"Shh-WIMW' " " ' “BR A R Y " Michigan Sm: University IllIlllmuzlyulllllmw"insulin This is to certify that the thesis entitled EDCPERD’IENI‘AL AND NUMERICAL TECHNIQUES RElA'I'ED TO 'I'I-IEMAXIMJMAIIUVABIEDEPI'HOFAPPIESINA BULK STORAGE presented by YmSEF SHAHABASI has been accepted towards fulfillment of the requirements for -—-—Ph-rDr——degl’ee in Agr—r-Eng-r— OVERDUE FINES: 25¢ per day per item RETURNING LIBRARY MTERIALS: Place in book return to remow charge from circulation records EXPERIMENTAL AND NUMERICAL TECHNIQUES RELATED TO THE MAXIMUM ALLOWABLE DEPTH OF APPLES IN A BULK STORAGE BY Yoosef Shahabasi A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Agricultural Engineering 1979 ABSTRACT EXPERIMENTAL AND NUMERICAL TECHNIQUES RELATED TO THE MAXIMUM ALLOWABLE DEPTH OF APPLES IN A BULK STORAGE By Yoosef Shahabasi The objective of this study was to develop experi- mental and analytical techniques to determine the maximum safe depth for apples in a bulk storage. The problem was perceived to consist of two components. One component involved the determination of the contact forces in a bulk bin while the other component related to the deter- mination of whether a specific loading would produce a bruise. The mechanical properties of Jonathan apples before storage and two periods during storage were deter- mined experimentally by compressing cylindrical specimens until a failure occurred. The modules of elasticity changed significantly between October 1 and November 15. A very small change occurred between November 15 and December 31, 1978. The respective averages were EOCT = 3279 Kpa, E = 2360 Kpa. The NOV = 2516 Kpa and EDEC Yoosef Shahabasi average maximum normal strain at failure was 0.14, 0.11 and 0.12 for three dates. The average normal stress at failure decreased from 444 Kpa on October 1 to 252 Kpa on November 15 and 235 Kpa on December 31. One hundred fifty apples were sampled on each date with four samples being removed from each apple. The distributions of the elastic modulus and failure strain were used in a computer model to predict bruising for a particular load. The model was based on the assumption that a bruise occurs when the maximum normal strain exceeds a specified value. The load which produced a bruise was converted to a depth by assuming a single column stack for the apples. Apple-to-apple contact was found to govern the allowable depth. The October 1 apples could be piled 5.14 meters without brusing but the November 15 and December 31 apples could be piled only about 1.8 meters. The sig- nificant decrease was attributed to the decrease in the modulus of elasticity between October 1 and November 15. As existing finite element type computer model which had been used to model the contact forces between small diameter steel balls was modified for use with large diameter low modulus materials such as apples. The validity of the model was established by experimentally measuring the contact forces between 6 cm diameter rubber Yoosef Shahabasi balls which were stacked in a rhombohedral fashion. There were seven layers with either four or five balls in each horizontal layer. All of the contact forces differed by less than 20 percent from the values calculated using the computer model. Approved: aj of sor Department C irman To Sattar and Masumeh ii ACKNOWLDGEMENTS The author sincerely appreciates the guidance and cooperation of his major professor, Dr. Larry J. Seger- lind (Agricultural Engineering) during the development of this research work. Also appreciation is extended to Dr. Haruhiko Murase (Agricultural Engineering) for his helpful suggestions. Equally, his gratitude is extended to Dr. D. H. Dewey (Horticulture), Dr. W. N. Sharpe, Dr. G. E. Mase (Metallurgy, Mechanics and Material Science), Dr. G. K. Brown (USDA) and Dr. R. H. Wilkinson (Agricul- tural Engineering). Thanks are due to his parents, brothers, sister and uncle for their encouragement throughout this study. The author is particularly indebited to the Iranian people from which, through a scholarship, this graduate work was made possible. iii TABLE OF CONTENTS LIST OF TABLES LIST OF FIGURES LIST OF APPENDICES Chapter I. II. III. IV. INTRODUCTION REVIEW OF LITERATURE 2 2 2. 2. .3 .4 MN 0 o l 2 0m Mechanical Injury Stress Analysis in Fruits Under Loading Criteria for Maximum Allowable Load Mechanical Properties of Granular Systems 2.4.1 The Arrangement of the Parti-o cles in a Stack 2.4.2 Systematic Packing of Spheres. of Different Sizes Contact Theory Contact Forces Between Granular Particles ANALYSIS OF THE PROBLEM CHANGES IN THE MECHANICAL PROPERTIES OF APPLE FLESH DURING COLD STORAGE 4 4. .1 2 .3 General Remarks Experimental Study 4. 2.1 The Test Fruit . . 4. 2. 2 Specimen Preparation . 4. 2.3 Uniaxial Loading of Cylindri- cal Specimens Experimental Results and Discussion . iv Page vi vii xi Chapter Page V. BRUISE MODEL; ALLOWABLE DEPTH . . . . 46 5.1 Maximum Normal Strain . . 46 5 2 Calculation of the Allowable Storage Depth . . . . . . . 50 5.3 Results and Discussion . . . . . 50 VI. CONTACT FORCE MODEL . . . . . . . . 54 6.1 Introduction . . . . . . . . 54 6.2 Model Formulation . . . . . . . 55 6.3 Calculations . . . . . . . . 61 6.4 Results . . . . . . . . . . 65 VII. EXPERIMENTAL INVESTIGATION OF THE CONTACT MODEL . . . . . . . . . . . . 72 7.1 General Remarks . . . . . . . 72 7.2 Equipment . . . 72 7.3 Calibration of Gages of Pressure Transducers . . . . . . . 77 7.4 Experimental Procedure . . . . 81 7.4.1 Pressure Transducer Place- ment . . . 81 7.4.2 Readjusting. Digital Strain Indicator . . . 84 7.4.3 Loading and Strain Reading . 86 7.5 Results and Discussion . . . . . 86 VIII. SUMMARY AND CONCLUSIONS . . . . . . 99 IX. SUGGESTIONS FOR FUTURE RESEARCH . . . . 104 APPENDICES . . . . . . . . . . . . . 106 LIST OF REFERENCES . . . . . . . . . . 126 Table 2.1 4.1 7.1 7.3 7.4 LIST OF TABLES Effect on Porosity of Spheres Inserted in Voids of Rhombohedral System . Mean Values and Standard Deviation of the Mean of Three Different Elastic Parameters for 150 Jonathan Apples Deflection of Nodes in x and y Direction of a Two Dimensional Rhombohedral Assem- blage for 45 N of Load at each Node Values of Strain (e) at Different Values of Load (N) and Gage Number 6 Average Measured Contact Forces for 157.5 N of Load Average Measured Contat Forces for 180 N of Load Average Measured Contact Forces for 202.5 N of Load . . . . . . . . . Mean Values of the Three Different Para- meters--Group I . . . . . . . . Mean Values of the Three Different Para- meters--Group 11 Mean Values of the Three Different Para- meters--Group III . . . . . . Compression Test of Apple Tissue at Three Different Deformation Rates vi Page 25 38 66 83 89 90 91 108 109 110 125 Figure 1.1 LIST OF FIGURES View of the Handling Tank System with a Conventional Apple Harvester . . . Silo Storage Being Filled Pressure Distribution on the Contact Sur- face of Sphere as Subjected to a Flat Plate Under Load . . . . . Hertz Theory of Contact for Two Spherical Bodies in Contact . . . . . The Angle of Intersection of the Sets of Rows in the Layer . . . . . A Simple Rectangular System The Orthorhombic System Rhombohedral System . Pyramid and Its Triangler Side A Rehombic Layer above Another .A Rhombic Layer above the Cusp of Another A Rhombohedral System with a Rhombic Form on the Top . . . . . . . . . . Two Cases of Exerted Load on the Bottom Spheres . . Stress Distribution Within an Elastic Sphere with Poissan's Ratio = 0.3 Com- pressed with a Flat Plate . . . . . Section View of a Model Particle Stack vii Page 10 18 19 19 20 21 23 23 23 27 29 31 Figure 2.14 Forces Acting on a Particle in the Parti- cle Stack The Failure Point on the Load Deformation Diagram . Modulus of Elasticity v.5. Storage Time Failure Stress v.5. Storage Time Failure Strain v.5. Stroage Time Distribution of the Modulus of Elasticity for 150 Jonathan Apples for the Three Storage Periods . . . Distribution of Stress at Failure for 150 Jonathan Apples for the Three Storage Periods . . . . . . . . . . Distribution of Failure Strain for 150 Jonathan Apples for the Three Storage Periods . . . . . . . . . 522 as Related to Depth. October 1 Data Height and Percent Bruise Relationship for Jonathan Apples in a Bulk Storage(Case 1) Height and Percent Bruise Relationship for Jonahtan Apples in a Bulk Storage (Case II) . . . . . . . . . Two Dimensional Rhombohedral Packing of Identical Spheres Subjected to Uniform Pressure Loading at the Top Planar Graph Representation of Figure 6.1 and Corresponding Contact Forces Representation of Member i Planar Graph Representation of Deflection of Nodes in x and y Direction when 225 N Load was Exerted viii Page 31 37 39 41 42 43 44 45 49 51 52 56 57 59 67 Figure 6.5 6.6 Planar Graph Representation of Calculated Contact Forces for a 225 N Load . . . Planar Graph Representation of Calculated Contact Forces for a 157.5 N Load Planar Graph Representation of Calculated Contact Forces for a 180 N Load . Planar Graph Representation of Calculated Contact Forces for a 202.5 N Load Test Box with Plexiglass Window and Connecting Wires Different Parts of Loading Piston Dimensions and Assemblage of the Pressure Transducers Terminal Box of Multichannel Digital Strain Indicator Multi-Channel Digital Strain Indicator with Model OD-1014 Printer Calibration of Pressure Transducer Calibration Curve for Gage Six Pressure Transducer Placement in Different Locations of the Assemblage A Complete Set-Up of the Simulated Force Distribution Experiment of Two Dimensional Rhombohedral Arrangement of the Rubber Balls . . . . . . . . Planar Graph Representation of Measured Contact Forces for a 157.5 N Load Planar Graph Representation of Measured Contact Forces for a 180 N Load Planar Graph Representation of Measured Contact Forces for a 202.5 N Load ix Page 68 69 70 71 74 75 76 78 79 8O 82 85 87 92 93 94 Figure Page 7.13 Planar Graph Representation of Calculated Contact Forces for a 157.5 N Load . . . 95 7.14 Planar Graph Representation of Calculated Contact Forces for a 180 N Load . . . . 96 7.15 Planar Graph Representation of Calculated Contact Forces for a 202.5 Load . . . . 97 Appendix A. B. LIST OF APPENDICES Experimental Data Computer Program for Calculation of Height and Percent Bruise Relationship A Computer Program for Calculation of Contact Forces and Nodal Deflections in a Two Dimensional Rhombohedral Assemblage of Spheres Calculation of Location of Maximum Strain (Z) in a Single Apple Under Contact Load . . . . . . . . . . Loading of Cylindrical Specimens at Different Deformation Rates xi Page 107 111 114 117 123 I. INTRODUCTION Apples utilized by the processing industry are usually stored in stacks of bulk bins in the plant yard. Apples received early in the harvest season are subjected to moderate daily temperatures that hastens ripening which increases the shrink due to weight loss and spoilage. Apples harvested late in the season may suffer from freez- ing damage around the stacks despite some protection from straw or plastic covering. Cold storage is being used with increasing fre- quency despite its cost, particularly by slice processors. The higher cost of storage and a trend to fewer and larger plants necessitates reducing the overhead by operating the plant for a longer period. One method of reducing the overhead is to construct less expensive types of storage which will protect the apples from excessive heat or freezing conditions. The USDA agricultural research group at Michigan State University have been developing a totally integrated system of equipment for mechanically harvesting, trans- porting, and storing apples. A description of the bulk storage silo and the procedure for its loading and unloading has been reported by Burton and Tennes (1977), and Tennes, et a1. (1978). The system consists of: a. A quick acting mechanical shaker designed to shake the trunks of trees spaced as close as 2.4 meters (8 feet) apart, while moving at 1.6 Km/h (l mile/h). b. A horizontally positioned reinforced fiber glass tank mounted on a heavy duty trailer with a filling- well at the top center of the tank. The tank holds 8300 liters of water and is used to transport the apples, Figure 1.1. c. A bulk silo storage system consisting of a storage facility, a reservoir that can provide a large amount of water and a conveyer and handling apparatus for the loading and unloading of fruit from the storage, Figure 1.2. Recommended management practices for bulk storage includes the avoidance of loading warm fruit into the storage (this can be done by hydrocooling of fruits before storage)znu1controlling the temperature during storage by prOper insulation and ventilation using cool night air. The depth of apples should also be such that little bruising occurs within the stack. The determination of the maximum safe depth for apples stored in bulk is a factor that has not been determined. A study of the tank .L I Figure 1.1 View of the Handling Tank System with a Conventional Apple Harvester. loads acting on apples and the allowable depth for safe storage is needed. The primary objective of this study was to develop some experimental and analytical techniques in determining the maximum depth for safe storage of Jonathan apples in a bulk bin. Specific objectives were: 1. To study the changes in the mechanical prOp- erties of Jonathan apples during refrigerated storage and use these in a simulation model to predict the allowable storage depth. 