic bin, when compared with a circular bin of the same hydraulic radius and height, affects grain pressures quite differently, primarily because of increased overall friction effect. Other factors are: type of hopper (if used), hopper's slopes, eccentricity; number, arrangement and locations of discharging orifices; devices such as central perforated emptying pipe or circumferential rings. b. The structural system relates, first of all, to the building material of the structure; usually concrete (staves or reinforced, but also prestressed), steel, or wood. The building material may affect pressures in several ways, ranging from affecting the coefficient of (*) Irregular shapes such as this are excluded by definition 1. a in section IV. A. 24 friction of the grain on the wall, to the rigidity of the walls and stability of the structure as a whole. As in the theory of retaining walls in Soil Mechanics, the supporting walls in deep bins may affect the pressures exerted on them by moving (outwards, inwards) tilting or deflecting. These changes are not only dependent on the type of building material alone, but as in other cases of structural design, are dependent on the structural system as a whole. For example, a bin wall is to deflect differently under the same external loads, according to whether the connection of the wall is hinged or rigid at the bottom and/or top, is a section of a continuous wall extending to neighboring bins (where each of these bins may be full, being filled or emptied), and so forth. The possible combinations of situations thus become quite involved in some cases and therefore call for repeated pressure calculations in order to take deflection effects into account. Figure 4 represents simple cases of possible deflections of (a) double-hinged wall, (b) bottom-rigid top- hinged wall, and (c) double-rigid wall. External initial pressure distribution is the same for the three cases, but redistribution of pres- sures is likely to occur after deflections take place. Process is repeated until equilibrium is achieved. 25 (a) (b) (C) Figure 4. Three typical cases of bin walls affecting deflection and redistribution of external pressures. 3. Operations Operations occurring in grain bins are: a. charging b. discharging c. simultaneous charging and discharging d. state of rest It was pointed out before (see section I. A. ) that operations (a) to (c) give rise to dynamic effects which may cause critical changes in pressures. a. Charging. The nature and method of the charging operation undoubtedly affect the grain pressures. A comparison between a grain bin and a similar liquid container is illustrative. A liquid container can be filled through an opening in the bottom or sidewall according to the law of communicating vessels, or by applying pressure. The kinetic energy of the incoming liquid is transferred to the mass already presented 26 in the tank and the surface of the liquid rises steadily without special accompanying dynamic effects. However, if the tank is filled through an opening at the top, waves are formed on the surface which descend gradually through the mass of the liquid. These waves increase in amplitude as the height of pouring and the rate of filling increase. In the extreme case, where the tank is filled by a sudden pouring of a large quantity of liquid through an opening at the top, intense dynamic effects are induced as a result of a sudden release of the kinetic energy of the liquid in falling. A grain bin cannot be filled through an orifice at the bottom by any means because of significant internal friction. Therefore, grain bins are filled through top orifices only. In filling by this manner, the grain falls and, as in the case of liquid, releases considerable kinetic energy upon impinging on the grain already present. Here too, it is possible to observe waves on the surface of the fill within a certain time of ”relaxation period, ” until the fill achieves a state of mechanical equilibrium. The amplitude of the oscillations and the time of damping depend on the kinetic energy and rate of charging. If a large quantity of grain is suddenly poured into a bin (a probable case where automatic scales are used), strong dynamic effects and fluctuations of grain pres- sures are observed. An eccentric stream of grain causes a concentration of dynamic action in the vicinity of the wall. In the extreme case, the flow may 27 impinge directly on the opposite wall. On the other hand, scattering the flow by simple mechanical devices, results in the so-called (in Russian sources) "rain filling. " In such a way, the dynamic effects are restrained. However, these methods increase the density of the fill by about 10 percent and hence cause a corresponding increase in pressures. The more important factors associated with charging are there- fore: point of charging--centric or eccentric, method of charging-- continuous or intermittent; rate of charging--magnitude, constant or varying; inclined-flow charging through the wall; charging by special mechanical devices to reduce kinetic energy, such as perforated pipe, lifted vertical pipe with screen at end. b. Discharging. The discharging operation causes the motion of very large masses of grain, much more so than charging. Accord- ingly the flow of grain during discharging has considerable influence which increases with the rate of flow. In general, both internal friction and wall-friction are affected. Not only the rate of discharging is important, but also its steadiness. Suppose that during a fast rate of discharging, the orifice is alternately opened and closed. , Under such conditions the flow undergoes severe interferences; instead of a rather continuous and regular flow, an irregular pulsating dynamic flow is resulted which causes a considerable increase in pressures. 28 Eccentricity. Before the influence of discharging on the increase of grain pressures was fully recognized, it was customary to take into account the effect likely to arise from eccentric discharging orifices. One of the recommendations required that for such cases grain pres- sures should be calculated according to hydraulic radius increased by a certain factor. Thus, for instance, in a square bin with discharging orifice located in the center of one of the side walls, the hydraulic radius used in calculations is that of a bin with central discharging orifice, however, with doubled area (Fig. 5a). _: , orifice/3L : orifice l ' l I .1 (a) (b) Figure 5. Recommended areas to be used for dynamic factors arising from eccentricity of discharging orifice. If the orifice is to be in the corner, the area is taken four times as big, as shown in Fig. 5b. (These recommendations are, in fact, equivalent to suggesting dynamic factors of l. 33 and 2. 00 for cases (a) and (b), respectively.) Eccentric discharging also engenders a greater increase of pressures on the wall opposite the discharging orifice. The slopes and shape of the hopper may also affect the flow-pattern and thereby grain pressures too. 29 As mentioned before (section III. A. l), the grain may conglomerate into large clods as a result of excessive moisture, heat, and pressure. A collapse of such clods during discharging flow causes irregular dynamic effects which create sudden fluctuating increases in pressures. Although it is difficult to trace these phenomena comprehensively, due to their randomness and irregularity, it is important to know, at least, the statistical probability of extreme points in the dynamic pressure- curves associated with this type. An additional phenomenon, very similar in its effect, is the arching of grain into domes and the collapse of these domes during discharging flow. The more important factors associated with discharging are therefore: point of discharging--centric, eccentric, several orifices (scattered over the whole surface of bottom, or along the perimeter); hopper's slopes and shape; perforated discharging pipe-~centric or eccentric; circumferential rings--various dimensions and spacings; method of discharging--continuous or intermittent; rate of discharging--magnitude, constant or variable. c. Simultaneous charging and discharging incorporate factors belonging to both of the previous groups. 