Development of computational method to determine maximum void ratio of sand-soil particles using gravitational sphere packing method

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(1)UNIVERSITI PUTRA MALAYSIA DEVELOPMENT OF COMPUTATIONAL METHOD TO DETERMINE MAXIMUM VOID RATIO OF SAND-SOIL PARTICLES USING GRAVITATIONAL SPHERE PACKING METHOD. MOHAMMAD MAHDI ROOZBAHANI. FK 2012 137.

(2) DEVELOPMENT OF COMPUTATIONAL METHOD TO DETERMINE MAXIMUM VOID RATIO OF SAND-SOIL PARTICLES USING GRAVITATIONAL. U. PM. SPHERE PACKING METHOD. T. By. ©. C. O. PY. R. IG. H. MOHAMMAD MAHDI ROOZBAHANI. Thesis Submitted to the School of Graduate Studies, Universiti Putra Malaysia, in Fulfilment of the Requirements for the Degree of Master of Science October 2012.

(3) PM. DEDICATIONS. ©. C. O. PY. R. IG. H. T. U. I would like to dedicate this project to all those who have helped me to complete it..

(4) Abstract of thesis presented to the Senate of Universiti Putra Malaysia in fulfilment of the requirement for the degree of Master of Science. PM. DEVELOPMENT OF COMPUTATIONAL METHOD TO DETERMINE MAXIMUM VOID RATIO OF SAND-SOIL PARTICLES USING GRAVITATIONAL SPHERE PACKING METHOD. U. By. MOHAMMMAD MAHDI ROOZBAHANI. H. T. October 2012. R. Faculty: Engineering. IG. Chairman: Professor Bujang Bin Kim Huat, PhD. Abstract: These days, there has been an increasing interest in virtual reality sim-. PY. ulation of the geotechnical field. More recently, literature has emerged that offers linear experimental Equation which positively correlates the maximum and minimum void ratio of sandy-soil particles. However, far too little attention has. O. been paid to virtually simulate one of this parameter in order to rapidly compute. C. these main parameters of the relative density without any need to do experimental works. Gravitational sphere packing which arranges non-overlapping spheres. ©. within a confined space has powerful capability to simulate the interaction of large number of particles. The goal of this research is to employ gravitational sphere packing to compute maximum void ratio of sandy-soil particles with particle size distribution as input data. In this study, pre-processing, processing, and post-processing codes of the opii.

(5) timized gravitational sphere packing were developed in MATLAB software. In pre-processing stage, the initial position of the spheres was determined in (x-y) plane by applying different random number distributions. At the stage of processing, three manners of rolling, collision, and stability conditions were considered. PM. to pack particles for each incoming sphere within the cylindrical container. The. post-process code computes porosity of packed particles through the cylindrical. container for both with and without boundary effects. The simulated model was. U. validated by ASTM D 4254 standard method for measuring the maximum void ratio in metal cylindrical container, in which soil particles with different types of. T. sorting are employed for comparison.. H. Results showed that maximum void ratio of dry sandy-soil particles was computed. IG. by considering wall-sphere interaction condition in gravitational sphere packing. However, deactivation of boundary effects led to calculate the porosity of the. R. unconsolidated wet sandy-soil particles. Both above mentioned results had close values in agreement with the experimental works. The study also revealed that. PY. log-normally particle size distribution provides the lowest value of the porosity. In addition, particles had high potential to be packed in square container with the. O. lowest value of the porosity.. All in all, the new presented method of optimized gravitational sphere packing. C. rapidly simulates arrangement of the soil particles in dilute suspension. The rapid code implementation of this method can compute the main parameters of the. ©. relative density with high accuracy which was verified by ASTM D 4254 standard test.. iii.

