THE ANTI-DYNAMIC TUBE IN MASS FLOW SILOS

Transcription

1 THE ANTI-DYNAMIC TUBE IN MASS FLOW SILOS A THESIS SUBMITTED TO THE UNIVERSITY OF WESTERN AUSTRALIA FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF CIVIL AND RESOURCE ENGINEERING DESIREE NORTJE

2 ABSTRACT Storage of granular solids in silos has been the practice for many years. Engineers have been faced with the problem of making the silos empty more efficiently and minimising the forces acting on the walls of the silo during material discharge. To this end the anti-dynamic tube was invented. The tube has a smaller diameter than the silo and consists of several portholes along its height and around its circumference. When the discharge gate of the silo is opened the granular material enters the tube through the portholes, flows down the inside of the tube and exits the silo through the discharge gate. Most tubes have been installed such that there was sufficient space between the base of the tube and silo bottom for the granular material to flow simultaneously through the discharge gate. The flowing material causes a down drag on the tube from the friction of the granular material on the walls of the tube. Previous research has underestimated the magnitude of these frictional forces resulting in catastrophic buckling failure of the tubes, blocking the discharge gate of the silo. A blockage of the discharge gate requires top emptying of the silo resulting in financial losses and down time of equipment. A steel model silo with an anti-dynamic tube was set up in the laboratory to measure the friction on the tube during material flow. From the results of these experiments, an equation has been derived to estimate the magnitude of the down-drag force. Furthermore, an empirical expression was developed for the effects of the speed of the flowing material on the magnitude of the down-drag force. To keep construction costs down, it is necessary to optimise the wall thickness of the tube. There is currently no theory for the buckling capacity of a thin walled cylindrical shell with multiple perforations around its height and circumference. Therefore additional experiments were undertaken on a cylindrical shell with multiple perforations subjected to a combination of an axial as well as an external lateral pressure. Following on from the experiments, finite element analyses were undertaken to compare with the experimental results. For each finite element analysis an out-of-roundness was introduced as an initial wall imperfection. From these analyses and the cylinder experiments, a method of producing interaction curves for tubes with varying ratios of open area has been developed.

3 DISCLAIMER No portion of the work presented in this thesis has been submitted in support of an application for another degree or qualification from this, or any other, university or institute of learning. Désirée Nortje, August.

4 ACKNOWLEDGEMENTS My sincere thanks are given to my supervisor, Dr Ken Kavanagh, for his encouragement, insight, informality, approachability, understanding and extreme patience. Without his motivation and kindness I would have given up in the early stages of the project. I would also like to thank Dr Kavanagh for supporting the application for my research scholarship provided by the Department of Civil and Resource Engineering. Thanks are also due to the workshop staff who assisted me in setting up both my model silo and cylinder experiments. Special thanks goes to Jim Carrol, Jim Waters, and Neil McIntosh for their assistance with the laboratory machinery. Without their help and guidance the experiments would have been cumbersome, difficult to execute and probably never would have happened. Thanks are also due to Wladyslaw Bzdyl and Sun Nichersen for maintaining the electrical equipment and their endless patience in explaining my queries. ACI Glass, Penrith Plant, kindly allowed me to print and bind my thesis at their offices. I am very grateful for their generosity in making space available for me and providing invaluable assistance for binding. Finally, there are not enough words which can describe my gratitude to my husband, Richard, and my two daughters, Stephanie and Jennifer, for their patience while I undertook this PhD research. Many of our decisions revolved around this project and it s eventual completion date. We can now finally start making our plans become a reality. Thank you to all. Desiree Nortje, August

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6 LIST OF SYMBOLS c r m t w c h J JH v vw v hopper half angle a constant stress ratio strain in the circumferential direction strain in the r direction strain in the q direction angle in the circumferential direction in the hopper internal angle of friction of the granular material anti-dynamic tube wall friction angle wall friction angle material bulk density angle in the material. w < < m angle subtended by the hopper axis and centered at the hopper vertex vertical stress on the silo axis horizontal stress Janssen static vertical pressure Janssen static horizontal pressure vertical pressure vertical stress adjacent to the hopper wall shearing stress shear stress in the vertical direction at an arbitrary distance from the silo axis vw shear stress in the vertical direction adjacent to the hopper wall angle of inclination of the major principal stress A A c A P silo cross sectional area surface area of a cylindrical shell plan area of a cut out in the wall of a cylindrical shell projected plan area of an object

7 B C c d t d g D E F F H G h h H H h h t factor derived from the Mohr circle in the circumferential section of the silo Reimberts characteristic constant of the silo constant diameter of the anti-dynamic tube diameter of the silo discharge gate silo diameter flexural rigidity of a thin walled shell Youngs modulus factor derived from the Mohr circle in the hopper a constant vertical pressure distribution factor in the cylindrical section of the silo a constant vertical pressure distribution factor in the hopper a constant height of cone of material above the top of the silo step size used in the Runge-Kutta method of numerical integration total height of the silo height of hopper height of the anti-dynamic tube i N K K K a K nn K p L m M N N x N P Q q R r r c ratio of z N to hopper height =z N / H h stress ratio at rest stress ratio active stress ratio factor used in the Runge-Kutta method of numerical integration passive stress ratio length of a cylindrical shell coefficient of friction bending moment axial force axial load at buckling of a cylindrical shell with cut outs axial load at buckling of a cylindrical shell with no cut outs silo perimeter stress ratio shear stress hydraulic radius = A/P radius of a thin walled cylinder radius of a cut out in the wall of a cylindrical shell

8 r r av r b r t s N S v radius of the hopper at the level of the transition average radius radius of the bottom of an element in the hopper radius of the top of an element in the hopper pressure normal to the hopper wall dimensionless pressure ratio = v /D t wall thickness of a cylinder u,v,w displacements in the cartesian plane U c strain energy in the circumferential direction U B x,y,z z Z z z N bending energy cartesian co-ordinates depth co-ordinate dimensionless depth ratio = z/h element thickness depth of the maximum pressure in the hopper

9 CONTENTS ABSTRACT DISCLAIMER ACKNOWLEDGEMENTS LIST OF SYMBOLS CHAPTERS INTRODUCTION. Introduction.. Silo Types and Flow Patterns.6.3 Silo Inserts.7.4 Introduction to Wall Pressures. CLASSIC WALL PRESSURE THEORIES. STATIC WALL PRESSURES.. CYLINDRICAL SECTION... Janssen.... Reimbert Janssen vs Reimbert s theory.7.. HOPPER SECTION... Walker.8... Jenike Radial Pressures.... Linear Normal Wall Pressure.... Radial Pressure Field in the Solid Position of the Maximum Pressure in the Hopper....3 Walters Static Hopper Pressures.3. DYNAMIC WALL PRESSURES.. CYLINDRICAL SECTION... Walters.5

10 .. HOPPER SECTION... Flow/Slip in the Hopper by Equilibrium of a Slice Walters Pressures in Converging Channels Jenike Radial Stress Field.49.3 SWITCH PRESSURES.3. CYLINDRICAL SECTION.3.. Jenike Upper Bound Pressures Walters Switch Pressure in the Cylinder HOPPER SECTION.3.. Jenike Switch Pressure in the Hopper Walters Switch Pressure in the Hopper.78 3 WALL PRESSURE MEASUREMENTS 3. LITERATURE SURVEY 3.. STATIC PRESSURES 3... Cylindrical Section Hopper Section DYNAMIC PRESSURES 3... Cylindrical Section STRESS RATIOS EXPERIMENTAL SET-UP 3.. Steel Model Bulk Solid material Data Acquisition Strain Gauge Bridges Floating Pressure Cells Ball type pressure cell Tube type pressure cell Plate type pressure cell Pressure Cell Calibration Multi-turn potential meters Gate Switches EXPERIMENTAL RESULTS 3.3. Description Static Tests 3.4

11 3.3.3 Dynamic Tests Switch Pressure Stress Ratios ANTI-DYNAMIC TUBE THEORY 4. LITERATURE SURVEY 4.. Pieper Reimbert Ravenet McLean Ooms and Roberts Kaminski and Zubrzycki Schwedes and Schulze EXPERIMENTAL SET-UP 4.. Anti-Dynamic Tube Model EXPERIMENTAL RESULTS MATHEMATICAL MODEL Tube Parameters Variable Vertical Pressure across a Slice BUCKLING OF THIN CYLINDRICAL SHELLS 5. ELASTIC SHELL BUCKLING THEORY 5.. Cylinder subjected to uniform external lateral pressure Cylinder subjected to axial pressure 5... Special Case General Case Cylinder subjected to combined axial and lateral pressure PERFORATED CYLINDRICAL SHELLS 6. LITERATURE SURVEY 6.. Tennyson Almroth and Holmes Starnes Jr Scutella 6.7

12 6. DISCUSSION PERFORATED CYLINDER EXPERIMENTS 6.3. Experimental set-up Experimental results FINITE ELEMENT ANALYSIS 6.4. Description Cylinder with 6.5% open area Cylinder with 36.6% open area Solid Cylinder COMPARISON WITH LABORATORY TESTS 6.5. Cylinder with 6.5% Open Area Cylinder with 36.6% Open Area Interaction Plots for Cylinders with Multiple Perforations CONCLUSIONS 7. SILO WALL PRESSURES 7.. Static pressures Dynamic pressures Switch pressures Stress Ratios ANTI-DYNAMIC TUBE FRICTIONAL DRAG PERFORATED CYLINDERS INTERACTION CURVES APPENDICES 8. APPENDIX A: IMPLEMENTATION OF THE RUNGE-KUTTA METHOD 8.. The Runge-Kutta Equations A. 8.. Equilibrium Slice Method A. 8. APPENDIX B: CALIBRATION CONSTANTS 8.. Pressure Cell Calibration B. 8.. Anti-Dynamic Tube Support Calibration B.3

13 8.3 APPENDIX C: CHECK LISTS 8.3. Pre-Static Test Check List C Pre-Dynamic Test Check List C. 8.4 APPENDIX D: MODEL SILO WALL PRESSURES TESTS 8.4. Static Test Results D Dynamic Test Results D APPENDIX E: ANTI-DYNAMIC TUBE TESTS Frictional Drag Test Results E. 8.6 APPENDIX F: SHELL THEORY 8.6. Uniformly Compressed Circular Ring F Flexural Rigidity of a Shell F. 8.7 APPENDIX G: PERFORATED CYLINDER TEST RESULTS 8.7. Cylinders with 6.5% Open Area G Cylinders with 36.6% Open Area G Lateral Pressure Tests G APPENDIX H: EIGENVALUE BUCKLING MODE SHAPES 8.8. Cylinders with 6.5% Open Area H Cylinders with 36.6% Open Area H Solid Shell H.7 9 REFERENCES

14 INTRODUCTION. CHAPTER INTRODUCTION. INTRODUCTION A silo is a structure well known to most people. It s use is for the storage of any bulk material which is of a granular nature such as grain, wheat, lupins, salt, sugar, cement, coal, etc While there are many silos in existence, this does not imply that all knowledge about silos has been determined and that very little is left to still be discovered. Silos in all sizes are being constructed all around the world, some of which operate very successfully, and others which do not. Silo discharge is classified into two main groups, either concentric or eccentric discharge. In eccentric discharge the gate is off-centre with respect to the centre line of the silo. Due to this eccentricity, flowing material causes large bending stresses on the walls of the silo. These bending stresses are erratic in nature and difficult to predict due to the erratic nature of eccentric flow. In the case of concentric discharge, the centre of the gate aligns with the centre line of the silo, or the centre of the group of gates aligns with the centre line of the silo. The research covered in this thesis focuses only on concentrically discharging silos. Generally speaking a silo consists of two main sections, namely a hopper and an upper bin. The joint between the upper bin and the hopper is referred to as the transition. A flat bottom silo has an effective transition which is formed within the stored material which does not exit the silo during discharge. This remaining material is referred to as the dead material. Figure.b shows a flat bottom silo with dead material forming an effective transition with the walls. Silos vary in shape from circular to square and rectangular. Depending on the shape of the silo, there may be one discharge gate in the hopper as for the circular case, or several discharge gates as for the rectangular silo. Figure. a,b,c shows some of the typical silos in use. In figure.d, a bank of silos has been shown where the interstitial areas between the silos have also been used for the storage of material, shown by the shaded area. Some silos also have their cones inverted as shown in figure.. This type of silo is used mainly for storing and blending bulk materials which are in powder form, such as raw meal, cement and lime. This type of silo has not been considered in this thesis.

15 . DES NORTJE: PhD THESIS: THE ANTI-DYNAMIC TUBE IN MASS FLOW SILOS (b) square or doubly symmetrical silo (c) rectangular silo with a group of outlet gates (d) a bank of circular silos Figure.: Typical silo geometries. Effective transition Dead material Figure.a: Inverted cone silo Figure.b: Flat bottom silo. Some of the problems associated with the storage of bulk materials are the segregation of particles of varying sizes, excessive wear on the walls due to the flowing material and cracked walls from the flowing material. The wall pressures are generally referred to as the static and dynamic pressures, ie the filling and emptying pressures, respectively. One of the methods proposed to alleviate these problems is a silo insert called the anti-dynamic tube. Other names for the anti-dynamic tube are the tremmie tube, decompression tube, discharge tube and static-flow pipe. This method consists of placing a tube centrally inside the silo, which has a smaller diameter than the silo. The tube may extend the full height of the silo with multiple perforations around its circumference and along its length. The material then flows into the tube through

16 INTRODUCTION.3 the holes, and down the tube to the discharge gate. This type of tube causes the silo to empty in successive layers resulting in a first-in-last-out situation. For materials which degrade with time (either biologically or mechanically) this is highly undesirable. An alternative arrangement is a shorter tube which extends only a portion of the silo height. The optimum length of the tube is determined from the internal friction angle of the material. In this arrangement the silo empties in two or three stages only. In some instances, port holes are accommodated at the base of the tube allowing material discharge to occur simultaneously from the bottom of the silo as well as through the tube at higher levels. This PhD thesis considers anti-dynamic tubes placed in mass flow silos storing free flowing granular materials, such as sand, grain, lupins etc. This introductory chapter gives a background to the various silo types and flow patterns. A brief background of the different types of silo inserts used to overcome material flow and wall pressure problems has been given, as well as a general description of the wall pressures acting on the silo during filling and material discharge. Chapter two consists of a study of the classic theories for static, dynamic and switch pressures acting on the wall of the silos. This includes the Janssen, Jenike, Walker and Walters theories for the pressures in a silo. From these pressure theories it has been established that one of the main factors affecting the determination of the horizontal wall pressures is the assumption of a suitable stress ratio. Thus a section of the literature survey has been dedicated to the stress ratio as recommended by other researchers. Chapter three gives an overview of the wall pressures as measured by researchers world wide in either model silos or full scale silos. This has been categorised into the static, and dynamic pressures for the cylindrical and hopper section of the silo. There is not much data available for the measurement of the stress ratio in silos and consequently the literature survey covering stress ratios is relatively short. Following the literature survey is a description of the steel model silo set up in the structures laboratory. To enable the measurement of the pressures during material flow, three novel types of floating pressure cells were developed. These pressure cells were inexpensive, easy to construct, easy to calibrate and were found to be very responsive to the instantaneous pressures found in the flowing material. The results of the test are discussed in section three of this chapter, while a full set of the data has been given in Appendix D in the form of graphs. Throughout this

17 .4 DES NORTJE: PhD THESIS: THE ANTI-DYNAMIC TUBE IN MASS FLOW SILOS thesis, reference is made to the appendices for further detailed results. Chapter four is dedicated to the anti-dynamic tube. The first section consists of the limited research work by others that has been done on tubes. In many cases the reports are merely descriptive, rather than relating to the actual frictional drag on the tube. There has been much verbal discussion with Professor Roberts, from the Centre for Bulk Solids and Particulate Technologies in Newcastle, Australia, about the difficulty of determining the friction force acting on the tube during material flow. In one such a discussion, recount was given of a tube installed in a full scale silo which started punching through the base of a flat bottomed silo when the discharge gate was opened. It was mentioned that several attempts were made to support the tube from the walls and the roof of the silo to prevent excessive loading on the silo bottom from the tube. This discussion gives a good appreciation of the extent of the drag forces on the tube during material flow. The second section of chapter four gives an in-depth description of the tube experiments undertaken for this PhD research, and the method of measuring the force on the tube during material flow. The experimental results of the drag force measurements with a full set of graphs have been given in Appendix E. From the literature survey on the pressures in the material, a mathematical model of the pressures exerted on the anti-dynamic tube has been presented in the last section of chapter four. Since the tube is a shell structure, chapter five of this thesis has been dedicated to shell theory, in particular, cylindrical thin shells. The topic of a thin shell is appropriate to the anti-dynamic tube, as the wall thickness of the tube would need to be a optimised for financial reasons. Furthermore, during material flow there is wear on the walls of the tube which results in thinning of the walls over time. There is no theory for the structural stability and strength capacity of a thin shell with multiple perforations around it s circumference. Consequently the classic theories of shells subject to axial, lateral and a combination of both pressures have been studied. There has been limited work from previous researchers who have conducted tests on thin cylindrical shells with either one or two cut outs, placed at the mid height of the shell. The shells considered had a varying ratio of radius to wall thickness as well as the diameter of the cut out in the shell wall. These shells were subjected to an axial load only and the results from this work has been presented in the

18 INTRODUCTION.5 literature survey of chapter six. Included in this survey is the work from an honours thesis, L Scutella, University of Western Australia, 998, which described tests on thin shells with multiple perforations subjected to an axial load only. In Scutella s work, four different percentages of open area were considered. The open area is defined as the ratio of the area of the cut outs to the surface area of the shell. Scutella s research presented a good basis for comparison with the experiments conducted in this PhD research on similar shells subjected to both axial and external lateral pressures. However only two percentages of open area were considered, 6.5% (a cut out radius of 5mm) and 36.6% (a cut out radius of 76mm). These tests consisted of subjecting the shells to a combination of a varying axial load and a constant external lateral pressure. The final section of chapter six describes the large deflection, non-linear finite element analyses undertaken on a solid shell and shells with the same open areas as the laboratory experiments. In this thesis the term solid shell has been employed to describe a thin walled cylindrical shell with no holes in the wall, hence solid. As there is no shell which has perfectly curved walls, the finite element analyses included initial geometric imperfections which were imposed on the walls. As there are an infinite variety of geometric imperfections which could be imposed on the walls in such analyses, it was considered reasonable to use the mode shapes from elastic eigenvalue buckling analyses. However, there is no way of predicting which mode shape as an imposed wall imperfection will result in the lowest failure load in the large deflection, non-linear buckling analyses. Consequently, the first thirty mode shapes from the eigenvalue buckling analyses were expanded and imposed as imperfections in an axial load analysis and an external lateral pressure analysis for each shell. From these analyses the final mode shapes were chosen as the required wall imperfection in the non-linear buckling analyses of the shells subjected to simultaneous axial loads and lateral pressures. The degree of wall imperfection imposed on each shell was varied till the finite element results reasonably matched the results from the shell experiments in the laboratory. Finally from these analyses, interaction curves have been presented as a design tool for shells with multiple perforations of varying open area ratios. Finite element analysis is a topic undergoing much research on a continuous basis. It is therefore necessary to stress that in this PhD research, finite element analysis has been used as a means to an end, similar to the use of pressure cells, strain gauges and other equipment in the laboratory tests. Consequently no attempt has

19 .6 DES NORTJE: PhD THESIS: THE ANTI-DYNAMIC TUBE IN MASS FLOW SILOS been made to give an in depth theory of finite element analysis. The final chapter in this thesis presents the overall conclusion of this PhD research including the work on silo wall pressures, the resulting frictional drag on an antidynamic tube during material flow and the recommended interaction curves for thin cylindrical shells with multiple perforations.. SILO FLOW PATTERNS Silos are defined according to the resulting flow pattern within the silo. A W Jenike (964) defined types of flow pattern, mass flow and funnel flow, as shown in figure.3a and b respectively. Figure.3c shows a combination of the two types of silos. Funnel flow cylinder D cr Mass flow hopper (a) (b) (c) Figure.3: Basic silo types: a) Mass flow ; b) Funnel flow ; c) Expanded flow.. MASS FLOW SILOS Mass flow silos are characterised by steep hopper half angles and smooth wall surfaces enabling all the material to flow when the discharge gate is opened. This is most desirable for materials which degrade or consolidate with time, as mass flow guarantees complete discharge of the material. If the material becomes segregated during filling, re-mixing during discharge can be ensured. Mass flow silos are classified according to the hopper shape; axi-symmetric silos have conical hoppers and plane flow silos have wedge shaped hoppers with long slotted openings or a group of discharge gates... FUNNEL FLOW SILOS In funnel flow silos the material forms a funnel within itself above the hopper outlet, causing a last-in-first-out situation. The silo does not completely empty when flow has stopped and an area of dead material remains inside the silo. This is

20 INTRODUCTION.7 undesirable for most materials, but has the advantage of protecting the walls against excessive wear. Funnel flow silos have shallow half angles or are flat bottomed and cause segregation problems and erratic discharge...3 EXPANDED FLOW SILOS A third type of silo is the expanded flow bin which combines the two types of flow. The critical pipe diameter D cr in the funnel flow cylindrical section determines the minimum dimension for the mass flow hopper below..3 SILO INSERTS Since the anti-dynamic tube is a silo insert a brief discussion of types of inserts has been given. Most silo inserts are discharge devices used as a solution for a poorly flowing hopper. Ideally, the silo should first be designed with a gravity flow hopper that would discharge the material satisfactorily, as gravity flow is the cheapest and most reliable method of discharge. If a gravity flow hopper cannot be installed then an appropriate discharge aid should be considered. Reed and Duffell (983) have categorised discharge aids into three types: ) Pneumatic: those which rely on the application of air to the material to initiate flow ) Vibrational: those which rely on the application of high frequency low amplitude vibrations to the hopper wall 3) Mechanical: those which rely on mechanical means to discharge/extract the material from the hopper..3. PNEUMATIC DEVICES These devices rely on controlled quantities of air at low pressure being applied to the material thereby reducing its strength and improving the flow characteristics. By introducing the air at the wall of the hopper, the wall friction is reduced and the material in the region of the wall becomes liquid. Reed and Duffell (983) state that this method works best with dry (or very low moisture content) materials less than 3microns in size..3. VIBRATIONAL DEVICES Vibrational devices rely on the ability of the material to transmit vibrations thereby reducing the strength of the bulk solid. This method should not be operated when

21 .8 DES NORTJE: PhD THESIS: THE ANTI-DYNAMIC TUBE IN MASS FLOW SILOS the discharge gate is closed as the vibrations cause densification of the material and a higher strength material than initially loaded results. Their effectiveness in handling sticky, flaky and fibrous materials is doubtful as they rely on the ability of the material to transmit the vibrations..3.3 MECHANICAL DEVICES The simplest mechanical device consists of chains suspended in the silo. If the material arches, an upward pull of the chain breaks the arch and the material starts flowing. Alternatively, paddles within the material which rotate about a horizontal or vertical axis maintain the material in a state of continuous motion. For hard coarse materials, wear on the chain and paddles can be significant. Screw feeders are a commonly used mechanical method of discharging and controlling flow rate of the bulk solid, but are also subject to high wear rates Binsert The binsert approximates the concept of the anti-dynamic tube except that it has sloping sides, not parallel like the anti-dynamic tube. Figure.4: Binsert The binsert is a patented device invented by JR Johanson (98) consisting of a cone-in-cone insert as shown in figure.4. The smaller hopper inside the silo hopper must be designed in accordance with mass flow principles. He states that the location of the insert relative to the outer hopper is such that the included angle, satisfies mass flow criteria. Johanson states that the outer hopper half angle, must be twice the angle required for mass flow Anti-dynamic tube The Reimberts (976) claim to be the inventors of anti-dynamic tube in France in the 95 s. Figure.5c is a photograph taken from the Reimbert s book (976) illustrating the use of their tubes. The main purpose of the tube was to alleviate the pressures on the walls during flow of the material. The Reimbert tube is a small diameter tube with multiple perforations along it s length, which is placed centrally in the silo, as shown in figure.5a. This design caused the silo to operate on a lastin-first-out basis. A modification to the Reimbert tube is reported in Ooms and Roberts (985) and was installed in full scale silos in Port Adelaide. The

22 INTRODUCTION.9 modification consisted of reducing the overall height of the tube and omitting all the holes such that the silo empties in two stages only, as shown in figure.5b. Material Flow (a) (b) Figure.5: a) Reimbert tube; b) Roberts tube Figure.5c: Photograph taken from Reimbert(976) showing the use of anti-dynamic tubes.