30 4. Surrounding: Direct and indirect influences on pressures are due to the surroundings which are listed as: a. foundation soil b. wind forces c. solar radiation d. humidity and temperature e. vibrations f. other. B. Discussion It is obviously impossible to explicitly include all the factors mentioned in the previous section within the framework of a consistent mathematical analysis. Fortunately, this is not at all necessary. How- ever, it is desirable that derived relationships in such analysis, do include factors such that each represents at least one major group and so that all four groups are representable. Such a relationship may be written symbolically in the form PV = fiG, B, O. S, Y) (III. B. l) where the capital letters stand for Grain, Bin, Operations, Surroundings, respectively. It is found that some factors are unique and more characteristic than others. Also, some factors have definite relationships 31 (experimentally or otherwise determinable) to others included in the same group. Bearing these facts in mind, it is proposed to use the following factor 5: G —> y, k G-B —-> x B a R, H, k 0 —-> k S ->k where y - density of grain X - grain-wall friction factor R .. hydraulic radius of bin cross-section H .. total height of bin k - a factor defining the ratio of lateral to vertical pressures at any point, under any possible conditions. (III. B. 1) can therefore be written py = m. x, R. k. y) (111. B. 2) where H is absorbed by k. The above choice, although appearing to be arbitrary, is of a "necessary and sufficient" type. To better understand the motivation for this choice, a further explanation is pursued. Problems associated with pressures are often quite complicated because there is lack of definite knowledge regarding the source of 32 pressures. Fortunately, in spite of the overwhelming complexity of the overall picture of pressure in grain bins, the sole source of these pressures is easily traced as the gravitational field, i. e. the weight of grain contained in the bin. Thus, the totality of forces involved is fully known, whereas it is their distribution throughout the system that concerns. This elementary approach suggests, first of all, a definite upper bound (y H) for all possible vertical pressures. The existence of a non-vanishing grain-wall friction effect guarantees that some non-zero portion of the grain weight is transferred vertically into the walls. Whatever this portion may be, the total magnitude of load carried ‘by the walls and the bottom is a constant and never exceeds nor falls behind the total weight of grain stored. (This simple fact is too often overlooked. ) On the other hand, friction effect reveals the existence of lateral pressures. The factor k specifies the relationship between lateral and vertical pressures and therefore predominantly controls the picture of pressure distribution. k is admittedly still somewhat vague at this point of the discussion, however is guaranteed to possess at least one useful intrinsic property--i. e. k is bounded below and above by zero and one, respectively. That is to say, the lateral pressures possess the same upper bound as the vertical pressures, whereas the least upper bound is much smaller. The frictional stress is a function of the lateral pressure, involving k as a parameter. Therefore, it seems that there exists a "mechanical chain reaction" in the direction: 33 weight—>latera1 pressure—>frictional stress. This chain reaction loops, so that it looks schematically as follows: , , reduces frictional \ stress ’7 0) (vertical 3 H o A pressure) 5 no , Y lateral generates weight pressure < k Since this looping is repeated infinitely many times throughout the depth of the bin, the total mechanism becomes considerably complex, especially where the factors involved are not constant. It is, therefore, of utmost importance to find a mathematical model corresponding to this mechanism. Once the model is established, the mathematical analysis of pressure distribution becomes feasible. PART TWO: ANALYSIS IV. DEFINITIONS A. General Definitions A 2.9 is a container that bounds geometric figures belonging to the set of all right cylinders that satisfy either of the following conditions: a. The directrix of the cylinderical surface is an arbitrary closed convex curve. b. The directrix is a pair of infinite parallel straight lines; the bin is then called an infinite bin. The cylindrical surface is called the wa11(s) or boundary of the bin. The lower base is called the (flat) bottom. The upper base is called the top, and physically may or may not exist. The hydraulic radius of a bin is in the respective cases above: a. The ratio of the area bounded by the closed directrix to the the. length of the directrix. b. The ratio of the area bounded by the rectangular cross- section of a finite segment of an infinite bin, to the total length of the walls included in the segment. 34 35 A bin is shallow or de ep according to whether the ratio of its height (generatrix) to the hydraulic radius is less than or egual to or larger than some arbitrary positive real number, respec- tively. (See section IV. B. ). A bin is called a grain bin if its interior is partially or com- pletely occupied by grain. (See def. 6.) Granular material is a mass consisting of a collection of solid particles and has the following properties: a. A fixed natural angle of repose. (See def. 7.) b. If bounded by a deep bin, it maintains gravity flow when some region of the bottom is removed. c. If bounded by a deep bin, it transforms to a geometric figure of constant slopes when the walls are gradually removed. G rain is a collection of plant seeds that satisfies the properties of granular materials. A single member of the collection is called a k e r n e l . The natural angle 2_f repose ofagranular material is the angle of the steepest slope of the right circular cone with horizontal base that the granular material may form under gravity. The lateral pres sure at a point on the wall surface is the sum of the horizontal forces exerted by the grain on a square neighbor- hood of this point, of unit area. 10. 11. 12. 36 The frictional stress at a point on the wall surface is the sum of the vertical forces exerted by the grain (due to lateral pressure at this point) on a square neighborhood of this point, of unit area. The wall friction parameter atapoint on the wall surface at a given instant, is the ratio between the frictional stress and the lateral pressure at this point, at the particular instant. The vertical pressure atapoint onahorizontal cross- section of the grain is the sum of the vertical downward forces exerted by the grain on a square neighborhood of this point, of unit area. The bottom pr e 5 sure is the vertical pressure on the bottom of the bin. B. Definition of a Deep Bin The wall friction effect under static conditions characterizes the main difference between deep grain bins and shallow grain bins, or containers of materials other than granular materials. It is therefore desirable to use the wall friction effect as a measure of the "deepness" of a bin. For this purpose Eq. (I. A. 4) may be employed. The asympto- tical behavior of pressures is due to the exponential function §(y) = exp {- (fikR—O) y} (IV. E. 1) where05§(y)Sl for cozyzo. 37 Define: A bin is de e E if its total depth H satisfies the inequality HzH , whereH d d is a depth such that §(Hd) = 0. 05. In other words, a bin can be considered to be deep if its depth is sufficiently large such that at least 95 percent of the wall friction effect is developed at some level above the bottom; and any load below this level is essentially carried by the walls alone, due to friction. H To solve for the ratio —, insert H R in (IV. B. l) d pk _ 0 _ §(Hd) - exp {-(T) Hd} - 0. 05 and thereby Hd 1 3- — . IV. B. 2 R “1‘0 ( ) For example, in terms of the ratio Hd/D for circular cylindrical bins, one obtains Hd '13" _3 1 -4 . (IV.B. 3) pk 0 Typical values of k0 and u for grain, are respectively 0. 333 and 0. 450, and therefore H 41:2. 4 D 1 0.450 x 0.333 ‘ 5. so that for practical considerations, a circular cylindrical grain bin is deep if (IV. B. 4) U131: 1V U1 38 Nate—s. This definition restricts the class of deep bins to include only considerably tall structures. It should be noted that it was the modern taller and larger structures that introduced the problems of excessive pressures. More lenient definitions may similarly be con- structed to include shallower bins in the class of deep bins. Silage silos apparently need not be nearly as tall to satisfy above definition. Typical values of k0 and u for silage are respectively 0. 575 and 0. 650, and therefore 2 2 (IV. B. 5) DIE for deep circular cylindrical silage silos. C. Classification of Materials Granular materials are sometimes referred to as semi-fluids or bulk-solids, probably to suggest their intermediate properties. It seems that granular materials, subject to gravity pressures, indeed possess properties whose ranges lie between those of fluids and mono- lytic solids. Table 2 presents a classification of materials with respect to gravity pressures in storage containers. From this classification there arises a rather compact formulative way to designate materials by their gravity-pressure properties. For example, a material satis- fying Eq. (I. A. l) for lateral pressure, is designated by I-Al-Bl-Cl-Dl-El, i. e. water. This designation is called a material-type. More than one material is associated with a material-type. In general, higher order 39 ”Emacs? find I > Godmosoo .. o omomou mo mamas Housed: u o Hot/3 .33 um oudmmoum H.232 can mmonum 12830:“ 553qu 03mm u x 1.53 ind um monummoum Hogan»; was Hagen—SH Googon 03mg .. x "0.353 O >n> DVD DVD 0H0 ONO wawo awAwo OHK wawo OHK awxwo Ewe .umcoo .umnoo Jmaoo Jmaoo O mmDA<> .mO HOZH mamwnoume Hmfidnmum ozm ofioo HHH 3.3.838 Hoadcmnm min 0:00 Icon HH omega H 1Hfi>mHm on «common £53 mfimwuouma mo GowumomfimmmHO .m 283. 