2. To study the contact force distribution and transmission in the bulk storage of large diameter spheres. Anuma .moccoev wofiafim mcfiom owmgoum oaflm--.N.H chamwm wasp oHnflonm wopmuomuom _fl_' ,, _._\ J! 17/, 7x/ ,, W ‘ vr I- hi H“! II IIII Ill llll III.- "IIICIIII— / oafim uco> hfim\pouwz \Illl\ . \\\ AU o>Hm> oumw uoau30\uoH:H Hfio>omom gho 00" .0 Au nu 0W o>Hm> aflmhm nu mafia oocmhpao HmW\\ _ it! no . oESHw onm on poacfi Lopez (3g,¢;i§; —~» :53; o =: /5 Q: .qazggZHZEZEZgZZZggggigé 55:1. 0 :> oumm oESHm II. REVIEW OF LITERATURE 2.1 Mechanical Injury Bruising and injury to agricultural commodities during mechanical handling Operations has been a problem of interest to agricultural engineers for several years. As a result, many investigations have been conducted to determine the mechanical behavior of agricultural products when they were subjected to various types of external forces. The increase in use of mechanical harvesting for agricultural products has generated a need for basic information on material properties. Gaston and Levin (1951) reported an extensive study of causes of apple bruising in handling Operations. Their study included the loading of apples under both impact and dead load conditions. For the impact test, apple samples were dropped from heights up to 24 inches onto various types of surfaces. Apples were subjected to an increasing dead load until the desired load of bruising has been achieved. They reported that a dead load of 8.5 pounds was required to produce a 3/8 inch diameter bruise in a 2.5 inch diameter McIntosh apple. No bruise occurred below 8.5 pounds. Above this load, the amount of bruising was proportional to the load applied. Their test included a very large number of apple samples and 5 the data reported were aimed more toward demonstrating the importance of minimizing loads on apples. Mohsenin and thlich (1962) carried out a more intensive study to determine some of the important engin- eering parameters involved in mechanical damage. They studied apple sections under impact and static loads. The static test was conducted by applying various dead loads on the apples for 100 hours at 34°F. The energy required to bruise under an impact load was found to be roughly twice that required under a static load. Fletcher et a1. (1965) studied the effect of vari- able loading rates to determine trends of the rate char- acteristics, rather than the properties at isolated rates of loading. They wanted to correlate the mechanical prOperties at one rate of loading with those at another. Their study brought out some important relationships between slow and fast rates of loading but they did not investigate maximum allowable loads. Hammerle and Mohsenin (1966) used a vertical drop tester for dynamic impact loading. The main objective of their study was to develop the apparatus and the method of testing. Bittner et a1. (1967) developed the concept of using a simple pendulum to simulate free fall of the fruit Specimens. They used the energy balance theory to evaluate the effectiveness of various cushioning mate- rials based on rebound energy, energy absorbed by the cushion and energy absorbed by the apple. Fridley and Adrian (1966) worked on resistance to mechanical injuries of apples. Their results, given in terms of compression yield force and impact yield energy, showed that in comparison with peaches, pears and apri- cots, apples had the least potential for mechanical har- vesting. Mattus et a1. (1960) showed that drop heights more than six inches onto a hard surface produces internal bruise in pears which developed brown spots in the flesh of the fruit. Location of the bruise has suggested that maximum shear stress can be a possible failure parameter (Fridley and Adrian, 1966). A dynamic triaxial compression test was conducted by Miles and Rehkugler (1971) at varying levels of compression stress, shear stress and axial strain rates. These investigators reported that shear stress was the most significant failure parameter. Dal Fabbro (1979) concluded that the maximum normal strain is the primary factor causing the failure of apple flesh. 2.2 Stress Analysis in Fruits Under Loading Knowledge of the stress distribution in fruits under static and impact load is limited because of the difficulty involved in determining material properties and the lack of analytical solutions valid for the irregular shapes involved. The load could be exerted from a flat body to the fruits or from one fruit to another. Finney (1963) reported a significant difference existing between certain potato varieties in their response to applied surface pressure. Mohsenin and Galich (1962) applied the same technique to apples, potatoes, pears and tomatoes concluding that the compression test appeared to offer the most promise of evaluation of mechanical behavior as related to bruising. The most common type of loading that fruits are subjected to is the contact load which can produce a bruise. Contact forces occur in harvesting, handling and storage. Contact stresses are caused by the pressure of two bodies having a point (small area) contact: common type of contacts are the sphere and a plane, Figure 2.1 or two spheres, Figure 2.2. Boussineq (1885) solved the problem of concentrated forces acting on the boundary of a semi-infinite body. Timoshenko and Goodier (1970) dis- cuss the contact problem as solved by Hertz. The maximum Figure 2.l.--Pressure Distribution on the Contact Sur- face of Sphere as Subjected to a Flat Plate Under Load. .511 V; 4 mm... ‘4 \ Bodies nnnnnnnnnn 11 pressure on two spherical bodies in contact is 1.5 times the average pressure on the surface of contact or 3P Zwa q0 = 2 (2'1) Assuming that both balls have the same elastic properties and taking u = 0.3 the corresponding maximum pressure is 2 P 3 2 (R1 + R2) qO - 3/2 3' - 0.388 PE R2 . R7 (2.2) 1 2 where q is maximum pressure on the surface of contact a is the radius of the surface of contact P is the applied load > E is the modulus of elasticity of spheres R1 is the radius of sphere one R 2 is the radius of sphere two The first known application of Hertz solution for contact stresses in agricultural products is reported by Shpolyanskaya (1952) for determination of modulus of deformability of the wheat grain compressed between two parallel plates. Finney (1963) used the Boussinesq solution for concentrated forces acting through a rigid die for potato, apple, portions of corn kernels, peaches and pears. The 12 Hertz and Boussinesq techniques have also been applied to apples (Morrow and Mohsenin, 1966). McIntosh apples were subjected to constant load (creep test) or constant deformation (stress relation test) by a flat rigid plate. In the case of McIntosh apples loaded with a l/8 inch cylindrical rigid die, they concluded that the stress was approximately zero at a depth of two inches below the surface of the fruit. Fridley, et a1., (1968) applied Hertz and Bous- sinesq theories to obtain force-deformation curves for peaches, pears and apples. They showed that the bruise in an apple usually occurs under the center of the area of contact at a small distance beneath the surface of the fruit. A flat plate loading was used in the study. Agricultural products are generally viscoelastic. Viscoelasticity comprises an irreversible energy trans- formation where the relation between stress and strain is governed by time effect. During the early experiments on mechanical behavior of fruits and vegetables, it was observed that force deformation relations includes the time effect (Finney, 1963; Mohsenin, 1963; Timbers et a1., 1966). Experiments conducted on McIntosh apples showed that apple flesh behaves as alinear viscoelastic material (Morrow and Mohsenin, 1966). Latter Chappell and Hamann (1970) studied the viscoelastic behavior of apple flesh, 13 but they found the material properties to be somewhat stress dependent and thus could not be characterized as linear. Hamann (1967 and 1970) also noted the nonlinear properties in apple flesh but he solved the apple impact problem for stress at the surface and in the interior of the apple considering apple flesh as a linear-viscoelastic material. In the later studies, the finite element method was used to determine stresses in apples resulting from contact with a flat plate. Apaclla (1973) considered the apple as an elastic material and used the finite element method. Rumsey and Fridley (1974) used the finite element method assuming a linear viscoelastic shear modulus and an elastic bulk modulus for the material. De Baerdemaeker (1975) con- sidered a material with time dependent bulk modulus and shear modulus to obtain the creep deformation and the stress distribution of a sphere in a contact with a flat rigid plate using the finite element method. He concluded that apples subjected to contact creep loads experience maximum stresses at the initial application of the force. Sherif (1976) solved the quasi-static contact problem for nearly incompressible agricultural products using finite element method. Other theoretical studies involving vegetative materials includes Gustofson (1974) who obtained a 14 numerical solution to the axisymmetric boundary value problem for a gas-solid-liquid medium and Murase (1977) who develOped stress-strain constitutive equations con- taining parameters necessary to describe the mechanical behavior of vegetative material including the water poten- tial term. 2.3 Criteria for Maximum Allowable Load One of the major reasons for studying the mechani- cal properties of fruits and vegetables has been to deter- mine the maximum allowable load to which these materials can be subjected without causing objectionable damage. Limited work has been done to understand the mechanics of bruising in fruits and vegetables. A recent work using a compression test on tissue specimens reported that shear stress is the most important failure parameter (Miles, 1971). The question of how the maximum allowable shear stress can be determined for a whole fruit was not answered in this work. The concept of "Bioyield Point," indicating initial cell rupture in whole fruits such as apples and pears, has been prOposed as the criterion for maximum allowable load that fruit can sustain without showing any visible surface bruising (Mohsenin, 1962 and 1965). In the case of apples the bruise is immediately below the skin and in most cases can be removed in the peeling process. In fruits such as peaches, bruising and 15 tissue failure can occur some distance below the skin and the damaged protion can be seen only in canned products by the consumers. In these cases, shear strength has been taken as the maximum allowable load (Horsfield, 1970). In recent studies the application of the theory of elasticity and importance of the modulus of elasticity of the fruit in single and multiple impacts suggested that the failure was due to excessive internal shear stresses. It is shown that maximum shear stress is prOportional to (a) the energy of fall, (b) the moduli of elasticity of the fruit, and (c) the radii of the fruit and the impact surface (Horsfield, 1970). Nelson and Mohsenin (1968) have determined a relation between bruise volume and load. They report that bruises caused by dynamic loads are larger than those caused by equivalent quasi-static loads. Fletcher et a1. (1965) reported that energy force and deformation to repture first decreases with increasing loading rate, then increases. Mohsenin (1971) reported that, the significance of certain viscoelastic properties of fruits and vegetables may not be understood but we should be ready to use such data as the maximum allowable load that these products can sustain under impact, dead loads, and vibration in designing handling systems. He also mentioned that these data should be available in the form which can be used by engineers. 