23 . DES NORTJE: PhD THESIS: THE ANTI-DYNAMIC TUBE IN MASS FLOW SILOS.4 INTRODUCTION TO WALL PRESSURES The two pressures which are of interest to researchers are the vertical pressures acting on the silo bottom, and the horizontal (or normal) pressures acting on the silo walls. Depending on which type of pressure exists in the silo, the vertical pressure is either larger than, or smaller than, the horizontal pressure. The pressures exerted by the stored material in the silo can be divided into three main types. The first type of pressures are those which develop during loading of the solid into the empty silo, and are generally referred to as the static pressures. Since the silo consists of two main sections, ie the cylindrical and the hopper sections, the static pressure distribution in each section is different. When the material is loaded into the silo, an active pressure field develops and the lines of major principal stress are nearly vertical. The associated minimum stress ratio possible is the active stress ratio, K a which is less than one. The static vertical pressures are greater than the pressures acting normal to the silo walls. The static normal pressures are the minimum pressures which can be expected in a silo. The second type of pressures are those which develop during discharge of the material from the silo, and are generally referred to as the dynamic pressures. Again, the distribution of the dynamic pressures in the cylindrical section are different to those found in the hopper. When the discharge gate is opened the relative motion of the solid with respect to the wall is the same as the wall moving in towards the solids, and hence the pressure field changes from an active to a passive state. The stress ratio now has a value greater than one, and the lines of major principal stress are approximately horizontal. The dynamic pressures acting normal to the silo walls are greater than the dynamic vertical pressures, as well as the static normal pressures, but are not necessarily the largest that can be expected. Based on the above discussion, a passive stress state cannot theoretically develop in the cylindrical section of the silo, since the walls are parallel and there is no relative motion in towards the material. However, it is doubtful whether a perfectly parallel wall can be achieved in practice, and consequently many researchers allow for the possibility of a small convergence in the walls of the cylindrical section. Thus in the theories which have been presented in this chapter, a passive stress state has been assumed in the cylindrical section, to determine the possible distribution of flow pressures.

24 INTRODUCTION. The third pressure on the walls of the silo also occurs during discharge of the material. This is the switch pressure, which is a transient pressure exerted over a small area of the walls for a small period of time. The switch pressure is caused by the change-over from a static to a dynamic stress state and travels quickly up the hopper the instant the gate is opened. The switch pressure may extend the full height of the silo, or it may become trapped at the transition from the cylindrical to hopper section. In both the hopper and the cylindrical section, the switch pressure is the largest expected pressure acting on the walls of the silo. Most researchers, and silo design codes, give the switch pressure as a multiple of the static pressure, the actual ratio being dependant on the theory developed. The direction of the major principal stresses have been shown by the lines in figure.6 for the three types of pressures found in the silo. Figure.6a shows the near vertical lines of the major principal stress during filling of the silo. Figure.6b shows the lines of the major principal stress as near horizontal during emptying of the silo, with the assumption that the passive stress state develops in the cylinder. Figure.6c shows the location of the switch at an instant in time when the switch has travelled up in to the cylinder and the stress changes from an active to a passive state. (a) (b) (c) Figure.6: Schematic representation of the lines of major principal stress.

25 CLASSIC WALL PRESSURE THEORIES: LITERATURE SURVEY STATIC PRESSURES. CHAPTER CLASSIC WALL PRESSURE THEORIES. STATIC WALL PRESSURES.. CYLINDRICAL SECTION... The Janssen Theory Prior to Janssen s experiments and his paper of 895, it was assumed that the material in a silo exerted a triangular hydrostatic pressure distribution down the height of the wall. Now the Janssen formula is used in engineering standards world wide for computing the initial filling pressures in the vertical section of the silo. Roberts(995) reports on Janssen s experiments which were performed on wooden model silos filled with wheat. The silos being of square cross-section with sides measuring approximately, 3, 4 and 6mm. The models were mounted on adjustable screws, while the bottoms were formed by close fitting movable boards connected to a weigh bridge. In this way the pressure on the bottom of the silo could be measured. The Janssen theory was developed by considering the equilibrium of the vertical forces acting on a horizontal elemental slice in the cylindrical section of the silo, as shown in figure.. s s = length of the side A = silo cross sectional area P = silo perimeter z = depth from the top z dz = element thickness v v = average vertical pressure h dz d v /dz = change in vertical pressure through the element thickness. = shearing stress between the material v + d v dz dz and the silo wall h = horizontal pressure = material density Figure.: Forces acting on a horizontal elemental slice = wall friction angle

26 . DES NORTJE: PhD THESIS: THE ANTI-DYNAMIC TUBE IN MASS FLOW SILOS Equilibrium of forces on the elemental slice is given by : dv v dz A P dz v A Adz dz (.) Simplifying the above equation gives : d v dz P A (.) In equation (.) P/A is the perimeter divided by the cross-sectional area and for a circular silo P/A = D/(D /4) = 4/D (.3) The shearing stress between the material and the wall is given by : h Tan h (.4) Janssen assumed a constant stress ratio of K= h / v =K a, through the depth of the silo. Substituting equations (.3) and (.4), and substituting for h = K/ v into equation (.) gives the following expression for the vertical pressure: z 4 h D z4k / D v dz e (.5) D 4K The derivation of Janssen s equation was based on the following assumptions: ) the vertical pressure is constant across a horizontal slice ) the stress ratio K = h / v is assumed constant at all depths in the silo 3) the bulk density, does not vary with depth 4) the wall friction is fully mobilised and the material is on the point of slip Roberts(995) reports that Janssen showed from his experimental results that the vertical pressure distribution across the cross-section was not uniform. He showed that the pressure was higher in the centre, at.5 times the average value, and lower in the corners of his models, at.8 times the average value. He concluded that the wall pressures could be estimated with sufficient accuracy by assuming a constant pressure distribution across a horizontal slice. In this thesis the horizontal pressure on the silo wall is of interest, which is given by D z 4K / D K h V e (.6) 4

27 CLASSIC WALL PRESSURE THEORIES: LITERATURE SURVEY STATIC PRESSURES.3 The graph in figure. shows the distribution of the horizontal pressure as a function of the height to diameter ratio of the silo. The values used in the graph are: =6.8kN/m 3, D=.96m, =Tan.44. The stress ratio used for the purposes of this graph is the at rest ratio given by: K=.9=-Sin45 material friction angle of 45). From equation.6, as the depth z, tends to infinity, the exponential term tends to zero, and the pressure tends towards a maximum given by the asymptote: D. 4 Therefore, in a very tall silo, there is no increase in the vertical pressure at the silo bottom when more material is loaded on top. The additional material weight is carried by the walls of the silo. H/D Ratio Horizontal Pressure (kpa) K Figure.: Graph of Janssen horizontal pressure distribution This asymptote is inversely proportional to the wall friction angle and therefore as the wall friction angle increases the horizontal pressure acting on the wall decreases, as shown by the curve labelled in figure.. The horizontal pressure is directly related to the stress ratio since the horizontal pressure is calculated from the vertical pressure by multiplying by the stress ratio. As can be seen in figure., increasing the stress ratio by a factor of two has the greatest influence in the upper regions of the silo.

28 .4 DES NORTJE: PhD THESIS: THE ANTI-DYNAMIC TUBE IN MASS FLOW SILOS... The Reimbert Theory Marcel and Andre Reimbert (976) developed a theory for the static pressures in the cylindrical section of the silo. They approached the problem by taking a horizontal cut through the silo and considering the vertical forces acting on the free body diagram of the upper portion, as shown in figure.3a. The depth of the silo, z, is measured from the top of the silo, while the cone of material on top has a height of, h. The Reimberts state that if there were no friction on the walls of the silo, the graph of vertical pressure would be a straight line as shown in figure.3b. The cone of material above the silo is shown by the offset, h/3 in figure.3b. h h/3 Stress w z v Figure.3a: Free Body Diagram Depth (z) z + h/3 Figure.3b: Graph of vertical stress However, since there is friction on the walls of the silo, the equation of vertical equilibrium, can be written as: h v A AP A Pz (.7) 3 where the cross-sectional area of the silo is given by A, the circumference is given by P, and the volume of the cone of material above the silo is given by Ah/3. In equation.7, v is the vertical reaction from the material below acting on the upper portion, is the shear stress acting on the sides of the walls and is the bulk density of the material. From their experimental results, the Reimberts state that the shape of the curve of the shear stress on the walls of the silo is as shown schematically in figure.4.

29 CLASSIC WALL PRESSURE THEORIES: LITERATURE SURVEY STATIC PRESSURES.5 Stress Line of hydraulic stress in the silo for zero friction Depth (z) (z) z+h/3 v max Figure.4: Graph of shear stress The Reimberts state that with increasing depth the curve of the shear stress approaches an asymptote, which is parallel to the line of the hydraulic stress, and has been shown dotted in figure.4. Therefore, as the depth of the silo increases the equation of vertical equilibrium becomes: h P v max P z (.8) 3 A The shear stress has the following expression, which was derived from their experiments: (z) z z C c (.9) The Reimberts define the characteristic constant, C c, as follows: v max h / 3 Cc (.) Now the unknown term, vmax, in the expression for the friction function is contained in the expression for the characteristic constant. As the silo is being filled, v reaches a maximum limit and any additional elemental slice of material of thickness dz, loaded into the silo is carried by friction on the walls. This can be expressed as: Pdz = W e = Adz (.) But the friction term can also be expressed as : = hmax (.) Equating equation. and. gives: hmax = (A)/(P) = (/) (.3) This expression is a constant since the vertical pressure in the silo becomes constant as the depth increases and therefore the horizontal stress also becomes constant.

30 .6 DES NORTJE: PhD THESIS: THE ANTI-DYNAMIC TUBE IN MASS FLOW SILOS The maximum vertical stress can be found from the maximum horizontal stress by using an appropriate stress ratio. The stress ratio they propose is the active stress ratio. Therefore, substituting for vmax into the expression for the characteristic constant gives: C c = (/(K a ) - h/3) (.4) And hence, the friction function is fully described as: = (z ) / (z+/(k a )-h/3) (.5) The Reimbert theory then considers the general expression for friction as given by equation., and state that since the horizontal stress varies with depth, the expression for the friction can be written as: (z) = h dz (.6) Therefore, the derivative of the friction function as given in equation.5 would give an expression for the horizontal pressure. The derivative of equation.5 is: d dz z C c ( z) ( z ) Cc z Cc z Cc (.7) Therefore, the final expression for the horizontal stress can be written as: h Cc (.8) z Cc From figure.4, the total weight of the material in the silo, z+h/3, equals the shear stress on the walls of the silo plus maximum vertical stress acting on the bottom. This is expressed as: vmax + =z+h/3 (.9) The final expression for the vertical stress on the bottom of the silo, can be found since the expression for the friction function is given in equation.5. Therefore, the expression for the vertical stress is: z h v z (.) Cc 3

31 CLASSIC WALL PRESSURE THEORIES: LITERATURE SURVEY STATIC PRESSURES Discussion of Janssen vs Reimbert Theory Briassoulis(99) states that both the Janssen and Reimbert s theories are unconditionally applicable for any silo geometry and stored material. For the silo model and material used in the laboratory, a comparison between the two theories is shown in figure (.5). The Janssen theory gives a greater vertical stress as the depth increases, while the Reimbert theory gives a greater horizontal stress in the upper parts of the silo. Both theories tend to the same value of horizontal pressure as the depth of the silo increases. Vertical Pressure 5 5 Horizontal Pressure Depth below surface Janssen Reimbert Depth below surface Janssen Reimbert Figure.5: Comparison between Janssen and Reimbert theories Re-writing Janssen s expression for the horizontal pressure, equation (..6) as: h 4Kz D e D (.) 4 K Noting that for a cylindrical silo A/P = D/4; Reimbert s expression for the horizontal pressure, equation (.) can be re-written as: D z h 4 Cc (.) Therefore, only the terms in square brackets need be considered when comparing equation (.) and (.). For a silo with no cone of material on top, the expression for the characteristic abscissa in Reimbert s theory can be re-written as C c = D/(K a 4) and substituting

32 .8 DES NORTJE: PhD THESIS: THE ANTI-DYNAMIC TUBE IN MASS FLOW SILOS this in equation (.) gives the term in square brackets as: -(4Kz/D + ) - The graphs in figure 6 show the influence of the wall friction angle and the stress ratio, K, on the terms in square brackets for both equations of horizontal pressure. The Janssen term has been shown with a blue line and the Reimbert term in a red line. The graphs show that as the wall friction angle increases, the Janssen and Reimbert expressions tend to coincide, while a smaller stress ratio causes them to diverge. Horizontal pressure (kpa) Horizontal pressure (kpa) Depth below surface (m) 3 4 Janssen Reimbert o 5 o Depth below surface (m) Janssen Reimbert 5 5 (a) Wall friction angle varies from to 5 degrees (b): Stress ratio varies from.5 to Figure.6: Comparison of terms in Janssen and Reimbert equations: (a): Wall Friction ; (b) Stress ratio.. HOPPER SECTION The theory for the wall pressure in a convergent hopper was presented by Walker in 96. The theory is included in this section for completeness only, since the results appear to be at a variance with physical intuition and with the results of Janssen theory in the cylindrical section... Walker Theory Walker (96) considered a smooth walled hopper with a hopper half angle (, filled with a nearly incompressible material. His initial premise was that the principal stress planes in the hopper were vertical and horizontal, so that the shear stress on a vertical plane is zero. Under the assumption of zero shear, the normal stresses must increase hydrostatically with depth.

33 CLASSIC WALL PRESSURE THEORIES: LITERATURE SURVEY STATIC PRESSURES.9 Therefore: V gz z (.3) h Vertical pressure (kpa) 3 4. Depth below top of silo (m) Cylindrical section Hopper Janssen Walker Figure.7: Graph of static vertical pressure in a model silo Therefore, plotting a graph of the static pressure in the silo using the Janssen theory for the cylindrical section, and the Walker theory for the hopper section, gives the diagram shown in Figure.7 for the model silo in the laboratory. This graph illustrates the vertical pressure distribution in the silo. If the material in the hopper lies within the yield surface, and only the wall is assumed to slip, Walker gives the Mohr circle for an element on the wall as shown figure.8. Point P on the circle represents the stresses at the wall which is tangential to the wall yield locus. The wall friction angle is given by w and the material friction angle is given by m. Walker assumes the horizontal stress to be equal to the minor principal stress, 3, on the circle. material effective yield locus hopper wall yield locus P m W 3 N C Figure.8: Mohr s circle for stresses at the hopper

34 . DES NORTJE: PhD THESIS: THE ANTI-DYNAMIC TUBE IN MASS FLOW SILOS The angle which the normal stress, N, acting on the hopper wall makes with the horizontal stress, is given by in Figure.8. The value of N is obtained from the geometry of the Mohr circle. N Sin Cos w C (.4) Sin Sin w w From equation.4, it can be seen that Walker gives the normal stress at the wall as a function of the hopper half angle and the wall friction angle. The material friction angle does not influence the value of N. Figure.9 shows the variation of the ratio given by C in equation.4 as a function of the wall friction angle. The four curves shown in figure.9 are for a hopper angle of 5 increasing to in 5 increments Constant C Wall friction angle ( w = =5 Figure.9: Variation of C as a function of the wall friction angle As can be seen, the shallower the hopper slope, the higher will be the normal wall stress according for the same wall friction. The lowest normal wall stress will be achieved using a shallow hopper half angle, ie 5, and a higher wall friction angle. However too high a value for the wall friction angle will inhibit mass flow and result in funnel flow.

35 CLASSIC WALL PRESSURE THEORIES: STATIC PRESSURES: HOPPER.... Jenike Theory Jenike has written several papers on the pressures exerted on the hopper walls as well as the pressures within the solid stored in the hopper. The theories developed were covered in several papers. To fully understand Jenike s derivations, a brief summary, in chronological order, describing the outline of each theory is given below. A similar outline has been given in chapter. for Jenike s theories of pressures due to the flowing material in the silo. ) Radial pressure field in the solid stored in a hopper. In this theory Jenike (96) considers an element of material, at an arbitrary location in the stored solid, and evaluates the forces acting on this element by considering equilibrium of all forces. The equations which result after extensive algebraic manipulation, are lengthy differential equations which are solved using numerical methods. Jenike shows from the results of these equations that the pressures in the solid increase linearly from zero at the vertex of the hopper. Jenike defines this as a radial stress field. ) Linear normal pressure exerted on the hopper wall. In this analysis, Jenike (968) states that the weight of the material in a hopper without a surcharge, is carried by the vertical components of the shear and normal wall stresses. By simple equilibrium of forces, Jenike derives an expression for the pressure normal to the hopper wall. This analysis was extended by Jenike (973) to include normal pressures acting on the walls of hoppers with a surcharge. 3) Position of the maximum pressure in the hopper. Having found the expressions for the normal wall pressure and the radial pressure field, Jenike(968) equates the two expressions to determine the location, i N, of the maximum normal wall pressure in the hopper. Jenike s theory for the linear normal pressures on the hopper walls has been dealt with first in this thesis, followed by the radial pressure field and the location of the maximum static pressure....) Linear normal pressure exerted on the hopper wall Jenike(968) states that the pressures in a hopper have been shown experimentally to decrease to zero at the vertex of the hopper. Therefore, Jenike assumes a linear pressure distribution acting normal to the wall during filling of a

36 . DES NORTJE: PhD THESIS: THE ANTI-DYNAMIC TUBE IN MASS FLOW SILOS hopper with no surcharge as shown in figure.(b) below. This pressure field cannot extend all the way to the free surface in a hopper without a surcharge, but will be topped off by a compatible pressure field, with pressures decreasing towards zero at the free surface. The maximum pressure in the hopper occurs at the interface of the two pressure fields. The distance of the maximum pressure from the vertex has been defined by Jenike using the ratio i N, as shown in figure.(c). r H h N max i= N z N z Depth i N =z N /H h (a) (b) Pressure (c) Figure.: (a) Hopper; (b) Pressure distribution; (c) Ratio i Jenike states that the total weight of the material must be supported by the walls and ignores any support gained from the material below. Therefore, the sum of the vertical components of the wall shear and normal pressure, and N, acting on an element of thickness dz, equals the weight of the element in the hopper. The vertical components of the wall support, acting over an area of rdz / Cos. are given by: ( w N Sin N Tan Cos ) r dz / Cos (.5) The radius of the hopper is given by: r = z Tan (.6) The volume of the element is approximately given by: V= z Tan dz (.7) Therefore, the weight of the element is given by: W Tan z dz (.8) By equating equation.5 and.8, re-arranging and integrating over the hopper wall from to H h gives the following, (where H h is the height of the hopper):

37 CLASSIC WALL PRESSURE THEORIES: STATIC PRESSURES: HOPPER.3 Hh 3 Tan H 6 (Tan Tan w ) h nz dz (.9) In Figure.(c), for < z <z N : N = z N max /z N And for z N < z <H h : N = (H h -z N ) N max / (H h -z ) Substituting for N in equation.9, and integrating gives the following expression: 3 Tan H h 3 z N 6 (Tan Tan w ) N 3z N H z h N 3 H h H z h N 3 H h 3 3 z N 3 (.3) Solving for N gives the expression for the normal pressure on the hopper wall as: N =D/[(Tan+Tan w )(+z N /H h )] (.3) Ratio z N /H h.8.6 z N /H h =.5 z N /H h =.5 z N /H h =.75 Locus of Maxima Figure. shows the maximum normal wall pressure in the hopper as given by equation.3, for various values of z N /H h The graph was drawn for Normal wall pressure (kpa) Figure.: Normal Pressures on the Hopper Wall for various values of the Ratio i N values of =8 kn/m 3, hopper diameter D=m, hopper half angle =5, and a wall friction angle w =. From equation.3, the material bulk density and diameter of the hopper have the greatest effect on the wall pressure. As can be seen, from figure. the maximum pressure decreases, as the position of the maximum increases within the hopper. The locus of the maxima has been shown in figure.. Since the maximum cannot be at the free surface, the locus has been

38 .4 DES NORTJE: PhD THESIS: THE ANTI-DYNAMIC TUBE IN MASS FLOW SILOS arbitrarily terminated at a value of i N =.875 The shape of the locus can be closely approximated by a cubic polynomial of the general form: i N = A 3 N + B N + C N +D. The constants in the equation are different for each hopper being analysed. Having derived an expression for the normal wall pressure in a hopper with no surcharge, Jenike(973) adopts the same approach in deriving an expression for a hopper with a surcharge. This surcharge, J, calculated from the Janssen equation at the level of the transition, increases the pressure at the top of the hopper, from zero to some value t at the transition. Jenike states that this surcharge is supported by the walls above the location of the maximum normal pressure, as shown by the shaded area in Figure. (b). J t r H h p p N max Additional pressure due to material in the cylindrical section N z N z Pressure distribution in hopper with no surcharge (a) (b) Pressure Figure.: (a) Hopper ; (b) Pressure distribution with surcharge By equilibrium, the additional pressure is given by: J Tan Tan w Tan Hh zn p p z dz (.3) where p is the pressure at some level, z, for no surcharge and p is the additional pressure at the same level, z, for a surcharge acting on the hopper and (p - p ) = t (z z N )/(H h -z N ) (.33) The ratio i is defined by Jenike as: i = z /H h Substituting equation.33 into equation.3, and substituting for i N =z N /H h gives:

39 CLASSIC WALL PRESSURE THEORIES: STATIC PRESSURES: HOPPER.5 J Tan Tan w Tan t in Hh ( i i N) i zn di (.34) Integrating and solving for t gives: t J i i Tan Tan N 3 Tan N w (.35) Figure.3 shows the normal wall pressure acting on a hopper with a surcharge. The Janssen pressure, J, for the hopper surcharge was calculated for a cylinder height of six times the hopper diameter, since the vertical pressure in a cylinder does not increase substantially beyond this height to diameter ratio. The graph has been drawn for values of =8 kn/m 3, a hopper diameter D=m, a hopper half angle =5, and a wall friction angle w =. As the position of the maximum pressure on the hopper walls increases, so the pressure at the transition also increases. Ratio in.8.6 z N /H h = z N /H h =.5 z N /H h =.5 z N /H h = Normal wall pressure (kpa) Figure.3: Normal Pressures on the Hopper Wall for various values of Ratio i N for a hopper with a surcharge The dotted lines in figure.3 indicate the lines of pressure for no surcharge in the hopper, as given in figure..