40 characters in a material-type indicate more complex materials. Table 3 presents some examples of material-types. Table 3. Examples of material-types of materials stored in bins and containers. Material-type Material a I-Al-Bl-Cl-Dl-El water, oil b II-Al-Bl-Cl-Dl-El dry sand, aggregate, Janssen's grain c II-A2-B2-C2-D l-E2 grain d III-A2-B2-C2-Dl-El Portland cement, flour e III-A2-B2-C2-D2-E2 silage f IV-Al-Bl-Cl-DO-El conglomerated grain, hardened (*) concrete (*I Materials may change material-types due to chemical, biological, organic and environmental factors. An extreme example is concrete. When poured into the forms it is a mixture of components (water, sand, aggregate and cement) belonging to material types (a), (b), (b), (d), respectively. Nevertheless, the first component is qualitatively dominant and poured concrete is closest to material-type (a). However, after a comparatively short period, the concrete hardens and gradually becomes of the material-type (f). Similar changes, although less critical, may occur in grain, flour or silage, under certain conditions. The concerns of this study are grain-types: II-AZ-BZ-CZ-Dl-EZ and simpler types. Specific grain-types will be specified when essential. V. THE ONE-DIMENSIONAL PROBLEM A. Analytic Solution 1. Derivation of the governing differential equation In the one-dimensional problem all variables are considered functions of the depth coordinate - y - alone. For simplicity, the infinite bin case is dealt with. Similar treatment can be applied to any other bin with a symmetrical cross-section, such as: a circle, a square, or a regular polygon. Consider a horizontal differential slice of height dy at depth y in a unit segment of an infinite bin, having width 2p. The vertical pressure acting on the top surface of the prism is q, the lateral pressure exerted by the prism of grain on the walls is p, and the re- sulting frictional stress is 8. Let dq denote the increase in vertical pressure along the interval (y, y + dy). Then the increase in the resultant of the vertical forces acting on the prism in a downward direction, along this interval, is Adq =yAdy- sLdy (V.A.l) where A - cross-sectional area L - that portion of the cross-sectional perimeter which is bounded by the walls y - unit weight of grain. 41 42 Rearrangement of (V. A. 1) yields = . V. A. 2 dy A v I ) Define k : E (V. A. 3) q s 1. — — (V. A. 4) P A R — '1: (V. A. 5) and thereby s=kkq. (V.A.6) Insert (V. A. 6) into (V. A. 2) to obtain 14 + .1. = v A dy kaq Y ( . . 7) and assume that R, k, x, y are constants or functions of y alone. Denote B=-E'. '(V.A.8) Then, the most general equation is of the form d 33 + mm = y(y). (V.A. 9) Eq. (V. A. 9) is an ordinary linear differential equation of the first order, where y and q are the independent and dependent variables respectively, and 8 and y are functions of y alone, or constants. 43 2. Boundary conditions and solutions The general solution of (V.A.9) is known to be (see Appendix A.l) q = C exphffi dy) + exphffi dy) v exptfs dy) dy (v. A. 10) where C is an arbitrary constant. In order to determine C uniquely, at least one boundary condition must be specified. In this case (a one- dimensional problem), it is necessary and sufficient to specify at the initial point y = 0, the following initial conditions: 0 without surcharge q(0) ={ (V. A.ll) 1/2 Yo h with triangular surcharge of height h; Y0 is constant. a. Without surcharge. Imposing the first condition on (V. A.10) one obtains C = — y eprfi dy) dyly=o (V. A. 12) and by (V. A. 3), (V. A. 10) and (V. A. 12), a complete solution for lateral pressure is of the form p = k eXPI-ffi dy) Y exflffi dy) dy - v eXP(]5 dy) dyly:o . (V. A. 13) b. With surcharge. Imposing the second condition on (V.A. 10) one obtains C = {1/2 Yo h exp(jfl dy) - y exp(f5 dy) dY>Iy=O (V. A. 14) and a complete solution is of the form 44 p :k exp(-v/'fl dy) Y exp(jS dy) dy +{l/2 Yo h exp(‘[fl dy) - - v eprffidy) dy>|y=o . (v. A. 15) Eggs. By the broad definition of L and A the solution is, in fact, applicable not only to the infinite bin case, but to any symmetrical bin. Since case (a) is less complicated, unless otherwise indicated the equa- tion (V. A. 13) will be hereinafter referred to as the analytic solution. (V. A. 13) affords a complete solution of the one-dimensional prob- lem, provided 8 and y are known and can be expressed in rather simple forms. It will be shown later (see section V. D.) that plausible assump- tions can be made regarding the functional behavior and form of 5 and y. B. Algebraic (Numerical) Solution 1. Construction of the fundamental model Before any attempt is made to investigate the applicability of the analytic solution, it seems desirable to supply an alternative algebraic (numerical) solution that is free from any analytic difficulties that may normally be associated with a complicated expression such as (V. A. 13). Another purpose of seeking an algebraic solution is to make practical calculations more readily applicable to programming and processing by digital electronic computers. The construction of the mathematical model of the grain pressure mechanism is essentially motivated by the discussion presented in 45 section III. B. The reader is referred to that section for fuller comprehension. Consider, again, a unit segment of the infinite bin having width and height 2p and H, respectively. For reasons of symmetry, treat one-half only. Divide H into u slices each of height Ay. Starting from the top, associate with each slice an ordinal number i, (l S i S u, i is an integer), and local values for various parameters, such as: unit weight y,, wall-friction parameter k, and ratio parameter K,. . 1 1 1 NOMENCLATURE: Yi- A - Hon I average unit weight of the i-th slice cross-sectional area, A = 1- p = p weight of the i-th slice, Gi = yiAAy resultant of the downward vertical forces acting on the i-th slice resultant of the lateral forces acting on the i-th slice resultant of the vertical frictional forces exerted by the i-th slice on the wall due to F)1 ratio between F_ and 0,, K, = F,/Q. 1 1 1 1 1 wall friction parameter at the i-th slice (ratio between Si and F,, 1 x. = S./F.) 1 1_ 1 average vertical pressure on a horizontal cross-section passing through the center of gravity of the i-th slice average lateral pressure exerted by the i-th slice on the wall average frictional stress at the wall, exerted by the i-th slice 46 k - ratio between pi and qi, ki = pi/qi T - accumulated load per unit length carried by the wall at the i-th slice p - hydraulic radius Isolate slice #1 and examine the forces acting upon it: (i) weight of slice - Gl (downwards) (ii) resultant of lateral forces - F1 = K1 G1 (inwards) (iii) resultant of frictional forces - S1 = 1.1 K1 G1 (upwards at the wall) The residual resultant of downward forces is therefore R1=G1(l- lel). (V.B.l) R is that portion of G that is transmitted to slice #2, the remaining 1 1 portion being carried by the wall. The same process can be applied to each slice as follows: # slice G, K, X, F,’ S,‘ R. __ __l __1 __1 _.1. .._l ._1 K .. 1 G1 1 )‘l K1G1 XIKIGI 01(1 XlKl) 2 62 K2 X2 KZGZ XZKZGZ Gan-XZKZ) i G. K. A. K. G. X. K. G. G. (l-X. K.) 1 1 1 1 1 1 1 1 I 1 1 u G K A K G X K G G (l-X K ) u u u u u u u u u u u where F'i and S'i are those portions of Fi and 51’ respectively, due to G. 1 alone. 47 A portion of the weight of each slice is transmitted to the wall by friction. The residual portion is transmitted downwards to the succes- sive slice. After Ri is transmitted to the i+l slice, it is again divided into two portions: one is transmitted to the wall and the residual is transmitted to the i+ 2 slice, and so on. (*) Thus, the complete process is illustrated as follows: Slice # Vertical Downward Forces 1. Slice #1 - QL ‘-' G1 2. Slice #2 - G2 Q : from slice #1 - 2 G1(l - lel) 3. Slice #3 - from slice #2 - Q3 = :-(1 XZK from slice #1 - G12 (1-112K122)(1-x K 2) r 1. Slice #i - Gi from slice #i-l -l-H1i-ilK I) from slice #1-2- Q = G:_12(l-).:_2K1_2)(1->.i_1Ki_1) fr om slice # l - (*) Note: 1. LG1(1-1)\ K1)(l-).2KZ)...(1- K 1.14 i-l) The derivation is made for a case without surcharge. Surcharge may be simply added to G 48 The sum of the vertical forces acting (downwards) on the n-th slice is therefore : A - - - Qn YPIYn+Yn_1I1 xn-lKn-l) + Vn-2(1 >‘n-ZKn-ZN1 xn-lKn-l) +' ' ' . + y1(l-k1K1)(l-XZK2) . . . (l-kn_1Kn_1)]. (V. B. 2) 2. A search for a compact notation The series involved in (V. B. 2) characterizes the model sought. It is desirable to find a more compact form for this series so that it can be handled and analyzed conveniently. The series can be represented as a series of determinants as follows: l 0 0 l 0 + 0 - 0 + YnIlI-+\(n-l Yn-2 (1 )‘n-1 n-1) O (l-xn-lKn-l) 0 0 (l-knn2 n-2)| l 1"kn-lKn-l . 1-)\n-2Kn-2 . + Y1 I (V. B. 3) I l-lel 49 The determinants involved in (V. B. 3) can be recognized as being associated with the sequence of n submatrices produced from the n-by-n diagonal matrix _ I I ‘1 1 . o o 3 ___I ‘ 5 I _ ‘ o ' o 1 kn-1 n-l : i q : o o 14. K I \ by successively taking the l-by-l upper left submatrix, 2-by-2 upper left submatrix, and so on. Let the expression "det Subi A" denote "the determinant of the i-th submatrix of an n-by-n arbitrary diagonal matrix A, produced by deleting from A all rows and columns whose ordinal number is larger than i. " Then the series (V. B. 3) can be written in the form of a compact sum as follows: n ZY i det Subi[I - kK] (V. B. 4) i=1 ' n+1 where I is an n-by-n identity matrix, and AK is a diagonal matrix of the form XK =D1agonal [0, xn_1Kn_1, xn_2Kn_2, . . . , lel]. (V. B. 5) Thus (V. B. 2) becomes 50 n = A E Qn y p. Yn -i det Subi [I - XK] (V. B. 6) i=1 +1 and other quantities are readily obtainable in terms of (V. B. 6) as follows: (i) Average vertical pressure at the n-th slice :A Z Y Vn det Subi [I - XK]. (V. B. 7) i=1 +l-l (ii) Average lateral pressure exerted on the wall at the n-th slice Knp pn 3 kn qn; kn = Z? (V. B. 8) n = K 2 - . . . np i=lYn+1-i det subi [1 1K] (v B 9) (iii) Average frictional stress at the wall at the n-th slice Sn : kn pn n 2 - . . . )‘n Kn pi:1 Yn+l det Subi [I XK] (V B 10) (iv) Accumulated load per unit length carried by the wall at the n-th slice n T = E s, n jzlj n j = E LK. >3 y.