16 Dal Fabbro (1979) studied the strain failure of apple material. Cylindrical and cubic apple specimens were subjected to uniaxial, biaxial and triaxial state of stress. Linear elastic and viscoelastic material prOp- erties were used to calculate the stress and strain com- ponents within the apple flesh. He reported that in the uniaxial loading of cylin- drical specimens the normal stress at failure varied for different strain rates. Triaxial loading of cylindrical specimens indicated that maximum shear stress and normal stress at failure vary for different levels of cylindri- cal stress. He also reported that the uniaxial, biaxial and rigid die loading of cubic and cylindrical specimens discards the maximum normal stress failure criteria. Experimental results from his studies indicate that the maximum normal strain at failure remains rela- tively constant for all the loading situations. The most significant conclusion of his research was that apple material fails when the normal strain reaches a critical value. 2.4 Mechanical Properties of Granular Systems Granular systems consist of cohesionless particles where the individual grains are independent of each other except for frictional interaction and geometric 17 constraints resulting from the particular type of packing. The two most important prOperties of granular materials are their strength and compressibility characteristics. The component particles in a granular system may be of any size from the smallest diameter of ICU (like powder) to pebbles, cobbles, or even boulders (several inches in diameter) (Brown, 1970; Farouki, 1964). Many studies of the packing of solid particles have been based on spherical or near spherical particles. Graton and Fraser (1935) discussed the geometry of vari- ous assemblages of discrete, ideal spheres. Also sys— tematic arrangement of spheres in connection with the flow of water through soil was first studied by Slichter (1899). 2.4.1 _The Arrangement of the Partic1es in aTStack The stacking arrangement of the particles in a mass of material determines the points of contact between the particles and the direction of the normal at contact points; this essentially establishes the force system which acts between the particles. 2.4.1.1 Systematic arrangement of uniform spheres. Ideal spheres may be packed in ordered layers of various types definable by the angle of intersection of the set of rows in the layer. Layers in which sets of 18 rows have angles of intersection of any value between the limiting values of 60° and 90° are possible. Since square and simple rhombic layers (Figure 2.3) represent the limiting types of systematic packing, only these two are considered. C Figure 2.3.--The Angle of Intersection of the Sets of Rows in the Layer. Three different systems may be formed by stack- ing square horizontal layers one above another. Case 1. A simple rectangular system; each sphere has its center vertically above that of the sphere below (Figure 2.4). Case 2. The orthorhombic system results when the center of the upper sphere is offset a distance R in the direction of one of the rows (R is the radius of the spheres) (Figure 2.5). 19 SQUARE LAYER TOP PLAN END ELEV- Figure 2.4.--A Simple Rectangular System. END ELEV. RVE .1 Figure 2.5.--The Orthorhombic System. 20 To get the vertical distance, d, between the center of a sphere in a row whith the one above or below it, consider triangle C10 C3 (Figure 2.5) which can be written: R2 + (d + R)2 = 4R2 d2 + 2dR - 2R2 = 0 -2R + g J3'R or d = -R . /3 R = 0.73 R (2 3) 0.. ll Case 3. Rhombohedral system. In this case each sphere is in contact with four spheres below, four above, and four in the same layer (Figure 2.6). ea c e C / 70 6 PLAN END ELEV. Figure 2.6.--Rhombohedral System. 21 To get the projectional distance between two spheres in two layers, consider the pyramid of C with the trianglar side of C1C CS (Figure 2.7). 3 Knowing y in this triangle which is: or y = /3 R will give x as 2 x = 3R2 - R or X = /2 R Figure 2.7.--Pyramid and Its Triangler Side. 1C2C3C4C5 (2.4) The rhombohedral system is the most important theoretically, and usually is the basis for calculations. It is also the most important from a practical viewpoint, because it gives the densest state. The three packing configurations discussed below may be formed by stacking simple rhombic layers one above another. 22 Case 4. When in the orthorhombic system, the spheres of the next rehombic layer are placed in such a way that the center of each sphere lies vertically above the sphere below it (Figure 2.8). Case 5. There is a rhombic layer at the bottom and each sphere in the next rhombic layer rests in the cusp between two Spheres in the layer below (Figure 2.9). The distance d is obtained by considering the triangle C1C2C3, giving d = R /3. Case 6: This system is similar to case three except each layer from the tOp is in rhombic form (Figure 2.10). To find distance d consider pyramid ClC C C dis- 2 3 4’ tance y can be calculated by considering triangle C4C3C2, then y would be y = /3 R. Having y, d can be calculated easily from triangle C4OH as: 0.. N u *< I to| r—a \1 o H a. II The angle which the side wall makes with the bottom influences the packing formation (Faruki and Winterkorn, 1964). 23 RHOMBIC LAYER PLAN END ELEV. Figure 2.8.--A Rehombic Layer above Another. END ELEV. Figure 2.9.--A Rhombic Layer above the Cusp of Another. PLAN END ELEV. Figure 2.10.--A Rhombohedral System with a Rhombic Form on the Top. 24 A 90° angle will favor cases 1, 2, 4, and 5, a 60° or 120° angle favors cases 2 and 3. The packing is also influenced by the angel which the side walls make with each other. A 90° angle favors formation of square pattern and hence cases 1, 2, and 3. Intersection of the side walls at 60° with themselves and at 90° with the bottom favors cases 4 and 6. The walls of the container give rise to a wall effect which causes the porosity in the vicinity of the wall to be greater than that in the body of the packing. This has been studied by Furnas (1929) who obtained an expression for the voids, VW, present in a ring at the wall of area nd-D/Z V = {V + K (1 - v)} (l_1_____ W - ———- (2.5) where is the diameter of the particle. is the diameter of container. is the voids present in the interior. W bfloflmgm mo Egghmé 0.53m mfg: . 25. 09985 mi OAu D 1,1 jj Door Door ooou oouu oovu coma coma coon ooun oovn coon coon ooov oouv (edxt 4413119813 30 snInPoW 40 of elasticity changed significantly between October 1 and November 15. The decrease in the modulus value was not as much between November 15 and December 31. Figure 4.3 Shows the failure stress had a very Sharp decrease in the first 1.5 months, but did not decrease with the same rate for the next 1.5 months. The failure strain as Shown in Figure 4.4 decreased in the first 1.5 months and increased Slightly in the next 1.5 months. A statistical investigation (x2 test) indicated that the modulus of elasticity and the failure stress depend on the storage period. Fail- ure strain did depend on the storage period for the first 1.5 months but not for the second 1.5 months. The distribution of the elastic modulus, failure stress and failure strain during the storage period is Shown in Figure 4.5 through 4.7. The higher modulus of elasticity and the higher failure strain values for the freshly picked apples indicates a higher resistance to bruising. The changing of the modulus and failure strain values toward smaller values as storage time increases Shows a lower resistance of apple material to bruising. 41 9:: 09395 .m.> mmonum ousfimm-.mé 0.53m manna: . 95L. ommuoum mi p Tat- CNN ova on“ can can can own can can oov Out 0'. (mix) 559.113 9.1111th 42 9:2. ommuoum .m.> 598m PBS—Eiéé 333mm 935: . 059 owwnoam o.” m... coo - D D ‘l 000 .0 2.0.0 09.0 9.3.0 as emitted 11118.1 43 .mwofiuom ommuoum mouse may pom mofimm< cmnumcow omH pom kpflufiummam mo m5aswoz ecu mo GOMusnwuumHm--.m.v owsmflm 38 bfluflmflm mo 838: «0; (v NC 0' on on ‘0 an 00 as on Ca «a 0« 0 n .J . (a 0 pl 2 3 J v .V : 3 .U ‘5 c c a I Page m .H: 80.5 I 328 m4 .HH 905 I figs o .H 98o E 2 D 30 aueoxed :.1 a satddv 1 O ' 44 .mcofiuom ommhoum vouch esp How mofimm< cmnuw:0h omH pom ousfiwmm um mmohum mo :ofiusnfluumfin--.o.v opzwfim I mficoe m .H: 98o oIIIIIIo mnumoa m.H .HH @5090 I mummy ohsflwmm um mmohum Hgshoz 000 0mm Own 0h“ 00“ 00¢ 00.. u p b L p u b b P p > t H3GQ=AV.HAWSLU 3 a 2 setddv go iuaoxed 0 P Or on 00w 45 .mvofiuom owmhoum 009:8 0:0 pom moama< 00:00000 omH wow cwmhum opsafiwm mo aofiusnfluumfla--.h.v ohswfim 5:35 onsfimm :0 00.0 00.0 3.0 9.0 «0.0 :..0 9.0 00.0 00H0 8.0 00.0 00.0 00.0 00.0 «0.0 8.0 0 .J . D b b b L h c t 0 so? 0 O n N O C T: no setddv go moored .00 +2. .00 I 9300.: m .HHH 90.5 .00 I gag: mA .3 90.5 .00. .all... 5:05 o J 050.5 V. BRUISE MODEL; ALLOWABLE DEPTH 5.1 Maximum Normal Strain The bruise model used here to predict the allow- able depth for bulk stored apples is based on the failure Strain criteria developed by Dal Fabbro (1979). It states that apple flesh fails when a maximum normal strain exceeds a critical value. The stress components for any point along the z axis were given in Chapter II (2.9). Since failure occurs because of the maximum normal strain, an equation for the maximum normal strain, 8 is needed. Assuming a homo- zz geneous, isotropic material, Hooke's law gives 5 = E {Ozz - u (Orr + 000)] (5'1) Substituting (2.9) produces 1 (PH = (1 + U)QO _ '1 822 E’ [Zu[l W Tan ( + (1 + 121*] (5.2) The most common contact which occurs in a bulk storage is apple-to-apple contact which can be modeled as the contact between two spheres. To Simplify the 46 47 problem, it was assumed that all apples have the same diameter, 0.0625 m, and a Poisson's ratio of 0.35. The modulus of elasticity and failure strain were treated as random variables. Using the assumptions on diameter and Poisson's ratio, the radius of contact, (2.7) become 1/3 El l E2 1/3 a = 0.21745 F -——————— (5.3) E1 52 1/3 E1 + E2 -2/3 while q0 = 10.097 F ——————— (5.4) E1 E2 The modulus of elasticity values E and E2 in (5.3) and 1 (5.4) are for the two apples in contact. Equation (5.2) gives the maximum strain as a function of V which is equal to Z/a. Given a load F and the modulus of elasticity of each apple, the radius a, (5.3), can be calculated as well as qo, (5.4). Knowing a and qo, a can be calculated for any value of z. 22 Theoretically the location of the maximum value of €22 can be obtained by differentiating (5.2) with respect to 2, setting the resulting equation to zero and solving for z. The resulting equation, however, is not easily solved and it was decided to calculate 822 for several loadings and moduli values to see how it behaved (see 48 Appendix D). Examples of some curves for €22 are shown in Figure 5.1. The average location of the maximum value was 3.65 mm (3.5 — 3.8 mm) for the October 1 data and 3.0 mm (2.8 - 3.2 mm) for the other two sets of data. These values of 2 were used in the Simulation model dis- cussed in the next section because the value of 522 at 3.65 mm or 3.0 mm differs very little from the maximum value when these values of z are not right at the location of 822 maximum. The apples at the bottom of the bin are in con- tact with a flat hard surface which could be either steel or concrete. In this case E1 for the steel (or concrete) is much greater than B while R is infinity. These prop- 2 l erties modify the equations for a and q0 to F(l715278 + 8.6 E2) 1/3 a - 82740000E2 (5'5) 1/3 1715278 + 8.6 E2 '2/3 and q0 = 0.4774 F W (5.6) The location of 522 maximum for this type of contact was determined in the same manner as for apple- to-apple contact. The location of the maximum values were at z = 3.88 mm (3.55 - 4.21 mm) for the October 1 data and 3.0 mm (2.88 - 3.12 mm) for the other two dates. 