(6) Abstrak tesis yang dikemukakan kepada Senat Universiti Putra Malaysia sebagai memenuhi keperluan untuk ijazah Master Sains. PM. PEMBANGUNAN SUATU KAEDAH PENGIRAAN BAGI MENENTUKAN LOMPANG MAKSIMUM PARTIKEL-PARTIKEL TANAH BERPASIR DENGAN MENGGUNAKAN KAEDAH PEMBUNGKUSAN SFERA GRAVITI Oleh. T. Oktober 2012. U. MOHAMMAD MAHDI ROOZBAHANI. H. Pengerusi: Professor Bujang Bin Kim Huat, PhD. IG. Fakulti: Kejuruteraan. R. Pada masa kini, minat terhadap simulasi realiti maya dalam bidang geoteknik se-. PY. makin meningkat. Baru baru ini, kesusasteraan menawarkan persaman eksperimen linear yang mana menghubungkaitkan nisbah lompang maksimum dan minimum partikel tanah pepejal secara positif. Walaubagaimanapun, perhatian yang amat. O. sedikit telah diberikan kepada kaedah simulasi maya untuk mengira salah satu parameter tersebut bagi secara pantas mengira parameter-parameter penting bagi. C. ketumpatan relatif tanpa menjalankan sebarang eksperimen. Kaedah pembungkusan sfera graviti yang menyusun sfera-sfera yang tidak bertindih di dalam ruang. ©. yang terbatas mempunyai kebolehan untuk menjalankan simulasi bagi sebilangan besar partikel-partikel. Matlamat penyelidikan ini adalah untuk menggunakan pembungkusan sfera graviti untuk mengira nisbah lompang maksimum bagi tanah zarah berpasir dengan menggunakan taburan saiz partikel sebagai data input. Di dalam penyelidikan ini, pre-pemprosesan, pemprosesan serta pemprosesan pasca iv.

(7) bagi pembungkusan sfera graviti yang mengoptimumkan telah diperolehi dengan menggunakan perisian MATLAB. Pada peringkat pre-pemprosesan, kedudukan awal sfera-sfera di plane (x-y) telah ditentukan dengan menggunakan nombor taburan rawak yang berbeza. Pada peringkat pemprosesan, tiga kaedah jatuh,. PM. perlanggaran dan keadaan stabil telah diambilkira bagi membungkus setiap sfera yang masuk ke dalam silinder. Kod yang digunakan pada peringkat pemprosesan. pasca pula mengira keporosan partikel-partikel yang dibungkus melalui silinder. U. bagi keadaan kesan dengan sempadan serta keadaan tanpa kesan sempadan. Model. simulasi telah disahkan oleh piawai ASTM D 4254 yang digunakan bagi mengukur. T. nisbah lompang maksimum dalam bekas silinder , di mana partikel-partikel tanah. H. di dalam susunan yang berbeza telah dibandingkan.. IG. Keputusan menunjukkan nisbah lompang maksimum bagi partikel tanah zarah berpasir yang kering, telah diukur dengan mengambil kira interaksi dinding sfera. R. di dalam pembungkusan sfera graviti. Walaubagaimanapun, penyahaktifan kesan sempadan telah membawa kepada pengiraan keporosan partikel tanah zarah. PY. berpasir yang basah dan tidak tepu. Kedua-dua keputusan yang dibincangkan, mempunyai nilai-nilai yang hampir sama dengan keputusan yang diperolehi melalui eksperimen. Penyelidikan ini juga mendedahkan taburan saiz partikel log-biasa. O. mempunyai nilai keporosan paling rendah. Partikel-partikel tersebut juga mem-. C. puyai potensi yang tinggi untuk dibungkus di dalam bekas segiempat sama dengan nilai keporosan yang paling rendah.. ©. Rumusannya, kaedah baru yang mengoptimumkan pembungkusan sfera graviti ini mensimulasikan dengan pantas kedudukan partikel-partikel tanah di dalam penggantungan yang cair. Kod pantas bagi kaedah ini boleh digunakan bagi mengira parameter-parameter penting bagi ketumpatan relatif dengan ketepatan yang tinggi yang telah disahkan oleh Ujian Piawai ASTM D 4254. v.