+1 idet Subi [I - m]. (V. 13.11) j=1J 31:1J 51 C. Investigation and Comparison of Solutions 1. Constant coefficients Start out with the simplest case where all parameters are con- stant. Let x = p, k, y, and p be constants. Also, let all the non-zero entries of the diagonal matrix ).K be equal, such that (kK)ii = pK, where k . K = FAY 1s constant. All manipulations are with respect to lateral pressure only. a. Analytic solution. Lateral pressure p is given by (V. A. 13). Substitute the constant coefficients into (V. A. 13) to obtain p = k eXPI-gfdy) v exp(bfdy) dy - %XP(bjdy) dylyzoI J k where b = E— is a constant. Then p = k)! eXPI-b YIIgl eXPIb Y) - 11;] =k—bx[1 - exp(-bv)] _ Y —%—[1 - exp {-(-p&)y}]. (V.C.l) (V. C. 1) turns out to be identical with the well-known Janssen's solution (compare Eq. (I. A. 4) ). b.. Algebraic solution. From (V. B. 9), the lateral pressure pn can be written n = Z) pn Knp _ v det Subi [I - XK]. (V. C. 2) 1:1 n+l-i Substitute the constant parameters in (V. C. 2) to obtain 52 pn = KPYI1 + (1'HK) +(1-pK)2+. . . +(l-|J.K)n-1]. The series included in the square brackets is a geometric series with a ratio factor smaller than 1. By summation of the geometric series n =va I1 ' (l-ILKII Pn 1 - (I’ll-K1 = I? [1 - (l-pK)n]. (V. c. 3) Using (V. B. 8) sz-l, (v.c.4) p n and (V. C. 3) finally becomes a g 1 n Pu: H [1-{1-(p)y‘£} 1. (v.c.5) c. Comparison to Janssen's solution. Compare (V. C. l) and (V. C. 5) to find out that except for the expressions exp{-(Ip$)y}and and {l - (1%))“:7} , the equations above are identical. However, using a limiting process where n—goo (Ay——)o), it is evident that . n 11m {1+_:_l(_ 1%”) =exp{-I%)Y}. n-—>oo (Av-9 0) It is thus shown that (V. C. l) and (V. C. 5) are practically identical (’1‘) for sufficiently small Ay and that (V. C. l) is identical with Janssen's solution. ”I For practical calculations of deep bins, Ay = 1 ft. to l yd. may be considered ”sufficiently small. ” 53 2. Variable coefficients a. Analytic solution. Explicit integration of the analytic solution (V. A. 13) is very sensitive to the nature of the functions y(y) and 8(y), especially the latter. Unfortunately, fly) is bound to be a very com- plicated function and therefore an analytic solution is seldom practical. However, considerations explained later (see section V. D. 2), suggest for the qualitative behavior of fly), under certain conditions, the nature of some decreasing hyperbolic function. The simplest form to satisfy this requirement is a rational function of the type (V. C. 6) x where r) = F, k and p are constant and so are B and C. This implies that (V. C. 7) where B and C are some arbitrary constants. The density-function is naturally an increasing function, and for simplicity it is chosen here to be a linear function (see section V. D. 1). Therefore \(Iy) = v0 + 6y (v. c. 8) where Y0 and 6 are constants. Inserting (V. C. 6) and (V. C. 8) into (V. A. 13) and evaluating, one has to carry out the following manipulations: 54 (i) Findjmy) dy = h—fg—dey B a): = 3&— ln(1+Cy)(). (ii) Find g(y) = y(y) exp(ffl(y) dy) dy = (yo + by) exp[N ln(l +C y)]dy where N = Hg is a constant. Expanding and integrating by parts, one finally obtains N+1 _(1+Cy) 6(1+Cy) g(y)“ C(N+1) [”0 + 6”) " C(N+2)]' (iii) Find g(o) “ ——1——[Yo - 57%;; —C(N+1) 1' Using the results (i), (ii) and (iii) on (V. A. 13), the solution is found to be B y N+l} 5 N+1 a {1-(1+cy) p(y)=—-3—[1- —(—y+—- ]. (v.c.9) C(N +1) (1+Cy)N+l yo N+2 yo C(N+2)(1+Cy)N+1 b. Algebraic solution. Use the same [3 and y functions. In terms of kn, (V. B. 9) becomes 11 = A 2 — A pn kn y 1:1 Yn+l-i det Subi [I nk y] (V. C. 10) where y . =y +6[n'+l-i]Ay' (n'=n-l) n+l-i o ' 2 >1: ( )Constants of integration may be deleted throughout this manipulation. 55 x . r) = 7)- IS a constant I is an n-by-n identity matrix B n = m, and k 13 the diagonal matrix k o o . . . 0 9 B o l+C(n'-1)Ay B k = O l+C(n'-2)Ay B o o u 1+c 1/2 A); c. Comparison. It seems that the only way to compare between Eq. (V. C. 9) and Eq. (V. C. 10) is by observing their graphs. Therefore, a numerical example was worked out, and the results are illustrated in Fig. 6. The algebraic solution was calculated first for large increments of Ay = 10'. Twelve discrete points of solution were obtained and joined by straight lines. When Ay was taken smaller, more points were produced, and the algebraic curve tended to approach the analytic curve. For suf- ficiently small Ay the curves practically coincided. This behavior, and the result of section V. C. 1. c. suggested, in general, the following relationship: Eq. (V. A. 13) 5 lim Eq. (V. B. 9). (V. C. 11) n—)oo (Ay—m) 56 Lateral Pressure (lb./sq. ft. ) 100 200 . 300 400 \\q-\\ h \\ \ \ \ \ \ \ Ay=3' \\1 \ Algebraic 20 A Ay=10' 40 Numerical Data: H = 120 ft. A p = 5ft. 2‘3, 60 — 1 = 0.425 E c) = 30° 0 G k = 0.333 I ° I yo = 451b./cu. ft. \ 5 = 0. 06 1b. /cu. ft. /ft. 1‘ 80 {—- . c = 0.03 \ Ay= 10ft.; 3ft. 1| I I 100 I I I l l I 120 Figure 6. Comparison between the analytic and algebraic solutions with variable coefficients. 57 3. Relationship of analytic solution to the Reimberts' semi-empirical solution Equation (V. C. 9), as clumsy as it may look, turns out to be a generalization of the Reimberts' semi-empirical solution. This can be readily shown if the appropriate choice of coefficients and constants is made. First, let y (y) = Yo be a constant (i. e. 6 = 0), which reduces (V. C. 9) to the form BY pIy) 1713,3130 - (1 + 030“”) ]. (v. C. 12) Let B = 2 k0, where k0 is the Rankine-Keenen-Caquot coefficient given by (I. A. 3). Let x :12 u, where p. is the coefficient of grain-wall friction, pk and let C = —p-9. Then 1 N_L.E:2p. Zko ‘1 p C p P-ko ' (p) Insert the values of N, B, and C into (V. C. 12) which thus reduces to PY o Pko '2 My) ~T- [1 - {1 + (—p-)y} ]. (V.C.13) It is now observed that, except for a difference in notation, Eq. (V. C. 13) is precisely identical to Reimberts' solution reported in their reference (p. 35, Eq. (11)) in the form __ z -2 pz _ pmax[l-( A +1) ] 58 where z - depth below the edge of the vertical wall p - lateral pressure exerted on the wall by the grain at depth 2 2 —-5-D—' where pmax 4 tg ¢' ’ 6 - density of grain D - interior diameter of cylindrical bin 4)‘ - angle of friction of grain on wall A = D - h is a constant 4 t 6' t 2 (3- - i 3 8 8 4 2 h - height of a conical surcharge (zero in this comparison) (I) - angle of internal friction of grain t2(1 $)=l-sin$ =k g 4 2 1+mn¢ d 4. Discussion The Reimberts' solution was derived on the basis of both theoretical considerations and empirical data achieved through many years of exten- sive experimental investigations of dynamic effects in grain bins. The derivation is different from the treatment of this study, which is strictly theoretical. However, it is most interesting to note that the present analysis supplies a fundamental answer to the understanding of the Reimberts' method and results, probably not known before. Assuming that the Reimberts' solution is in good agreement with their experimental findings, the result (V. C. 13) determines the exact 59 behavior of the ratio-function and wall-friction parameter during those experiments. The essence of the Reimberts' method can be summarized as follows: (i) the density-function Yo is a constant .. . . 1 . (11) the wall-friction parameter 2 p IS a constant (iii) the ratio-function is a decreasing hyperbolic rational function of the form (iv) the lateral pressure is calculated according to the general analytic solution in one-dimension, Eq. (V. A. 13), using the above factors as coefficients. These conclusions are unexpectedly sound. Unfortunately, it was next to impossible to achieve explicit expressions for k(y) and X other- wise; and it is the generalized solution that made it possible to gain an insight into some of the interrelationships of factors involved in this complex system. Furthermore, it was established that both Janssen's solution and the Reimberts' solution--hitherto thought of as independent-- are not independent but are special cases of the same prototypical solution-model. This eventually leads to the idea that there exist more solutions that can be derived from this prototype. Adherence to this idea yields the following table: 60 .N .< xmpcommda mom At :3 2 .N :3 .sm 3883 .8662 a .xI u a a: u x a as > u > ~m4o-~o-~m-~<-: s Laue: am 3845s $6.28 3? u x 3: n .A. 3 r n r ~m4.o-~o-~m-~<-= .m m s I + 5 .o .3 am Leia H m o _mtmnfism o n Ex 1 m. u A > u r Hm-5-ao;m-~<-: .4 x N $1.1: am 0 0 9:34.82. 