49 NN .0000 H honouuo .cuaoa ou woumfiom mm 0--.H.m owsmfim . 96 r .10 96 tfif' 96 r I' Y0 «6 Tfi’ fit I 06 (mu) 2 Indaa r ”N . 9N . 1N . ”N z HVHNM r I O& 1 0; 000 000 '00 000 551.5 NN... 50 5.2 Calculation of the Allowable Storage Depth The calculation of the allowable storage depth was carried out using (5.2) while treating El’ E2 and the failure strain of the pair of apples (or apple in the case of flat plate contact) as a random variable. A normal load F was selected. A random generator was used to select the values of E1, E2, efl,and efz from the data discussed in Chapter IV. The maximum strain in each apple was calculated. If this strain exceeded the failure strain for the apple, the apple was said to be bruised. This calculation was repeated 500 times for each loading. A percent of apples bruised was then calculated. The normal force was increased and the process repeated. The equivalent depth was calculated assuming a Single column stack where each apple weighed 0.85 N (Appendix B). 5.3 Results and Discussion The percent of apples with bruise for each storage group are shown in Figures 5.2 and 5.3. It is immediately obvious that fresh apples can be piled much deeper than apples which have been in storage 1.5 to 3 months. The fresh apples have a Significantly larger modulus of elas- ticity value which means that it takes more force to produce a fixed amount of deformation. There is not an appreciable difference for the results for apple-to-apple 11.0% 10. 29‘ 9-55‘ 2-94‘ 2.20‘ Apple Depth, meters 51 “£51, Case I : Two Apples in contact 61009 ‘ D‘so >140 >130 >120 >110 >100 P90 >60 r50 .40 130 20 147 Percent of.Apples with Bruise Figure 5.2.--Height and Percent Bruise Relationship for Jonathan Apples in 3 Bulk Storage (Case I). lknmalload,hbwtnw 52 -uS;Lr Case II : Apple in contact with a flat steel surface 11.02 - 150 1oa9‘ .140 955‘ - 130 8-82 - - 120 5.08J v 110 3 235 'IMJ g £3 561- .90 ‘E Q) E :z .. 5.884 ' 80 .. 5 “5% E? 514- '3 0 431‘ " H E E 3.... a 234. 220- 147‘ '20 0.73 - f 10 1" fiw ' V T V T I T V V i r o 0 5 1o 15 20 25 so 35 40 45 so 55 so 65 70 Percent of Apples with Bruise Figure 5.3.--Height and Percent Bruise Relationship for Jonathan Apples in a Bulk Storage (Case 11). 53 contact and the results for apple-to-flat surface con- tact. In Figures 5.2 and 5.3 it can be seen that curve for Group III apples is above that of Group II because the value of strain at failure as the failure criterion for Group II was lower than Group III (Figure 4.4). The curves in Figures 5.2 are used as follows. If an individual wants to store Jonathan apples for three months and is willing to accept 10 percent bruising, these apples can be stored to a depth of about 3 m or 10 feet. This is a conservative estimate, however, because single column contact is not what occurs within a stack and the contact forces on the apple are smaller, allowing a greater depth. VI. CONTACT FORCE MODEL 6.1 Introduction The allowable depth for apples stored in bulk that was calculated in the previous chapter assumed a single column stack. This is not what occurs in the actual pile. One apple will contact several others and the actual contact force probably is less. A computer model for calculating the contact force between spherical bodies is presented in this Chapter. An experimental verification using rubber balls is discussed in the next chapter. The computer model developed here is based on the model developed by Davis (1974). His model is basically a two-dimensional truss analysis where the center of each sphere is considered as the node and the members connect- ing the nodes have the nonlinear property of two spheres in contact. Davis used small diameter approximately one- half inch diameter, steel balls in his study. Six centi- meter diameter rubber balls were used in this study. The sphere is much larger and softer. The following simplifications were made. 54 55 a. The spheres are assumed to be identical and only two-dimensional packing problem is considered, i.e., each sphere in the assemblage of Spheres has its center on a common plane. b. Only normal forces at the Sphere contacts are considered (Shear forces are neglected due to small magnitudes) and the Hertz theory is used to relate the magnitudes of such forces to the corresponding sphere compression. c. Arrangement of spheres in a sample is assumed to be in a geometrically stable configuration. The ini- tial configuration for this analysis, therefore, must be taken so that no gross movement or rearrangement of the Spheres will occur during uniform pressure loading. e. The change in the contact area is due to the small compressions at the Sphere contacts rather than the gross change in the packing geometry. 6.2 Model Formulation Considering a small stable sample of identical Spheres arranged in two-dimensional rhombohedral system and loaded at the top by a uniform pressure, Figure 6.1. The information in Figure 6.1 is shown in Figure 6.2 in the form of a planar graph which is obtained by connecting the center of the spheres. The nodes of the graph in 56 L/////////////////////////////// //// Figure 6.1.--Two Dimensional Rhombohedral Packing of Identica Loading at '4A8’ 11 WV 699 /K W 9099 (8191 = QYVYV ////////////////////////// Spheres Subjected to Uniform Pressure the Top ,//////////////////////////////////// / 57 (3 Node numbers 1 IX -————+kanactlknces Figure 6.2.--Planar Graph Representation of Figure 6.1 and Corresponding Contact Forces. 58 Figure 6.2 represent the sphere center and branches of the graph correspond to the contacts between adjacent Spheres. Since Shear components of the contact forces are neglected, forces are transmitted throughout the packing along the branches of the planar graph. The task is to find the forces at each of these branches and study their distribution in the bin where they are located. AS mentioned before, since the initial configuration of the assemblage is geometrically stable and no sphere will be moved or dislocated, only a small compression at the contact points of the spheres will occur during the load- ing. In structural analysis, members (branches) are idealized as lines which meet at points (nodes) which are called joints, so the problem formulation can be employed and the displacement of the Sphere centers are introduced as unknowns. Assuming that the rigid-body degrees for the assemblage have been removed by the introduction of appropriate supports, the equilibrium equations for each movable joint of the assemblage may be written as: [N] {F} = {P} (6.1) where {F} is the (B x 1) matrix of contact force magnitude and it is customary to assume the contact forces positive when they are in compression (B is the number of bars). 59 {P} represents (2J x 1) matrix of applied node forces where J is the number of joints or nodes. [N] is a generalized branch node incidence matrix or simply an incidence matrix. To understand the construction of [N] consider Figure 6.3. displaced sition T‘— node A (+end) arbitrary aSsumed “Ode C (‘9 d) direction Figure 6.3.--Representation of Member 1. Figure 6.3 shows a typical truss bar and the associated with which is the bar force F1 and the bar length change Ai, chosen so that positive Pi and A1 corresponds to tension or stretching within the bar and a unit vector ni. Knowing the displacement of the ends of a bar, it is possible to compute the change in length of the bar by the following relationship A. = n. (6A - 6C) (6'2) 60 which involves projecting the joint displacement vector along the original position of the bar. Equation 6.2 can be written for the entire structure as [A] = [N] {<5} (6.3) where [N] is a (B x J) = (row x column) matrix whose elements N.. are 1) 111 if node j is the positive end of branch i -ni if node j is the negative end of 13 branch i 0 otherwise B and J are the number of bars and joints, respectively. {6} is the node displacement matrix. In the circular assemblage of Figure 6.1, B is the number of contact points and J is the number of mov- able nodes. From elementary mechanics of solids it is known that the bar forces and displacement are related through Hooke's law, in which F. = Ki A. (6.4) The Hertz contact theory applied to the ith contact yields (Davis, 1974): 61 F. = (6.5) where = 20 m /3 (1-0) K k is a constant which depends on the radius and elastic properties of spheres G and u are shear modulus and Poison's ratio, respectively. Equations (6.1), (6.3) and (6.5) can be combined in the usual manner for the node formulation and written (Zienkiewicz, 1971) [N]T[K(5)][N]{6} = {P} (6.6) where [N] is (B x B) transformation matrix [K(0)] is the (B x B) diagonal Hooke's law matrix {6} is (ZJ x l) deflection vector {P} is (ZJ x 1) external load vector Equation (6.6) represents a set of 2J simultaneous, non- linear equation for the unknowns {6}. 6.3 Calculations The objective here is to use the formulation of the previous section to calculate the joint deflections and then the forces between the uniform spheres arranged in the two-dimensional rhombohedral system shown in Figure 6.1. 62 The displacements for all nodes are not the same. Nodes l, 5, 10 and 14 can move only in the y-direction while nodes 19, 20, 21, 22 and 23 cannot move in either direction. Each of the rest of the nodes can move in both the x and y directions. Using the appropriate degrees of freedom for each node, there are 32 force equilibrium equations. Since there are 50 contact forces between spheres, the trans- formation matrix has a dimension of (32 x 50) and can easily be constructed from the force equilibrium equations. The transformation matrix is in fact the direction cosines of the branches in the assemblage which for rhombohedral system would be either cos 60° or cos 30° depending on the location of the nodes and branches. It is necessary to have values for k and A1 to calculate F1 in (6.5). A value for A1 is necessary because (6.5) is nonlinear in Ai. An arbitrary value of -0.254 cm was assumed for each component of {6}. Equation (6.3) was then used to calculate each Ai' Assuming Cos 30° for each component of transformation matrix in first iteration, the value of A1 will be 0.254 cm. To determine the value of k in (6.5) a Single rubber ball, 6.09 cm diameter, was marked with a small drop of black ink. A Sheet of white paper was placed between the ink Spot and the rigid flat circular head of 63 the Instron machine. After exerting a load of 72 Newtons the radius of the contact area of the ball on the white paper was measured to be a = 1.2 cm. Timoshenko gives the radius of contact area as 3 3nF(K1 + K2) RlRZ a = (6.7) 4 (R1 + R2) l-ui l-ug where K = and K = l 0E1 2 0E2 F is the applied load R1 and R2 are the radius of the two spheres in contact. E and u are modulus of elasticity and Poisson's ratio. Assuming the rigid plate of the base of Instron has E = w when compared with rubber ball and its radius is equal to infinity, (6.7) reduces to 3 30FK1R1 a = ——T— (6.8) Substituting the appropriate values in (6.8) K1 would be K1 = 0.003345 cmz/N 64 To find the value for “1’ a cubical specimen of one centimeter in dimension was carefully cut from the rubber ball, and was loaded in the Instron machine and the Strain perpendicular to the load and parallel to the load were measured. Poisson's ratio was determined to be u = 0.13. Using 0 = 0.13 and K1 = 0.003345, E1 was calcu- lated as E1 = 93.55 N/cm2 and the shear modules becomes _ E _ 2 G ~ 2 +U — 41.39 N/cm Since k 26 /2R / [3(1 - u)1 Substitution gives k = 78.26 N/cm:”/2 The diagnonal terms of [K(6)] in (6.6) are each given by 1/2 k.. = k (A1 11 ) or k.. = -39.44 N/Cm 11 Since (6.6) is a nonlinear equation, it must be solved using iterations until the calculated displacements on two successive iterations differ by less than a spe- cified amount. Convergence to the solution was obtained in about 18 iterations for the problems discussed in the last section (see Appendix C). 