(8) ACKNOWLEDGEMENTS. My utmost gratitude goes to God. I would like to express a special thanks to. PM. my academic advisor Professor Bujang Bin Kim Huat. It was a well rewarding experience to have been under his supervision. I deeply appreciate his patient instructions and continuous support he has given me during my study and research.. U. I am also graceful to my supervisory committee member, Dr. Haslinda Nahazanan. I warmly thank Dr. Afshin Asadi for his generous contributions and effective. T. advices to my research. I indeed appreciate Professor Hasanah Mohd Ghazali for. H. her kind supports during the difficult moments of my research.. IG. On a personal note, I am thankful of Professor Lori Graham-Brady for her kind, supportive, and valuable helps during my research.. R. I would especially like to thank my wife, Elnaz Younessian, for her continuing support and patience. I owe my loving thanks to my parents Hoeseinreza and. ©. C. O. PY. Fatemeh Roozbahani.. vi.

(9) PM. I certify that a Thesis Examination Committee has met on 11 October 2012 to conduct the final examination of Mohammad Mahdi Roozbahani on his thesis entitled “Development of Computational Method to Determine Maximum Void Ratio of Sand-soil Particles using Gravitational Sphere Packing Method” in accordance with the Universities and University Colleges Act 1971 and the Constitution of the Universiti Putra Malaysia [P.U.(A) 106] 15 March 1998. The Committee recommends that the student be awarded the Degree of Master of Science. Members of the Thesis Examination Committee were as follows:. U. Ahmad Rodzi Mahmud, PhD Associate Professor Faculty of Engineering Universiti Putra Malaysia (Chairperson). IG. H. T. Abang Abdullah Abang Ali, PhD Professor Faculty of Engineering Universiti Putra Malaysia (Internal Examiner). PY. R. Zainuddin Md. Yusoff, PhD Senior Lecturer Faculty of Engineering Universiti Putra Malaysia (Internal Examiner). ©. C. O. Lori Graham-Brady, PhD Professor Faculty of Engineering Johns Hopkins University Country USA (External Examiner). SEOW HENG FONG, PhD Professor and Deputy Dean School of Graduate Studies Universiti Putra Malaysia Date: vii.

(10) This thesis was submitted to the Senate of Universiti Putra Malaysia and has been accepted as fulfilment of the requirement for the degree of Master of Science. The members of the Supervisory Committee were as follows:. PM. Bujang Bin Kim Huat, PhD Professor Faculty of Engineering Universiti Putra Malaysia (Chairperson). ©. C. O. PY. R. IG. H. T. U. Haslinda Nahazanan, PhD Senior Lecturer Faculty of Engineering Universiti Putra Malaysia (Member). BUJANG BIN KIM HUAT, PhD Professor and Dean School of Graduate Studies Universiti Putra Malaysia Date: viii.

(11) DECLARATION. I declare that the thesis is my original work except for quotations and citations. PM. which have been duly acknowledged. I also declare that it has not been previously, and is not concurrently, submitted for any other degree at Universiti Putra. IG. H. T. U. Malaysia or at any other institution.. R. MOHAMMAD MAHDI ROOZBAHANI. ©. C. O. PY. Date:. ix.

(12) TABLE OF CONTENTS. Page ii iv vi vii ix xii xiii xvii. T. U. PM. ABSTRACT ABSTRAK ACKNOWLEDGMENTS APPROVAL DECLARATION LIST OF TABLES LIST OF FIGURES LIST OF ABBREVIATIONS. IG. 1 INTRODUCTION 1.1 Motivation of the Research 1.2 Research Objectives 1.3 Scope of the Research. H. CHAPTER. 1 1 3 3 5 5 5 5 6 7 13 13 14 16 21 22. 3 METHODOLOGY 3.1 New Gravitational Sphere Packing Method 3.1.1 Spatial Binning 3.1.2 Computational Method 3.2 Rectangular Container 3.2.1 Computational Method in Different Random Numbers Distribution 3.3 Cylindrical Container 3.3.1 Computational Method in Cylindrical Container. 23 23 23 24 28. ©. C. O. PY. R. 2 LITERATURE REVIEW 2.1 Introduction 2.2 Soil Inner Structure 2.2.1 Fabric of the Soil Particles 2.2.2 Particle Associations in Soils 2.2.3 Porosity 2.3 Particle Packing 2.3.1 Different types of packing 2.3.2 Equal-sized Sphere Packing 2.3.3 Gravitational Sphere Packing 2.3.4 Close Sphere Packing 2.4 Conclusion. x. 30 39 39.