40 .6 DES NORTJE: PhD THESIS: THE ANTI-DYNAMIC TUBE IN MASS FLOW SILOS...) Radial pressure field in the solid stored in a hopper. Jenike [96] defines a radial stress field as a field in which all the stresses increase linearly with the radius, r, for the initial loading of material in the hopper. Youngs modulus for the material is eliminated during the process of derivation and Poisson s ratio is assumed constant. The material is assumed to be non-linearly elastic and is assumed to slip at the walls. In his derivation Jenike considers an element in the hopper with a set of spherical co-ordinates (r,,) with origin at the vertex of the hopper. Jenike does not give a complete diagram of the element in his papers. The element shown in figure.4 has been drawn to fully understand the shape of the element and the directions of the forces acting on it. The areas of each face of the element have been shown in figure.5, with the expressions for the length of each side and the area of each face. Considering equilibrium in the r-direction (refer figure.4): + r r r dr A bott d r drd d Sin rabott r Aright Cos... r d d... + r d A right r dr d d Cos Cos Aright d d d A r dr dd Cos Sinr drd right c... d + Cos r dr ddsin (.36) Divide throughout by:( r drddsin ) r r r d r Cot Cos (.37) r r r r r r c Re-arranging gives the following final equation for equilibrium in the r-direction: r r r r r d Cot Cos r c r (.38) This expression differs from Jenike s expression by the term Cos(+d/) instead of Cos(). By small angle approximations, this difference in the equations has little effect on the final solution and the term d/ has been ignored. The expression for equilibrium in the direction is given after figures.4 and.5

41 CLASSIC WALL PRESSURE THEORIES: STATIC PRESSURES: HOPPER.7 c d Plan View d/ c Section B-B r r r dr r r r dr r d/ r r d r d r d B Section A-A A A Figure.4: Element in the stored material in a conical hopper Element in the stored solid B Plan on hopper

42 .8 DES NORTJE: PhD THESIS: THE ANTI-DYNAMIC TUBE IN MASS FLOW SILOS S 4 S 3 S 6 S 5 Area back d The position of the element within the hopper is defined as follows: r = radius from hopper vertex dr = small change in r = angle of rotation of r d = small change in d = small change in the horizontal angle Area front The lengths of the element sides are defined as follows: S = r d Area top S (r+dr) Sin Area right S = (r+dr) d S 3 = r Sin(+d) d S 4 = (r+dr) Sin(+d) d S 5 = r Sin d S 6 = (r+dr) Sin d (r+dr) Sin(+d) dr Area left Area bottom S r Sin(+d) Figure.5: Element in the Stored Material in a Conical Hopper r d r Sin The areas of the element faces can be estimated as follows: Area bottom = S (S 3 +S 5 )/ =r d d Sin Area top = S (S 4 +S 3 )/ r d d Sin + r dr d d Sin = A bott + r dr d d Sin Area right = dr (S 5 +S 6 )/ r dr d Sin Area left = dr (S3+S4)/ r dr d Sin + r dr d d Cos =A right + r dr d d Cos Area front = Area back = dr (S +S )/ r dr d The volume of the element is given by: V = dr (A top + A bott )/ r dr d d Sin

43 CLASSIC WALL PRESSURE THEORIES: STATIC PRESSURES: HOPPER.9 Considering equilibrium in the -direction : Cos d d dr r A d Cos d A d Cos right right.. - Cos d d dr r A d d A d right r r right r.. + bott A r Sin d d dr r d Sin.. - d Cos d d r dr c Sin d d r dr A dr r r r bott (.39) Dividing throughout by:( r dr d d Sin ), collecting terms and re-arranging gives : d Sin Cot 3 r r c r r (.4) This expression differs from Jenike s expression by the term Sin(+d/) instead of Sin(). By small angle approximations, this differences has little effect on the final solution, and has therefore been ignored. To complete the solutions of equations.38 and.4, Jenike(96) first finds expressions for, r, c and r. As the material is loaded in the hopper of the silo, it contracts both vertically and horizontally. Therefore, the material does not reach the limiting state of stress but is in an elastic-active state of pressure. The elastic stress strain relations in the hopper are therefore given by: E c r r E c r (.4 a, b, c) E r c c The strains given in equations.4 a,b,c can be written as functions of the radial displacement as follows: r = -u/r = c = -u/r + (.4 a,b)

44 . DES NORTJE: PhD THESIS: THE ANTI-DYNAMIC TUBE IN MASS FLOW SILOS From equation.4b it follows that = c Jenike defines the stress ratio in the hopper as: k = / r (.43) This is similar to the definition used by Janssen for the stress ratio in the cylindrical section of the silo. In a Mohr circle for the stresses on the element, the average stress is given by: = ( r + )/ (.44) The general equation for the radial stress is given by Jenike as: = rs (.45) where S is the radial stress field and is a function of the co-ordinates r and Therefore, from equations.43,.44 and.45: r = k /(+k) = k rs /(+k) (.46) = c = k /(+k) = k rs /(+k) (.47) The shear stress on the wall is given by: r = Tan w = k rs Tan w /(+k) (.48) The derivatives of r, and r are as follows: r r r r S r k ds dr (.49) r k r k ds d (.5) r r d d r w w k ds Tanw r k d krs dw kcos w d (.5) r r r d dr r w w r k ds Tanw S r k dr krs dw kcos w dr (.5) Now the derivatives of r, and r given by equations.49,.5,.5 and.5 can be substituted into the equations of equilibrium given by equations.38 and.4. In his analysis, Jenike has assumed ds/dr and d w /dr to be zero. Therefore, collecting terms in ds/d and d w /d and dividing throughout by (+k)/(k ) gives the following: ds S dw k Tanw S Tan wcot Cos (.53) d Cos d k k w ds k Sin 4STan w (.54) d k

45 CLASSIC WALL PRESSURE THEORIES: STATIC PRESSURES: HOPPER. Substitution of ds/d from equation.54 into equation.53 gives the following for d w /d: d w d 3 k 4 Cos k w k Sin Cot SinwCosw k S k Cos Cos k S w (.55) Assigning a constant value to k, the stress ratio in the hopper, equations.53 and.55 can be solved by numerical integration. The value of S, for the given value of k, is then substituted into equation.5 to determine the value of the stress normal to the wall,, which is a linear function of the hopper radius, r. From equation.5, the stress field, S, must be unitless since the density,, has units of kn/m 3 and radius, r, has units of m. This results in the units of kpa for the stress,, normal to the wall....3) Position of the Maximum Pressure in the Hopper The locus of maximum pressures is unique for a given hopper and stored solid, as shown in figure.. This locus has been drawn by assuming values for the ratio of z N /H h. Jenike states that the position, z N, of the peak pressure is determined from the intersection of the locus of maximum pressures with the linear (radial) pressure field shown in figure.. This has been shown schematically in figure.6. D Hopper Locus of maxima Radial pressure field z N Peak pressure Figure.6: Position of Peak Pressure in a Hopper The normal wall pressure acting on the hopper due to the radial pressure field, has been given by equation.47, and is repeated as follows : = krs/(+k). Referring to figure.5, the radius, r, in equation.47 is measured from the vertex of the hopper, and does not denote the radius of the hopper. Therefore, at the level of the maximum pressure, let r=r N, which can be written in terms of i N as follows:

46 . DES NORTJE: PhD THESIS: THE ANTI-DYNAMIC TUBE IN MASS FLOW SILOS rn zn Cos in Hh Cos Then equation.47 becomes: ks in Hh (.56) Cos k The equation for the maximum pressure has been given by equation.3 and is repeated as follows: N =D/[(Tan+Tan w )(+z N /H h )] In equation.3, D is the diameter of the hopper as shown in figure.6, and can be re-written as : D=r=H h Tan And by definition i N =z N /H h HhTan Therefore, equation.3 becomes: N (.57) Tan Tan i w N By letting equation.56 equal equation.57, and re-arranging, an equation for the position of the peak pressure, ratio i N, is given as follows: in in k Sin k S Tan Tan w (.58) where S is the radial pressure field in the solid and is determined from equations.54 and.55. The position of the maximum pressure is directly related to the stress ratio, k, and the hopper half angle,, and is indirectly related to the stress field, S, and the hopper wall friction angle, w. Re-arranging equation.58 in terms of w and gives the following: k Sin Tanw Tan (.59) in in ks A graph of equation.59 for k=.8, S=.4 and varying the hopper half angle has been plotted in figure.7 below. Unlike the graph given in Jenike (968), the contours of i N in figure.7, decrease from i N =.8 closest to the horizontal axis to i N =. closest to the vertical axis. This is the reverse of the graph given by Jenike. Increasing the values of both the stress ratio, k, and the stress field, S, from.3 to.9 has the effect of decreasing the required wall friction angle for a given hopper half angle. Since these graphs represent filling conditions in the hopper, the stress

47 CLASSIC WALL PRESSURE THEORIES: STATIC PRESSURES: HOPPER.3 ratio, k, cannot have a value greater than one. Increasing the stress field, S, beyond a value of gives a negative value for the wall friction angle, which is not possible. Wall friction angle w i N =. i N =. i N =.4 i N =.6 i N = Hopper half angle Figure.7: Contours of i N for a conical hopper for k=.8 and S= Walters Static Hopper Pressures. In his paper Walters(97) gives the same analysis for the static vertical pressure in the hopper as for the dynamic vertical pressure in the hopper. The full analysis of the equilibrium on a horizontal elemental slice in the hopper section, by Walters, has been given in chapter... of this thesis. The difference between the static and the dynamic case, is that in the static case, the shear stress at the wall is the minimum value. This minimum value is determined from the intersection of the wall yield locus and the Mohr circle representing the stresses at the wall, as shown graphically in figure.8. Walters analysis of a horizontal elemental slice in a hopper results in the same differential equation (equation.3) for both the static and dynamic conditions, except that the constants E and F H are as follows: Sin Sin m D E Sin Cos m D (.5 repeated)

48 .4 DES NORTJE: PhD THESIS: THE ANTI-DYNAMIC TUBE IN MASS FLOW SILOS Sin Cos Sin Sin m m F (.8 repeated) H Cos Sin ysin m m where the +ve sign refers to static conditions and the ve sign refers to dynamic conditions. Material yield locus Wall yield locus Dynamic shear value Static shear value + + Figure.8 General Mohr circle for stresses in the material adjacent to the wall.

49 CLASSIC WALL PRESSURE THEORIES: DYNAMIC PRESSURES: CYLINDRICAL SECTION.5. DYNAMIC WALL PRESSURES.. CYLINDRICAL SECTION... The Walters Theory Walters (973a) follows the same assumption made by Jenike, that the lines of major principal stress are approximately vertical during initial filling of the solids into the silo, as shown in figure.9(a). When the discharge gate is opened and the material starts flowing, the lines of major principal stress become nearly horizontal, as shown in figure.9(b). In both cases, the angle which the major principal stress makes with the wall is S and D, for the static and dynamic cases. v vw S D H dz v v z dz (a) (b) (c) Figure.9: Lines of major principal stress (a) Static, (b) Dynamic; (c) Force balance on an elemental slice In the same manner as Janssen, Walters solves for the equilibrium of vertical forces acting on an elemental slice of thickness, dz, in the cylindrical section of the silo. The result is stated again as follows: dv dz 4 vw (.6) D where v is the average vertical stress acting across the elemental slice and vw is the shear stress on the silo wall. To solve equation.6, the shear stress at the wall, vw must be related to the average vertical stress, v acting across the slice. First vw is related to v and then vw is related to vw. Since the vertical stress acting across a horizontal elemental slice is not constant, Walters assumes the average vertical stress, v, is related to the vertical stress adjacent to the wall, vw, by a distribution factor, as follows: vw = F v (.6)

50 .6 DES NORTJE: PhD THESIS: THE ANTI-DYNAMIC TUBE IN MASS FLOW SILOS The distribution factor has been determined by Walters from the mohr circle given in figure., which has been greatly enlarged in this thesis for clarity. The distribution factor has been given the symbol F and not D (as in Walters), to avoid confusion with D used for the silo diameter. The calculations have been continued after figure. m vw v C w H vw v Material yield locus Wall yield locus Stresses at the wall Average stresses Stresses at the centre x Figure.: Mohr Circle for Stresses at the Wall and in the Material for Flow Conditions

51 CLASSIC WALL PRESSURE THEORIES: DYNAMIC PRESSURES: CYLINDRICAL SECTION.7 The green circle in figure., represents the stresses at the centre of the silo, the blue circle represents the average stresses, and the red circle represents the stresses at the wall. All three circles are tangential to the material effective yield locus. The vertical stress, vw, and the shear stress, vw, at the wall, are determied from the intersection of the wall yield locus with the circle for wall stresses. Walters assumes that the horizontal stress, H, remains constant across the slice as shown in figure.. The circle for average stresses has a radius, x, and the centre of the circle is given by ( H + v )/ The centre of the circle is related to the internal angle of friction of the material as follows: H v x Sin m x which can be re-written as: v H (.6) Sinm The radius of the Mohr circle, x, for average stresses is given by: x v H v H (.63) Substituting equation.6 into equation.63, solving for x and re-arranging gives the following expression for the radius of the circle: Tan m x H H Sin m Cos vw m (.64) Cos m Silo centre line v Silo wall vw =max The shear stress at the centre of the silo is zero, while the shear stress is a maximum at the silo wall. Walters assumes that the shear stress varies linearly as shown in figure.. The shear stress at an arbitrary distance, r, r D/ from the centre of the silo is given by v, and is related to the shear stress at the wall, vw, as follows: Figure.: Shear stress variation across the silo v = vw r / (D/) (.65) Substituting for x from equation.64, and from equation.65 into equation.63 gives the following expression for the distribution factor, F given in equation.6,

52 .8 DES NORTJE: PhD THESIS: THE ANTI-DYNAMIC TUBE IN MASS FLOW SILOS as follows: vw = F v where F is given by: Cosw Sin m Sin m Sin w F (.66) 3 / 4Tan m Sinm Tan w Cos w Sin m 3 Tan w Tan m The graph of the distribution factor, F, as a function of the material friction angle, m, and the wall friction angle, w, has been shown in figure. below. 3. w =89 Distribution Factor (F) w =4 w = Material Friction Angle ( m ) Figure.: Distribution factor as a function of both material friction angle ( m ) and wall friction angle ( w ) The curves of the wall friction angle in figure. have been given in increments of. From equation.66 for F, a wall friction angle of 9 has no meaning as this would require a division by zero, therefore a maximum value of w = 89 has been shown in figure.. From the figure it can be seen that for all values of material friction angle and wall friction angle, the vertical stress at the wall of the cylinder, vw, is greater than the vertical stress, v, at the centre of the cylinder. Now that vw has been defined as a function of v the next step in the solution of equation.6, is to define the shear stress at the wall, vw as a function of vw.

53 CLASSIC WALL PRESSURE THEORIES: DYNAMIC PRESSURES: CYLINDRICAL SECTION.9 To find an expression of vw as a function of vw the Mohr stress circle for the stresses at the wall, as given in figure., has been repeated in figure.3 below. From the Mohr circle Walters derives an expression for the ratio of vw / vw. Material yield locus Wall yield locus P Stresses at the wall M N vw H D W vw m w O Figure.3: Mohr circle for stresses at the wall Let the radius of the circle be denoted by a = PW = NW Let B= vw / vw = P H / (OW W vw ) And P H = a Sin D ; OW = a / Sin m ; W vw = W H = a Cos D (.67a,b,c) The subscript D in D refers to the angle shown in figure.9b for the case of material discharge ie the dynamic case. Therefore, B = Sin D Sin m / ( - Sin m Cos D ) (.68) From triangle MWO: + D = / + w (.69) Therefore, D = / + w - (.7) From triangle PMW: = Arc Cos (MW/a) (.7) And MW = OW Sin w (.7) Substituting equation.67b for OW into.7 gives: MW = a Sin w / Sin m (.73) Substituting equation.73 into equation.7 and re-arranging gives : D / + w - Arc Cos( Sin w / Sin m ) (.74)

54 .3 DES NORTJE: PhD THESIS: THE ANTI-DYNAMIC TUBE IN MASS FLOW SILOS Substituting equation.74 into equation.68 and re-arranging gives the following expression for B: Sinw Cos m B (.75) Cosw Sin m Sin m Sin w Therefore, vw = B vw (.76) The graph of B as a function of the material friction angle has been given in figure.4 below. Walters gives the plot of the factor B as a function of the wall friction angle, therefore the shape of the graph in figure.4 does not correspond to that given by Walters. The curves of varying wall friction angle have been given in increments of. The first two curves for wall friction angles of 5 and have been plotted in dotted lines for clarity only. 3 w = w =5 w =.5 Ratio (B).5 w = Material Friction Angle ( m ) w =8 Figure.4: Ratio B as a function of material friction angle. Substituting equation.6 into equation.76, and then substituting this into the equation of equilibrium of forces (equation.6) gives the following: dv dz 4BF v D (.77) where F is given by equation.66 and B is given in equation.75. Plotting the product of the distribution factor, F, and the ratio, B, gives a set of curves similar in shape to the curves in figure.4. This product has been shown in

55 CLASSIC WALL PRESSURE THEORIES: DYNAMIC PRESSURES: CYLINDRICAL SECTION.3 figure.5 below. The curves for wall friction angles of 5, and have been plotted in dotted lines for clarity only. For most silo problems the wall and material friction angles lie within the range shown by the red dotted line in figure.5. 3 w = w =.5 w =4 Product BF Material Friction Angle ( m ) w =8 Figure.5: Product of distribution factor, F, and ratio, B. Before Walters gives the solution to equation.77, he first puts the variables in dimensionless form by dividing throughout by D and letting S V equal v /d and Z=z/D as follows: dsv dz 4BFSV (.78) The solution to equation.78 for the vertical pressure in a silo has the form : S V 4BFZ e (.79) 4BF Walters gives the horizontal pressures acting normal to the silo wall from the relationship given below: S H = S V *B*F/m (.8) As Walters points out these equations are of similar form to the classical Janssen equation, with the factor BF in place of Janssen s K a, where Tan w and K a is the active stress ratio.

56 .3 DES NORTJE: PhD THESIS: THE ANTI-DYNAMIC TUBE IN MASS FLOW SILOS The vertical pressures acting across a slice have been given in figure.6a, and the horizontal pressures acting normal to the silo wall have been shown in figure.6b. The static pressures acting in the silo have been calculated using the Janssen equations as given by equation.5 for the vertical pressures, and equation.6 for the horizontal pressures. The vertical dynamic pressures have been calculated using Walters equation.79, and the horizontal pressures have been calculated using equation.8. To keep the equations consistent, the Janssen equation has been divided throughout by D to present it in dimensionless form. Vertical pressure Horizontal pressure Janssen static Janssen static Depth within silo Z=z/D Walters dynamic Walters dynamic 7 8 (a) (b) Figure.6: Static and Dynamic Pressures: (a) Vertical pressure (b) Horizontal pressures From figure.6 it can be seen that the vertical dynamic pressures are approximately 3 times less than the static pressures for values of Z greater than 5. However, the dynamic horizontal pressures are greater than the static pressures by a factor of approximately.6. This excludes the effect of a switch pressure which has been discussed in chapter.3. The dynamic pressures approach a constant value at a height to diameter ratio of approximately, whereas in the case of static pressures, the asymptote is only reached at height to depth ratios of approximately 5.