65 6.4 Results Table 6.1 gives the values of the nodal deflections for the configuration of Figure 6.2 with a concentrated load of 45 N applied at each of the upper nodes. These values are also shown in Figure 6.4. The contact forces are Shown in Figure 6.5. The contact forces for other loading situations are given in Figures 6.6, 6.7, and 6.8. 66 00.00 0 00 0 0 00 00.00- 00.0 00 0 0 00 00.00- 0 00 0 0 00 00.00- 00.0 0 0 0 00 00.00. 00.0 0 0 0 00 00.00- 00.0. a 00.0 - 00.0 00 00.00- 00.0. 0 00.0 - 00.0- 00 00.00- 0 0 00.0 - 00.0- 00 00.00- 000.0 0 00.0 - 00.0- 00 00.00- 0 0 00.00- 0 00 00.00- 000.0- N 00.00- 00.0- 00 00.00- 0 H ”000 fissv ..fieeu. 5000 000000000 » 000000000 x .02 000000000 w 000000000 x .02 00 0000000000 00 0000000000 0002 .00 0000000000 :0 0000000000 0002 A0000000000 way 000: £000 00 0000 mo 2 m0 000 0w00050000 00000:0050gp 00000050500 030 0 mo 500000000 5.000 x :0 0000: mo 0000000009 .0.0 00009 67 0 Node number Y "’ Deflection of node in x direction (an 00:x ‘ Deflection of node in y direction (nmo $ 13.10 1310 3.61 12.77 13-61 @0‘45 943 o . 4 3‘ 0.15 r r * 6.47 5'8 5-3 6.47 Figure 6.4,.— I‘Janar Graph Repremntutim of Deflection of Nodes in x and y direction when 225 N Load was exerted 68 ‘5 N ‘5 N 45 N 45 N 45 N Fl' 0.069 5' 0.069 ‘3' 0-039 ‘4' 0-0'9 0, § "5. of ’4. °» :- ‘10 0.3 «N “f F13'1'31 '?' 4? "3° 9' 3: {’5 g: '9‘?» 1° 6; Q ‘o ’9‘ '9' - '0': 9f 9‘ ‘0' w o O F - . F2" 0.76 F25 0.76 F26 0 76 27 0 7‘ s ‘3._ a? '9“. b? '1’ «a? 4‘ A? ‘0 b. ‘9 o r :V ‘4‘ o 0 z w}- ‘c ' ‘9 «‘5 9* g‘? ‘1:- “A V 9' F36' 0.071 137- 0.39 F3; 0.071 0 9‘. a - J 0 9 e r $ ‘ 6 3 b. .9 b. ‘6 A. ‘4 O ‘ o ‘09 0 90 :V 9‘ 00° 9‘ 4‘” 9' Q“ 4“ V Figure 6.5.--P1anar Graph Representation of Calculated Contact Forces for a 225 N Load. A11 Values in Newtons. 69 31.53 '«1.SN 31.“ 31.55 31.5% Fl'0.31 F2'0.31 F's-0.31 F"0.31 % 0 '6 9. . ‘3'"- ° °‘- 9 :‘P ‘9 o t , 9v ’ N V \ 9v FISM'SP {ls-4.59 ’\ ? ck” .— I c ~o (4 ’4 63 P :3 (:3 1:" 7' “ «4‘ .‘3‘ :‘2 "is w R 4 V FB'ZJDS FA':.6S F,..'2.65 g o) f? g a? r’ ,x' . Ac (.3 A, t; " “i i? 0% ' ‘9 D" .x . A (5 A? (a k. ‘3'; ‘ ‘53 cf v "~ ‘v «w . ‘f- a V .n 9' or a " I F3- 0... F38 0.24 a 9'- 6 r‘ :p g d, ._ ‘5» "‘ - 4 W a '2‘ hr- 0: . e A ' ‘ :4 V “ ‘6‘ :0, V ~ 0’ g 4“ _ Q w? gs Figure 6.6.--Planar Graph Representation of Calculated Contact Forces for a 157.5 N Load. All Values in Newtons. 70 36 N 36 N 36 N SON 36 N Fl' 0.35 F2-0.35 [73-035 F" 0.35 o, N 5," 9:: Q: ‘2” 0?. '9' '9' 7’- ~‘ (‘0 e“ / f o 0 ' t" o l g I" r3 A . Q ‘9 «‘4 . ~ ‘0 a r; Q‘N / 9, w 0‘ V I . u . I 2 F” S 22 F ‘ 3 1905 F15 5 2 ‘0 Q~ 5 ‘\ «o .— ' 0 - A Q '3‘. ,3 '5' ‘5. '3? j~ 42' 9~ N A ‘3- ' " «A Q ' Q g 3 ‘73 «IN ‘d: Q‘ / '0 e / 9. «N w 9’ F"- 3.01 F26. 3.01 F.7' 3.01 ‘3 I - 9 fl '0. cc, '9'. .5“, 5° .0 0,? r . Jr" 0 9‘ “I '09 we.) 0'» ‘ ‘I\ ‘f‘ ' ,Q ‘J‘ % Q v 9’ F36- 0.28 F37- 1.5 F‘B' 0.28 3~ 8 ‘é‘ 0? r‘ x 55- '5, '3’- v “.3 '0' 9 . é" ‘.'- 0. ~ . ‘Q A}. Q o . 7° ‘43. g V 2» w .09 ‘3» Q‘ V Q‘ ‘s‘ w Figure 6.7..-- Planar Graph Representation of Calculated Contact Forces for a 180 N Load. All values in Newtons 71 50.5 .\ 30.5 N 40.5 .‘é 40.5 .\ 40.5 N r - 0.30 F2: 0.40 F330.“ F“ 0.40 Figure 6.8.--P1anar Graph Representation of Calculated Contact Forces for a 202.5 N Load. All the Values in Newtons. VII. EXPERIMENTAL INVESTIGATION OF THE CONTACT MODEL 7.1 General Remarks To simulate the force distribution between bodies in contact in a bulk storage of apples, a series of tests were conducted in the laboratory. In order to have some uniformity in the granular material, it was decided to use rubber balls with an average diameter of 6.25 to 6.85 cm which is close to that of Jonathan apples. A simple case of two dimensional packing with a rhombohedral assemblage was used. The total of twenty-three balls was used with five balls in a row and five rows in a rhombic manner. In order to insure that the rubber balls were similar in size and stiffness, the balls were sorted by diameter and then by stiffness. The stiffness was meas- ured using a flat plate test on the Instron Testing machine. The stiffness criteria was that the balls should require between 54-63 N to produce a 1.7 cm deflection. Several portions on each ball were tested. 7.2 Equipment 1. Test Box: A wooden box 53 cm high, 32 cm wide and 6.8 cm thick was constructed. The box was made 72 73 with these dimensions so that it could be loaded using an Instron testing machine and could hold five rubber balls in each layer with a little gap for the installation of the pressure transducers. The perforated plexiglass win- dow of the box was easily assembled or removed so that the connecting wires from the pressure transducers could extend from the box to the strain indicator, Figure 7.1. 2. Loading Piston: To exert the load in a uni- form manner from the head of the Instron to the balls in the testing box, a loading pistion was constructed, Fig- ure 7.2. The piston consisted of a metal strip to which the supporting bars were connected. A wooden layer (1.5 cm thick) and a foam layer (2.5 cm thick) were attached on the bottom side of the metal strip. The piston had rectangular cross-section with dimension of (31.5 cm x 6.5 cm) which would fit into the top of the testing box. 3. Pressure Transducer: A special pressure transducer was designed for the experiment and is shown in Figure 7.3. The pressure transducer consisted of a 3.2 x 2 cm upper plate 0.5 mm thick. It was made from spring steel. A lower plate, with the same dimensions except it was thicker (1 mm), was made from hard steel. Two millimeter diameter rollers, two cm long, were glued to the bottom plate. The upper plate rested on the 74 Figure 7.l.—-Test Box with Plexiglass Window and Connecting Wires. 75 I“! \ thread attached on the head of the Instron r-————~—> SUpporting bars - -___, _____-_ ~> . .1737;gr232r71mU§7§77U%%::::;:i metal strip ‘m__ =f\\2\\"‘ wooden layer 30.5 cm ,. foam layer Figure 7.2.--Different Parts of Loading Piston. 76 -vzygwr Educ - \Imin .,:‘_;L‘ ' Ix ..cr plate fa r. ‘ - roller / 7 strain 334:0 a Umn‘l‘ I‘IUIC chr I ‘ 1mm plate Figure 7.3.——Dimensions and Assemblage of the Pressure Transducers. 77 rollers. A strain gage, (Micro Measurement EA-O6-250BG- 120) was attached to the lower surface of the upper plate. The upper plate acts as a simply supported beam and was used to determine the magnitude of a load once it was calibrated. 4. Multi-Channel Digital Strain Indicator 161- mini-system: The strain gage transducer was attached to a model 161 (B G F Instrument, Inc.) digital strain indi- cator. This apparatus had ten channels and a terminal box where the pressure transducers and their compensating gages were connected, Figure 7.4 and 7.5. 5. A model OD-1014 printer was connected to the multi-channel Digital strain indicator. Total of twenty pressure transducers were made, of which eight were used compensating gages. 7.3 Calibration of Gages of Pressure’Transducers Each pressure transducer was assembled and con- nected in a singel active arm bridge form to the strain indicator and calibrated. Calibration was done by gluing one of the rubber balls on the head of the Instron machine and locating the transducer beneath the ball on the load cell of the Instron, Figure 7.6. Different levels of load were exerted on the transducer and the corresponding strain was read from the strain indicator and a 78 -0 . [0 printer 0 1 O O 00 O 0 train indicator ch9 ch8 ch7 ch6 ch5 ch4 c113 chZ chl I I I | 0 a 2 I 3 ’ , .., ., ~ . - .1 l‘ '1' :;‘.k‘ 4 4 S 4 compansating two active arm strain 8389 Figure 7.4.--Terminal Box of Multichannel Digital Strain Indicator. 79 Figure 7.5. Mild-Channel Digital Strain Indicator with Midel 0D-1014 Printer 80 Figure 7.6. Calibration of Pressure Transducer 81 calibration curve for each gage was obtained. An example of such a curve is given in Figure 7.7. This process was repeated at least five times for each gage in order to obtain an average curve. The average calibration curve had a variation of i 3% from the other curves at a maxi= mum load of 90 N (20 lbs.). Example force and correspond- ing strain data are given in Table 7.1. 7.4 Experimental Procedure First step was to connect the eight pressure trans- ducers and the eight corresponding compensating gages to the digital strain indicator. Digital strain indicator was adjusted for Rcal = 1474 Q from the table of cali- bration set points based on 120 0 single active arms input and gage factor of 2.03. Since there was just one active gage in the pressure transducer, a compensating gage was placed on the steel plate, outside the text box and a two arm-bridge hook-up was used. 7.4.1 Pressure Transducer Placement The objective was to measure the contact forces at as many contact points in the assemblage as reasonably possible. The eight pressure transducers allowed the contact force at eight different points to be measured at a time. The attachment of the transducers on the exact contact points in the assemblage was very important. 82 3500 ‘ 3300 ‘ 3100 ' 2900 1 2700 ‘ 2500 ‘ 2300 i 2100 * Strain (um/mm) x 10-6 3 75 8 8 a U1 0 0 L 1300 1100 4 700‘ 300 . j . . . . . . 0 9 18 27 36 45 54 63 72 Load (N) Figure 7.7.--Ca1ibration Curve for Gage Six. 83 TABLE 7.l.--Values of Strain (e) at Different Values of Load (N) for Gage Number 6 Load F (N) Strain Load p (N) Strain 4.5 308 49.5 2390 9 606 S4 2552 13.5 872 58.5 2686 18 1110 63 2804 22.5 1344 67.5 2920 27 1540 72 3018 31.5 1724 76.5 3118 36 1892 81 3210 40.5 2072 85.5 3310 45 2232 90 3410 84 Transducers were located at the proper points and were glued on the surface of the balls at the designated con- tact points. The contact points should seat exactly in the middle of the upper-plates of the transducers in order to obtain a correct force value. Three loading replications were made at each contact point considered. This process was completed for eight different transducer positions in the assemblage, Figure 7.8 a-h. The two top rows, 1 and 2, are added to the arrangement to create a uniform loading effect on the balls of row three and below. To locate the pressure transducer in Figure 7.8b, for example, the balls of the bottom row were carefully placed in the test box while the plexiglass window was removed. Then transducers were positioned and glued on the surface of the balls of the bottom row (layer 7) on the lower plate side of the transducer. Balls in the other rows were placed in a manner that it was made sure the balls were in contact at the right points. As the balls of the rows were placed and arranged from bottom to tOp, the plexiglass window was slided in piece by piece To reducethe friction between the balls and the sides of the test box, a lubricant (vaseline) was used. 7.4;2_ Readjusting Digital Strain Indicator Once the balls and the transducers for a particle arrangement were in palce, the test box was placed on the 85 pressure transducers different locations of _ Figure 7.8.--Pressure Transducer Placement in Different Locations of the Assemblage. 86 load cell of the Instron and the initial strain resulting from the weight of the balls were removed by setting the strain indicator to zero. 7.4.3 Loading and StrainReading The load was applied using the Instron testing machine located in the Wood Technology Laboratory of the Forestry Department. The Instron was zero balanced and the load was applied using a head speed of 0.508 cm/min (0.2 in/min). The Instron chart recorder was moving at 1.225 cm/min (0.5 in/min). Each location required about 12 minutes to print all the strain values. A maximum load of 202.5 N was applied. Strain values were printed every one-half minute and a mark was made on the loading curve of the Instron strip-chart recorder to indicate the correspond- ing load at that moment. The strain indicator printed one channel at the time with a two second time interval (20 seconds for all channels), Figure 7.9. 