(13) 3.4. Experimental Works 3.4.1 Test Sieves 3.4.2 Specific Gravity 3.4.3 Maximum Void Ratio. 43 44 44 46 52 52 52 55. PY. R. IG. H. T. U. PM. 4 RESULTS AND DISCUSSIONS 4.1 Model Validation 4.1.1 Validating the elimination of the boundary effect 4.1.2 Mono-sized sphere packing 4.2 Effect of rectangular container’s sides on porosity for equal-sized sphere packing 4.3 Effect of Different Random Numbers Distribution on Porosity of Spherical particles 4.3.1 Mono-sized sphere packing with different distributions 4.3.2 Multi-sized sphere packing with different distributions 4.4 Total Porosity Calculation of Unconsolidated Sandy Soil with New Gravitational Sphere Packing Method 4.4.1 The occurrence of variations in porosity by getting spheres away from walls edge 4.4.2 Porosity variation in packing of bio-sized spheres through the height of container 4.4.3 Porosity of unconsolidated sandy soil in different sorting coefficient 4.5 Calculation of Maximum Void Ratio of dry sandy soils Based on Developed Gravitational Sphere Packing Method 4.5.1 Results of experimental works 4.5.2 Maximum void ratio. O. 5 CONCLUSIONS AND FUTURE WORK 5.1 Conclusions 5.2 Future Work. 60 60 62 67 67 80 88 90 90 97 98 98 100 101 107 109. ©. C. REFERENCES BIODATA OF STUDENT LIST OF PUBLICATIONS. 57. xi.

(14) LIST OF TABLES. Table Characteristics of different ideal packing structure. (Adapted from Dullien (1992); Mitchell and Soga (2005)). 15. PM. 2.1. Page. Sorting specifications. (Adapted from Beard and Weyl (1973)). 51. 4.1. Porosity values with the wall effects in the cylindrical container. 56. 4.2. Porosity values without the wall effects in the square container. 57. 4.3. Packing porosity for mono-sized spherical particles. 62. 4.4. Porosity for different distributions of multi-sized sphere packing. 67. 4.5. The statistical mode of filtered porosity results in different inner cylinder containers for mono-sized sphere packing. 72. 4.8 4.9. T. H. IG. 88. Porosity of sandy-soil particles in the different domains of the sorting coefficient. 89. Sieve analysis. 90. Specific gravity parameters. 92. R. 4.7. The minimum porosity results through the height of cylindrical container for the biosized sphere packing. PY. 4.6. U. 3.1. O. 4.10 Characteristics of different particle size distributions. ©. C. 4.11 Comparison between the results of the simulated model and experimental works. xii. 96. 97.

(15) LIST OF FIGURES. Figure. Page. Fabric of soil particle (Adapted from Mitchell and Soga (2005)). 6. 2.2. Particle association (a) Individual clay platelet interaction, (b) individual silt or sand particle interaction, (c) clay platelet group interaction, (d) clothed silt or sand particle interaction, and (e) partly recognizable particle interaction. (Adapted from Mitchell and Soga (2005)). 8. U. PM. 2.1. Different pore structure (Adapted from Korkut (2003)). 10. 2.4. Structure of the MIP.(Adapted from John I. D’Souzae (2008)). 12. 2.5. Ideal packings of uniform spheres: (a) simple cubic, (b) cubic tetrahedral, (c) tetragonal sphenoidal, (d) pyramidal, and (e) tetrahedral. (Adapted from Dullien (1992); Mitchell and Soga (2005)). 3.1. Elimination of boundaries by periodic boundary condition for Square Container. (Adapted from Jodrey and Tory (1979)). 21. Spatial binning of the container. 25. The geometrical method for making one sphere tangent to the top of three spheres. 27. O. 3.2. 20. R. 2.7. 3.3. Spherical cap. 30. 3.4. Plan view of boundary effect elimination via square container. 31. 3.5. Side view of boundary effect elimination via square container. 32. 3.6. Program flow chart of rectangular container. 33. 3.7. Program flow chart accompanied by applying different random numbers distribution. 34. 3.8. Normal distribution histogram. 35. 3.9. Exponential distribution histogram. 36. C ©. 16. Elimination of boundaries by periodic boundary condition for Cylindrical Container (Adapted from Mueller (1992)).. PY. 2.6. IG. H. T. 2.3. xiii.