57 CLASSIC WALL PRESSURE THEORIES: STATIC PRESSURES: HOPPER.33.. HOPPER SECTION... Flow/Slip in the Hopper by Equilibrium of a Slice In order to examine the variation of stress within the hopper, a slice equilibrium model similar to the Janssen model has been investigated, as shown in figure.7. The model assumes that the wall slip and the material flow occur simultaneously, so that the Mohr circle is tangential to the yield surface and the wall stress is defined by the friction angle w. A circular hopper with a half angle (, a radius of r at the transition has been assumed. The depth (z) is measured from the transition, and the stress at the transition is given by Janssen theory. The vertical stress v is assumed to act uniformly over the slice, and the pressures normal and tangential to the wall are given by N and respectively. The material in the hopper has a bulk density of. It will be shown that the slice model leads to a first order differential equation with non-constant coefficients, and that a solution can be obtained numerically. Importantly, the first order differential equation allows only one boundary condition for stress at the top of the hopper. The variation of stress with depth and the stress at the hopper bottom are dependent on the material properties and the hopper half angle, and are obtained from the numerical equation solution. V The radius of the top of the element is given by: rt r z Tan (.8) r The radius of the bottom of the element r t dz is given by: rb r (z dz) Tan (.8) N r b z And the average radius of the element is given by: rav r (z dz / ) Tan (.83) Figure.7: Elemental slice of material in the hopper The shear stress acting on the wall of the hopper is given by: v NTan w Tanw (.84) Q The pressure normal to the hopper wall N, has been substituted by v / Q, where the expression for Q will be derived later.

58 .34 DES NORTJE: PhD THESIS: THE ANTI-DYNAMIC TUBE IN MASS FLOW SILOS By taking the sum of the vertical forces equal to zero, and considering the downwards direction as positive, gives the following equation for equilibrium: vrt v v z dz rb NSin rav dz Cos Cosrav dz Cos rav dz (.85) Substituting equations.8,.8,.83 and.84 for r t, r b, r av and N in equation.85, and re-arranging, collecting and canceling common terms, gives the equation below. v z r ztan Tan r ztan Q Q Tan Q w r ztan v (.86) The boundary value problem given in equation.86 is a first order differential equation and can be re-written as: v c v z where c is the term given in the square brackets in equation.86. (.87) The term (r -ztan) in the denominator of c, was found to make the solution of equation.87 unstable. Therefore, the differential equation was re-written as: v ztan z v r ztan C r (.88) Tan C (.89) Q Q Q where w Tan r ztan r ztan N D - V - + Hopper wall Figure.8: Stresses acting on an element - To find the expression for Q in equations.84 and.86, the stresses acting on an element of material, adjacent and parallel to the hopper wall, has been shown in Figure.8. The corresponding Mohr circle has been shown in Figure.9. The sign convention used is that of Gere and Timoshenko (996), where compressive stresses and shear stresses acting in a clockwise direction are negative, and shear stresses acting in a counter clockwise direction are positive. A positive or negative symbol has been placed next to the arrowhead of each

59 CLASSIC WALL PRESSURE THEORIES: STATIC PRESSURES: HOPPER.35 stress in Figure.8. The hopper wall has a half angle inclined to the vertical, and a wall friction angle of w. The stress acting normal to the wall is N and the stress acting at right angles to N is D. The vertical stress is V and acts on a horizontal plane as shown in Figure.8. In Figure.9 below, the arrowheads on the axes indicate the positive directions of normal and shear stresses. The circle has been drawn tangential to the material yield locus since the material has not yet yielded. The maximum shear stress which can occur between the element and the wall, is determined by the point of intersection of the wall yield locus and the circle. This point locates the normal stress N on the circle. Using the method of origin of planes, the plane of the hopper wall has been plotted at an angle, to the vertical and has been shown by the dotted line through N in Figure.9. The intersection of this line with the circle locates the origin of planes labeled O P. Since the vertical stress acts on a horizontal plane, a line has been drawn through O P to locate the point of the vertical stress on the circle, labeled V. Thus the conjugate pairs of stress D and H are located, since the other points have been established. Material yield locus Wall yield locus N H C W M O + D O P V + Figure.9: Mohr circle for an element adjacent to the hopper wall. The radius of the circle can be given by: r N D N W where W =Tan W. (.9)

60 .36 DES NORTJE: PhD THESIS: THE ANTI-DYNAMIC TUBE IN MASS FLOW SILOS Let = D / N which is less than since D < N on the Mohr circle. Then the radius of the circle can be re-written as: r N 4w (.9) The vertical stress can be given by: N D v r Cos (.9) where ATN N Tan N D w Tan ATN w (.93) Substituting equation.9 and.93 into.9, and re-arranging gives the following expression for the vertical pressure acting in the hopper under filling conditions: v Q N (.94) where Q Tanw 4 w Cos ATN (.95) The only unknown in the expression for Q is the stress ratio given by D / N. To solve the boundary value problem given by equation.88, the Runge-Kutta method of numerical mathematics has been used. This is stated as follows: h n n 6 Kn Kn Kn3 Kn4 (.96) where: K n =ƒ(z n ; Nn ) K n =ƒ(z n +h/ ; Nn +K n h/) K n3 =ƒ(z n +h/ ; Nn +K n h/) (.97 a,b,c & d) K n4 =ƒ(z n +h ; Nn +K n3 h) The solution of equation.88 using equation.96 and.97a,b,c and d, has been done using an excel spreadsheet. The full set of calculations showing the implementation of the Runge-Kutta method on the differential equation.88, has been given appendix A.

61 CLASSIC WALL PRESSURE THEORIES: STATIC PRESSURES: HOPPER.37 Depth of hopper Vertical Pressure The solution of equation.88 gives a curve as shown in Figure.3. The shape of this curve is in contrast to that given by Walker, refer Figure.7. According to the theory given in this thesis, the maximum vertical pressure is at the transition from cylinder to hopper, and not at the base of the hopper, as suggested by Walker. Furthermore, the pressure decreases rapidly with depth in the hopper. Figure.3: Typical vertical pressure curve in a hopper To determine the effect of changing the variables on the solution of equation.88, a sample silo was analysed. In this example, the silo has a height of 7m, a diameter of m, a hopper depth of.6m and hopper half angle of 5, a wall friction angle of and a bulk density of 7kN/m 3. The value of the stress ratio, K, used to determine the vertical pressure at the level of the transition was.4. These results have been shown respectively in Figures.3 a,b,c,d and e, for varying hopper radius, material bulk density, wall friction angle, hopper half angle and the stress ratio,. Vertical Pressure (kpa) 4 6 Vertical Pressure (kpa) Depth (m) r= r= r=3 Depth (m) = =5 = 7 (a): Changing silo radius (b): Changing material bulk density Figure.3 (a & b): Effect of variables on vertical pressure in the hopper.

62 .38 DES NORTJE: PhD THESIS: THE ANTI-DYNAMIC TUBE IN MASS FLOW SILOS In Figure.3(a), the silo radius has been varied from m to 3m. It can be seen that at depths equal to one silo diameter, there is an increase in the vertical pressure. For a radius of m, the vertical pressure at a depth of m is 3.34kPa, while for a radius of 3m the vertical pressure at a depth of 6m is4.76kpa. This is an increase of 4.5%. In Figure.3(b), the material bulk density has been increased from kn/m 3 to kn/m 3. At a depth of m, there is an increase in the vertical pressure of 9.7kPa to 8.54kPa for an increase in bulk density from kn/m 3 to kn/m 3. Thus when the bulk density is doubled, so too is the vertical pressure. Since the bulk density of a material can vary with time, moisture content and source, a range of densities needs to be specified when designing a silo. In Figures.3 c & d below, the effects of changing the stress ratio and the wall friction angle have been shown. From Figure.3 (c) it can be seen that at a depth of m, the vertical pressure is 5.76kPa for a stress ratio of.3, while for a stress ratio of.9 the vertical pressure increases to kpa. This is an increase of approximately 45%. Therefore, using the correct stress ratio has a significant effect on the value of the calculated vertical pressure. From Figure.3(d) at a depth of m below the transition, it can be seen that changing the wall friction angle from to has the effect of decreasing the vertical pressure from.53kpa to 5.kPa. This is a decrease of approximately 33%. Vertical pressure (kpa) Vertical pressure (kpa) Depth (m) =.3 =.6 =.9 Depth (m) = =5 = 7 7 (c): Changing hopper stress ratio (d): Changing wall friction angle Figure.3 (c & d): Effect of variables on vertical pressure in the hopper.

63 CLASSIC WALL PRESSURE THEORIES: STATIC PRESSURES: HOPPER.39 Depth (m) Vertical Pressure (kpa) = = =3 From Figure.3 (e) it can be seen that increasing the hopper half angle from to 3 has the effect of increasing the vertical pressure, up to a depth of.75r below the transition. Beyond this point, the vertical pressure decreases with increasing hopper half angle. (e): Changing the hopper half angle Figure.3 (e): Effect of variables on the vertical pressure in the hopper... Walters Pressures in Converging Channels Walters (97b) developed his theory for pressures acting on the walls of axially symmetric hoppers in the same manner as his theory for pressures in vertically sided silos. Figure.3a below shows the lines of the major principle stress in the hopper when the material starts flowing. The stresses acting on a horizontal elemental slice have been shown in figure.3b. D D z Area A w v dz Area (A+dA) w v + d v (a) (b) Figure.3: (a) Lines of major principle stresses (b) Elemental slice

64 .4 DES NORTJE: PhD THESIS: THE ANTI-DYNAMIC TUBE IN MASS FLOW SILOS Walters gives the results of the vertical force balance on the elemental slice in the hopper as follows: dv dz A da v dz P A Tan g w w (.99) where A is the cross sectional area of the element at a depth z from the top of the hopper, P is the perimeter at that depth, v is the uniform average vertical stress acting across the slice, w is the shear stress acting along the wall, w is the stress normal to the hopper wall and is the hopper half angle. To solve equation.99, Walters finds a relationship between the third term in equation.99, ( w + w Tan), and the vertical stress at the hopper wall, vw, from the geometry of the Mohr circle given in figure.33. As before, the green circle represents the stresses along the centreline of the hopper and the red circle represents the stresses at the hopper wall. The stresses at a distance r from the centre of the hopper have been shown in figure.33 by the blue circle. Since Walters makes the assumption that the horizontal stress, H, remains constant across the slice, the points P and N can be located on the Mohr circles. Walters further defines the angle as that angle which the line through point P makes with the horizontal axis of the graph in figure.33. Let the radius of the circle for the stresses at the wall be denoted by a, then: w = asin D (.) Similarly w = OC + C w = a/sin m + acos D (.) Therefore, substituting equations. and. into the third term of equation.99 and rearranging gives the following: w + w Tan = a ( Sin m Sin D + ( +Sin m Cos D )Tan ) / Sin m (.) To eliminate the radius, a, from the expression in equation., Walters finds a relationship between the vertical shear stress at the wall and the vertical stress at the wall, as follows: Let vw = E vw (.3)

65 CLASSIC WALL PRESSURE THEORIES: DYNAMIC PRESSURES: HOPPER SECTION.4 Wall yield locus Material yield locus P N a x D w vw v vc H C m w Stresses at the centre Stresses at distance r from hopper centreline Stresses at the wall vw w vr O Figure.33: Mohr circle for stresses in the hopper during discharge where both vw and vw are found from the Mohr circle to be: vw = a Sin ( + D ) and vw = a /Sin m a Cos( + D ) (.4a,b)

66 .4 DES NORTJE: PhD THESIS: THE ANTI-DYANMIC TUBE IN MASS FLOW SILOS Substituting.4a,b into.3 and rearranging gives the expression for E as follows: SinmSin D E Sin Cos m D (.5) The expression for angle D in equation.5 is found from the Mohr circle shown in figure.34 as follows: In triangle MCO, MC = OC Sin (.6) In triangle NCO, OC = a / Sin m (.7) Therefore, MC = a Sin / Sin m (.8) In triangle PCM, = ArcCos(MC/a) (.9) Substituting for MC from equation.8: = ArcCos (Sin / Sin m ) (.) From triangle MCO, ++ D = / + (.) Therefore, + D = / + -/ + - ArcCos (Sin / Sin m ) (.) The expression for + D in equation. is similar to the expression derived for D in equation.9, except w is replaced by. It can be seen from figure.34 that as the hopper half angle tends to zero, ie a vertically walled silo, the angle tends to the wall friction angle w. The method for determining the angles w and has been given at the end of this section. Material yield locus Wall yield locus P N vw w a D M H C vw m w O Figure.34: Mohr circle for stresses at the wall

67 CLASSIC WALL PRESSURE THEORIES: DYNAMIC PRESSURES: HOPPER SECTION.43 Now the radius, a, can be eliminated from the expression in equation. by dividing ( w + w Tan ) by vw to give the following: w w vw Tan Sin m Sin D SinmCos Sin Cos m D D Tan (.3) To simplify equation.3, Walters adds and subtracts TanSin m Cos(+ D ) from the numerator on the RHS, which after rearranging gives the following: w wtan vw SinmSin D Sin Cos m D Tan E Tan (.4) Substituting equation.4 into equation.99 gives: dv dz A da v dz P A E Tan g vw (.5) Equation.5 still cannot be solved due to the unknown term of vw, therefore Walters defines the relationship between the average vertical stress, v, and the vertical stress at the wall, vw, as before (refer chapter...): vw = F h v (.79 repeated) where F h is the distribution factor associated with the hopper stresses. To determine the expression for the distribution factor F h, Walters defines the relationship between the average vertical stress, v, and the vertical stress at a distance, r, from the centreline of the hopper, vr, as shown in figure.33. Line of average vertical stress, v Vertical stress, vr at distance r Horizontal elemental slice r C L Figure.33. Average and general stress distribution across an elemental slice in the hopper. Figure.33 shows a typical horizontal elemental slice in a hopper, with the line of average vertical stress acting on the element. The line of the real vertical stress has been shown arbitrarily by the curved line in figure.33. The vertical stress at a distance, r, has been shown by the dotted vertical line. In Walters paper, the relationship between v and vr has the general form:

68 .44 DES NORTJE: PhD THESIS: THE ANTI-DYANMIC TUBE IN MASS FLOW SILOS v r rw r vr dz (.6) Walters assumes that the vertical shear stress, vr at a distance r from the centre of the hopper is a linear function of the vertical shear stress at the wall of the hopper, vw, as before, so that: vr / vw = r/r w vr = vw r/r w (.7) where r is an arbitrary distance from the centre, and r w is the radius of the hopper at the wall. Let the radius of the circle through point N be x, then: x = vr + (x / Sinm - r ) (.8) And hence: Sinm x H H Sin m vr Cos m (.9) Cos m In the Mohr circle for stresses at the wall, + D < / + m. If the hopper half angle is greater than this limit, mass flow in the hopper can not occur. From the geometry of the Mohr circle: v = x / Sin m - H (.) Substituting for x from equation.9, and for vr from equation.7 into equation., and re-arranging, gives the following: v H Cos m Sin m Sinm cr / r w (.) where c=(tan / Tan m ) Substituting equation. for v into equation.6, the average stress across the slice can be integrated to give the following: v H Cos m H Cos m sin Sin c m sin m m 3c y Sinm 3 / (.)

69 CLASSIC WALL PRESSURE THEORIES: DYNAMIC PRESSURES: HOPPER SECTION.45 where y = [-(-c) 3/ Tan m ] /3c = Tan 3Tan Tan m 3 / (.3) From the Mohr circle in figure.3, H = vw /Tan (.4) Substituting for vw from equation.3 into equation.6 gives: H = E vw /Tan where E is given in equation.5 (.5) Substituting for H from equation.5 into equation. gives the following: v Sin m ysinm E vw Tan Cos m (.6) Now substituting for E from equation.5 into equation.6, and re-arranging gives the following: Sin m ysinm vw F Sin Sin Sin h Cos vw v (.7) Cos m m where FH Cos Sin m Sin m Sin (.8) Cos Sin m y Sinm Substituting for vw from equation.7 into equation.5 gives the following: dv dz da v A dz P A E Tan F g da 4Tan For a conical hopper: A dz D ztan H v (.9) (.3) and P A 4 D ztan (.3) Substituting equation.3 and.3 into equation.9, and re-arranging gives the following: dv dz 4v D ztan E F Tan F g H H (.3) The equation for the vertical force balance on the elemental slice given in equation.3 can now be solved. As before Walters puts this equation into dimensionless form; Let S v = v /(gd), Z=z/D, Z = z /D (z = at the top of the hopper) E FH and let M FH (.33) Tan

70 .46 DES NORTJE: PhD THESIS: THE ANTI-DYANMIC TUBE IN MASS FLOW SILOS Then equation.3 can be re-written as: dsv dz MTan Sv (.34) MTan The solution to the linear first order differential equation,.34 is: M ZTan ZTan S v (.35) Tan M ZTan To find the vertical pressure in the hopper, the unknowns in the expressions for E and M need to be determined. From equation. for + D, the graph in figure.34 has been plotted for determining the value of D and. The curves of material friction angle varying from to 8 has been shown in increments of. To demonstrate the use of figure.34, a curve has been plotted for a silo with a wall friction angle of a material friction angle of 45 and a hopper half angle of 5. The value of for the wall friction is entered along the x-axis. The intersection of this vertical line with the curved line gives a value of D,for this example, on the y-axis as 54. To this value of D a value of is added = 5 which gives + D as D or ( D + ) m = ( w ) or () Figure.34: D + as a function of w and

71 CLASSIC WALL PRESSURE THEORIES: DYNAMIC PRESSURES: HOPPER SECTION.47 4 m =45 D + =4 + 8 D or ( D + ) 6 4 D =54 w = = ( w ) or () Figure.35: D + as a function of w and for a material friction angle, m =45 The intersection of this horizontal line with the curve gives the value of as 39 along the x axis of the graph. Having thus found the value of, this can be used in equation.8 to evaluate F H Substituting equation.3 for y into equation.8 shows that equation.8 is the same as equation.84 for F, except that w is replaced by. The graph for F H is therefore, the same as figure.6 for various curves of angle. All the unknowns in the expression for the average vertical pressure in the hopper given by equation.35 have been defined and the stress distribution in a hopper without a surcharge can be plotted, as shown by the graphs in figure.36. For all three graphs the scales of the vertical pressure, on the horizontal axis, and the depth, on the vertical axis, are the same. Figure.36a was calculated for a material friction angle of 5, wall friction angle of and the hopper half angle varied from 5 to 5 in increments of 5.

72 .48 DES NORTJE: PhD THESIS: THE ANTI-DYANMIC TUBE IN MASS FLOW SILOS Dimensionless Average vertical pressure S v =5 = =5 m =5 m =3 m =35 w =5 w = w =5 Z=z/D (a) (b) (c) Figure.36: Average vertical pressure in conical hopper : (a) Effect of changing hopper half angle, (b) Effect of changing material friction angle (c) Effect of changing wall friction angle Similarly figure.36b was calculated for a wall friction angle of, a hopper half angle of and the material friction angle varied from 5 to 35 in increments of 5. In figure.36c the material friction angle used was 5, the hopper half angle was and the wall friction angle varied from 5 to 5 in increments of 5. From figures.36a and b it can be seen that a larger hopper half angle and higher material friction angle both have the effect of lowering the vertical pressure acting in the hopper, while from figure.36c it can be seen that the wall friction angle has a negligible effect on the vertical pressure in the hopper. For a hopper with a surcharge pressure due to the cylindrical section above the solution to equation.34 would be as follows: Sv M M ZTan ZTan ZTan S v Tan M ZTan ZTan (.36) where all variables are as previously, Z is the dimensionless depth ratio at the transition, S v is the vertical pressure acting at the transition due to the material in

73 CLASSIC WALL PRESSURE THEORIES: DYNAMIC PRESSURES: HOPPER SECTION.49 the cylindrical section of the silo. The value of S v can be calculated from equation.97 (the Walters expression for the vertical pressure in a cylinder). Dimensionless average vertical pressure, S v Figure.37 shows the Height to diameter ratio, Z Cylinder Hopper dynamic average vertical stress acting in a silo filled with material with a friction angle of 5, a wall friction angle of and a hopper half angle of. The dotted line in figure.37, shows the pressure distribution in the hopper with no surcharge acting in the cylindrical section. 3 Figure.37: Dynamic average vertical stress acting in a silo...3 The Jenike Radial Stress Field Jenike (968) defines the condition of flow as a particular case of failure, which occurs when pressures are such that shear occurs without destroying the isotropy of the material. During flow, the bulk density of the material is a function of the pressures. When the pressures are constant, the solid shears at a constant density. When the pressures increase, the solid compacts and the density increases, and when the pressures decrease, the solid expands and the density decreases. Thus, flow can proceed indefinitely. In the hopper the mass of solid contracts laterally and expands vertically which implies horizontal, or nearly horizontal, major principal stresses and a plastic-passive state of pressure exists in the hopper. This state may extend to the top of the bin. Because the solid slides along the walls as it flows, the vertical pressure at the wall, vw is accompanied by a frictional stress at the wall w. There is a change in the wall pressure at the transition from the cylindrical section

74 .5 DES NORTJE: PhD THESIS: THE ANTI-DYANMIC TUBE IN MASS FLOW SILOS to the hopper section, with the pressures decreasing to zero at the vertex of the hopper. The speed of the flowing material is generally sufficiently slow and close to the steady state condition for the inertia forces to be negligible. Therefore, the conditions of equilibrium are satisfied and the vertical force supported by the walls is equal to the weight of the stored material. The element in the hopper is the same as that shown in figure.8 and figure.9. The symbols used in this chapter are the same as those used in chapter...3. resulting in the same equations of equilibrium as previously. These equations have been repeated here for continuity. Considering equilibrium in the r-direction + r r r r r d Cot Cos r c r (.56 repeated) Considering equilibrium in the -direction + r r r 3 r d Cot Sin c (.58 repeated) In Bulletin 8, Jenike (96) gives the following relationships from the Mohr circle as follows: Sinm Cos (.37) r c Sin Cos m m (.38) Sin (.39) from the assumption that in axial symmetry the circumferential stress is equal to either the major or minor principal stress of the median plane. r = Sin m Sin (.4) In Bulletin 8, Jenike(96) gives the following relationship for the average vertical stress in the hopper: v =rs (.4) where is the density and is assumed a function of r and as follows: = (r,) s is the radial stress field and is also function of r and as follows: s=s(r,) The angle between the major principal stress and the r co-ordinate in figure.8 is given as ; = (r,)