7.5 Results and Discussion The measurement of the contact forces was repli- cated at least three times for each location and the average of these results are reported here. The contact forces were obtained for axial loads of 157.5, 180 and 202.5 N (35, 40 and 45 lbs). The applied forces and 87 Figure 7,9“ A Conplete Set-Up of the Sinulated Force Distribution Experiment of Two Dimensional Rhombohedral Arrangement of the Rubber Balls 88 correspoding strain values are given in Tables 7.2 through 7.4. Loads greater than 202.5 N were not applied because the pressure transducers were not as reliable in this loading range. Corresponding planar graph of the applied loads are shown in Figures 7.10 through 7.12. For comparison between calculated and measured values of contact forces the planar graph representation of calculated contact forces are repeated in this section and are shown in Figures 7.13 through 7.15. Comparing planar graph representation of calcu- lated contact forces with the measured ones for three different loadings (157.5, 180 and 202.5 N) the following was concluded. All the measured contact forces are within 20 percent of the calculated ones. A possible reason for this is that in the theory which was used to calculate the contact forces, the frictional forces were ignored. Actually there is friction between adjacent spheres and between the spheres and the walls of the test box. There is an approximate constant ratio of 1.14 between measured and calculated contact forces. If the frictional forces in the model were taken under consideration, then this constant ratio would be very small. In other words, the measured contact force values would be very close to the 89 Table 7.2-:Average measured contact forces for 157.5 N of load 157,5 Newtons of Applied Load (31.5 Newtons on each Node) Channel No. Ave. Observ Ave. Measured Contact Location of Strain -x 10 Forces (N) Transducers 0 1420 27.9 5 1560 27.9 7 1582 27.9 8 1668 27.9 9 1520 27.9 7 1150 18 O 1040 18.45 3 1050 18.45 9 1200 20.7 ' 5 1230 20.25 8 1200 18 4 1150 18 6 1100 17.55 3 1050 18.45 7 1140 17.55 8 1380 21.6 S 1050 16.65 4 1070 17.1 0 1180 21.6 6 1100 17.55 9 1100 18.45 7 1170 18.45 3 1390 26.1 5 870 13.27 4 1070 17.1 0 960 16.65 8 960 13.95 9 1430 25.65 6 1100 17.55 7 1665 30.15 3 910 15.3 S 980 15.3 4 980 15.52 0 885 15.07 8 1040 15.3 9 910 14.62 6 1635 29.25 0 9 0.22 9 8 0.23 3 4 0.21 7 200 2.44 3 160 2 38 4 215 2.38 5 195 2.43 0 290 3.94 9 170 2.3 3 265 3.92 9() Table 7.3-€Average measured contact forces for 180 N of load 180 Newtons of Applied Load (30 Newtons on each Node) Channel No. Ave. Observ Ave. Measured Contact Location of Strain x 10 Forces (N) Transducers 0 1530 30.5 5 1680 31.05 3 1792 31.05 9 1523 27.90 7 1512 30.6 7 1280 20,7 0 1175 21.37 3 1150 20.7 ' 9 1430 23.4 5 1390 23.85 8 1340 20.7 4 1250 21.15 6 1245 20.25 3 1155 20.92 7 1280 20.7 3 1550 25.2 5 1210 19.80 4 1180 19.35 0 1270 23.85 6 1245 20.25 9 1230 21.15 7 1280 20.7 3 1525 29.47 5 1020 16.2 a 1290 19.8 0 1080 19.35 4 990 15.52 g 1610 30.15 6 1280 21.15 7 1805 34.2 3 950 16.2 5 1130 18.22 4 1100 17.55 0 980 17.1 8 1185 17.55 9 1000 16.2 . 6 1815 33.75 0 12 0.27 9 105 1.38 3 4 0.24 7 215 3 3 180 2.99 ‘ 205 3.01 5 210 2.92 0 340 4.98 9 240 3.09 3 330 4.89 Egégggl 921 Table 7-4-fAverage measured contact forces for 202.5 N of load 202.5 Newtons of Applied Load (40.5 Newtons on each Node) Channel No. Ave. Observed Ave. Measured Contact Location of Strain x 10 Forces (N) Transducers 0 1574 51.95 5 1822 35,1 8 1911 34,2 9 1750 55.75 7 1849 35,55 7 1360 22.5 0 1270 23.85 5 1510 24.5 9 1450 26.1 5 1500 26.55 3 1490 25.85 4 1570 25.85 6 1550 22.05 3 1290 25.85 7 1370 22.95 8 1685 28.55 5 1295 21.5 4 1280 21.6 , 0 1420 27,90 5 1365 22.95 9 1350 23,35 7 1400 25.4 5 1715 55.1 s 1080 17.1 3 1400 22.05 0 1180 21.6 4 1075 17.1 9 1725 55.52 6 1590 25.4 7 1965 51.15 5 1070 18.90 5 1250 20.7 4 1280 21.6 0 1180 21,5 8 1510 20.25 9 1200 20.7 6 2000 58.7 o 11 0.29 9 115 1.68 3 6 0.51 1 270 -5.26 3 220 5.5 4 '250 5.5 5 240 3.28 o 400 5.83 9 235 3.48 5 370 5.46 31.5 N F.‘ 0.225 31.5 N 31.5 N F3. 0 92 F2. 0 31.5 N FI'O 31.5 N / .nr.w w. ”I «VONN e5 at F13' 3.94 F36'0.22 Fz"2.44 #4 w o - on? for a 157.5 N Load. Figure 7.10.--P1anar Graph Representation of Measured Contact Forces 93 36 N 36 N 36 N 36 N 36 N no '3' 0 '3' o I:4' ° 0, ‘0 's q ‘9 '6 . b? e. K? ’- -° 6. Q $ ~ 5 N r o 0 o - l ’ 4x ’ *4 1.. o ‘ o ,' 9. 9.9 *4 8g; «4" 0 «Q 9‘ 9‘ V V Fu’l.” I34' 3.09 1:15 4.89 Figure 7.11.--P1anar Graph Representation of Measured Contact Forces for a 180 N Load. 40.5 N 40.5 N 40.5 N 94 l0.5 N 40.5 N 2"° ’37- 1 o 6' 1"56" 0.29 Contact Forces for a 202.5 N Load. Figure 7.12.-~P1anar Graph Representation of Measured 95 31 5 N 31.5 N 31.5 N 31.5 N 31.5 N Fl-O.31 F -O.31 F3-0.31 F‘-0.31 Figure 7.13.--Planar Graph Representation of Calculated Contact Forces for a 157.5 N Load. All Values in Newtons. 96 Ff 0.35 ? ~ 0‘ ‘9 (“a 9‘ § 9, {a ‘9 V F210 3.01 ‘9 0x r ‘3' ‘9 0 5? 0". § ’9 a ‘i-‘I ~ 9‘ ~ 6 ‘? ,5 0"" O 3V ‘6\ 0 1'. '9 ‘J' Q ' ‘9‘ Q” 9‘ «a? ‘9 « V 9- E33- 0 23 1:37- 1.5 F”. 0 23 (.8 0 '5 a ' - fi e rm q 9- Q .9. ‘5‘} $5 y 1" . o Q’ ~ f. 6. '3’ W; .9 4 o ‘ ' “5 0 9' ' 9' Q§° 9‘ g§ 9' (‘5‘ g,“ V Figure 7.14.--P1anar Graph Representation of Calculated Contact Forces for a 180 N Load. All Values in Newtons. 97 60.5 N 60.5 N 40.5 N 40.5 N 40.5 N I I F I Fl- 0.40 F2 0.40 F3 0.46 4 0.40 Figure 7.15.--P1anar Graph Representation of Calculated Contact Forces for a 202.5 Load. All Values in Newtons. 98 calculated ones and error would be quite small. This indi- cates that the presented model could be more accurate than what was concluded earlier, if the frictional forces were considered. The distribution of the contact forces in the members both in calculated and measured ones were very similar. These will conclude that there is an agreement between the calculated values and the measured ones. VIII. SUMMARY AND CONCLUSIONS Development of experimental and analytical tech- niques to determine the maximum safe depth for apples in a bulk storage was the primary goal in this study. The problem was studied in two different components. One component involved the determination of the contact forces in a bulk bin while the other component related to the determination of whether a specific loading would produce a bruise. Cylindrical specimens with a height of 1.27 cm and a diameter of 1.27 cm and a cross-sectional area of 1.266 cm2 were prepared. These Specimens were then com- pressed using a deformation rate of 1.27 cm per minute until a failure occurred. Tests were conducted on three different dates, October 1, November 15 and December 31. These were denoted as Groups I, II and III. Each group consisted of 150 apples and four samples were removed from each apple. The failure strain, cf, the stress at the failure, of, and the elastic modulus, B, were deter- mined for every sample. The modulus of elasticity changed significantly between October 1 and November 15. A much smaller change occurred between November 15 and 99 100 December 31. The respective averages were EOct = 3279 Kpa, ENov = 2516 Kpa and EDec = 2360 Kpa. The strain at failure showed a small decrease between October 1 and November 15. The reduction rate was lower between Novem- ber 15 and December 31. The average maximum normal strain at failure was 0.14, 0.11 and 0.12 for the three dates. The average normal stress at failure decreased from 444 Kpa on October 1 to 252 Kpa on November 15 and 235 Kpa on December 31. The distribution of the elastic modulus and fail- ure strain were used in a computer model to predict bruis- ing for a particular load. Since this model was based on the assumption that a bruise occurs when the maximum normal strain exceeds a specific value, an equation for the maximum normal strain was determined and used in a computer model. The average location of the maximum normal strain for apple-to-apple contact was 3.65 mm for October first data and 3.0 mm for the other two sets of data. For apple in contact with a flat hard surface these values were 3.88 mm for October 1 data and 3.0 mm for the other two data. To calculate the allowable storage depth, the modulus of elasticity of an apple in contact with another (E E2) and the failure strain of the pair of apples 1’ 101 (EFl’ ch) or apple in the case of flat plate contact were treated as random variables. A normal load F was selected and a random generator was used to select the values of E E and e 1’ 2’ 5131 the experimental part. The maximum strain in each apple F2 from the data obtained in was calculated. If this strain exceeded the failure strain for the apple, the apple was said to be bruised. The load which produced a bruise was converted to a depth by assuming a single column stack where each apple weighed 0.85 N. Apple-to-apple contact was found to govern the allowable depth. The October 1 apples could be piled 5.14 meters without bruising but the November 15 and December 31 apples could be piled only about 1.8 meters. The sig- nificant decrease is due to the decrease in the modulus of elasticity between October 1 and November 15. A finite element type computer model used for small diameter steel balls was modified for use with large diameter low modulus materials such as apples. This model is basically a two-dimensional truss analysis where the center of each sphere is considered as the node and the members connecting the nodes have the nonlinear property of two spheres in contact. The validity of the model was established by experimentally measuring the contact forces between 6 cm diameter rubber balls which were stacked 102 in a rhombohedral fashion. Specially designed pressure transducers were located at the proper points and were glued on the surface of the balls in the assemblage. There were seven layers in the assemblage with either four or five balls in each horizontal layer. A maximum load of 202.5 N was applied by Instron and strain values were printed. All of the contact forces differed by less than 20 percent from the values calculated using the computer model. The following conclusions were drawn from this study. 1. The maximum allowable height for freshly har— vested Jonathan apples in contact with each other in a mono-column arrangement is 5.14 m (17.13 ft). This value decreases as the storage period increases. It was about 1.85 m (6.1 ft) after three months of storage at 2.2°C. As for apple in contact with a flat surface the maximum allowable safe depth was around 7 m for the October 1 apples and decreased to 2.2 m by November 15. 2. The modulus of elasticity of Jonathan apples decreased 23.3 percent between October 1, 1978, and Novem- ber 15, 1978. The average of 600 samples was 3279 Kpa on October 1 and 2516 Kpa on November 15. The modulus value decreased another 6.2 percent between November 15 and December 21 to 2360 Kpa. 103 3. The failure strain of Jonathan apples averaged 0.14 on October 1, 1978, decreased to 0.11 on November 15 and then increased slightly to 0.12 by December 31. 4. The average failure stress of Jonathan apples decreased significantly between October 1 and December 31, 1978. It averaged 444 Kpa on October 1, 1978, and decreased to 252 Kpa by November 15 and further decreased to 235 Kpa by December 31. The total decrease of 209 Kpa was 47.1 percent of the original value. 5. The structural model formulation for calcu- lating the contact forces within an arrangement of rubber balls disagreed with the experimental by a constant ratio. The experimental results were about 0.83 of the calculated values. 6. The susceptibility of two apples in contact to bruising is higher than that of an apple in contact with a rigid flat surface because maximum allowable safe depth for any of three groups of apples in Case 11 (apple in contact with rigid flat surface) was always higher than Case I (two apples in contact). IX. SUGGESTIONS FOR FUTURE RESEARCH Suggestions for future research based on the experiences and results of this study include: 1. The significant variation of the mechanical prOperties of Jonathan apples during the first 1.5 months of storage indicates that these prOperties should be studied on a weekly basis or at least every two weeks. 