(16) 37. 3.11 Arbitrary distribution of data in the square container. 38. 3.12 Arbitrary distribution histogram. 38. 3.13 Intersection volume with cylindrical container. 41. PM. 3.10 Log-normal distribution histogram. 42. 3.15 The main sample of soil provided in UPM Geotectonical Lab. 45. 3.16 The order of the sieves according to Table 4.8 in UPM Geotectonical Lab. 46. 3.17 Mechanical shaking the sieves. 47. T. U. 3.14 Complete program flow chart within the cylindrical container. H. 3.18 Evacuating the pyknometer. IG. 3.19 Pyknometer inside of desiccator containing anhydrous silica gel. 48 48 49. 3.21 Loosely pouring sandy soils in the standard mold. 49. R. 3.20 Metal cylindrical container. 4.1 4.2. 54. Average coordination number of square containers from Fig. 4.1. 55. x-y plane of rectangular boundaries. 57. Effect of container size on porosity. 58. ©. C. 4.4. 50. 100 square containers within the main cylindrical container. O. 4.3. PY. 3.22 Carefully trimming the soil surface with a straightedge. 4.5. Increase proportion of porosity according to ratio of containers sides 59. 4.6. x-y plane of the model. 59. 4.7. x-z plane of the model. 60. 4.8. x-y-z plane of the model. 61. 4.9. Mono-sized sphere packing with normally distributed initial (x − y) coordinates of spheres. 63. xiv.

(17) 64. 4.11 Mono-sized sphere packing with log-normally distributed initial (x− y) coordinates of spheres. 65. 4.12 Mono-sized sphere packing with arbitrary distributed initial (x − y) coordinates of spheres. 66. 4.13 Multi-sized sphere packing with a normal distribution. 68. 4.14 Multi-sized sphere packing with an exponential distribution. 69. U. PM. 4.10 Mono-sized sphere packing with exponentially distributed initial (x − y) coordinates of spheres. 70. 4.16 Multi-sized sphere packing with an arbitrary distribution. 71. T. 4.15 Multi-sized sphere packing with a log-normal distribution. H. 4.17 Mono-sized sphere packing for spheres with radius 1. IG. 4.18 Mono-sized sphere packing for spheres with radius 0.5. 73 74. 4.20 Porosity variations of inner containers for un-filtered data of spheres with radius 1. 75. 4.21 Porosity variations of inner containers for un-filtered data of spheres with radius 0.5. 76. 4.22 Porosity variations of inner containers for un-filtered data of spheres with radius 0.25. 77. O. PY. R. 4.19 Mono-sized sphere packing for spheres with radius 0.25. C. 4.23 Porosity variations of inner containers for filtered data of spheres with radius 1. ©. 73. 78. 4.24 Porosity variations of inner containers for filtered data of spheres with radius 0.5. 79. 4.25 Porosity variations of inner containers for filtered data of spheres with radius 0.25. 80. 4.26 Bio-sized sphere packing with radius 1 and 0.5 in (x-y-z) plane. 81. 4.27 Bio-sized sphere packing with radius 1 and 0.5 in (x-z) plane. 82. xv.