3 u x a n .x r u > 54640392-: .m 34.: .wm o o 8333 £3... a u x o u a r u r Hm-5-ao;m;<-: .N :31: .wm o 033.8%: H” x o u .x > u > 5-5-5-5434; .L no» coca—pom GOMHUSMIOBMH IMMMWWMHW WNW—WWW“ L2 .4 .3 $82 I mnoflocah oflmwuouomHmau oghuadiopmz .mcoflgoE .350 mo mgr—gong o no GomudHOm 33394 .w 3an 61 So far, the discussion was related to the analytic solution only. Nevertheless, the same results could be achieved relying on the algebraic solution instead. Relationship (V. C. 11) already suggested that the analytic and algebraic solutions are identical. It is important to clarify this point. Both analytic and algebraic solutions were motivated by precisely the same physical mechanism and basic assumptions, although they were mathematically constructed in different manners. It is intuitively and practically justifiable to suppose a relationship such as (V. C. 11); however, it should be admitted that by no means can this be considered a mathematical proof. What was shown up to this point is that Analytic Physical Algebraic corres onds t corres onds to p 07‘ I p \ Model Model \4 Phenomenon ‘ 7 There is no mathematical evidence that these relationships are necessarily transitive. Anyhow, it is still suggested to accept intuitively the identity. A rigorous mathematical proof is apparently not trivial and is beyond the scope of the present study. D. The Characteristic Functions The characteristic functions are: l. density-function Y (Y) 2. ratio-function k (Y) 3. friction-function X (y). 62 Their knowledge is essential to determine the L3 and v functions necessary for the solutions (V. A. 13) (see notes in section V. A. 2. b.) and (V. C. 2). 1. The density-function The appar ent den siiy of grain stored in bins is defined as the weight of grain bounded by a unit volume. Most types of grain have rather definite density values, although the density value of a particular type of grain may vary within a range of some i 10 percent. Variations in grain density may be due to factors such as: moisture content, compactness (looseness), pressure, vibrations, settlements and so on. Under normal storage conditions, density is found to be bounded between some minimum and maximum values which differ by 10 to 20 percent. If not considered merely as a constant, the one-dimensional density- function is expected to be: (i) continuous and smooth (ii) monotonically increasing with depth of bin (iii) strictly positive (*) (iv) bounded betweeny , =v and V mm 0. max The simplest function to satisfy these requirements is a linear function of the form (*) v and y are experimental values. 0 max 63 lyo+6y for OSySH y(y) = (V.D. 1) f > H Ymax or y d where 6 is a constant defined by If H is considerably larger than H it is preferable to choose a d, curved function. It seems reasonable to expect that increase in density versus depth is mainly proportional to vertical pressures. Accordingly, v(y) is assumed to increase exponentially, namely uk My) = vo+ (Vmax- v0) [1 - exp {-(To-W} ]. (V.D. 2) Notes. For most practical cases in grain bins it is sufficient to assume that the density-function is a constant = Y = - (y + v ), or ave 2 max 0 a linear function such as (V. D. 1). Equation (V. D. 2) seems to satisfy well the density-function in silage silos. 2. The ratio-function The one-dimensional ratio - func ti on k(y) is defined as the ratio of the lateral pressure to the vertical pressure at any depth y, namely k(y) =P—(fln (V.D. 3) 64 k(y) is essentially an empirical factor. In order to investigate the nature of k(y), it is necessary to rely on experimental findings reported from various sources. It seems fair to summarize these findings as follows: (i) Under static conditions, actual lateral pressures closely agree with Janssen's curves. (ii) Under dynamic conditions, actual lateral pressures may exceed Janssen's curves (by tens to hundreds percents) until they achieve a maximum at some depth above the bottom, and then decrease rapidly. The behavior is experimentally not clear below that region (see Figures land 2). a. Static conditions. In addition to the general definition given in section I. A. , static conditions are the case where the walls do not yield appreciably under the action of the lateral pressures. Janssen (1895) who first introduced the ratio-factor, assumed that it is an empirical constant. Koenen (1896) suggested to use the theoretical "coefficient of earth pressure at rest” to represent this constant. Thus Rankine's coefficient was borrowed for use, namely _1-sin¢ ko_l+sin<)> (v. D. 4) where q) is the angle of internal friction of grain, treated as an empirical constant. The above form of k0 is adopted in most calculation methods 65 and is found to be sufficient for the case of static pressures. It is therefore proposed to leave it so in the present analysis. However, there arise some important notes. By (V. D. 4), k0 is a function of 4), the angle of internal friction of grain which is defined: "The angle whose tangent equals the ratio between the shearing resistance per unit area to the corresponding normal stress in a non-cohesive grain. " <1) is actually a measure of the stability of the grain. This factor is frequently confused with the angle of repose 9 which indeed often possesses the same value and also serves as a measure of stability. The following quotations reflect this confusion: Ketchum (1919): ". . . The angle of internal friction is the angle whose tangent is the coefficient of internal friction of the particles upon one another, and is nearly always larger than the angle at which the material will stand if poured in a pile on a level floor. The angle of internal friction is commonly referred to as the angle of repose. ” Terzaghi (1941 and 1943): ". . . The angle of repose is equal to the angle between the horizontal and the slope of a heap of soil produced by dumping the soil from some elevation. . . . For perfectly clean and dry sand or gravel, the angle of repose is fairly independent of the height of the heap and the method of dump- ing, and is approximately equal to the angle of internal friction of the sand in its loosest state . . . . " "Early investigators of soil problems generally assumed that the angle of internal friction of sand is identical with the angle of repose. However, laboratory experiments have shown that the angle of internal friction of sand depends to a large extend on the initial density. In contrast to the angle of internal friction, the angle of repose of dry sand has a fairly constant value. It is always approximately equal to the angle of internal friction of the sand in its loosest state. " Huntington (1957): ". . . For many years it was an almost universal practice to compute the earth pressure against retaining wall on the assumptions that the soil was cohesionless and that the value of 66 angle of internal friction could be considered equivalent to the angle of repose. The common form of Coulomb's earth pressure theory, the theory of Rankine, and the various formulas, pro- cedures, and constructions--such as those of Poncelet, Culmann, and Rebhann as commonly used--are based upon cohe sionless soil. The assumption that the angle of repose is a satisfactory measure of the shearing resistance is untenable, as pointed out by Terzaghi. For cohesive soils, the angle of repose has no physical meaning; for cohesionless soils the value of the angle of internal friction should be preferred. " Brandes (1961): ". . . It should be noted that the angle of internal friction is often called, erroneously, the angle of repose of the filling and thus is confused with the angle of natural slope. The angle of internal friction of a material without cohesion is approxi- mately equal to its angle of repose or natural slope; for most materials, although not all, it is slightly larger than the angle of natural slope. " It is thus understood that ¢ is not a constant and therefore k0 is not. Whenever 4) is taken as a constant, it becomes practically the same as the angle of repose. This practice is however always on the safe side, since the resulting k0 is larger than it would have come out otherwise. Therefore —l_'_s_in_9 omax—l+sin9° (V'D'S) It is worth-while to investigate the influence of the variable 4>(y) on k0. First, establish an approximation to the function <1) = ¢(y). <1) is primarily dependent on the density (and vertical pressure). If the rela- tionship may be assumed to be linear (proportional), then, qualitatively, (*) ¢(y) behaves like v (y). By the use of (V. D. 2), (My) can be written in the form * ( )This was, in fact, established also by the experimental findings of Platonow and Kowtun (1959) from full-scaled structures. 67 ¢<)-¢ +(¢ -¢)[1-ex GEE—338)] (VD6) where (1)0 - angle of repose, 8 ¢ = a4) , where a is an empirical coefficient depending, among max 0 other factors, on H; i. e. , a = 2 approximately in deep grain bins, according to the findings of Platonow and Kowtun (1959). is defined by Eq. (V. D. 5). 0 max Then, substituting (V. D. 6) in (V. D. 4), k0(y) is finally obtained in the form - ' G 1 - sin [9(2- exp<-%fi+::: JV)” _ ' 6 1 ”mm - exp{- 215:: 9;.” k'o(y) = (V. D. 7) and ko(y) is thereby expressible as a function of y, involving 9, H and p as parameters. The behavior of the function ko(y) may be seen by using the trigonometric identity = tan (—-%). (v.13. 