75 CLASSIC WALL PRESSURE THEORIES: DYNAMIC PRESSURES: HOPPER SECTION.5 The derivative of the shear stress given in equation.4 therefore becomes: r r r r r r Sin m s Sin r rs s SinmCos (.4) r r r The derivative of the average vertical stress with respect to is as follows: s Sinm Cos r rs SinmSin (.43) The derivative of the average vertical stress with respect to r is as follows: r r r r r r s SinmCos r s SinmSin (.44) r r Substituting equations.4 and.43 into the equation of equilibrium in the direction given by equation.58 gives the following: s Sin msinr r rs r s SinmCos r s m m r m Sin Cos r rs Sin Sin 3Sin Sin... d Cot Sin Cos Sin Sin... m m (.45) r Now substituting for =rs into equation.45 and collecting terms in s/r, s/, s and the remaining terms gives the following: s A r s B Cs D (.46) where A = r Sin m Sin (.47) B = (-Sin m Cos (.48) C Sin m Sin r r r Sin Cos Sin rcos... m m

76 .5 DES NORTJE: PhD THESIS: THE ANTI-DYNAMIC TUBE IN MASS FLOW SILOS... Sinm Sin 3SinmSin SinmCot Cos (.49) D = -Sin (.5) Solving for s/r in equation.46 gives : s r D A C s A B A s (.5) Similarly in the r direction, substituting equations.4 and.44 into the equation of equilibrium given by equation.56, and collecting terms in s/r, s/, s and the remaining terms gives the following: s s E F Gs H r where: E r Sin Cos m (.5) (.53) F Sin Sin m (.54) G r r Sin Cos r Sin Sin r Sin Cos... m m Sin Cos Sin Cos Sin Sin SinCot... m m m m (.55) H Cos (.56) Substituting for s/r in equation.5 into equation.5 and collecting terms in s/, s and the remaining terms, gives the following: EB s EC ED F G s H (.57) A A A m Due to the lengthy size of equation.57, each term has been evaluated separately. To maintain an overview of the solution, equation.57 has been repeated several times during the following calculations. The first term of equation.57 is as follows: EB F A SinmCos Sinm Sin r SinmCos rsinmsin... Sin m SinmSin Cos m SinmSin (.58)

77 CLASSIC WALL PRESSURE THEORIES: DYNAMIC PRESSURES: HOPPER SECTION.53 EB s EC ED Equation.57 repeated: F G s H A A A EC G A Sin Cos r Sin Sin r... m r m r... SinmCos m m m... SinmSinCot Sin Cos Sin Cos Sin... Sin mcos Sin Cos r m r SinmSin r SinmCos... Cos m m Sin r... Sin Cos 3 Sin Cos... Cot... Sinm CosCos (.59) Sin Now multiplying out all the terms, replacing Cos with(-sin ), cancelling out, and after many lengthy algebraic manipulations, the second term of equation.57 can be simplified to: EC G A Cos SinmSin r m Sin 3 Cos Sin... m Sin Cot... Cos Sin m Sin (.6) r m EB s EC ED Equation.57 repeated: F G s H A A A ED SinmCos Sin H Cos A SinmSin (.6) Solving for s/ in equation.57 is as follows: s EC G A s EB F A ED H A EB F A (.6) As described previously, solving the terms in equation.6 is a lengthy process, equation.6 has again been repeated throughout the calculations to maintain an overview of the solution progress. To evaluate the first term in equation.6, substitute equations.6 and.58 to give the following:

78 .54 DES NORTJE: PhD THESIS: THE ANTI-DYNAMIC TUBE IN MASS FLOW SILOS EC G A EB F A Sin 3 Cos m m m Sin Sin... m Sin Sin Cos m rsinm CotSin... m m Cos r m Cos m m Cos Sin Cos Sin This can be simplified to give the following: EC G A EB F A SinmSin Cos m r Cos Sin Cos m m Sin m... r Sinm... Sin Sin 3 Cot Sin Cos Cos m m Cos m (.63) This is the same as the f(r,) term given in Jenike s Bulletin 8 (96). EC ED G H s A A Equation.6 repeated: s EB EB F F A A To solve the second term in equation.6, substitute equations.6 and.58 to give the following: ED H A EB F A Sin Cos Sin Sin m m m Sin Sin Sin mcos Sin Sin Cos Sin m which can be simplified to give: ED H A EB F A Sin Cos m m Sin Sin Cos m (.64) This is the same as the g(r,) term given in Jenike s Bulletin 8 (96)

79 CLASSIC WALL PRESSURE THEORIES: DYNAMIC PRESSURES: HOPPER SECTION.55 Thus, equation.6 has the form: s f(r, )s g(r, ) Therefore, (s/ f(r,)s - g(r,) (.65) where f(r,) is given by equation.63 and g(r,) is given by equation.64 Having solved for s/, equation.65 can be substituted into equation.5 to solve for s/r as follows: s r B GA EC A FA EB B C s A HA ED D FA EB (.66) As previously, the evaluation of both terms in equation.66 is a lengthy process and equation.66 has been repeated throughout the following calculations to maintain an overview of the solution progress. Evaluating the first term in equation.66 : B A GA EC Cos C FA EB r SinmSin rsin SinmSin Cos m... r Cos Sinm Sinm Sinm... Sin Sin m Cos m r Cos m 3... Sin m... Cot Sin Cos Cos Sin Sinr... m m Cos m r... m m r m Sin Cos Sin rcos 3 Sin Sin Sin m Sin Sin Cot Cos m This can be simplified by multiplying out all the terms and then collecting terms in /r, /, /r, / and the remaining terms. After considerable mathematical manipulation the result is as follows:

80 .56 DES NORTJE: PhD THESIS: THE ANTI-DYNAMIC TUBE IN MASS FLOW SILOS B GA EC A FA EB C r r Sinm Cos m Sin m Cos Sin Sin... m Cos m r Sinm... Sinm CotSin Cos (.67) r r Cos m Equation.66 repeated: s r B GA EC A FA EB B C s A HA ED D FA EB To evaluate the second term in equation.66, substitute equations.47,.48,.5, and equation.63 as follows: B HA ED A FA EB D SinmCos r SinmSin Sin m Sin Cos m Sin Cos m Sin Multiplying out, cancelling and collecting terms, the above equation can be further simplified to give: B HA ED A FA EB D Sin m r Cos m Cos Cos Cos m (.68) Both equations.67 and.68, can be multiplied throughout by the radius, r, of the hopper. This then gives the final expressions as found in Bulletin 8. Finally the two differential equations have now been defined as follows: s f (r, ) s g(r, ) s r h(r, ) s r j(r, ) where f(r,) is given by equation.63, g(r,) is given by equation.64. (.69) (.7)

81 CLASSIC WALL PRESSURE THEORIES: DYNAMIC PRESSURES: HOPPER.57 h(r, ) Sin Cos m m Sin m Cos Sin r Sin... m Cos m r r Sinm... Sinm CotSin Cos (.7) r Cos m Sinm Cos j(r, ) Cos (.7) Cos m Cos m Jenike further simplifies his calculations by making the assumption that is only a function of and that the density g is a constant. Then the terms /r, / /r become zero and equations.69 and.7 become: s f ( ) s g( ) (.73) s r r h( ) s j( ) (.74) Both equations.73 and.74 are first order linear partial differential equations. Sin Sin m m where f( ) Sin Sin 3 Cos Sin m Cos m m... Sinm... Cot Sin Cos Cos m Cos m (.75) Sin Sin g( ) Sin ( as before ) (.64 repeated) Cos Cos m m Sin h( ) Cos m m m m m Cos Sin m m Sin CotSin Cos m (.76) Sin Cos j( ) Cos ( as before ) (.7 repeated) Cos Cos In Bulleting 8, Jenike(96) gives the solutions to the differential equations given in equations.73 and.74. Then Jenike assumes s/r to be equal to zero. This simplifies the solution process considerably to give only two unknowns in equations.74 and.75. These unknowns are / and s/.

82 .58 DES NORTJE: PhD THESIS: THE ANTI-DYNAMIC TUBE IN MASS FLOW SILOS The unknowns / and s/ can easily be solved for. Equation.74 now becomes: h()s = -j() (.77) Multiplying equation.77 throughout by Cos m /Sin m and dividing throughout by s, re-arranging and collecting terms,gives the following expression for /as follows: Cos Sin m s Sin Cos m Cos Sin m scos m... Sinm CotSin cos ssin Cos Sin ssinm... (.78) m m Equation.78 represents the variation of with respect to along a given ray of radius, r as shown in figure.38. r On the axis of the hopper, equals zero and equals 9. At some arbitrary distance in the hopper, when varies from to, varies from to on the same ray. Thus a family of Figure.38: Variation of with respect to within the hopper. solutions can be plotted for equation.78 for a range of material friction angles, m, and stress field values of s. Substituting for / from equation.78 into equation.73 and solving for s/ gives the following: s s Sin Sin s Sin Cot Cos m Cos Sin m Sin (.79) Equations.78 and.79 can be solved using numerical methods and applying the physical boundary conditions of = on the axis of the hopper, and = at the wall.

83 CLASSIC WALL PRESSURE THEORIES: DYNAMIC PRESSURES: HOPPER.59 The solution to equation.79 is substituted into the expression for the average vertical stress given by equation.4, to give a radial stress field as follows v = r s() (.8) w w Figure.39: General Hopper geometry In Bulletin 8, Jenike(96) considers the general case of a hopper as shown in figure.39, where the walls of the hopper have different friction angles, the slopes of the hopper walls are not the same on either side of the axis and the wall friction angle approaches the material friction angle, ie rough walls. In this thesis only the solution of an axially symmetric hopper (ie one hopper half angle) and a wall friction angle less than the material friction angle has been considered. Since the density in equation.8 does not equal zero, the radius does not equal zero and the stress field s() does not equal zero, the radial stress field given by equation.8 cannot extend upwards to a free surface. Therefore, the stress field in a hopper without a surcharge deviates significantly from a radial stress field in the upper part of the hopper. The solution of equation.78 for can be plotted on a graph with as the vertical axis and as the horizontal axis. Jenike (96) states that the boundary conditions of equation.78 are not mathematically uniquely defined, but can be determined (-,) (,) (,/) (,) (,) Figure.4: General shape of the function / in (,) co-ordinates from the physical boundary conditions. There are two physical boundary conditions imposed on the equation for /which are symmetrically located as shown in figure.4. On the axis of the hopper, = / and = ; and at the wall =. Since there is no direct way of finding a solution which connects two boundary points, Jenike has computed several solutions and interpolated the required functions.

84 .6 DES NORTJE: PhD THESIS: THE ANTI-DYNAMIC TUBE IN MASS FLOW SILOS Having thus solved for the radial stress field, Jenike derives the normal wall pressure acting on the hopper during flow from equation.37, which has been repeated below: Sinm Cos (.37 repeated) D In the derivation of N, the normal pressure on the hopper wall, Jenike uses the (x,y) coordinates as shown in figure.4 below. B The hopper has a diameter, D, at the top, a N -x H h width of B=y at some distance x above the vertex, and a hopper half angle of. Figure.4: Co-ordinate system in the hopper x y The polar co-ordinates used in the derivation of the radial stress can be expressed in terms of (x,y) as follows: r = Sin(B/) at the wall. (.8) Substituting equation.8 into equation.8 and in turn substituting this into equation.37 gives the following expression for the normal stress acting on the wall: B s( Sin SinmCos N ) which Jenike has written in the following form: N B Sin s( ) SinmCos (.8) Putting equation.8 as a function of the depth of the hopper gives the following equation for the normal stress acting on the hopper wall: N x Sin s( ) D SinmCos (.83) H h where B has been replaced with Dx/H h. In equation.83 above, the term x/h h is the depth within the hopper as a ratio of the hopper height. From equation.83 it can be seen that N is a linear function of the radial stress s(), the bulk density of the material,, the diameter of the hopper at the transition, D, and the depth ratio, x/h h. The graph in figure.4 shows the hopper normal wall stress, and has been plotted for s() = D = =.

85 CLASSIC WALL PRESSURE THEORIES: DYNAMIC PRESSURES: HOPPER.6 Depth ratio (x/h h ).9 =5.9 =5 m = = Normal wall pressure N (kpa) Normal wall pressure N (kpa) (a) (b) Figure.4: Normal wall stress N, acting in the hopper during flow Figure.4(a) shows the effect on the value of N, of varying the hopper half angle from 5 to 5, for a material friction angle of 45and an angle of 6. Using the same scale, figure.4(b) shows the effect of changing the material friction angle from 45 to 5 (=6 as in figure (a) and =5). The third curve in figure.4(b) shows the effect of changing the angle from 6 to 45 ( m =45 as in figure (a) and =5). From equation.83, the greatest effect on the hopper normal wall pressure is due to the radial stress field s(), the material bulk density,, and the diameter of the hopper at the transition, D. The graphs in figure.4(a,b) show that the hopper half angle has a greater effect on the normal wall pressure compared to the material friction angle and the angle of the major principal stress,.

86 .6 DES NORTJE: PhD THESIS: THE ANTI-DYNAMIC TUBE IN MASS FLOW SILOS.3 SWITCH PRESSURES.3. CYLINDRICAL SECTION.3.. Jenike Upper Bound Pressures The pressures exerted by a solid stored in a silo are affected by the wall imperfections of the silo (ie deviations from cylindrical) and the boundary layers of material which form at the walls. Therefore, Jenike only considers the bounds on wall pressures, the minimum being described by Janssen (equation.5) and the critical upper bound which Jenike has based on the considerations of strain energy of the stored material.. During flow of a mass solid, Jenike(973b) states that the energy is lost at the maximum rate possible. Therefore, the recoverable strain energy tends towards a minimum, which is approached as closely as the wall imperfections will allow. Using Janssen s assumption of the stresses being independent of the horizontal coordinates, Jenike solves for the strain energy in the cylinder in one dimension. He neglects the strain energy due to shear stresses and assumes the cylinder walls to be rigid. Hence in his analysis, the vertical co-ordinate, z, is the independent variable. Figure.43 shows a typical horizontal element in a cylinder, of cross sectional area A. In his derivation, Jenike assumes the vertical pressure, V, to be the major principal stress, v z dz the horizontal pressure H, is the minor principle stress and the circumferential pressure, C, is the intermediate stress. Therefore: = V = RS (.84) = 3 = C = H = KRS (.85) Figure.43: Horizontal Element in the cylinder where K= H / V is the stress ratio. Since Jenike(973b) does not give a definition of the symbol S, it has been assumed in this thesis that S has been used to denote a stress field. Furthermore, while equation.84 appears to be similar to equation.63, the symbol R denotes the hydraulic radius of the cylindrical section and not the radial co-ordinate as previously. There is no explanation for this change of symbols in either Bulletin 8 (Jenike 96) or in the paper relating to the switch pressure, Jenike (973b)

87 CLASSIC WALL PRESSURE THEORIES: SWITCH PRESSURES: CYLINDRICAL SECTION.63 The recoverable strain energy in an element of thickness dz is given by: U A dz V d V H dh (.86) V H The recoverable parts of the energy expression given in equation.85 are as follows: dv dh dh d V d V, dh (.87a,b) E E E E In equation.87, Young s Modulus of the stored solid, E, and Poison s ratio,, are assumed constant. Substituting equations.87a,b into.86 gives the following: U A z E RS RS KRS dz KRS Collecting terms gives: z E KRS RSdz U A R E z 4K K S dz (.88) As the switch proceeds up into the cylindrical section, it is located at some arbitrary level, z, as shown in figure.44. Above this level the stress ratio K is equal to the Janssen stress ratio, and below this level the stress ratio varies and therefore is a function of S. Vertical Pressure z Janssen static pressure Static pressures: K=K Janssen Cylindrical silo Depth Switch pressure Dynamic pressures: K varies Figure.44: Location of the switch pressure during flow

88 .64 DES NORTJE: PhD THESIS: THE ANTI-DYNAMIC TUBE IN MASS FLOW SILOS Jenike (973b) assumes that the switch propagates slow enough so that acceleration terms are negligible, and therefore the equation of equilibrium of a horizontal elemental slice as shown in figure.43 is as follows: d dz K R v v (.89) The derivative of equation.84 is as follows: d v dz ds R dz (.9) And re-arranging equation.84 in terms of S gives: S = v /R (.9) Substituting equations.9 and.9 into equation.89 gives the following differential equation in terms of the stress field S: ds dz K S (.9) R R Equation.9 integrates to: Kz / R e S K (.93) Substituting the expression for S given by equation.93, into the equation for the strain energy, given by equation.88, gives the following; U 3 K 4K K... X... z Kz / R Kz / R... K e e (.94) R Re-arranging equation.9 gives K as a function of the stress field S, as follows : K ds R dz (.95) S By applying variational calculus to equation.88 for strain energy, the minimum energy can be obtained by letting U =. h AR U { S S 4 SKS SSK 4 SKSK} dz (.96) E o Both S, (S+S), and SK, (SK+SK) must satisfy equation.95, which is the equilibrium equation re-arranged. Substituting into equation.95 gives:

89 CLASSIC WALL PRESSURE THEORIES: SWITCH PRESSURES: CYLINDRICAL SECTION.65 RdS/dz + SK = Rd(S+S)/dz + (SK+SK) (.97) Cancelling out SK and RdS/dz and re-arranging equation.97 gives SK as follows: R d( S) SK (.98) dz Substituting equation.98 into.96 and integrating by parts gives the following: U A R E R h 4S 4 KS S... z... h R d S S K 4 S 4 KS dz dz (.99) z There is a boundary condition on S such that at z=z, S=S J (the Janssen static pressure) and therefore no variation in S is admissible. At z=h, any value of S is possible to force ΔU to be zero for any value of S. Therefore: 4 S 4 KS at z = h must always hold. (.) Thus Jenike(973) has set the second term in the integral sign of equation.99 equal to zero. Jenike re-arranges equation. to give an expression for the stress ratio in the material as follows: K = /(-) (.) Jenike then states that the integrand in equation.99 must vanish to zero for any admissible value of S, and since S is arbitrary, this requires that: R d S 4KS 4S 4 KS in the range z < z < h (.) dz Substituting for K from equation.95 into equation. gives the following: S RS R 4 d RS 4S 4 dz Cancelling terms and dividing throughout by gives: R S S (.3) Jenike now introduces a new co-ordinate system such that: z z x R (.4)

90 .66 DES NORTJE: PhD THESIS: THE ANTI-DYNAMIC TUBE IN MASS FLOW SILOS Then a general solution to equation.3 is as follows: S= Ae x + Be -x + / (.5) where A and B are constants determined from boundary conditions. From the boundary condition at z=z ; S=S it follows that x = and substituting this into equation.5 gives: A + B = S / (.6) At the bottom of the silo: z=h and x=x; and from equation.: K=/(-) gives the following solution for A: A x KM S N e M KN (.7) x x e KM e KM where M = ((-)) and N = / (.8a,b) Substituting equation.7 for A into equation.6 gives the expression for the constant B. Three curves of the switch pressure envelope, as given by equation.5, have been shown in dotted lines in figure.45 for Poisson s ratio,.3. The solid curves are the Janssen static pressures in the silo for the same H/D ratios and wall friction angles. The first two curves were calculated for a H/D ratio of (D=) and 5(D=), both with a wall friction angle of. The third curve was calculated for a H/D ratio of, but varying the wall friction angle to 5. Also shown in figure.45 is the Janssen horizontal static pressure distribution in dimensionless form, ie H /D, for a static stress ratio of K=.4. The values of the switch pressure at three different levels in the silo (ie.5, 3.5, 5, and 7m) have been given as a multiple of the static horizontal pressure next to the graph for all three curves. The graph of Jenike s upper bound of switch pressures shows that the envelope of switch pressures tends towards an asymptote, and the greatest changes in pressures occur in the top half of the cylindrical section. It can also be seen that the switch pressure varies from approximately two to four times the static pressure value. The greatest horizontal pressures acting on the wall of a silo during flow would therefore occur on a very tall silo with a low wall friction angle.

91 CLASSIC WALL PRESSURE THEORIES: SWITCH PRESSURES: CYLINDRICAL SECTION.67 Dimensionless horizontal pressure.5.5 S=3.S JH S=4.S JH S=3.53S JH S=.S JH S=.89S JH S=.45S JH S=.S JH S=.48S JH S=.7S JH 6 7 S=.95S JH S=.S JH S=.S JH Depth 8 9 H/D=; w = H/D=5; w = H/D=; w =5 Figure.45: Jenike switch pressure envelope As the switch reaches the top of the cylindrical section the stress field, S, tends to a minimum. From equation.95 this implies that the stress ratio, K, approaches infinity, which makes Jenike s analysis less reliable for positions close to the top of the silo. Jenike states that the switch pressure stops at a height approximately one silo diameter below the top, which Jenike states has been observed experimentally. Although the expression for the switch pressure does not give an area of influence, Jenike states that the maximum pressure acts over an area of approximately one third the diameter of the silo, (D/3). Jenike emphasises that the curves given in figure.45 are only an upper bound to the wall pressures which can be expected during flow of the material. Small deviations in the shape of the cylinder as well as thin boundary layers of material on the wall, will reduce the maximum pressure.