2. Allowable depth values should be calculated for other varieties of apples which may be stored in bulk, These calculations cannot be performed without the dis- tributions of the mechanical properties thus these prop- erties must be studied as a function of storage time. 3. The presence of skin was neglected in this investigation and apples were assumed to be an isotrOpic, homogeneous mass. Rumsey and Fridley (1974) found that the presence of an elastic skin produced no significant change of the internal stress distribution. Gustafson (1974), however, showed that the restraint created by the skin can cause increased stresses in the body if the turgor pressure is accounted for. The effect of skin properties on the maximum allowable contact force from one apple to another or from the apple to a rigid flat 104 105 surface, however, has to be investigated. Consideration of apple skin and core might lead to dealing with aniso- trOpic materials, which in reality is considering the whole apple not just one part of it. 4. The measurement of strain at failure under numerous deformation rates. APPENDICES 106 APPENDIX A EXPERIMENTAL DATA 107 TABLE A.l.--Mean Values of the Three Different Parameters (cf, 0 and E)--Group I 8 hte : Oct 1, 1978 (Crow 1) Samle high! 1 1.2” On Variety : Jonathan Swle Diameter : 1.2" 01: ~ bad Speed of Instron : 1.27 On/mm Suplc Cross~5ection Area : 1.266 00‘ Chart Speed : 25.4 ()V20n 0 II? a 0 0 3'73 .1 03'7“ «a Q; 8"“ 8,15 55"" a .6. 1 0.140 476.06 3528.97 41 0.15 453.42 3144.85 81 0.148 417.43 3000.52 121 0.143 448.00 3302.30 2 0.148 427.68 3048.02 42 0.144 455.74 3474.35 82 0.138 405.43 3019.34 122 0.149 463.81 3678.26 3 0.140 470.83 3713 83 43 0.151 470.45 3312.19 83 0.138 428.07 3182.77 123 0.153 453.78 3194.04 4 0.144 451.29 3443.49 44 0.146 492.37 3783.65 84 0.150 453.03 3262.02 124 0.147 371.56 3111.96 5 0.147 431.16 3081.01 45 0.150 443.74 3292.57 85 0.145 415.10 3117.08 125 0.160 526.38 3706.14 6 0.145 429.23 3098.63 46 0.150 465.78 3336.76 86 0.152 453.23 3239.8 126 0.157 441.23 3075.39 7 0.134 434.84 3453.68 47 0.146 506.83 3900.24 87 0.146 442.2 3452.48 127 0.153 459.61 3373.17 8 0.145 526.76 4259.15 48 0.153 448.97 3115.37 88 0.152 - 436.39 3451.07 128 0.163 522.45 3478.11 9 0.150 432.29 3234.06 49 0.163 416.71 2684.34 89 0.162 478.00 3170.93 129 0.162 469.24 3224.50 10 0.150 456.32 3544.02 50 0.136 439.10 3353.12 90 0.148 447.02 3469.38 130 0.145 455.71 3495.63 11 0.147 447.23 3492.96 51 0.154 419.39 3063.90 91 0.151 409.30 2990.99 131 0.148 412.16 3057.47 12 0.137 448.50 3520.85 52 0.138 453.62 3523.46 92 0.158 510.90 3487.77 132 0.157 382.16 2605.50 13 0.159 439.87 2842.89 53 0.154 425.75 3141.25 93 0.148 464.45 3273.91 133 0.150 424.75 3027.36 14 0.140 403.11 3112.12 54 0.148 484.19 3653.93 94 0.160 428.65 2950.33 134 0.147 409.30 3008.24 15 0.140 453.23 3560.57 55 0.148 424.30 3109.03 95 0.148 450.50 3441 33 135 0.162 480.90 3178.79 16 0.139 451.48 3478.90 56 0.145 472.77 3686.52 96 0.142 463.48 3341.37 136 0.153 461.54 3313.22 17 0.146 431.55 3155.89 57 0.146 449.16 2717.82 97 0.137 418.39 3169.96 137 0.160 443.16 3265.93 18 0.129 423.04 3371.31 58 0.147 440.65 3196.05 98 0.150 458.05 3359 40 138 0.155 455.74 3222.45 19 0.146 463.68 3508.34 59 0.148 415.49 3103.42 99 0.151 423.17 3229.33 139 0.153 436.39 3135.51 20 0.138 453.42 3540.18 60 0.143 408.33 3155.89 100 0.151 445.48 3236.18 140 0.145 432.52 3282.56 21 0.149 490.38 3568.23 61 0.147 473.55 3400.27 101 0.147 407.34 3247 41 141 0.146 415.10 3122.08 22 0.148 465.23 3423.26 62 0.143 467.44 3522.28 102 0.162 424.78 3186.28 142 0.144 468.32 3412.51 23 0.144 462.32 3532.53 63 0.153 431.03 3089.14 103 0.142 456.71 3393.92 143 0.145 478.97 3714.24 24 0.144 445.29 3297.57 64 0.151 450.85 3175.97 104 0.166 394.78 2843 71 144 0.146 462.50 3344.71 25 0.163 491.93 3232.02 65 0.159 471.22 3226.03 105 0.151 477.03 3483 74 145 0.161 466.39 3260.06 26 0.141 470.44 3576.53 66 0.153 469.10 3247.92 106 0.146 383.17 3117.20 146 0.151 448.97 3113.30 27 0.140 472.39 3755.21 67 0.141 404.45 3110.45 107 0.150 430.34 2976.19 147 0.153 441.94 3052.15 28 0.156 476.52 3364.4 68 0.146 411.43 3258.69 108 0.143 349.50 2949 75 148 0.137 456.69 3546.30 29 0.152 456.90 4256.27 69 0.146 423.81 3190.62 109 0.153 418.96 3121.02 149 0.143 403.49 2991.45 30 0.153 460.58 3182.95 70 0.140 425.73 3150.19 110 0.145 440.26 3507.10 150 0.150 433.49 3214.43 31 0.152 450.50 3186.36 71 0.141 441.42 3358.16 111 .0.151 392.84 2972.82 32 0.148 399.43 2944.23 72 0.151 463.29 3157.83 112 0.155 477.99 3507.06 33 0.143 441.81 3207.90 73 0.148 460.77 3159.80 113 0.141 403.49 3161.03 34 0.148 439.68 3253.27 74 0.151 420.90 2952.00 114 0.143 412.20 I 3024.48 35 0.150 424.58 3807.18 75 0.154 446.84 3122.30 115 0.147 382.58 3400.55 36 0.148 430.97 3265.71 76 0.161 439.87 2981.56 116 0.150 411.23 3117.32 37 0.143 520.77 4043.57 77 0.151 434.46 3087.90 117 0.155 433.48 3354.16 38 0.151 427.68 3205.80 78 0.138 392.65 2945.95 118 0.147 407.75 3144.50 39 0.150 456.32 3274.89 79 0.146 465.40 3366.23 119 0.145 392.65 2950.50 40 0.143 440.13 3362.80 80 0.148 484.77 3699.15 120 0.147 431.92 3321.96 108 109 TABLE A.2.--Mean Values of the Three Different Parameters (cf, 0, and E)--Group II Date : Nov 15. 19"! (Grow 11) Sanple Height : 1.27 00 Variety : Jonathan Sample Diameter 1 1.27 Cm . Head Speed of Instron : 1.27 Git/min Simple Cross-section area : 1.266 On“ Dart Speed : 25.4 Ora/min male ‘74 °n ‘4 Apple ‘774 “n ‘1. mp1. ‘11 °n ’11 mp1. ‘74 °n ‘3 8b. (“5.) mp.) ND- (x94) 06:.) lb. mu) man) No. 08:0 an) 1 0.115 258.35 2867.15 41 0.121 248.68 2506.14 81 0.125 243.84 2486.79 121 0.116 250.61 2457.76 2 0.116 271.90 3252.83 42 0.121 239.97 2296.49 82 0.120 226.42 2167.48 122 0.130 292.22 2464.21 3 0.096 241.90 2014.60 43 0.123 253.51 2670.64 83 0.108 227.39 2409.38 123 0.123 202.23 2238.43 4 0.112 297.06 3206.06 44 0.121 269.06 2534.25 84 0.113 236.10 2338.42 124 0.134 270.93 2644.38 5 0.115 249.64 2502 92 45 0.119 248.68 2429.20 85 0.111 222.55 2032.01 125 0.133 249.64 2206.18 6 0.113 254.16 2644 84 46 0.118 239.97 2406.16 86 0.123 240.93 2270.69 126 0.121 261.25 2451.31 7 0.147 323.18 2640 S4 47 0.110 271.90 2673.87 87 0.110 216.74 2183.60 127 0.128 267.06 2515.82 8 0.117 308.67 3119.59 48 0.120 316.41 2808.92 88 0.116 245.77 2290.45 128 0.121 228.36 2219.08 9 0.118 276.74 3031 88 49 0.125 238.03 2125.55 89 0.116 229.32 2306.17 129 0.104 212.87 2353.63 10 0.117 275.77 2919 00 50 0.119 278.67 2757.73 90 0.105 233.19 2509.37 130 0.123 239.97 2115.87 11 0.112 274.80 2773.85 51 0.113 257.38 2831.91 91 0.115 244.80 2535.17 131 0.126 224.48 2156.42 12 0.116 302.86 3193.16 52 0.118 223.52 2164.25 92 0.114 223.52 2244.89 132 0.116 265.12 2725.47 13 0.115 303.83 3048 01 53 0.108 234.16 2459.76 93 0.126 233.19 2090.07 133 0.129 230.29 2154.57 14 0.107 287.38 3096 40 54 0.113 306.73 3067.37 94 0.108 241.90 2451.31 134 0.123 262.22 2599.91 15 0.123 245.77 2560 o8 55 0.116 284.48 2878.68 95 0.094 243.84 3096.40 135 0.124 242.87 2238.43 16 0.125 291.25 2924 07 56 0.109 290.28 2862.55 96 0.121 235.13 2273.91 136 0.125 256.42 2361.00 17 0.111 291.55 3077 04 57 0.119 285.44 2626.41 97 0.113 232.23 2341.66 137 0.135 245.77 2061.04 18 0.114 249.61 2574.34 58 0.121 262.22 2486.79 98 0.118 249.64 2399.71 138 0.111 204.16 2118.63 1° 0.126 297.06 2739.30 59 0.133 195.46 1699.79 99 0.100 220.61 2467.44 139 0.128 214.81 2051.36 20 0.123 322.21 2955 01 60 0.121 227.39 2144.90 100 0.120 265.12 2464.21 140 0.120 252.55 2361.00 21 0.100 214.80 2372.06 61 0.113 248.6 2454.54 101 0.111 247.71 2409.38 141 0.124 226.42 2181.76 22 0.118 246.74 2528 ‘2 62 0.113 229.32 2215.40 102 0.113 231.26 2298.10 142 0.129 250.61 2423.50 23 0.113 257.38 2738.37 63 0.108 252.55 2496.47 103 0.118 205.13 2051.36 143 0.125 235.13 2148.12 24 0.125 259.32 2305 71 64 0.106 248.68 3141.55 104 0.125 290.28 2902 87 144 0.145 260.29 2315.84 25 0.120 266.09 2443.25 65 0.121 269.96 2406.16 105 0 108 227.39 2386.80 145 0.135 263.19 2352.71 26 0.133 281.57 2354.55 66 0.118 243.84 2317.46 106 0.106 243.84 2580.33 146 0.138 254.48 2354.55 27 0.114 239.00 2696 44 67 0.127 247.71 2225.53 107 0.118 254.48 2419.06 147 0.128 248.68 2379.18 28 0.129 311.57 2823.62 68 0.113 244.80 2361.00 108 0.123 223.52 2061 04 148 0.126 221.58 2161.02 29 0126 237.06 2113.56 69 0.128 270.93 2419.06 109 0.125 241.90 2277.14 149 0.133 231.45 2302.94 30 0.125 275.77 2509.37 70 0.111 219.65 2273.91 110 0.109 231.26 2457.76 150 0.139 251.58 2115.74 31 0.114 282.54 2846 43 71 0.113 249.64 2346.49 111 0.124 219.65 2080.39 32 0.119 285.44 2889.97 72 0.118 278.67 2609.36 112 0.125 238.03 2151.35 33 0.119 251.58 2788.37 73 0.104 276.74 3028.66 113 0.115 262.22 2657.74 34 0.118 222.55 2161.02 74 0.116 231.26 2219.08 114 0.120 248.68 2509.37 35 0.121 325 12 2994 79 75 0.120 263.19 2541.62 115 0.124 250.61 2247.65 36 0.121 267.06 2515.82 76 0.113 256.42 2660.96 116 0.131 228.36 2115.87 37 0.105 270.93 3212.51 77 0.126 233.19 2157.80 117 0.123 226.42 2212.63 38 0.125 248.68 2222.08 78 0.121 230.29 2132.92 118 0.121 224.48 2112.64 39 0.113 210.94 2075.99 79 0.113 239.48 2396.48 119 0.125 266.09 2477.12 40 0.119 310.6 2967.38 80 0.111 256.42 2612.58 120 0.114 217.71 2264.24 110 TABLE A.3.--Mean Values of the Three Different Parameters (sf, 0, and E)-—Group III Date imc 30, 1978 (mp 111) Samle fright : 1.2’ On \ariety : Jonathan Samle Dunner : 1.2‘ On -. Head Speed of Instron : 1.27 CIUHUn Sample Cross-eection area 1.266 Cn' Chart Spa-d : 25.4 Cthun 8831: cu °n 9,. lune ‘n °n I, up). , ‘n °n 3, Apple Ham on 2‘ '80- are» man) '40- an) an) "°- WI) ow "0- 0:9.) aw 1 0.124 270.45 2443.24 41 0.138 274.80 2603.62 81 0.141 208.52 1915.27 121 0.121 234.06 2412.15 2 0.130 220.03 2242.24 42 0.124 228.74 2139.83 82 0.121 271.41 2706.2” 122 0.128 229.81 2262.39 3 0.121 199.81 2056.20 43 0.109 253.90 2552.68 83 0.130 188.97 2381.04 123 0.129 216.74 2438.26 4 0.105 219.65 2556.14 44 0.129 202.23 2073.22 84 0.121 223.90 2184.95 124 0.134 229.81 2550.22 5 0.135 240.84 2180.38 45 0.121 230.68 2290.68 85 0.128 262.71 2641.25 125 0.121 224.00 2583.55 6 0.124 240.45 2162.64 46 0.133 221.10 2026.36 86 0.123 252.55 2577.10 126 0.120 234.64 2484.72 7 0.123 200.87 2161.02 47 0.116 258.84 3019.60 87 0.124 241.42 2434.95 127 0.135 240.93 2631.01 3 0.125 207.07 2019.11 48 0.121 209.97 2052.74 88 0.110 283.51 5096.40 128 0.115 241.90 2741.60 9 0.120 201.76 1949.76 49 0 130 233.00 2102.63 89 0.131. 205.13 2077.32 129 0.121 216.74 2333.77 10 0.143 234.46 2038.48 50 0 126 223.03 2297.82 90 0.121 225.84 2409.23 130 0.135 244.03 2532.41 11 0.113 209.39 1911.05 51 0.120 222.55 2380.35 91 0.119 201.65 2061.77 131 0.131 228.84 2218.62 12 0.125 233.68 2006.20 52 0.118 264.16 2589.08 92 0.120 231.64 2519.28 132 0.120 190.62 2411.25 13 0.116 233.98 2136.83 53 0.118 285.44 2733.54 93 0.125 216.74 2273.15 133 0.111 235.67 2609.36 14 0.125 194.49 1902.99 54 0.120 239.97 2429.04 94 0.124 222.55 2303.86 134 0.111 195.46 2068.29 15 0.136 229.'1 1957.44 55 0.120 244.32 2407.54 95 0.125 218.68 2322.00 135 0.125 250.13 2525.04 16 0.138 225.94 2068.29 56 0.119 220.42 2280.95 96 0.134 230.19 2182.91 156 0.124 248.19 2773.85 1" 0.12 229.23 2128." 