8) Let (E- - 22) = w and examine tan (0. By (V. D. 6), g is an increasing function versus y, and therefore (I! is a decreasing one. The tangent function of the decreasing argument (0, where 0 < ((1 < E, is decreasing versus y and its square is even more so. It then turns out that k0(y) 68 is a decreasing function of y, convex toward the y-axis. Such be- havior can be well represented by the appropriate choice of a hyperbola such as the type dealt with in section V. C. 2. b. Dynamic conditions, preliminary investggition. It is generally accepted that dynamic effects occur during charging and especially discharging. The discussion below is restricted to dis- charging alone. It cannot now be specified in detail when, how or why dynamic effects actually occur. Nevertheless, some intuitive guesses may prove suggestive. Suppose that as a result of a discharging operation, the walls de- flect to the extent that the contact between grain and walls is reduced to a minimum. At this particular instant two main phenomena may occur: (i) Since the friction effect approaches zero, vertical pressures-- and thereby lateral pressures as well--rapidly increase. If the deflection of the walls is sufficiently large, so that the system can be looked upon as if the walls were completely removed (for a moment), the column of grain tends to collapse and transform into the characteristic pile (see the discussion in section VI. A. 1). (ii) At the same instant the walls, being released from pressure, tend to return. The assumption is that maximum lateral 69 pressures occur when the returning walls and the nearly- collapsing grain meet. Hereinafter, “dynamic conditions" will be regarded as any conditions under which the two following requirements are satisfied: (i) Lateral pressures are, qualitatively at least, in agreement with the experimental findings described in the previous section. (ii) The walls deflect appreciably. Proceed to determine the character of the dynamic ratio-function. Eq. (V. C. 10) is a model representing the behavior of lateral pressures under any conditions. A typical empirical curve of dynamic lateral pressures is _n_o_t_ strictly increasing, but possesses an extremum point. Examine (V. C. 10) to find out which particular factors can be responsible for this fact, and in what manner. It is observed that kn alone can possibly cause a decrease in the lateral pressure curve. Suppose that p(y) has a maximum at the i-th slice. Then the only possible way that the lateral pressure curve decreases in the region immediately below the i-th slice is that ki > > ki+l > ki+2 > . . . , which means that k(y) must decrease from the i-th slice, where the rate of the decrease depends on the rate of the decrease of p(y). 70 The behavior of k(y) from the origin to the i-th slice is still questionable. However, two plausible assumptions may clarify the situation: (i) k(y) is continuous on (O,H) (ii) k(y) is monotonic on (O, H). Relying on assumption (ii), k(y) must be either a constant, an in- creasing or a decreasing function of y. The first possibility, although satisfying the requirements above, is too trivial for this case, and any- how is treated separately under the case of the static conditions. An increasing k(y) is unlikely in the light of both of the above assumptions and the already established decreasing nature of k(y) below the i-th slice. The third possibility thus remains, which satisfies both con- tinuity and monotony, as well as the experimental findings. No attempt has been made yet to determine the rate of the decrease or the shape of k(y). This can be done by analyzing the shapes of reliable empirical lateral pressure curves. Such an analysis was made and it was found that k(y) is convex toward the y-axis, rather than con- cave or a straight line. No preliminary investigation of k(y) can be made at a region in the neighborhood of the bottom, since the pertinent experimental find- ings are insufficient. The discussion above established roughly the general nature of 71 the dynamic ratio—function. Similar results, although relying on an entirely different approach, will be obtained in section VI. A. 4. a. 3. The friction-function The friction-function X(y) is defined as the ratio of the frictional stress at the wall to the lateral pressure, at any depth y, namely My) =—X-f;)y; . (V. D. 9) The coefficient of friction of the grain on the wall (1 is basically dependent on the type of the grain, the material of the wall, and the degree of smoothness (roughness) of the wall surface. Other factors such as: variations of the moisture content of the grain, and the motion of the grain, may affect the coefficient of friction. Nevertheless, this factor is normally considered as a constant and it seems convenient to leave it so. The wall-friction parameter k (see section IV. A.) was introduced in order to make allowance for variations in the friction effect, either as a function of the depth coordinate y, or as a constant with respect to y, but as a variable parameter depending on other factors. Therefore, X = Up (V. D. 10) where V is either (i) a constant, (ii) a function of y, or (iii) a variable, but independent of y. The flexibility in the definition of 12, allows to take into account many possible variations in the friction effect. 72 In any case, V is bounded between 0 and I, such that X is bounded between 0 and p. a. Static conditions. The friction effect attains its maximum influence under static conditions. Therefore, let V = 1 such that x=x =p. (V.D.ll) b, Dynamic conditions. Since V is bounded between 0 and 1, it may prove useful to express V in the form of a trigonometric function, i. e. the sine function. Let V (y) = sin cy (V. D. 12) where c is a parameter. This representation may be appropriate especially for a case where the wall vibrates under dynamic conditions. When the wall vibrates, it is likely to deflect in a form of a sine curve. The mode of the vibrations is not known (involved in c), but can be estimated for simple cases. It is assumed that the wall is much more flexible than the mass of the grain is, and that the mass of the grain cannot continuously take on the form of the deflecting wall. Therefore, the grain is in very loose contact with the wall, or in no contact at all, along the regions where the wall is con- vex outwards. Thus, define V (y) as follows: sin cy for sin cy > 0 V(Y) = . (V. D. 13) 0 for sin cy S 0 From the nature of the sine curve and definition (V. D. 13), it is evident 73 that regardless of the mode of the vibrations and (assuming that the deflection of the wall can be approximated by a simple sine curve), V(y) vanishes intermittently over one-half of its domain of definition; or, in other words, there exist segments along the height of the wall whose accumulative length is one-half of the total height, along which V(y) is zero. Since the friction effect is important as an accumulative factor, it is not necessary to find the exact mode of the deflections. The mode can be taken, for convenience, as the arbitrary interval Ay, according to which all the numerical calculations are carried out. This choice produces a V ( y) function that vanishes at every other interval, and varies between 0 and 1 according to a sine curve, at every other interval. For numerical calculations of sufficiently small Ay, V (y) approaches the form of a step-function. Therefore V (y) can be written as follows: ave. sin cy for sin cy > 0 V (y) = (V. D. 14) 0 for sin cy S 0 where 2 ave. sin cy = .1? . Therefore, under ”simple" dynamic conditions and a choice of suffi- ciently small Ay, V (y) may be chosen to be a step-function of the form 74 f? for i odd integer V(y) = (v. D. 15) o for i even integer where the index i is related to Eq. (V. B. 9), i. e. i = y/Ay. Accordingly the friction-function can be expressed as a step-function of the form 2 g u for i odd integer My) = ' (v. D. 16) 0 for i even integer c. Practical recommendations. Eq. (V. D. 16) may suggest to choose a constant friction-function which is the average of the intermittent 2 values 0 and EH’ namely 1 X: pwgp, =l|)—* The choice of constants smaller than ~1§p is justified for cases where the assumption of a simple sinusidal deflection is not valid. In any case of uncertainty it is advisable to choose smaller values for the parameter V, including the possibility of a zero value. When V = O, the friction effects are completely neglected and thus, the lateral and bottom pressures attain maximum values. Nevertheless, to calculate the vertical load on the walls, xmax = u should be used. In summarizing, there may exist three cases of significant difference (among many possible intermediate cases): (i) Static conditions. V is a'constant =1. The friction parameter attains its upper bound, X = )1. max (ii) ”Simple" dynamic conditions, V(y) is a step-function of the 75 form given by Eq. (V. D. 15); or, more practically, V can be taken as l . 1 an averaged constant, namely, V = 3 . . X = 3 p. (iii) Complex dynamic conditions. u = 0 .