92 .68 DES NORTJE: PhD THESIS: THE ANTI-DYNAMIC TUBE IN MASS FLOW SILOS.3.. Walters Switch Pressure Walters(973) considers an instant in time during the discharge of the silo when the switch pressure is at a height z below the top of the silo as shown in figure.46. Horizontal wall pressure Janssen static pressure h Switch pressure Depth Walters dynamic pressure Figure.46: Location of the Switch Pressure according to Walker Walters assumes that above the switch, at depth h, the stress field is undisturbed and therefore the static pressure field applies. Below the switch, dynamic pressures exist with a surcharge pressure equal to the static pressure at that point. Below the switch the dynamic pressures are given by the differential equation as follows: d v dz 4BF D v (.95 repeated) where F is given by equation.84 and B is given by equation.93. With a uniform surcharge pressure acting above the level of the switch, the limits of integration for equation.95 become = J (the Janssen static pressure) at z=z. The solution to equation.95 then becomes: D 4BFz / D 4BFz / D e e (.9) V 4BF J In dimensionless form: S V 4BFZ 4BFZ e S e (.) 4BF J where S=/D and Z=z/D as before

93 CLASSIC WALL PRESSURE THEORIES: SWITCH PRESSURES: CYLINDRICAL SECTION.69 Therefore, below the switch, at depth Z h, the pressures are as follows: S V 4BF Z h 4BF Z h e S e (.) 4BF J The normal wall pressures at the level of the switch are determined as follows: S H = BF S V /Tan w = BF S V / for dynamic pressures (.) and S JH =K S J = (-e -4 Kz/D )/(4) for static pressures (.3) Multiplying equation. by equation. and substituting equation.3 for S J gives the following expression for the horizontal wall pressures: S H 4BF Z h BF 4BF Z h e S e (.4) 4 K JH At the level of the switch, Z=h, and the first term in equation.4 becomes zero Therefore, at the level of the switch, Walters gives the horizontal pressures as: S H BF S (.5) K JH which is simply the ratio (BF/K) multiplied by the static horizontal pressures. Figure.47 shows the ratio (BF)/(K) for a Janssen stress ratio of.4, for material friction angles varying in increments of. 6 m =8 5 Ratio (BF)/(K) 4 3 m = Wall friction angle w ) Figure.47: Walters Switch Ratio for horizontal wall pressures: for material friction angle varying from to 8

94 .7 DES NORTJE: PhD THESIS: THE ANTI-DYNAMIC TUBE IN MASS FLOW SILOS Equation.4 gives the height over which the switch pressure acts from the term Z-h in the exponent. By changing the co-ordinate system as shown in figure.48 below, the distance over which the switch acts can be determined. The horizontal axis is at the level of the switch and the vertical axis lies on the dynamic pressure graph. Horizontal wall pressure Janssen static pressure Depth x Walters dynamic pressure Switch pressure Figure.48: New co-ordinate system to determine height of switch For the new co-ordinate system, as S H tends to zero the value of x can be determined as follows: 4BF x BF 4BF x e S e (.6) 4 K JH Multiplying equation.6 by 4 and re-arranging gives the following: A=e -4BFx (.7) where A BF S (.8) K JH The solution to equation.7 is n (A) = -4BFx and therefore x is as follows: x = n (A) / (-4BF) (.9) Equation.9 depends on the factors B and F, which in turn, are functions of the material friction angle and the wall friction angle. Therefore, the depth over which the switch pressure acts varies for each silo.

95 CLASSIC WALL PRESSURE THEORIES: SWITCH PRESSURES: CYLINDRICAL SECTION.7 Figure.49 shows the switch pressure at various levels in a tall silo with a wall friction angle of and a material friction angle of 4. The silo height to diameter ratio is :. The pressures have been plotted in dimensionless form according to equation.97 and.98 for the dynamic pressures and equation.6 divided by D for the Janssen static pressure. The switch pressures have been plotted using equations.4 and.5. The calculations were done for the switch at level.5, 3.5, 5 and 7m below the top of the cylinder. As can be seen from figure.49, the maximum value of the switch pressure decreases from 5.S JH to 3.S JH as the switch moves up the cylinder. However, the area over which it acts remains constant, in this example the area of influence of the switch is approximately. silo diameter. Dimensionless horizontal pressures S H = 3. S JH S H = 4.6 S JH S H = 5 S JH 6 Depth Static pressures (S JH ) Dynamic pressures Switch pressures (S H ) Locus of switch S H = 5. S JH Figure.49: Walters Switch pressure at various levels in a silo of H/D= Figure.5 shows the switch pressure in a silo with a height to diameter ratio of 5. The wall friction angle, the material friction angle and the levels of the switch are the same as in the previous example. From figure.5 it can be seen that the area of influence of the switch is still approximately silo diameter. However, the

96 .7 DES NORTJE: PhD THESIS: THE ANTI-DYNAMIC TUBE IN MASS FLOW SILOS maximum values of the switch pressure have decreased by.5%=(5.-4.6)/5. at the lowest level to 39%=( )/3.8 at the highest level. Dimensionless horizontal pressures S H =.87 S JH S H = 3.38 S JH S H = 4. S JH 6 7 S H = 4.6 S JH Depth 8 9 Static pressures (S JH ) Dynamic pressures Switch pressures (S H ) Locus of switch Figure.5: Walters Switch pressure at various levels in a silo of H/D=5 Figures.5 and.5 show the switch in a silo with a H/D ratio of, as in figure.49. However, the material friction angle has been varied from 4 in figure.49, to 5 in figure.5, while the wall friction angle has been varied from in figure.49 to 5 in figure.5. By reducing the material friction angle by 37.5%, the magnitude of the switch pressures is decreased by approximately 44%. A reduction in the wall friction angle of 5% has the effect of increasing the switch pressure by 79% at the lowest level to 36% at the highest level in the silo. Therefore, using the Walters equation for determining the switch pressures during flow, the worst case pressures would occur in a silo with a high H/D (height to diameter) ratio combined with a high material friction angle and a low wall friction angle. A short silo with a rough wall would experience lower switch pressures, but consequently may not undergo mass flow of the material.

97 CLASSIC WALL PRESSURE THEORIES: SWITCH PRESSURES: CYLINDRICAL SECTION.73 Dimensionless horizontal pressures 3 S H =.7 S JH 3 S H =.56 S JH 4 5 S H =.78 S JH 6 Depth S H =.89 S JH Static pressures (S JH ) Dynamic pressures Switch pressures (S H ) Locus of switch Figure.5: Walters Switch pressures in a silo of H/D=, m =3, w = Dimensionless horizontal pressures S H = 4.3 S JH 3 S H = 6.6 S JH 4 5 S H = 7.5 S JH 6 Depth Static pressures (S JH ) Dynamic pressures Switch pressures (S H ) Locus of switch S H = 8.8 S JH Figure.5: Walters Switch pressures in a silo of H/D=, m =4, w =5

98 .74 DES NORTJE: PhD THESIS: THE ANTI-DYNAMIC TUBE IN MASS FLOW SILOS.3. HOPPER SECTION.3.. Jenike Switch Pressures in the Hopper The material starts to flow at the instant the discharge gate of the silo is opened. vertical support of the solids has been removed and the material above the outlet starts to expand vertically downwards. This reduces the vertical pressure within the material and causes a change from a static to a dynamic stress field. The major principal stresses now arch across the outlet of the silo. As more material expands the region of flow extends upwards into the hopper and the switch travels upwards. Jenike(969) states that this change in stress fields results in a deficiency in the wall support during flow. Figure.53 shows an instant when the switch is at level z above the vertex of the hopper. Above the switch the material is still in the static state while below the switch the material is in the passive state of stress. The shaded volume of solid between the two stress fields does not belong to either, but is in transition from an active to a passive state. The area under the dynamic pressure curve represents the total weight of the solid which has not changed significantly compared to the curve for the static pressures. Therefore, there is a deficiency in the wall support as shown by the shaded area between the static and dynamic pressure curves. Equilibrium is maintained by a switch pressure which is exerted on the walls of the hopper and travels upwards from the discharge gate to the transition, where it can become locked in position, or move up into the cylindrical section of the silo. The force of the switch pressure is equal to the shaded area under the pressure curve. N t Pressure J Static pressures Dynamic pressures Switch pressure z Depth Deficiency in wall support Figure.53: Deficient wall support in the hopper during flow

99 CLASSIC WALL PRESSURE THEORIES: SWITCH PRESSURES: HOPPER SECTION.75 Since both the static and dynamic curves are known, Jenike (968) states that the magnitude of the concentrated force needs to be superimposed over the dynamic pressures to obtain the envelope of the design pressures for flow conditions. Jenike(969) assumes that the switch pressure has a triangular distribution acting normal to the hopper wall over a depth of.3d parallel to the hopper wall as shown in figure.54. During flow, the mass of the material in the hopper remains constant for both static and dynamic conditions. When the switch is located at the transition, the deficiency in wall support is due to the difference between the static hopper pressures, t, and the dynamic hopper pressures, N. This difference in pressure ( t - N ) is also equal to the difference between the initial surcharge acting on the hopper given by the Janssen pressure, J, and the radial flow pressure,, as shown in figure.53. J t S Pressure dz SZ SZ.3D z s h z b Depth Figure.54: Switch pressures acting on the hopper wall. The vertical components of the switch pressure, SZ and the shear stress, SZ at level z s from the vertex of the hopper, acting on an elemental slice of thickness dz, given in figure.54, are as follows: Sin SZ and Cos SZ (.) In figure.54 the change in pressure during flow varies from at level z b to a value of S, at level h. Therefore, SZ at a level, z s, can be given by: z z s b (.) SZ S h z b

100 .76 DES NORTJE: PhD THESIS: THE ANTI-DYNAMIC TUBE IN MASS FLOW SILOS Therefore, the additional force due to the switch acting normal to the wall can be given by: h h dz A Cos P (.) SZ Cos z b z b dz Sin P J h SZ Cos where A h is the area of the hopper at the level of the transition and is given by: A h = r = h Tan and P is the perimeter of the elemental slice given by: P = r = ztan (.3) (.4) Note that in the integration of the area under the curve, Jenike has ignored the error as shown by the red shaded area in figure.5, due to the approximation of the area as a right angle triangle. The shear stress along the wall is given by SZ = SZ Tan w (.5) Substituting for SZ from equation., SZ from equation.5, and for A h and P given by equations.3 and.4, into equation. gives the following: s Tan Tan Tan Tan z z z J h dz (.6) w s s b h z b z b h In equation.6, h-z b =.3D Cos. Therefore, integrating equation.6 and cancelling out Tan, gives: h Tan Tan Tan J w S.3 DCos 3 z s 3 h z z s b z b z s b Tan Tan h h z 3 w b.3dcos 6 h (.7) Equation.7 can be simplified to give: z s b h z 3 J w b.9dcos h Tan Tan Tan (.8) Solving for the additional pressure at the transition due to the switch, S, gives:

101 CLASSIC WALL PRESSURE THEORIES: SWITCH PRESSURES: HOPPER SECTION.77 s.9d Cos Tan J z b Tan Tan h z w b h 3 (.9) where h is the height of the hopper from the vertex (not the gate), and z b is the bottom of the area of influence of the switch, refer figure.54, and z b =h-.3dcos Therefore, all the variables in equation.9 are known and S can be calculated. Jenike then adds this additional pressure at the transition to the dynamic static pressure in the hopper at the level of the transition as follows: = t + S (.3) where t is given by equation.53 in chapter...3. and S is given by equation.9 above. The shape of the curve for the switch pressure acting at the transition between the cylindrical and hopper sections, is as is shown in figure.54.

102 .78 DES NORTJE: PhD THESIS: THE ANTI-DYNAMIC TUBE IN MASS FLOW SILOS.3.. Walters Switch Pressures in the Hopper. In his derivation of the switch pressures in the hopper, Walters(97) adopts the same approach as for the cylindrical section given in chapter.3... Figure.55 shows the lines of the major principle stresses above the switch and below the switch during flow of the material when the discharge gate of the silo is opened. Static pressures z s Switch Dynamic pressures Therefore, below the switch, the dynamic pressures are given by equation.36 for a hopper with a cylindrical section above, resulting in a surcharge pressure acting at the level of the transition. Figure.55: Switch pressure at depth z from the transition. S v ZTan ZTan Tan M Z Tan M S ZTan Z Tan M (.36 repeated) E FH where M FH (.33 repeated) Tan F H m m Cos Sin m Sin m Sin (.8 repeated) Cos Sin y Sin SinmSin D E Sin Cos m D (.5 repeated) In equation.36, Z =Z s the level of the switch, and the variable Z now starts from below the switch. (Note Z=z/D). S in equation.36 is the Janssen vertical pressure in dimensionless form, ie S J = v /D. However, to determine the dynamic vertical pressure below the switch, at depth Z s, the value of S should be replaced with the static value of the vertical pressure in the hopper, S v.

103 CLASSIC WALL PRESSURE THEORIES: SWITCH PRESSURES: HOPPER SECTION.79 Walters(97) gives the expression for the dimensionless normal wall stress as a function of the dimensionless average wall stress as follows: S N =EF H S v /Tan w (.3) This was determined from the general expression (in dimensionless form): T=Tan w S N, where T is the dimensionless wall shear stress = w /D Therefore, S N = vw /D = T/Tan w (.3) In chapter... it was shown that Walters(97b) relates the wall shear stress vw to the vertical pressure at the wall, vw, by equation.3. Substituting equation.3, in dimensionless form, into equation.3 gives: S N = ES vw /Tan w (.33) where S vw is the dimensionless form of the vertical wall pressure acting at the wall. Walters then relates vw to the average vertical pressure acting across the hopper slice by equation.7. Substituting the dimensionless form of equation.7 into equation.33 results in equation.3 Substituting equation.3 into equation.36 gives the dimensionless normal pressure during flow of the material as follows: S N EF ( ZTan ) H Tan Tan w M S v ZTan Z Tan M EF H Tan w S v ZTan Z Tan M (.34) The envelope of the switch pressure is found by letting Z=Z in equation.34. Since the term (-ZTan)/(-Z Tan) becomes equal to, the first term equals zero, and equation.34 becomes: S N =EF H S v /Tan w (.35) where S v is the dimensionless form of the static vertical pressure in the hopper. Substituting the dimensionless form of the Walters equation for the static normal wall pressure, into equation.35 results in the following expression: (EF ) H D S S (.36) N NS (EF ) H S

104 .8 DES NORTJE: PhD THESIS: THE ANTI-DYNAMIC TUBE IN MASS FLOW SILOS where (EF H ) D is the dynamic value of the constants E and F H ; and (EF H ) S is the static value of the constants E and F H. This ratio is as follows: Sin Sin m D Sin Cos m D Sin Sin m s Sin Cos m s Sin Cos Sin Sin m m Cos Sin ysin m m Sin Cos Sin Sin m m Cos Sin ysin m m (.37) Since the variables in equation.37 are all constant for a given hopper geometry, the ratio (EF H ) D /(EF H ) S for the envelope of the switch pressures is also a constant. The shape of the switch pressure envelope is therefore dependant on the shape of the static pressure curve. In his paper, Walters substitutes the following expression for + in equation.37: + = / + ArcCos(Sin/Sin m ) (. repeated) where the +ve sign refers to static conditions and the ve sign refers to static conditions. The value of can be determined as shown in figure.35 of chapter... Walters states that the ratio given in equation.37 gives a value of 3.3 for a material friction angle, m =5, a wall friction angle, w =5 and a hopper half angle, =4. However, this was checked on a spreadsheet, where the value of was solved by trial and error, and found to be 8.79 for dynamic conditions and.58 for static conditions. These values of in equation.37 give the following numerical values: (F H ) D =.6 ; (E) D =. and (F H ) S =. ; (E) S =.4 Substituting these values in equation.37 gives a switch pressure ratio of 447.8, which is excessively large. The spreadsheet was checked several times by comparison with hand calculations and found to be correct. It is therefore suggested in this thesis that the switch pressure ratio as given by Walters in equation.37 is not reasonable.

105 CLASSIC WALL PRESSURE THEORIES: SWITCH PRESSURES: HOPPER SECTION.8

106 WALL PRESSURE MEASUREMENTS: LITERATURE SURVEY: STATIC PRESSURES 3. CHAPTER 3 WALL PRESSURE MEASUREMENTS 3. LITERATURE SURVEY 3.. STATIC PRESSURES 3... Cylindrical Section Bishara et al (98) undertook finite element (FE) analyses of the pressures in the cylindrical section of a silo of 7.3m internal diameter and 4.4m tall (H/D=3.3). The material used in their simulations was a granular cohesionless sand. They state that the FE horizontal pressures were shown to be % larger than the calculated Janssen horizontal pressure. However, on closer inspection of their graphs for the horizontal pressure, the amount by which the FE solution is larger than the Janssen formula, varies from 8% at. diameters from the top, to % at.6 diameters from the top. This is higher than they have reported. From the graphs of their results for the vertical pressures, the finite element solution gives values higher than Janssen for a depth of to.4 diameters, while below this level the Janssen formula gives higher values. At full depth, the Janssen formula gives a 6% higher vertical pressure than the finite element solution while at a depth of.9 diameters, the Janssen formula gives a 4.6% lower vertical pressure than the FE solution. They reported that the distribution of the vertical pressure across a horizontal plane was about 5% higher in the centre of the silo than in the vicinity of the walls. The stress ratio Bishara et al (98) used in their calculations was not given, so it has been interpolated from their graphs for the purposes of this thesis. The stress ratio used in their calculation of the Janssen horizontal pressure, was found to be approximately,4 taken from three points: (9m depth: 47/=.39, 4m depth: 43/6=.4, 9m depth: 33/84=.39). The graphs from their finite element results suggest a stress ratio of approximately.5 taken from the same three points. (53/3=.5, 5/97=.5, 46/87=.53) Suzuki et al (985) conducted tests on small and medium sized model silos. The dimensions were.3m internal diameter by.7m tall for the small model (H/D=5.7), and.4m internal diameter by 6.4m tall for the medium sized model (H/D=4.6). The test material used in the small model was Milo ( ) and in the medium model tests were done using Milo ( ), maize, soybean meal and alfalfa

107 3. DES NORTJE: PhD THESIS: THE ANTI-DYNAMIC TUBE IN MASS FLOW SILOS meal pellets were used. They report that their test results from both models compare favourably with the Janssen horizontal pressure, particularly at depths of less than half the model diameter (ie in the upper part of the silo). From their graphs of the measured horizontal pressure in the smaller model, their plotted test results are 6%, 5% and 37% less than the Janssen equation. As part of their experiments, Suzuki et al also measured the vertical pressure at six points across the bottom of the model for various levels of fill. Their graph has been reproduced in this thesis by scaling the points off their report and replotting them, as shown in figure 3.. The average values of the vertical pressure are shown in Vertical pressure on model bottom kg fill 6kg fill 4kg fill kg fill Distance from centre to wall: r/r o Figure 3.: Radial Distribution of Vertical pressure taken from Suzuki et dotted lines in figure 3.. As can be seen from this graph, Suzuki et al show that the central vertical pressure is the highest, and rapidly diminishes to a minimum within one third of the radius from the centre. The central pressure is approximately 6% greater than the average, while the minimum is approximately 5% less than the average. Their measured value of the stress ratio which they calculated from the measured average vertical stress and the measured horizontal stress was K=.5. Blight et al (989) conducted a set of tests on two identical full-scale silos containing cement. The silos were m internal diameter and 65m overall height and were strain gauged across the height with six temperature compensated gauges. For the calculation of emptying pressures, a stress ratio of was used to calculate the horizontal pressures. Their internal angle of friction was measured at 4, (which is higher than the design values of 8). They measured the stress ratio as being.35 to.37 while the calculated value of the at-rest stress ratio (given by K o =-Sin) was.9. They showed that the measured pressures were within the envelope of Janssen s pressures using a K o ratio of,35. However, the test results also showed the measured pressures in areas of low overburden were larger than the calculated values. This is equivalent to areas close to the top of the silo.

108 WALL PRESSURE MEASUREMENTS: LITERATURE SURVEY: STATIC PRESSURES 3.3 Molenda et al (993) have conducted experiments to determine the effects of filling method on the wall loads. They tested concentric, eccentric and uniform sprinkle filling methods using soft and hard wheats. From their experiments they found that the grains aligned themselves parallel to the free surface during filling. For the concentric filling methods, the bulk density of the material after filling was approximately 6% lower than for the cases where the grains were uniformly sprinkled in the silo. It is well established that the Janssen equation for the static vertical pressure in the silo, gives a good initial estimate of the minimum loads to be expected in the cylindrical section of the silo. The vertical pressure in the centre of the cylindrical section has been shown to be greater than the average value by approximately 6% to 5%, while Janssen only found it to be 5% greater. For the purposes of this thesis, a value of 3% greater than average will be used for the vertical pressure in the centre of the silo (where 3% is the average of the three values quoted). The shape of the individual particles affects the bulk density of the material in the silo due to the filling method employed. This effect does not show up in the finite element analyses, as can be seen by the average vertical pressures being higher than those calculated from the Janssen equation Hopper section Blair-Fish and Bransby (973) conducted tests on a model silo.5m square and.45m in height, with a hopper half angle which can vary between 3 and, filled with dry sand. From their report, the normal pressure on the hopper wall is approximately constant throughout the depth of the hopper. Van Zanten and Mooij (977) conducted tests on a model silo,.5m in diameter and 6m tall, fitted with a hopper half angle of 5. Two types of fill materials were used, viz PVC powder and sand. The graph of their results shows a large scatter of data, which has the average minimum value in the lower portion of the hopper and the maximum occuring at approximately half the hopper height. At the transition, the pressure in the hopper is shown to be approximately three times greater than the pressure in the cylinder. In their graph, Van Zanten and Mooij also show the calculated line of pressure in the cone according to Jenike. Of their nearly 5 data points for normal wall pressure in the hopper, only the four maximum points fall outside the limit given by Jenike.