57 0.124 251.87 2546.04 97 0.125 241.32 2451.31 137 0.131 268.51 2482.03 18 0.125 268.41 2335.20 58 0.116 245.77 2999.63 98 0.130 237.06 2410.99 138 0.139 283.32 2921.48 19 0.118 246.74 2185.22 59 0.125 243.35 2113.80 99 0.125 262.71 2584.88 139 0.136 206.58 2135.22 20 0.118 220.61 2215.86 60 0.124 260.29 2563.68 100 0.125 217.23 2414.58 140 0.129 195.99 1998.99 21 0.125 213.89 1858.64 61 0.121 214.81 2096.52 101 0.130 232.71 2318.84 141 0.121 214.81 2130.56 22 0.121 195.46 2075.55 62 0.125 237.06 2483.57 102 0.128 219.16 2292.47 142 0.114 218.19 2528.72 23 0.126 164.98 1678.82 63 0.120 266.09 2882.14 103 0.138 255.93 2455.54 143 0.126 234.16 2489.33 24 0.124 224.19 2197.31 64 0.128 239.00 2267.46 104 0.124 239.00 2603.37 144 0.125 241.90 2516.74 25 0.124 261.16 2477.92 65 0.121 235.61 2459.38 105 0.131 251.00 2403.70 145 0.129 244.32 2591.62 26 0.130 237.55 2386.80 66 0.131 252.55 2559.59 106 0.134 264.16 2538.86 146 0.136 239.97 2607.05 27 0.130 217.13 2179.26 67 0.136 291.25 2716.90 107 0.125 236.39 2361.46 147 0.125 239.97 2689.99 28 0.124 209.49 1969.80 68 0.120 222.07 2303.86 108 0.131 211.91 2175.42 148 0.115 231.26 2791.98 29 0.140 221.58 2028.78 69 0.131 243.84 2393.56 109 0.128 209.97 2235.24 149 0.125 275.77 2912.55 30 0.141 253.51 2685.15 70 0.120 238.03 2491.63 110 ' 0.128 234.16 2295.36 150 0.120 199.33 2004.36 31 0.120 228.74 2289.12 71 0.121 229.81 2305.26 111 0.118 236.10 2690.91 32 0.145 245.96 2478.96 72 0.116 224.48 2332.78 112 0.119 221.58 2509.37 33 0.118 240.93 2483.57 73 0.126 287.86 2890.43 113 0.111 247.71 2659.68 34 0.124 231.46 2092.48 74 0.128 259.80 2502.51 114 0.129 232.23 2394.58 35 0.149 230.77 2061.57 75 0.129 254.00 2554.53 115 0.126 250.03 2589.54 36 0.133 229.81 2239.36 76 0.120 252.55 2451.31 116 0.109 215.58 2316.78 37 0.136 243.84 2148.74 77 0.128 255.93 2422.67 117 0.123 203.68 2083.54 38 0.129 245.58 2179.56 78 0.129 242.87 2522.73 118 0.135 214.32 2153.92 39 0.129 211.32 2013.98 79 0.123 230.29 2488.17 119 0.131 249.64 2518.40 40 0.129 241.90 1981.51 80 0.115 223.03 2540.01 120 0.118 209.97 2105.16 APPENDIX B COMPUTER PROGRAM FOR CALCULATION OF HEIGHT AND PERCENT BRUISE RELATION- SHIP 111 Computer Program for Ca1culation.of Height and Percent Bruise Relation-ship, Case I, Group I : OUTPUT) L. \I F. n p C. L 1 TI 9 ‘ Cl ‘1) T .. e o “L T: 33 \I D. [I 2 N ' 1 O O 2 11 o 1 4 1.1 A. : V- I 2 . . t C; 7. 1| VI (It \I r. I S e a t 2 .n D. C D 9 t t) \I I 22:. u b I .E 9 Q 8- . . .’1H.. A Tl Tlfltl‘ 0.. r O 7J‘I1V 113CCP 4.: Q 0.11 )- til. 0/ o 1.//T T A!» 8 3.9 )2 01 o in. as C U)LL.\I .4 O 01’». D .I- .L T D «JETrU (4.. \l 31 . O— 4),).3 Tr. Sc. r. :1» //.\1( 7 OLA .U UIF 11 31.9.1111 002.40 In...) F. 08.0.2. 9 013(222. t 9 Gal‘ o 01. 98 R11 R (a )(‘101 0.115 \a TPSU: : §F966111 (9680 ? HUT U157... 9.1.9 5.7... t—.H1.c.. 9.7 t *r C... o C 9LT. 9.1 VI 7... 911*”. t A617. C rt M 2“,!) Q : 97.1.1.2. .) Turbcc o rt. 1. Tit. PwH-.PIVJ.. 1107EEKL O) O A.l/IT‘“ O] I. (.53 (r. ...-.(..+»DE(**1J 03 14771.4.10 Y4... Rluétrl .. 1;. F./1.2/1./ 11 o o oTLT Nun F.‘ 4.4. Q r4 v F. \Jrzf. 0’ o 1 91‘! nu. on. “V; .-r u. .l c 6 o) a. agt 61.. 0..” A 4.1109942U73U n. r P H ) . u. 1.1.6.4.... .133 . I. .9 ...rn.u.v..r .2... CC. 331.01.... R: . J 4 Cr... (I. r..L+AAa1r.LC .1... n. 1106 _. '1;(9127¢7v1.*i*9, Q 0.. :V 17.01;? :p .1115 9 o: t F o...?F1...1.t e a: 3.5.11.1 S :5 :UT. 9 9.5127. 011.... . ‘G.w211(77n .4.Q.(XXLT.0 INN. 0.“... (4 in..7|.a ARLI‘II‘J..RRRC:1C../AAC.LNEP\IET r. E:¢.CD.H4UM\IP.1F.F: o o: z :24.“ 0 0M H‘UI\UTCM.¢ n. M. D p4 .. ., 7. z .. : :11: :17713 5 «11.59 a. . WETAL 9 1.15:. a n... C: . 1.. .1. a 11.319? A : : : 6.....C C, Cr... 7h.-. FF, Gq1.PT.D-'UYT;£EF CC4.ADVA7.ALLEEIIIIP..E p?» CC .. q. t. “ A18 112 Conputer Program for Calculation of Height and percent Bruise Relation-ship, Case II, Group I 113 \ g e. L) 0 e H \v \ o o— I A A A O A '- 6". U D \ \ c o o 0- H H D v w \J * (. II «t q .,‘ A A [-— LJ .." A q u. u. (u 4: q '"4 L. \J O- 0 O I O H 4‘) O 8" ll Cu I— .J H g. 4.9 .3. .J c .' 0 4r r H A F 0 ll 6-1 1‘] u U V a ‘l k! U. «so A I. .1 (\g\ C; ('3 A ‘1 u] "A 1. La 0 8'- ” UCJ O H L‘. O V—‘LL.A A U! 6— 0 £1,804 41' C 0-4 'JAA 1.1.1) o M 8- v u. a 1 nouq o \ P- ‘L' 0.9 U H \J A JHF‘ mu 1 U L? 0 ‘4. O {4. .. v -.‘ 0.1)” [-4.04 6 A .3 8'0." +v a N H .JLIJH C'\ N U) V Ln. O O wn 1. D. c 2"" fit. \ 5...] A o-u—H .' u-cw 01L 0 OH :— vtLVu't 4) 4 NC H P9 2,)- LLHLJH ID (\I NM v .1:— F New 0 H >- N03 V 04: Old Ola.- ‘30 On! hic- '3' V’ N.) UIJ Cl. Q78 (PH-'3‘ Hub 8 £11 X4.) HQ? Gaza-nu.“ an 80‘ ha- Lu a «U find 00 "(:1'50 o—tvC‘msv-H‘P'iza II a Hung. 22H O 0|! 4: IL onva o N qur- o <1U)u.qu- ago-no ‘1' OH xnu Pure. a‘zvt‘acdv-L’!<¢vct OQ'*~HudQQNUuJCUHHULCLu H J L1 (.1 - -'- Hm APPENDIX C A COMPUTER PROGRAM FOR CALCULATION OF CONTACT FORCES AND NODAL DEFLECTIONS IN A TWO DIMENSIONAL RHOMBOHEDRAL ASSEMBLAGE OF SPHERES 114 APPENDIX C A COMPUTER PROGRAM FOR CALCULATION OF CONTACT FORCES AND NODAL DEFLECTIONS IN A TWO DIMENSIONAL RHOMBOHEDRAL ASSEMBLAGE OF SPHERES The following computer program was written in the basic language for the CDC 6500 to solve Equation (6.6). The given data are the transformation matrix [N] = (32 x 50) and the force matrix {P} = (32 x l). The program compares the contact forces as well as nodal deflections after each iteration with those calculated previously to determine when to half the solution. The covergence to the solution was obtained when both contact forces and nodal deflections after each iteration with those calculated previously to determine when to halt the solution. The convergence to the solution was obtained when both contact forces and nodal deflections had the same last three decimal points and it was achieved in the 18th iteration. 115 116 10053951 llODIH E(32950)yAl<32p50193<50y50)9C(50o32)9F(32932),G(SO)yF1(32932) IzonIN P<32).n<32).F2<50).N<50) 13001N0 260IRINT -c<';J;°>' 27uHEMHAT PRINT 0; :eonnr £20148 29orn1NT -E<';J;->- SQOFEHMAT PRINT E; 310HAT P=rtc 3POPRINI -P<-;J;°)- BROFEHMAT PRINT P; 300HA1 F1=1NU(F) beFHINT 'F1('$J;')' 3600IHHAT PRINT F1; 370MAT D:F1¥P xaoxr lNIlJ/3)£§ J/3 THEN 41o 390PPINT-n<-;J;->' 400nfiI PRINT 0; aloNAT G=C40 420FOR 1:1105 430F?(I)=“8.?6KSQR(ABS(G(I)**3)) 440HEXT I 4SOIF INT(J/3)£} J/3 THCN49O 460PFINT -P2<-;J;°)' 470flAT PNINT P2; 480PPINT-G(-;J;')' avOPENNAT PPTNT G; sooroP IaITuso 5100(Ivl)=78.26480R(ABS(G(I))) 520NEXT I SIOPRINT -a<-;J;°)' uaoPINNAT PRINT 8; 550J=J+1 56060 To 250 570PEN NOTHING APPENDIX D CALCULATION OF LOCATION OF MAXIMUM STRAIN (Z) IN A SINGLE APPLE UNDER CONTACT LOAD 117 APPENDIX D CALCULATION OF LOCATION OF MEXIMUM STRAIN (Z) IN A SINGLE APPLE UNDER CONTACT LOAD Case I: Two Apples in Contact In this case two apples are in contact with two different modulii of elasticity, E1 and E2. Measured values of strain at failure (cf) and the modulus of elas— ticity of 150 apples were used to determine the depth at which the maximum strain occurs (Z). The following computer program calculates the value of Z for different loads (F). The program is set up in such a way that each time two modulus of elasticity values and their coupled corresponding strains at failure is randomly chosen (out-off 150 available data for each testing date. This is done by RANF (-l) in the program). Substituting randomly chosen El and E2 in the place of E in Equation 5.2 will give a and e 2, respectively. maxl max The program compares these two values (calculated maxi- mum strain at failure) with the actual (measured) values of strain at failure and counts the value of calculated strain if it is larger than the measured one. This proc- ess is repeated for different depth (2) from 0.1 mm up 118 119 to 5 mm. Each time maximum strain and its corresponding Z value under a given load (P) will be printed. Figure 5.1 shows the relation between depth (Z) and maximum strain Emax under different loads (F). 'It was found that the maximum strain for apples group I occurs aroundZ = 3.56 mm and for group II and III it is around Z = 3.0 mm. 120 UTPUT) A... : I». \l r- . D r. 2.. 1 T 9 Q 1 \l.) T .. o o “J. T! 13‘ \J \I \I p I, n...— 3 7\. N 9 1. o o 2 o o I o \l .9 111 L a n _. Y. T. n . . i .1 .1. C T ’\ VI (l\ ’ a». P. L T. S a t t a. . 8 9 D. r... F 9 a t 2 1. l 2):. .1 . 4 L I or. G. .0. .w o o 021.? : . : T T.-.( .4. a D3)... IICCC 1.2. A. o (1. )0 *6/01. :1. I? .1 v T. AA 9 1:4; \I) .1. O *p,DK.DL AG 64 U1LL 1 .4 0 0.1/5. .32.. . ...... MT. v. nu. rr.?l., (1.. \l 1.31!- O- I.»\l\:)C. S Q. T. CI» C _. )n» lull 1 ... 7 e'Hb. p {u C. ”1.? 1 .4 9.17 I of I. Q 1A.... -\l EC. rt 0(va 8711(229tt a 2‘ o 01 t t 4?. R1. 9.. (“52402.1 n 1. 09.1.15 9) 9 T C 1v: 2 abut a! i i .1131 f 9 ((rr VAA... \A 6. .10 r::.* 625:... V- F... 61107.. 25. 1.11.126) .. a... 8.. «.1. c. a-.. £916.. ./ 9... v. 9 TSCC 9.2. F35... \ Cbmputer Program f0r Calculation of Z in Case I, Group I 8‘1DLIP‘I \o r T a PT. 11n.r.::.L 0\ o fill/Tr] or] 1.: .8 (C. T. €..+C.P( t is: 01.... i“77|—S 8T3 0 F1.P..C_FI$ 9 IPr./10,._/1.,/c. 11 o o 0L... 1:...» ”LII ..4 I\ Q *1 Yr... \1 LC. 0, I Q 1 00 I\ r... or... o. ._ Tl A I. Z O\. i a: 6 i 1 0. 41¢ 4L1.......Q1 9.1.). .1 t H. C . :1..1.r r. a...57.... . z . o..¢:.u..u-.T.T1v/T..4 T... T5... :f . .1: .3. 4;... I .. ..I_......H4L.P37 .3... u Ir. 9 co. I F 01.»..r :2... t a * 3.71.13 (.1 («Uh (UUU ' 6.31:... .,1. T. w ”7317.21... fit...... ....nu’l\xv..t. T E. YTV. N.. N On I. 07.!— 14((( RFR. *c./..LAHLTI..EYLT.T.T. P. .h :80“ TIM.“ IRIEV... 0 02:27.41... 0 0M»...I...8V.I\Hu .\ 7......TI r» H. u. L :91... 2 Z : z 3.: :1». l 7. an 1C; C» T._rr.. vim.» u.\nil .J :— Ti. .57 n. n... 92.1.». 1.1.17. 21.4-1... .1..- 2 2 2... HQ! F (LP. C Jun. ambit. r ...- R r i" Pr. DYTLLLPLC CCAITEAW7LA R....r.E.19 F IPFF.(#L.C.. CC Al. 2.;.. “a... .PI C. . p . 1.412 5.2 and and the Mpa (30 3.88 mm 121 Case 11: Apple in Contact with Flat Surface In this case the computer program uses Equation considers R1 of the flat surface to be inifity modulus of elasticity of enamel steel as 206850 6 x 10 Psi). Z for group I was determined to be and the other groups were around 3.0 mm. Conputer Program for Calculation of Z in Case II, GmLIp I : 122 ._ U \ o H I A A A 0 A i"- P’) U 2 \ \ C. o 0 V" '- v-I D v v C a :- H c q (a A 5 P U u"; A q a m N o q H .J U D- o i l O H c. o '- II C? 1"! J H O v A C. J «a v". .2 0 3' F o u-q A [s o c: H a— (J v F- J1 v 1" . LL “J W “U A O J. G. f\\ D D ‘ q L...‘ --Oa\ :- C. II F'- v ~ (4 d ‘0 3'4 0 thA A x O- .» gca 21‘ 0 4 Jan L14) 0 P) f- 2. man coo-m o u O-L'TU) (‘JS'V F1 :3 L. Zv-vH UM." v 0 LA CV o C. 0. V c 099—. b 54 C A 9 Fun +~ A N x :JLJH _C‘\ 0: mm") CL 0 O Q‘A u. 2.x 0 2AA n h? \ man H-JHC L‘) 919(- .x o 0“." VLF-VLF H O BOD H PU) O zfiu—n .- N h‘fibfi V .11 i "i'vv 0 «ed >- N32 0 U ad" 2.. .‘ut N va- 4: :1: ~13 v) w 0—4 a: C: I Zv-tHC‘ sun 0 C. x—-- 32’~1:;.J;’:—1H ION {tr-v.4 < run.) Uc‘"? ’Ci-Vsrfiv-‘Y‘H‘D :I z 01!ka 31H. 03.0w I. crew's ' 0 N LOQVZDD «wane—u t—ozcu—ccao—q act-no. t—ZL’; adv;«~<fic¢~(orv1cngh<~~w 220N222~1~u1~fi~\9~v21hbb CL: lit-vz- n I: Must HP~.V)A H; «4.2:: QHULZQLDCNHHH' :.'-II I11 on. mascoz kgzazguL3>-~.J<:.L.'gu:.a'J-4uu.uuuza - H N can: :dk (L1 F:HC‘J APPENDIX E LOADING OF CYLINDRICAL SPECIMENS AT DIFFERENT DEFORMATION RATES 123 APPENDIX E LOADING OF CYLINDRICAL SPECIMENS AT DIFFERENT DEFORMATION RATES To determine the effect of deformation rate on strain and stress at failure, a separate test was con- ducted. This time 20 apples were chosen and rates of deformation, 2.5 cm/min (1.0 in/min), 1.25 cm/min (0.5 in/min) and 0.5 cm/min (0.2 in/min) were used. The test was done on four different groups of apples. Group I were tested on October 1 (immediately after harvest), Group II were tested on November 15 (1.5 months of2.2°C storage), Group III were tested on December 30 (3 months of 2.2°C storage), and Group IV were tested on Febaruary 15 (4.5 months of2.2°C storage). All apples which were stored at 2.2°C were removed from storage 24 hours prior to testing in order to get to the equilibrium with the laboratory temperature. Table E1 gives the mean and standard deviation of stress and strain at failure for this test. 124 12215 603.269 3.983 «33:05 a U 002:0; .360 8%.3 8830 mo 05.8... .0... 5A: A0303. A: 0:80 .3032 032. 0%.: 8230 no 20:2. n 5A: 258... AA: 996 602:2 0300 Sofia 8830 mo 05:0.- m.A 5A3 «.33... A: .596 60262 .850 33.325 v33... AA 996 A 020: AAA0A000A0 00 00A00020000000 A00x0 0000A_ AAA AA A 0A AAA AA A >A AAA AA A 00000 51?. as. 62353 v.32 5325830 unopommflo. 085 um 30:00:02 03mm? 0390 mo «mop 5038950 11.3 3an LI ST OF REFERENCES 126 LIST OF REFERENCES Apaclla, R. 1973 Stress analysis in agricultural products using finite element method. Unpublished technical research report. Agricultural Engineering Department, Michigan State University. bittner, D. R., H. B. Manbeck and N. N. Mohsenin 1967 A method for evaluating cushing material used in mechanical harvesting and handling of fruits and vegetables. Transactions of ASAE, 10(6):7ll. Boussineseq, J. 1885 Applications des potentiels a L'Etude de I'Equilibre et du Movement des Solids Elasti- ques, Paris. Brown, R. L. and J. C. Richards 1970 Principles of powder mechanics, essays on the packing and flow of powders and bulk solids. Pergamon Press, London. Burton, C. L. and B. R. Tennes 1977 Forced-air cooling of apples in a bulk stor- age. 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