° . k = 0. For design purposes, the following is suggested: (i) For rigid structures with special discharging devices (such as a central perforated pipe) use (ii) For rigid structures without special discharging devices, or for semi-rigid structures with special discharging devices, use 1 x = — . 3 M (iii) For non-rigid structures with hazardous possibilities for dangerous dynamic effects, use X—9 0, or X = 0. Notes. The following are general examples of different types of grain bins that may occur in practice, with respect to the degree of ”rigidity” as used in the above recommendations. For more accurate determination of any specific design case, a complete structural analysis should be made. Rigid structures are structures such as clusters of monolytic reinforced concrete bins with rigid connections to a rigid slab foun- dation, based on a high-quality soil and with rigid connections at the tops. 76 Semi-rigid structures are single, exceptionally tall reinforced concrete bins; clusters of steel bins. Non-rigid structures are single, tall, steel bins with a non-rigid connection to the foundation, based on an inferior soil and subjected to frequent gusts. Proper discharging devices are such as those reported in the literature and tested in operation in many countries. However, the design and installation of such devices, do not necessarily guarantee that the dynamic effects will be eliminated. Therefore, a conservative choice of the values of the parameter V is encouraged. VI. RECOMMENDATIONS FOR FURTHER STUDY A. Grain-Pile Transformations The discussion presented in this section is by no means complete. Its primary aim is to indicate a possible method involving topological considerations, for the solution of the particular engineering problem discussed in the preceding sections. Since some of the ideas are un- common in this branch of engineering, further explanations and moti- vations will be introduced as the discussion proceeds. l. The associated characteristic pile It is a well known fact that a mass of grain supports itself in certain unique shapes (cone, pyramid, etc.) having constant slopes, under natural conditions. An accumulation of grain in a deep bin may be considered as a geometrical distortion of the "natural formation” (characteristic pile) which the same mass would have taken, if not constrained by the walls. Thus, a stored mass of grain in a deep bin has a potential tendency to transform into a rather simple predictable geometric figure. The pressure exerted on the bin walls may be regarded as a result of this intrinsic tendency. The geometry of a given bin and the properties of the grain stored in it are uniquely associated with some characteristic pile. For example, 77 78 a cylindrical bin is associated with a right circular cone, the dimen- sions of which are dependent on the dimensions of the circular cylinder, whereas its slope is dependent on the intrinsic properties of the grain. Factors such as the angle of internal friction, the natural angle of repose, and the moisture content, are apt to be of prime importance. Since the actual transformation never does take place, the angle designating the slope of the characteristic pile is called the imaginary angle of repose, and is denoted by 8i. Furthermore, each kernel can be associated with two unique locations: its location in the bin, and its (imaginary) location in the associated characteristic pile. This evidently leads to ideas related to set theory and top010gy, and from now on the discussion will be carried out more technically. 2. Example of a set-theoretic attack on the problem a. The initial set. Consider a two-dimensional vertical section of the infinite bin, and (from reasons of symmetry) treat one-half only. The H by p rectangle R thus formed, contains a finite number of ker- nels. The centers of gravity of the kernels will be called elements, or points. The collection of all points under their particular ordering in R, in a given discussion, is called the initial set A. The set A is a m finite set of discrete points and is assumed to be scattered uniformly * . . . . . ( )For 51mp11c1ty, the den51ty IS assumed to be a constant through- out this discussion. 79 over R. Form a p by q net (with the origin at the lower right-hand side, p and q in horizontal and vertical directions, respectively) over R, in a precise manner such that each a GA is uniquely associated with two integers i, j, where l S i S p, lSj S q and pq is the cardinal number of the set A denoted by card A. An example of such a net would be a net of uniform partitions in both directions consisting of intervals of length d, where d is the diameter of the spherical idealized kernels. Hereinafter this net is adopted. b. The image set. The associated characteristic half-pile is obviously a right triangle T. It is to be noted that certain regions of the triangle and rectangle overlap. Call the collection of the images of all a EA (included within the boundaries of T) the image set B. The boundaries of T can be established from two assumptions: ( ) 2:: (i) The equality of areas bounded by R and T (i. e. , card A = card B). (ii) The imaginary angle of repose equals the natural angle of repose. The height h and the base r of T are therefore h=V2Hptan 91 (VI.A.1) r =-\/2H pcotfiel (VI. A. 2) * ( )An area-preserving mapping, implied by assumption in footnote on page 78. 80 where H - height of R p - width of R 9 - natural angle of repose. Thus T is determined in terms of the bin geometry and the grain properties. Extend the net to include T too, such that (i) the lower right-hand side vertex is double-indexed by 1,1 (ii) the lower left-hand side vertex is double-indexed by r', 1 (iii) the upper vertex is double-indexed by l, h' where and h' = Daltr r'= .1; d c. Definitions. All terms used in this section are to be under- stood precisely as defined below, and not necessarily as interpreted in general usage. 1. Let aij 6A. A neighborhood of aij’ designated by N[aij]' is a non-empty subset X of A, such that (i) a,, 6X 1.] (ii) there exists an x 6X 3 x ¢ a.. mn mn 13 (iii) xmn€X©x'uv€X, where Ii - ulfli - m| Ii ‘ VISIJ ' nl" 2. A neighbor point of a point a.. 6A is a point x -7 a,,€N[a,,] ij mn ij ij such that 81 li-m'l=lor0 Ij- n|=1or0. 3. An inte rior point of A is a point ai.€A that has at least four neighbor s . 4. A boundaLy point of A is a point aijeA that has less than four neighbor s . 3. Grain-pile transformations Since card A = card B, there exists a non-empty finite setm (cardTIL = (pq)!) such that if Mam, then (i) M is a mapping, M: A—->B (ii) M is one-to-one (iii) M is onto. (_N_o_t_e_: A mapping M from a set A to a set B is said to be one-to- 9n_e and o_nt_o, if for every element belonging to A there exists a unique element belonging to B, and vice versa. ) Among all M 6m, seek a unique mapping Mo such that the par- ticular physical constraints are satisfied. The physical constraints are presented in the form of propositions as follows: PROPOSITION l: The physical ordering of the initial set A over R is preserved in some unique manner (to be discussed later, see sec- tion VI. A. 4) in the image set B over T. 82 PROPOSITION 2: Given any pair (aij, M0(aij) ) E MO, then it is possible to find a continuous curve Cij (of non-negative length) joining aij and its image point Mo(aij)’ Cijee, card 6 = pq, such that if all aij's physically travel along the corresponding Cij's when a physical * **) transformation actually occurs , some physical factor is a mini- mum with respect to any other set 6' (card 6' = pq) of continuous curves C',, each joining a,, and M (a..). ij ij 0 ij 4. Some intuitive intermediate results a. The decreasing character of the dynamic ratio-function. In line with proposition I, assume the following properties of the mapping M : o (i) Given a,, 6A, thenV N[M (a. .)]: 13 o 13 (ia) if aij is a boundary point, :3 a point t in every N[Mo(aij)] 3 M -1(t) is a neighbor of a... o 1.] (*)Such a physical transformation never does occur in reality during the life-time of the structure. However, when the walls yield to some sufficient extent, the transformation is "nearly” started. The forces acting on each kernel at this instant are tangent to the Cij curves at the initial points aij- From the point of view of these forces, it is merely a rather immaterial coincidence that at the successive instant the walls--rather than proceeding to yield--tend to return. This situation is typical to the dynamic effects during discharge (see section V. D. 2. b). (**) . _ In example, work or time. The latter ch01ce suggests a notable resemblance to the classical brachistochrone problem. This one, however, seems to be much more complicated and may be termed therefore as a “finite multi-brachistochrone" problem. 83 (ib) if a,j is an interior point, 3 points t1 and t2 in every 1 N[M (a..)] 3 M -1(t ) and M -1(t ) are distinct neighbors o 13 o l o 2 of a. ,. 1.] (ii) GI a non-empty proper subset ICMo 3 (aij' 1‘40“;in€I z) a.. = M (a..). 