109 3.4 DES NORTJE: PhD THESIS: THE ANTI-DYNAMIC TUBE IN MASS FLOW SILOS Suzuki et al (985) measured the wall pressures in the hoppers of two mass flow silos filled with milo. Their results show the pressures in the hopper to be the greatest at the base decreasing towards the transition between hopper and Dimensionless parameter P/P max Dimensionless parameter z/zh a b position of pressure cells Figure 3.: Distribution of normal pressure on hopper wall (a) no surcharge, (b) silo fully loaded cylindrical section. Their results show the static pressure in the hopper at the transition is approximately.3 times greater than in the cylindrical section. A diagrammatic representation of the pressure distribution in the hopper has been taken from the graph of their experimental results and shown in figure 3.. In this thesis, the vertical axis is the ratio below the transition, z, to the total depth of the hopper,z h. The horizontal axis is given as the ratio of the pressure, P, to the maximum pressure, P max, at the base of the hopper when the silo is fully loaded. From figure 3. it can be seen that the pressure distribution varies from nearly linear for no surcharge in the hopper, to a curved distribution for a fully loaded silo, with a minimum at approximately one third the depth. Kmita (99) gives a very different pressure distribution for the normal pressure on the hopper wall. The tests were conducted on a plane flow silo.8m wide and 3.6m overall height, filled with rinsed grit of particle diameter ranging between 3mm and 5mm. The silo has a hopper half angle of 5 and hopper height of.m. Kmita conducted tests for the case of the silo being filled from empty as well as partially emptied and then re-filled to the same height. The pattern of the pressure distribution is shown to be the opposite of that given by Suzuki et al in figure 3., with a maximum occurring at approximately one fifth the depth of the hopper below the transition. There is no difference in the pressure distribution for the partial emptying and re-filling case compared to filling from completely empty. Kmita shows the maximum to be five times greater than the maximum wall pressure in the vertical section of the model. It is clear from the varied results for the pressures in the hopper, that a large variety of factors influence these pressures and the exact pressure distribution

110 WALL PRESSURE MEASUREMENTS: LITERATURE SURVEY: STATIC PRESSURES 3.5 cannot be predicted. However, all the research reports studied indicate the static pressures generally do not exceed those given by Jenike, which is therefore a good initial assessment of the maximum static pressures to be expected in the hopper. 3.. DYNAMIC PRESSURES 3... Cylindrical Section Pieper (969) conducted tests on three model silos filled with a quartz sand. The two cylindrical models were.6m in diameter and 3m tall, and.8m in diameter and 6m tall. The third model was a square silo of.7m cross section and 5m tall. This gives H/D ratios of 5., 7.5 and 7. respectively. In the graph of flow results, Pieper shows the flow pressures in all three models to be approximately.3 times the static value. Blair-Fish and Bransby (973) conducted tests on a sand filled mass flow silo, 5mm square cross section and 375mm tall (H/D=.5). with a 3 hopper half angle. They presented their results for the measured pressures in the form of a bar graph at the point measured and the flow results were presented for each increment of emptying Ratio of flow/static pressure 3 Their flow results have been scaled and given as a ratio of the static pressure at the point, as shown in figure 3.3. This graph shows their flow results only reached a maximum of 3 times the static pressures in the cylinder, while in the hopper, the maximum did not exced.5 times the measured static value. Figure 3.3: Test results scaled from Blair-Fish and Bransby Richards ( 977) conducted experiments on a model silo.6m in diameter and m in height with a 5 hopper half angle, filled with sand (wet sand to study minimum opening dimensions and dry sand to study flow pressures). Richards reports that as soon as the gate was opened only slightly, the normal wall pressures just below the

111 3.6 DES NORTJE: PhD THESIS: THE ANTI-DYNAMIC TUBE IN MASS FLOW SILOS transition increased substantially above the static pressure, although the maximum pressure did not always occur at the start of flow when the silo was full. The flow rate was varied by a factor of 6 in these experiments and had no effect on the measured wall pressures. When the flow was stopped, the overpressures remained at the points where they were recorded. In the tests conducted by Van Zanten and Mooij (977) on model silos filled with PVC powder and sand, they considered flow in a perfect silo as well as flow in a silo with irregularities on the walls. The following discussion does not include the measured pressures at the points of irregularities in their model. The geometric aspect ratio of their model silo is shown in figure 3.4 below. First they measured the circumferential distribution of the overpressure at four points, 9 apart, at the transition during flow. They report that the distribution was highly assymmetrical and simultaneous peak pressures at two points occurred only occasionally. 4 D.5 D.9 D Figure 3.4: Geometric aspect of the model silo For the discussion in this thesis, their test results of the vertical distribution of the overpressures for sand, have been divided into two sections. Those between the transition and.5d above the transition, and those at levels greater than.5d above the transition. In the lower section of the cylinder, the flow pressures are greater than the pressure envelope given by Jenike Strain Energy. Their test results show these pressures to be 5.8 times the maximum Janssen static value and twice the maximum Jenike Strain Energy value. In the upper portion of the cylinder, their test results are 4.3 times the Janssen static pressure, but are within the envelope given by Jenike Strain Energy. At the transition, the maximum pressures are 5% less than the value given by hydrostatic pressure at a depth of 4D, and 39% less than the Jenike Peak Pressure. At a depth of.7d below the transition, the maximum test results are.4 times greater than the Jenike static value in the hopper. In the hopper, nearly all the test results lie outside the line of the Jenike peak Pressure. (Note: In their report, the Jenike peak pressures are smaller than the Jenike static value in the hopper) For their tests with PVC powder, the same division in their results has been made in this thesis. In the lower section of the cylinder, the pressures are approximately 3.7 times the maximum Janssen static value and twice the value given by Jenike Strain

112 WALL PRESSURE MEASUREMENTS: LITERATURE SURVEY: STATIC PRESSURES 3.7 Energy. In the upper portion, the results are twice the Janssen static value, and are also within the envelope given by Jenike strain energy. At the level of the transition, nearly half the test results lie beyond the maximum value given by hydrostatic pressure. In the hopper, the results again lie beyond the Jenike peak pressure envelope. At a depth of.7d below the transition the test results are. times the Jenike static value in the hopper. Nielsen and Andersen (98) conducted tests on full scale silos, 7m in diameter and 46m tall (H/D=6.6), filled with barley. They conducted several tests for various arrangements of filling and emptying but only their tests for central emptying have 46m 7m Depth below surface Flow/Static 3 Figure 3.5: Ratio of flow pressures to static pressures taken from Nielsen and Andersen s test results been considered in this thesis. The pressure cells were placed at four points, 9 apart, around the circumference, and at seven different vertical levels, giving a total of 8 pressure cells. From the results of their filling and emptying tests, the ratios of flow vs static pressures have been plotted in figure 3.5. These results show that in the upper half of the silo, the flow pressures are only.5 times the static pressures whereas in lower portion, the ratio is approximately,5 to 3. Rombach and Eibl (995), conducted finite element tests on material flow in the hopper. The hopper half angle in their model was, and the wall friction angle was.8. Their results show the dynamic pressure,. seconds after flow was initiated, was.4 times the static pressure.

113 3.8 DES NORTJE: PhD THESIS: THE ANTI-DYNAMIC TUBE IN MASS FLOW SILOS 3..3 STRESS RATIOS In his derivation of the vertical pressure in a silo, Janssen assumes a constant value of.4 for the stress ratio. Jenike and Johanson(969) recommend a constant value, assuming Poisson s ratio for the material does not vary and the material remains isotropic. They recommend a minimum value of.4 in the cylinder and.8 in the hopper. In their finite element analysis of the storage and flow pressures in silos, Bishara et al (98) give the following expressions for the horizontal and vertical pressures on the walls, immediately after filling: h.75 D H.46 and.5.5 v.8 D.55 H.6.33 Since the stress ratio is defined as : K= h / v : an equation for K can be found by substituting in their expressions for h and v to give the following: D H K.53 (3.).58 Equation (3.) implies that the stress ratio is independent of the internal friction angle of the material. It would be reasonable to expect the stress ratio to have a term for the internal friction angle in the expression. Furthermore, substituting for (D, H, and ) the values m, 3m,.4, and 6kN/m 3 respectively in equation (3.), gives a value of K =.7. Irrespective of the material used (whether cohesive or free flowing), this value is too low to give reasonable results. m: K=.4 5m: K=.67 m: K=.8 5m: K=.89 Figure 3.6: Stress ratios in silo from Reimbert s tests Ravenet (983) reports of tests done by the Reimbert brothers in 943 on full scale, flat bottomed silos in France which were exhibiting signs of being overstressed. According to Ravenet, the Reimbert brother s strain gauged these silos to measure the horizontal and vertical pressures on the walls. From the graph of Reimbert s results, given in Ravenet s report, the stress ratio at various

114 WALL PRESSURE MEASUREMENTS: LITERATURE SURVEY: STATIC PRESSURES 3.9 points along the height of the silo has been determined by scaling off the graph. These values are shown in figure 3.6. These results show the stress ratio increases with increasing depth in the silo. Briassoulis (99) derives an expression for the stress ratio in the Reimbert formula from the ratio of the horizontal stress to the vertical stress, and gives this as follows: K(y) y / p Ka (3.) y / p In the above expression, K a is the active stress ratio, p=r/(k a ), and y is the depth below the surface of fill. Briassoulis states that the Reimberts assume the stress ratio, K, to decrease with depth, which is shown in equation 3.. However, the results given in figure 3.6 from Ravenet s (983) report of the Reimberts tests, show the stress ratio to increase with depth. Referring to the Mohr circle for stresses at the hopper wall, as shown in figure 3.7, an expression for the stress ratio at the hopper wall under static conditions can be derived as follows: Wall yield locus P O W 3 H C V O p plane on which the horizontal stress acts plane on which the vertical stress acts Figure 3.7: Mohr circle for stresses at the hopper wall under static conditions. OC=O H + H C (3.3)

115 3. DES NORTJE: PhD THESIS: THE ANTI-DYNAMIC TUBE IN MASS FLOW SILOS But OC= r / Sin W and H C=rSin( W +) (3.4a&b) Substituting these values into equation (3.3) gives the expression for the horizontal stress as follows: Sin Sin W W H r (3.5) SinW Similarly, the expression for the vertical stress is given as follows: V Sin Sin W W r (3.6) SinW Therefor the expression for the stress ratio in the hopper becomes: Kh Sin W W Sin W W Sin (3.7) Sin As the hopper half angle varies to a minimum, in the limit, this would give the expression for the stress ratio at the wall in the cylindrical section of the silo as: K=(-Sin w )/(+Sin w ) (3.8) These expressions (3.7 and 3.8) imply the stress ratio is a constant value and not dependant on the level of overburden material in the silo.

116 WALL PRESSURE MEASUREMENTS: EXPERIMENTAL SET-UP EXPERIMENTAL SET-UP 3.. STEEL MODEL SILO Nielsen and Askegaard (977) studied the effects of model scale on the results of pressure measurements. Their experiments were done in a centrifuge, using a sample of 4mm in diameter and 5mm in height (H/D ratio of 3.75), filled with a cohesive material (silica gel) and dry sand. They concluded that test results on models filled with a cohesionless material can be transferred to a geometrically similar full scale silo, provided the model is not too small.they do not give a definition of what is considered not too small. However, they stated that it was not necessary to test cohesive materials in a centrifuge, if the model diameter was at least times larger than their model. This value of 48mm (=x4mm) was therefore used as an indication of their description not too small, for the purposes of this this research. The available sections in the laboratory used to set up the model, were just more than twice the minimum requirement, defining it as a large model. Therefore, it has been assumed in this thesis that there should be no scaling errors applicable to the test results. BUCKET ELEVATOR Figure 3.8 : Steel Model Silo A steel model silo of height 3.m,.98m in diameter (H/D=3.8) and outlet opening of.8m in diameter, was set up in the laboratory as shown in figure 3.8. The cylindrical section of the model was made up of four equal semicircular sections,.m in length. The sections were bolted together through the outside flanges, so that there were no obstructions to the material flow. The hopper was made up in two halves, and fitted to the cylindrical section through matching flanges at the transition. A hopper half angle of 5 was chosen from Jenike (967) as the maximum angle which would still cause mass flow of the material. The silo was filled by means of a bucket elevator.

117 3. DES NORTJE: PhD THESIS: THE ANTI-DYNAMIC TUBE IN MASS FLOW SILOS 3.. BULK SOLID MATERIAL The pressures exerted on the silo walls during flow are affected by obvious criteria such as wall friction angle, hopper slope, shape and the number of outlets in the hopper. The less obvious influences are the material properties such as cohesiveness, the ability to segregate and the degree of segregation due to the filling technique. Arnold (99) states that particle segregation influences the flow pattern in the silo and hence the wall pressures during flow as well as during storage. Coarser materials have a better degree of flowability than finer material, as well as a lower coefficient of wall friction. Based on this, a uniformly graded, dry cohesionless sand of particle diameter between.8mm and.6mm was chosen as the fill material in the experiments. The internal friction angle of the material was determined in a triaxial cell and was found to be 45. The wall friction angle between the material and the silo (also determined in a triaxial cell), was found to be. The material density was determined in the laboratory by allowing a sample of the material to fall from a height of m into a container of known volume and self weight. Two different methods of filling the container were used to simulate the filling operations of the model. The material was allowed to rain into the container and also to fall in a constant stream. There was very little difference between the density as determined from the two different filling methods and an average density of 6 kn/m 3 was used DATA ACQUISITION Data acquisition was obtained from a 6 channel AD card (Metrobyte Dash6) driven by a fortran computer code (K.Kavanagh 986). The original code was written for a cycle internal timing clock, which had to be hand calibrated in these experiments for the a/d card with a single cycle clock. The program was structured around background filling and emptying of 5 word buffers. Calculations were post-processed from data transferred to hard disc. Maximum data rates of 5kHz were obtainable without data interuption due to disc transfers. The input file has the four entries: number of channels, number of combinations, number of buffers and the input frequency. Therefore, if eight different types of pressure cells were used, the number of channels would be eight. As there were no combinations of pressure cells, this entry was entered as zero. The number of buffers and the number of channels entered in the input file, affected the time

118 WALL PRESSURE MEASUREMENTS: EXPERIMENTAL SET-UP 3.3 taken for data acquisition and size of the output data file. The output data file had the form of columns and rows, where the columns represented the individual channels and the rows represented the next data point for each channel. As the number of data points in the output file was used as a time reference, the exact sampling frequency of the single cycle clock in the computer, had to be determined. This hand calibration was done by timing a series of blank runs for an Actual frequency Input frequency 8 Figure 3.9: Actual frequency of computer data acquisition input file of channel, combinations, 5 buffers and input frequency. The input frequency was varied from 5 to 8 Hz in steps of 5. The number of 5 buffers was chosen so that at higher input frequencies, the time could be measured with a stop watch. If too few buffers were used, the test duration was too short to enable timing on a stop watch. For each input frequency, ten tests were run and the average was taken as the sampling time for that input frequency. The results of these tests are shown in figure 3.9. The slope of this graph gives a factor of applied to the sampling frequency in the input file. The total time, in seconds, taken for the computer to record data, was required to ensure data acquisition was not cut short before the end of a test, resulting in a loss of results. The computer s total sampling time was found to be given by: Sampling Duration=(5N B )/(F i.95369) (3.9) The time between rows of data, t r, was determined from the following equation: t r = N C /( F i.95369) (3.) where N B is the number of buffers, N C is the number of channels and F i is the input frequency. The size of the output data file was determined from: Number of data rows = 5 N B /N C (3.)

119 3.4 DES NORTJE: PhD THESIS: THE ANTI-DYNAMIC TUBE IN MASS FLOW SILOS 3..4 STRAIN GAUGE BRIDGES Figure 3.: Location of Strain Gauge Bridges From the literature on wall pressures in silos it was recognised that the pressures at the transiton during flow, were likely to be the greatest. Therefore, the first attempt at measuring the pressures was concentrated in the area just above and below the transition, in three groups of four points around the circumference of the model, as shown in figure 3.. The concept was to make small holes in the silo walls and to place an instrumented bridge across each opening. The strain in each bridge was recorded as a measurement of the applied pressure during flow of the material. The holes were made in the silo walls by cutting 5mm disks at each point shown in figure 3.. These disks were then fixed to the middle leg of the bridge as shown in figure 3.. Due to the fact that the disks were curved in one direction only, the bridges were placed on the silo wall so that the curvatures of the disks and silo were aligned. The holes were covered with a clear plastic sheeting on the inside of the silo wall. The dimensions of the aluminium bridge is shown in figure 3., with two strain gauges glued to the top and bottom of the bridge. The strain gauges used were Kyowa type KFG-5--C-, with a temperature compenstion for steel and a gauge length of 5mm. After the gauges were connected in series to give an average reading of deflection, they were tested for specified resistance to ensure no gauges had become damaged in the soldering process. Tape was placed over the gauges to reduce the risk of damage during handling.

120 WALL PRESSURE MEASUREMENTS: EXPERIMENTAL SET-UP 3.5 5mm disk cut out of silo wall mm 4mm Strain gauges mm mm Aluminium bridge mm 4mm mm Figure 3.: Aluminium strain gauge bridge Each bridge was loaded and unloaded three times to ensure the gauges had adhered to the bridge before they were calibrated. The gauges were then connected to the amplifier, and the bridges loaded up to 5 kilograms in increments of kilogram. Readings were taken after each increment. The bridges were then unloaded to 3 kilograms, a reading taken, and allowed to stand for 3 minutes after which another reading was taken. This was done to check for drift in the gauges. The results for each bridge were plotted on a graph of load versus voltage output and the slope of the graph gave the calibration factor for each bridge. Static readings were taken after the silo had been completely filled. The silo was emptied in stages into drums which were placed under the hopper. Emptying stopped when the drum was full and reloaded into the silo. Readings were taken for

121 3.6 DES NORTJE: PhD THESIS: THE ANTI-DYNAMIC TUBE IN MASS FLOW SILOS each stage of emptying. The results from the bridges did not record any strain for the static pressures or for the first stage of emptying, the output being attributed to electrical noise. The likely cause was that the bridges were too flexible relative to the model wall so that the material arched over the bridges. One bridge became dislodged during testing, and the plastic sheeting managed to hold the material in place without rupturing. This lockup of material is a strong indication that arching will occur over small diameters, where the opening is softer than the bin wall. This phenomenon may be more critical, the smaller the diameter of the opening. Due to the failure of these experiments, a radically different pressure measuring cell was developed. Rather than being mounted to the silo wall, the cells were placed in the material during filling, and the cells travel with the material during flow PRESSURE CELLS Richards (977) reports of vibrations being felt and heard during discharge from a mass flow model. The frequencies of the individual fluctuations were reported to be in the range of 5Hz to 85Hz, the frequency increasing with flow rate and being independent of particle size. Therefore, a responsive measurement and fast recording system was needed to investigate the material pressures during flow. Three types of material pressure cells, as shown in figure 3., were developed in the laboratory to allow continuous measurement of the pressures during filling and emptying of the silo contents. The concept of the cells was to place a standard pressure sensor in a small hollow object with a flexible wall, which could then be filled with an incompressible, low viscosity oil. The most important criteria was to ensure that there were no air bubbles retained in the oil or pressure sensors to affect the incompressibilty. (a) Y (b) Y (c) Z X Z X Figure 3.: Floating Pressure Cells: a) Ball type, b) Tube type, c) Plate type

122 WALL PRESSURE MEASUREMENTS: EXPERIMENTAL SET-UP The Ball Type Pressure Cell The first type of cell was the ball cell, where a pressure sensor (psi absolute) of capacity 5psi, was modified to fit inside a hollow thin walled rubber sphere, as shown in figure 3.3. The front ports and the side connection holes were clipped off, leaving only the shell of the sensor which was then fitted in the sphere. A large container filled with silicon oil was placed inside a vaccuum chamber to remove any air bubbles in the oil. The sphere and sensor were then placed in the vaccuum chamber and the air bubbles removed. The opening in the sphere, through which the sensor was inserted, was sealed off with a glue suitable for elastomeric materials. Removing the air bubbles from the inside of the sensor proved to be very difficult, and in some ball cells the prescence of an air bubble showed up during calibration. Electrical pins and cables Connection holes Pressure sensor shell Absolute port Pressure port Thin walled rubber sphere Figure 3.3: Pressure sensor modified to fit inside rubber sphere The Tube Type Pressure Cell The second type of pressure cell was the tube type cell which consisted of a 4mm diameter hollow Tygon ( ) tube, mm long, as shown in figure 3.4. The pressure sensor used in these cells was a standard 5psi differential pressure sensor, which was chosen above the 5psi sensor, as the sensor s capacity produced a larger signal-to-noise ratio. The smaller capacity sensors were not available in the absolute form, so that one port was blocked off from the atmosphere with a plug, leaving the remaining port as the active port. A solid end cap to close off the end of the tygon tube, and a plug with a central hole, were fabricated in the laboratory. The separate components for this cell were filled with oil by placing them in the container of oil in the vacuum chamber. Once the air bubbles were removed, the cell was then assembled while submerged in the oil, taking care not to introduce new air bubbles while handling the tube and components.

123 3.8 DES NORTJE: PhD THESIS: THE ANTI-DYNAMIC TUBE IN MASS FLOW SILOS flexible tubing pushed over port and plug, and fastened with a tie Pressure sensor open port blocked off 4mm diameter tygon plug with centre hole End cap Electric pins and cables Figure 3.4: Tube type pressure cell The sensor and the plug were connected using a flexible plastic tubing which was pushed over both the port and the plug and fixed on with a plastic tie. The plug and end cap were pushed into the tube and also fixed in place with a plastic tie. This system proved very successful as the air bubbles from the active port of the sensor were easily removed. Since the Tygon ( ) tube was translucent, a visual check of the existence of air bubbles was easily made. The tube cell was left to stand overnight in a vertical position resting on the sensor, and any trapped air bubbles would then float up into the tygon tubing The Plate Type Pressure Cell As shown in figure 3.3, the ball cell measured the average pressure acting in three dimensions, where as the tube cell measured the average pressure acting in a plane. The third type of pressure cell was therefore developed to measure the average pressure acting in one direction only. This was the plate type cell which consisted of an aluminium plate with a slot cut out the middle connected to the pressure sensor as shown in figure 3.5. port blocked off standard pipe connection threaded end cap pressure sensor mm wide slot Electric pins and cables rigid plastic tubing mmx3mm aluminium plate covered with rubber membrane on both faces Figure 3.5: Plate type pressure cell

124 WALL PRESSURE MEASUREMENTS: EXPERIMENTAL SET-UP 3.9 Again, a standard 5psi differential pressure sensor was used which had one port plugged off. Two threaded holes were made at either end of the aluminium plate extending into the slot. Both faces of the plate were sealed by a rubber membrane which was glued on to the plate surface. One hole was closed off with a threaded end cap, and a standard pipe fitting was used to screw into the other hole. A short length of stiff plastic tubing was pushed into the pipe fitting and then heated up in boiling water to fit over the active port of the sensor. The individual components of the pressure cell were placed in the container of low viscosity oil and a vacuum applied to remove all the air bubbles. Once the air bubbles were removed the cell was assembled while submerged in the oil PRESSURE CELL CALIBRATION The cells were placed in an air tight vessel with an absolute pressure sensor, first to determine their responses to an instantaneous pressure and then to calibrate each cell. The inlet and outlet to the vessel were sealed off with silicon sealant and allowed to stand for five hours before applying air. Compressed air was supplied to the vessel by means of a valve and regulator, which gave a digital read out of the pressure supplied to the vessel. The set up is shown in figure 3.6. compressed air supply inlet and outlet sealed with silicon valve and digital readout unit ii iii cables from pressure cells air tight container i iv amplifier To computer Figure 3.6a: Calibration of pressure cells (i) Ball cell, (ii) Tube cell, (iii) Plate cell, (iv) Plain sensor

125 3. DES NORTJE: PhD THESIS: THE ANTI-DYNAMIC TUBE IN MASS FLOW SILOS Figure 3.6b: Photograph of pressure Cell Calibration Before calibration, the cells were tested to ensure there were no air bubbles trapped inside and also to determine their responsiveness to an instantaneous pressure. The instantaneous pressure was applied to the cells by first regulating the pressure to a known value with the valve closed and once the pressure had been reached the valve was suddenly opened. The cells were then subjected to a series of instantaneous pressures and the output recorded on the computer. When the cells contained trapped air bubbles, the output of the cell compared to the plain pressure sensor exhibited a lag in the peak reponse, as well as a damped response in the decay curve of the graph. As shown in figure 3.7, the dotted shows there is no lag in the peak response of the cells. No lag in peak response Cells very responsive to instantaneous pressures Ball cell Plain sensor Voltage output Data point Voltage output Data point Figure 3.7: Typical response of a ball type cell

126 WALL PRESSURE MEASUREMENTS: EXPERIMENTAL SET-UP 3. Once the responsiveness of each cell had been determined, they were calibrated by applying a series of pressures, in increments of one, up to thirty two kilopascals. Before the cell output was recorded, the air pressure in the vessel was first set at a constant for each increment, thereby ensuring good data readings. For each increment of pressure, 7 data points were recorded. The average voltage output for each pressure increment was plotted as a point on the graph, and a linear least squares regression analysis applied to the data. The graph for each cell has been shown in Appendix B, with the calibration constant. The pressure cells were connected to the data logger by means of 6m long cables which enabled the cells to flow freely during material discharge. The cells were placed at various depths in the silo during the filling operation, and their positions measured MULTI-TURN POTENTIAL METERS multi-turn potential meter pulley inextensible wire mm flat plate Figure 3.8a: Multi-turn potential meters To determine the vertical position of the pressure cells during material flow, two multi-turn potentiometers were each connected to a pulley, supported at the top of the silo. Two flat plates, mm in diameter each, were fixed to an inextensible wire which connected the plates to the pulleys, as shown in figures 3.8a and 3.8b. The multi-turn potentiometers were connected to the data logger and calibrated by recording the voltage output for each quarter, or half turn of the pulley. The pulleys each had a circumference of 685mm, giving a reading for every 7mm travelled. The graph of results and the calibration constants for each multi-turn potential meter has been shown in Appendix B.