(18) 83. 4.29 Bio-sized sphere packing with radius 1 and 0.25 in (x-z) plane. 84. 4.30 Porosity variations through the height of container for un-filtered data of spheres with radiuses 1 and 0.5. 85. PM. 4.28 Bio-sized sphere packing with radius 1 and 0.25 in (x-y-z) plane. 86. 4.32 Porosity variations through the height of container for filtered data of spheres with radiuses 1 and 0.5. 87. U. 4.31 Porosity variations through the height of container for un-filtered data of spheres with radiuses 1 and 0.25. T. 4.33 Porosity variations through the height of container for filtered data of spheres with radiuses 1 and 0.25. 91. H. 4.34 Particle size distribution. 88. 4.35 Particle size distribution for well sorted soil particles with Sc = 1.19 93. 94. 4.37 Particle size distribution for poorly sorted soil particles with Sc = 2.35. 95. 4.38 Particle size distribution for very poorly sorted soil particles with Sc = 3.27. 96. ©. C. O. PY. R. IG. 4.36 Particle size distribution for moderately sorted soil particles with Sc = 1.68. xvi.

(19) LIST OF ABBREVIATIONS. Volume of voids. VS. Volume of solids. VT. Total volume. φ. Porosity. Dr. Relative density. emax. Maximum void ratio based on a very loose state of the particles. emin. Minimum void ratio based on a very dense state of the particles. e. In situ void ratio. P. Pressure. γ. Surface tension of the liquid. θ. Contact angle of the liquid. d. Diameter of the capillary. Vb. U. T. H. IG. R. O. P2. Bulk volume of the sample Volume of the vessel containing the sample. PY. VB Va. PM. VV. Volume of the evacuated vessel Initial pressure Final pressure. N. Coordination Number. C. P1. Pre-settled sphere centers. CS. Coordinate of the center of the incoming sphere. ©. C1 , C2 , C3. r1 , r2 , r3. Radiuses of pre-settled sphere. rS. Radius of incoming sphere. C1s. Distance of the two touched spheres. h. Height of spherical cap. a. Radius of base circle xvii.

(20) Variance of data. µ. The mean of data. r. Sphere radius. R. Container radius. b. Smallest distance of the axis of the cylinder to the center of the sphere. Vi. Volume of the intersection when b < R. Vc. Volume of the intersection when b > R. A. Largest root. B. Second largest root. C. Smallest root. K(k). First complete elliptic integral. E(k). Second complete elliptic integral. Π(k, −α2 ). Third complete elliptic integral. z. The sine of the amplitude. α. The modular angle. ri. T. H. IG. R. O. MAD. The parameter. PY. k wi. U. PM. σ. Residual of the ith data produced by regression smoothing method Robust weight function. Median(|ri |) Density of the liquid (M g/m3 ). m1. Mass of bottle(in g). m2. Mass of bottle and dry soil (in g). m3. Mass of bottle, soil and liquid (in g). m4. Mass of bottle when full of liquid only (in g). Gs. Specific gravity. ©. C. ρL. xviii.

(21) CHAPTER 1 INTRODUCTION 1.1. Motivation of the Research. PM. These days, simulations of the experimental works in the virtual environment. have been dramatically increased in the geotechnical field because of the rapid data access and observing inner variations of the soil particles finely. To this end,. U. many methods have been extended such as sphere packing techniques (Tory et al.,. 1973). Such these methods can simulate laboratory works in geotechnical field.. T. Sphere packing is an arrangement of non-overlapping spheres within a container.. H. This method is widely employed in two different structures which are statistical geometric and numerical. The former one is used in models with a small number. IG. of the elements, whereas the latter one is applied for a large number of elements. The random loose sphere packing was exercised in this research with the statistical. R. geometric procedure which is founded by Tory and Church (Tory et al., 1973).. PY. This method has been updated to be utilized in different fields such as powder and nano-technology, chemical, and geological engineering.. O. The content of this research is concerned with loosely packing of the sandy-soil particles with gravitational sphere packing method. However, different applica-. C. tions of this method have been used to compute an ideal system with low value of the porosity. Thus, a sever optimization is needed to get close this method with. ©. reality conditions in the experimental works of the geotechnical field. Maximum void ratio, emax , of the sandy-soil particles is defined by this method. emax is one of the main parameters of the relative density, Dr , based on the. Equation 1.2. Void ratio, e, is the ratio of the volume of voids to the volume of solids according to Equation 1.1. More recently, literature has emerged that.