13 o 13 (iii) Boundary conditions for the image set: (iiia) (a1 Mo(a11) ) e I 1! (iiib) (a M (a )) 61 1h" 0 1h' (iiic) Mo(apq) = br'l' (iv) If (i) is not satisfied for some Mo(aij) E B, then such an image point is called a point of discontinuity, or a singular point of B. These properties almost determine the mapping Mo. The second proposition will be used to complete the determination of Mo. In light of proposition 2, G is a family of mutually non-intersecting catenary-type curves. Knowing Mo and (3, it is possible to find the tangent of the angle (aij) between the tangent line to Ci' at aij and the vertical direction. The ratio-function at aij is directly proportional to tan aij' The assumed properties of Mo imply the following: a: (i) 61(1)( ) is a non-empty proper subset of B which is in the immediate vicinity of the bottom and the right-hand boundary of R. (*)Ran ge of I. 84 (ii) There is a proper subset SC B consisting of points of dis- continuity of the approximate form as shown in Fig. 7, with a ”center of singularity" at the middle. (iii) The remainder of B is a region where the first property of M0 is satisfied. For illustrative purposes, reduce the two-dimensional problem to a one-dimensional one. Thus the set A is condensed into a set A' of equally- spaced points along the vertical center line of R (designated by R'), and the set B is similarly condensed into a set B' along the median of T drawn from the left-hand side vertex of T (designated by T'). The mapping Mo implies that the upper point in A' goes to the far left-hand side of T', the next point in A' goes to the next location on T' and so on, where the spacing between the points along the median is determined by the equality of the areas of the trapezoids associated with the points of B'. Let a' be any point in A' and let k(a') define the (one-dimensional) ratio-function at a'. Joining the pairs (a', Mo(a')) by a family of mutually non-interesting catenaries, it is observed that tan a' is monotonically decreasing versus the depth, until the point of intersection (s) between R' and T' is reached. At this point, tan a' is undefined. The decrease of tan a' is emphasized as the point 5 is approached from above, starting at a height of approximately 2 hS above the bottom (where hS is the height of the center of singularity 5, above the bottom). 85 H—p ml m 1 7)" ! l I | O | a.. “4| ‘3 S: m I ".3. . PEI R. 5' 13 I l i Image Set B p: F.— mad—’4 L__ [_— Figure 7. Geometrical representation of a two-dimensional grain-pile transformation of the infinite bin. 86 Since k(a') 0:: tan a', k(y) possesses the same qualitative be- havior as described for tan a' and this result is with agreement with the discussion in section V. D. 2. b. b. The location offithe center of singularity. By geometrical considerations r -H p tan 9 - tan 9 (a) for the 2 4 . . . . infinite bin hs = ( (VI. A. 3) 2 ' D 3 23—2 DZH tan 9 - 8 tan 9 (b) for a circular \ cylindrical bin No comparison between the infinite bin and practical cases can be made; however, a comparison with the circular cylindrical bin is illuminating. Consider a typical circular cylindrical grain bin: H H — : 0 D :— 3 3 9 = 30 tan 9 = O. 577. Insert these values in (VI. A. 3b) to obtain hs = 0.127 H. (VI. A. 4) That is, the lateral pressure decreases rapidly toward an elevation which is about 12% above the bottom. The maximum pressure occurs above this elevation along a region whose length is ll to 2 hs’ or 0.18H Z to O. 24 H which is altogether O. 30 H to O. 36 H, or one-third of the total 87 height above the bottom. A comparison with Figures 2 and 3 shows close agreement. B. Recommendations for Further Study 1. Theoretical a. General. Conduct theoretical studies of the mechanical behavior of grain, as well as stress distribution within granular masses, using applied mathematics, analytical mechanics, rheology, analogy with flow theories, etc. Develop the mathematical theory of grain-pile transformations, by use of tools and ideas from topology and the calculus of variations. b. Specific. Find relationships, so that, given factors from (i) bin geometry such as: (ia) total height (ib) cross-sectional shape and dimensions and (ii) grain properties such as: (iia) moisture content (iib) natural angle of repose, it is possible to obtain the associated Mo mapping, and thereby the dynamic ratio-function in one, two, or three dimensions. 88 2. Experimental a. General. Conduct laboratory investigations of geometric, mechanical and rheological properties of grain, by methods analogous to those of soil mechanics, rheology and related fields. Investigate the special properties of grain due to organic and biological factors, as affecting grain pressures in storage structures. b. Specific. Determine the imaginary angle of repose of various types of grain under various conditions, in model studies. Determine the density-function (one and two dimensions), ratio- function (one and two dimensions), and the friction-function (one dimension) of various types of grain in various full-scaled deep grain- bin systems, under static and dynamic conditions. APPENDIX A. 1 General Solution of the General Ordinary Linear Differential Ejuation of the First Order The most general ordinary linear differential equation of the first order is of the type (see Eq. (V. A. 9), section V. A. l) d 5% + My) q = V(Y) (A1. 1) where {3 and y are functions of y alone. Consider first of all the homo- geneous linear equation icil+§q=0. dy Its variables are separable, thus 93+ fldy = 0 and the solution is q = c exp(-ffidy) where c is a constant. Now substitute in the non-homogeneous equation, the expression q = w exp(-ffidy) in which w, a function of y, replaced the constant c. The equation becomes dw _ dy exp(-j13dy) - v, whence w = C + y exp(f)3 dy) dy. 89 90 The general solution is therefore q =C exp(-jf3dy) + exp(-;[8dy) Y exp(ffidy) dy. (A. l. 2) The expression eprfldy) involved in (A. l. 2) is sometimes called an "integrating factor. " The analytic integration of the integrating factor is not always possible, for usually depends on the nature of the function p(y). (Compare notes in section V. A. 2. b.) A. 2. AnOutline of the Two-dimensional Problem l. Analytic and aljebraic solutions The two-dimensional problem is merely a generalization of the one-dimensional case. The derivations and the results become more complicated and therefore lose some of their practical value. However, they bear much importance for the purpose of further investigations in this field. a. Analytic solution. In the two-dimensional problem all the variables are considered functions of either the depth coordinate y alone, or both y and the transverse coordinate x. By a similar development to that presented in section V. A. 2, a solution parallel to Eq. (V. A. 13) is obtained in the form p p(p.y) = %J{k(X. y) [MM-[Mm y) dy) (fHX. y) exp(jfi(X. y) dy) dy - - y(x, y) exp(fi3(x, y) dy) dylyzo] ] )dx (A. 2.1) 91 where (3(x. y) = “3” 1:" Y) . (A. 2. 2) b. Algebraic solution. Similarly to the development in section V. B. , a net is formed over one-half of the rectangular vertical section of the infinite bin. Starting from the axis of symmetry, v subdivisions, each of width Ax, are made along the x-direction; whereas the vertical par- tition remains the same as in the one-dimensional case. Each smaller rectangle belonging to the net is double-indexed by i and j where lSi S u, l Sj S v, and so are the corresponding values of the local parameters. Thus, the parameters are not represented as continuous functions, but as arrays of discrete points which, in turn, can be represented in the form of v-by-u matrices. Then the two-dimensional solution comes out in the form v n = 2 K 2 S - PM 9 ._ nj ._ Yn+l-i,j det “bin ”mj] j—l 1—1 or, in terms ofk . nJ v n A :A 2: k 2 s -—Y . . pnv yj-1 nj ,_1Yn+l-i,j det ubi[I Ax (Xk)j] (A 2 3) _ 1_ where 92 )i k n-l,j n-1,j O X k n-2,j n-2,j 0 ol (Xk). = J x,k. 13 1L 2. The characteristic functions a. The density-function. The behavior of the density-function along the y-direction was discussed in section V. D. l. The distribution of the density along the x-direction may be assumed to be uniform for most practical cases. However--to be exact--it is probable that the density is larger towards the center line of the bin. If the density distribution in the x-direction is assumed to behave parabolically, whereas the density distribution in the y-direction is according to Eq. (V. D. 2), then the resulting two-dimensional density-function is illus- trated in Figure 8. The derivation of an analytic expression for this function is too complicated and does not seem to contribute much. The geometric illustration is qualitative. A more precise diagram can be constructed in any particular case and the entries of the density-matrix can be measured directly from the diagram and used when evaluating Eq. (A. 2. 3). 93 b. The ratio-function. The behavior of the ratio-function along the y-direction was discussed in section V. D. 2. The behavior of the ratio-function along the x-direction is even more complicated. There is no experimental data, nor theoretical knowledge of this factor. If assumed constant with respect to the x-direction, a typical two-dimen- sional ratio-function is illustrated in Figure 9; otherwise, it may become as complicated as Figure 10 illustrates. c. The friction-function. The friction-function is always one- dimensional, a case treated in section V. D. 3. 94 noun: 8. 1'”va MHTY— FWTIG‘ OVER A VERTICAL SECTIOIW TIC WHITE BIN . rm 9. 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