(22) offers linear experimental Equation which positively correlates the maximum and minimum void ratio, emin , of sandy-soil particles (Yilmaz and Mollamahmutoglu, 2009; Cubernovski and Ishihara, 2002). However, far too little attention has been paid to virtually simulate emax in order to rapidly compute these main parameters. PM. of the relative density (emax and emin ) without any need to do experimental works.. Therefore, this project comes to this end to simulate one of these parameters, emax , with particle size distribution as input data in order to rapid data access to these. U. parameters in a virtual environment.. VV VT. T. VV φ V = ;φ = e= V = VS VT − VV 1−φ. PY. VT = total volume. R. VS = volume of solids. IG. VV = volume of voids. H. where;. (1.1). O. φ = porosity. Dr =. emax − e emax − emin. (1.2). C. where;. ©. emax = maximum void ratio in correspondence with a very loose state of the particles. emin = minimum void ratio in correspondence with a very dense state of the particles e = in situ void ratio 2.

(23) In the gravitational or loosely sphere packing, very little research has been reported on the geotechnical behavior of the elements. This research is going to relate this method to the geotechnical field.. Research Objectives. PM. 1.2. Two main objectives of current research are categorized as follows:. U. • To find minimum porosity of the unconsolidated sandy soils after mechanical rearrangement of the sand grains.. T. • To compute maximum void ratio of dry sandy-soil particles based on the. H. statistical geometric approach of the gravitational or loosely sphere packing. IG. method.. It should be noted that above objectives are achieved according to the preliminary objectives mentioned below which are directly related to the fundamental. R. concepts of the random loose sphere packing (gravitational sphere packing) de-. PY. veloping this method for different applications such as geotechnical field. For example, better perception to choose an efficient container or random numbers for the post-processing computations in porosity.. O. • To investigate the different porosity values of the identical particles in dif-. C. ferent rectangular containers.. ©. • To find the lowest value of the porosity by examining different distributions. 1.3. of random numbers into loosely packed multi-sized spheres.. Scope of the Research. The main scope of this research is divided into two parts which are as follows:. 3.

(24) Simulation part: Vital necessities of optimization in the sphere packing method would play crucial role to provide logical results in agreement with experimental works. In Chapter 2, the different methods of the sphere packing are. packing has also been described in mentioned Chapter.. PM. briefly introduced, and exclusive literature review of the gravitational sphere. Sphere packing method has been optimized in the rapid collision detection. U. and container interaction with soil particles which are explained in Chapter 3. K Nearest Neighbors based on the KD-tress method and binary search. T. are applied to find the neighbors of every incoming sphere in the system, and also 3D Apollonius is applied to break 2 orders equations in simple ones. H. so as to rapid making tangent of spheres. Two conventional containers are. IG. considered for this method which are rectangular and cylindrical container. Interactions of the spheres with cylindrical container were solved by applying complete elliptic integrals. These intersections were resolved by spherical caps. R. for rectangular container.. PY. It should be noted that some outlier data of the porosity values through the height and diameter of the container are mathematically filtered by Lowes. O. Local Regression Smoothing procedure explained in Chapters 3 and 4.. C. In Chapter 4, The results of the simulated model according to the mentioned objectives are verified by last simulation models of the literatures and current. ©. experimental work.. Experimental works: Experimental works were done to find the maximum void ratio for sandy-soil particles with different sorting coefficients which are classified from very well sorted to the very poorly sorted. In addition, specific gravity, and sieve analysis are the Lab works which are discussed in Chapters 3 and 4. 4.

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