127 3. DES NORTJE: PhD THESIS: THE ANTI-DYNAMIC TUBE IN MASS FLOW SILOS The plates were placed in the material during filling for each test, and their positions from the top of the silo measured. When the material started flowing, the position of the pressure cells relative to the multi-turn potential meters was therefore easily determined. Figure 3.8b: Photograph of Multi-Turn Potential Meter connected to the Pulleys on top of the Model Silo 3..8 GATE SWITCHES To determine the exact time or data point when the gate was opened, two on/off switches were positioned on the gate. One switch was triggered by the closed gate and the second switch was triggered by a leverarm which had been placed on the sliding gate. When the gate was opened the voltage output changed from zero volts to five volts, and when the gate was fully opened the leverarm triggered the second switch, which changed the voltage output from five volts back to zero volts. This also gave the time taken and hence the number of data rows, to open the gate.

128 WALL PRESSURE MEASUREMENTS: EXPERIMENTAL RESULTS EXPERIMENTAL RESULTS 3.3. DESCRIPTION A total of twenty four tests were performed on the silo, of which four tests gave no useable results due to problems with the data logger. The first five tests were performed on the silo without an anti-dyanmic tube installed, to measure the static and dynamic pressures in the material. The remaining fifteen tests were undertaken to determine the frictional drag on tubes of varying lengths and diameters. In the drag tests, the pressure cells were placed in the silo to gain additional data about the speed with which the swich pressure travels up the silo, as well as to measure the effect of the tube on the wall pressures. The discussion of the drag tests on the anti-dynamic tube is given in chapter four. Owing to the large amount of effort involved in filling the silo for each test, a check list was made to ensure correct preparation was carried out. An example of the check lists for both the static and dynamic tests has been given in Appendix C. The static test check list included checking the electrical signals from all the cells, the multi-turn potential meters and the switches on the gate. The positions of the pressure cells from the top of the silo were measured and recorded, as well as the cell orientation relative to the silo wall. The check list also noted the calibration constants, the input data file and the method of filling the silo for later reference. During the filling stage a stopper was placed in the pulleys of the multi-turn potential meters to avoid the falling material causing the pulleys to turn and reach their full rotation before the dynamic tests were performed. The dynamic test check list included the input data file, calibration constants and checking electrical signals from the data loggers. At the end of each test, the cone of material at the top was levelled off and the silo was filled to it s capacity. The sampling frequency for each test was set approximately seven seconds longer to ensure the start and finish of each test would be recorded. Each test was timed with a stop watch to compare the test duration with the output from the multi-turn potential meters. The cables from the pressure cells were laid out individually next to the silo to ensure they would not become entangled. It was essential to leave enough spare cable for the pressure cells to flow freely down the silo. The stoppers from the pulleys were removed before the start of each dynamic test. However, due to the slip-stick nature of the material flow, the pulleys overshot their turning giving exaggerated flow rates. The friction of the pulleys was increased to avoid this problem.

129 3.4 DES NORTJE: PhD THESIS: THE ANTI-DYNAMIC TUBE IN MASS FLOW SILOS A complete list of the tests performed is given in table 3.. TABLE OF SILO TESTS Static and Dynamic pressures Test Number - Ball Ball 4 tubes 3 3 tubes 4 3 tubes 5 tubes 3 plates 6 to 9 Pressure cells with good test results No useable results: Elec tric al problems with the data loggers Mtpm Anti-Dynamic Tube Drag Force Test Tube Tube Hopper Number Diameter Length Gate diameter (mm) (mm) (mm) Tube support broke sections 8 Middle section broke off ten 895,84,845 seconds after flow started sections 8 895,84, sections 8 895,84, sections 4 895,84,845 Smaller gate opening sections 4 resulted in a funnel 895,84,845 flow pattern. 9 4 sections 4 895, Flow down the inside 8 8 and outside of the tube Table 3.: Static and Dynamic test list 3.3. STATIC TESTS The silo was filled by means of a bucket elevator without any attachments at the outlet, for the first five tests. From figure 3.9 it can be seen that the position of the cone of material varied with the filling process, making it a combination of eccentric and central filling. To determine the effect of this on the density of the material several samples were taken from various positions in the silo by placing a

130 WALL PRESSURE MEASUREMENTS: EXPERIMENTAL RESULTS 3.5 Bucket elevator Cone of material changes position during filling small bucket of known weight and volume in the material during filling. Each sample was then weighed to determine the density at that level. The bucket elevator was also equipped with a flexible hose which was positioned in the silo to cause a central filling situation. Again samples were taken from various depths in the silo and the density calculated. Table 3. shows the densities at various levels for both filling methods. The difference in the material density between the two methods was.7% which was not considered large enough to have a noticeable effect on the results. Therefore, an Figure 3.9: Position of the cone of material average value of 6kN/m 3 as determined in the laboratory has been used throughout this thesis. Test No Depth () Mass Density Test No Depth () Mass Density (m) (g) kn / m 3 (m) (g) kn / m 3 Free fall from bucket elevator Flexible hosing fixed to bucket elevator Average 6. Average 5.78 OVERALL AVERAGE = 6, kn/m 3 () Depth of container from top of silo () Container self weight: 87.9g and volume:.345m 3 Table 3.: Density measurements during filling of the silo The results from the first five static tests have been shown in table 3.3 and plotted in figure 3., with the equivalent Janssen horizontal pressure. The Janssen pressure was calculated using a stress ratio of.4. The pressure cells used in these

131 3.6 DES NORTJE: PhD THESIS: THE ANTI-DYNAMIC TUBE IN MASS FLOW SILOS tests were tube type cells, which were all placed vertically in the material approximately mm from the wall. The tube cells measured the average of the circumferential and horizontal pressures acting in the plane, and is given by: tube C H From Jenike (964), the horizontal and circumferential pressure acting in the horizontal plane of the axially symmetric cylindrical section are equal in magnitude. Therefore, the horizontal pressure acting normal to the wall is given by: H tube Test Depth Pressure Janssen below c ell (kpa) horizontal surface (H +C )/ pressure Depth below surface (m) Hopper Cylinder Horizontal Pressure (kpa) Table 3.3: Static Test Results Figure 3.: Static test results The pressure cells were placed in the silo at various depths. The exact orientation and depths of the cells from the top of the silo were recorded on a drawing which formed part of the check list, as shown in Appendix C. The filling process was therefore interrupted to place the pressure cells in the material. The total time taken to fill the silo, place the cells and note the depth and orientation, was approximately three to four hours. Thus, there was not enough time between successive layers for the material to be affected by the process, and filling has been considered as a continuous operation.

132 WALL PRESSURE MEASUREMENTS: EXPERIMENTAL RESULTS 3.7 Although most of the static test results are slightly greater than those shown by the line of the Janssen horizontal pressure, these results are deemed to be in good agreement with the Janssen theory. Therefore, the static results will be used to compare the ratio of the dynamic to the static pressures. This ratio gives an indication of the factors applied to wall pressures, and hence to the pressures exerted on the anti-dynamic tube in the material. From the literature survey, the vertical pressure in the centre of the silo has been shown to be approximately 3% greater than the average value. Therefore, the vertical pressure exerted on the anti-dynamic tube can be estimated from the ratio of dynamic to static results by multiplying the ratio by a factor of DYNAMIC TESTS The dynamic tests all followed directly after the static tests with no waiting time between tests in which the material had time to settle or de-aerate. At the end of each static test, the computer program was shut down and a new input data file was entered for the dynamic test. Zero offsets were recorded with the static pressure acting on the cells. Hence, the dynamic tests show a negative pressure when the cells passed through the gate of the silo with the moving material. This negative value is equivalent to the static pressure at the depth the cell was placed during filling. Figure 3. is a typical sample output showing the general trend of the dynamic tests. 8 First peak Second peak Pressure (kpa) 4 zero offsets -4 gate opened 3 4 Time (sec) 5 6 initial static pressure 7 Figure 3.: Sample output of a typical dynamic test

133 3.8 DES NORTJE: PhD THESIS: THE ANTI-DYNAMIC TUBE IN MASS FLOW SILOS Nearly all the tests showed a first peak pressure at the instant the discharge gate was opened and the material started moving. The pressure then reduced to approximately the static value (zero on the graph), although in some cases the pressure was considerably higher. The second pressure peak shown on the graph occurred when the cell passed through the transition and entered the hopper. As the cell left the silo the pressure reduced to a negative value. When the cell was placed at, or just above, the transition, the first and second peak curves in the output graph, merged to form a single curve. However, when the cells where placed below the transition only a single peak appeared in the dynamic output curve, followed by a rapid decrease in pressure as the cell moved down the hopper. A list of all the tests performed on the silo, as well as a complete set of the output results for each test, is given in appendix D. A table of the results for the first five dynamic tests is given in table 3.4, showing the values of the first and second measured peak pressures. The total pressures acting on the silo wall during flow are equal to the dynamic value plus the static value, which have also been given in table 3.4. The values plotted in figure 3. are the total pressures acting on the silo walls. Test Depth Static Dynamic pressures Total pressure No from top pressure First Second First Second of silo (m) peak peak peak peak Table 3.4: Dynamic Test Results

134 WALL PRESSURE MEASUREMENTS: EXPERIMENTAL RESULTS 3.9 The ratios of the measured dynamic to static pressures have been shown in figure 3.(b). This ratio was calculated by dividing the value of the first peak pressure by the measured static pressure at the depth the cell was placed. To calculate the ratio of the dynamic to static pressure at the transition, the value of the second peak in the output curve was used for the dynamic pressure, and an average value of 7. kpa was used for the static pressure at the transition. This average value has been calculated from the results of six static tests. Shown in figure 3.(c) is the ratio of the measured dynamic pressure to the calculated Janssen static pressure. For test numbers three to five, tube type pressure cells were used. These tubes were placed horizontally parallel to the silo wall and thus measured the average of the pressures acting in the meridian plane. Therefore an equivalent average static pressure has been calculated as follows: Average pressure at the point: AV = ( H + V )/, and stress ratio: K= H / V Therefore: AV = V (K+)/ where V is the Janssen vertical pressure given in equation..5, using =6.8kN/m3, D=.98m, =.44 and K varies. A varying stress ratio has been used in the above calculation of the Janssen pressure. For a depth of fill of zero to one diameter K=.5; from one to two diameters fill level, K=.3; and for two to three diameters fill, K =.. These values have been determined from the static tests and have been explained in chapter Depth below surface (m) (a) (b) (c) Figure 3.: Dynamic test results: (a) Measured horizontal pressures (kpa) ; (b) Ratio of measured dynamic to measured static pressure ; (c) Ratio of measured dynamic pressures to Janssen static. In all three graphs: * First peak ; Second peak

135 3.3 DES NORTJE: PhD THESIS: THE ANTI-DYNAMIC TUBE IN MASS FLOW SILOS The experimental results show the dynamic pressures to be between one and two times greater than the Janssen static pressures, and one to four times greater than the experimental static pressures. This is smaller than the values taken from the tests by Blair-Fish and Bransby (973), as well as Van Zanten and Mooij (977) Speed of the Switch Pressure The height of the pressure cells above the gate and the sampling rate of the computer, give an estimate of the speed with which the switch pressure travels up the silo. The calculation of the time between columns of data and the time between rows of data has been given in equations 3.9 and 3., and has been used to determine the time when the cells registered a change in pressure. A sample output data file has been given in table 3.5. Seconds Tube Tube 4 Tube 5 Plate 6 Plate 7 Plate 8 Switches Table 3.5: Sample Output Data File For this test the sampling frequency was 53 Hz, therefore the time between columns and rows of data was.98 and.37 seconds, respectively. It is reasonable to assume that the pressure wave may have passed over a pressure cell before the computer was able to record the data change. In the spreadsheet the cell number represents both the channel number of the data logger and the column number in the data output. Tube cell and plate cell 6, which were placed together at a height of.87m above the discharge gate, registered a pressure change at.8 seconds after the gate was opened. This gives the pressure wave speed was.74m/s for this sample output. By the time the pressure wave reached tube cell 4 and plate cell 7, which were both placed at

136 WALL PRESSURE MEASUREMENTS: EXPERIMENTAL RESULTS 3.3 the same level higher in the silo, the computer had already recorded channel 4, with no pressure change. However since plate cell 7 registered a pressure change, the time taken for the wave to reach these cells has been taken from channel 5. This is reasonable, since tube cell 5 was placed together near the top of the silo and therefore registered no pressure change at the instant in time when the pressure wave reached plate cell 7. Tube cell 5 and plate cell 8 were placed together at a height of.5m above the discharge gate. The pressure wave passed over these cells at a time of.6 seconds after the gate was opened. This calculation gives the speed of the switch as.69m/s. Due to the possibility of large errors being introduced when trying to determine when the pressure wave passed over the cells, only those cells which were placed in the upper sections of the silo have been used. This gives the average speed of the switch pressure travelling up the silo. Table 3.6 gives the average speed from 3 tests using only those cells which were placed in the upper third of the silo. T est Sampling Height of Switc h number frequency cell above pressure (Hz) gate (m) speed (m/s) Average speed:.3m/s Table 3.6: Switch Pressure Speed The average speed from all the tests has been shown in table 3.6 and is found to be.3m/s. The results from test numbers 7, 8, and have been excluded from the calculation of the average due to the large scatter of their results

137 3.3 DES NORTJE: PhD THESIS: THE ANTI-DYNAMIC TUBE IN MASS FLOW SILOS STRESS RATIO C H V Figure 3.: Directions of the stresses in the cylinder By placing two types of cells at the same level in close proximity to each other, the stress ratio can be determined for filling as well as during flow as shown in figure 3.. When a tube cell is placed horizontally and parallel to the silo wall the average of V and H is measured. Therefore by placing a plate cell to measure V or H, the stress ratios can be determined Static Stress Ratio The results of the static stress ratio from test number to 4 is given in table 3.7. Test Depth Average Measured Stress No. from Pressure pressure Ratio top(m) (kpa) V H H V Depth from top (m) Stress Ratio K K o K a Figure 3.3: Graph of static stress ratios to D to D Hopper Table 3.7: Static Stress Ratios The data from test numbers and 3 did not give reasonable results and have therefore been ignored. Thus, the overall average of the measured stress ratio is.3. This is approximately equal to the stress ratio for the material at rest namely: K o = Sin = Sin 45 =.93

138 WALL PRESSURE MEASUREMENTS: EXPERIMENTAL RESULTS 3.33 The active stress ratio given by : K a = ( Sin )/( + Sin ) =.7, is considerably less than the overall average. However from the graph in figure 3.3, it can be seen that the stress ratio decreases with the depth of material in the silo. It is proposed in this thesis, to divide the silo into three sections to calculate the stress ratio. At a depth of zero to one diameter below the top of the silo the ratio can be approximated from: K = Sin =.5. From one diameter to two diameters below the top, the ratio approaches the value for the material at rest: K o =.9. At depths greater than two diameters and in the hopper, the stress ratio is approximately equal to the active stress ratio, K a =.7. From Briassoulis (99) the Reimberts give the stress ratio as decreasing with depth, which is in agreement with these test results Dynamic Stress Ratio For each pair of pressure cells placed at the same level, two ratios have been determined; one at the start of flow and one when the cells passed through the transition. These results are shown in table 3.8 and plotted in figure 3.4. There is a large scatter in the measured stress ratios. Test Depth Stress No. below Ratio surface H V (m) Table 3.8: Dynamic Stress Ratios Depth below surface (m) 3 3 Stress Ratio Figure 3.4: Graph of Dynamic stress ratios Hopper Cylinder

139 ANTI-DYNAMIC TUBE THEORY: LITERATURE SURVEY 4. CHAPTER FOUR THE ANTI-DYNAMIC TUBE 4. LITERATURE SURVEY Other names for anti-dynamic tubes are decompression tubes, discharge tubes, static flow pipes and tremmie tubes. 4.. PIEPER Pieper (969) conducted tests which measured the force exerted on a horizontal bar during flow. A mm diameter bar was placed in the material,.5m above the outlet of a.8m square silo filled with quartz sand of 5kN/m 3 bulk density. The bar was supported by a frame hanging from the ceiling, as shown in figure 4.. The graph shown in figure 4., has been taken from the results by Pieper, as the average of the two values for the left hand and right hand support. In the paper, Pieper does not say if the deflection of the bar itself was taken into account in the output. Furthermore, no description is given of the method to take into account any downward movement of the bar supports, which may have caused it to derive additional support from the walls of the model. It must therefore be assumed that these results may have an experimental error in the value of the force recorded. Pieper states that the output of the vertical force on the bar during filling is.8m mm bar P b.5m 3.35 Pressure (kpa) Time (min) filling emptying Figure 4.: Horizontal tie placed in silo during flow. (Tests by Pieper) Figure 4.: Average values taken from Pieper s test results

140 4. DES NORTJE: PhD THESIS: THE ANTI-DYNAMIC TUBE IN MASS FLOW SILOS approximately the same as the Janssen static value for that depth. As can be seen from figure 4., the force suddenly increases fourfold when the gate is opened, decreasing slowly with the decrease in the height of the material. The maximum pressure on the bar recorded by Pieper was 65. kpa approximately one and a half minutes after the gate was opened. 4.. REIMBERT The Reimbert brothers (976) claim to be the inventors of the anti-dynamic tube as a means to produce a homogeneous outflow from a silo. Initially, this consisted of perforated tubes fitted to the side walls of the silo, connected to inclined tubes in the hopper, as shown in figure 4.3(a), or alternatively placed centrally as shown in figure 4.3(b). The Reimbert s state that the centrally placed tube caused the material to flow down the tube in successive layers, resulting in a first-in-last-out situation. In both cases, material flow along the walls is eliminated, thereby reducing the wall pressures as well as wear on the walls. The Reimberts claim to have successfully installed the tubes as a retro-fit in many existing concrete silos which were cracked due to excessive wall pressures. (a) (b) Figure 4.3: The Reimbert s tube: (a) Section and Top View of Tube down the side walls of the silo; (b) Centrally placed tube. However, according to Ravenet s report (983), the Swedish specialist Bergau, records that anti-dynamic tubes were first used at the beginning of the 9 s by Miersch in the Frankfurt/Main Silos; Duhle used them in the Alexander Dock Silo, in Liverpool; and Huart and Kvapil also made use of them in the early 9 s RAVENET Ravenet (983) conducted tests with an anti-dynamic tube in a transparent model to emulate flow in layers as reported by the Reimbert brothers. Ravenet states that

141 ANTI-DYNAMIC TUBE THEORY: LITERATURE SURVEY 4.3 the tube failed to discharge the material as the holes frequently became blocked. After several attempts, layered flow was achieved. He states that when the tube operated succesfully, the dynamic to static pressure ratio was approximately MCLEAN McLean (985) reports on general arrangements of the anti-dynamic tube being successfully installed in silos to allow safe side discharge. These arrangements alleviate the bending stresses in the wall associated with an eccentric outlet. The diagrams shown in figure 4.4 have been taken from McLean s report by scaling his drawings. While it is uncertain if McLeans diagrams were drawn to scale, the ratio of the tube to silo diameter in his diagrams is.. This is approximately twice the ratio given by other researchers. (a) Single outlet (b) Multiple outlets Figure 4.4: Side Discharge Outlets: (a) Single Outlet (b) Multiple Outlets The material is drawn from two or three levels in the case of the silo with multiple outlets. McLean reports on the importance of adequate support given to the side discharge chutes which protrude into the path of the flowing material. McLean (985) gives the following equations to determine the vertical forces acting on objects placed in the flowing material: D For a tall silo: J F v.5ap e (4.) 4K K where J 4z D H s (4.) and A p is the projected plan area of the object and H s is the height of surcharge above the silo. If there is no cone of material above the top of the silo, equation 4. reduces to a modification factor of.5 times the Janssen equation multiplied by the projected area of the object. Both Pieper and McLean report that the pressure on an object submerged in the

142 4.4 DES NORTJE: PhD THESIS: THE ANTI-DYNAMIC TUBE IN MASS FLOW SILOS material is approximately equal to the Janssen static pressure during filling of the silo. However McLean s formula (equation 4.) gives the force during flow of the material as.5 times the static value, whereas Pieper s test results show the flow force to be nearly four times the static value. Thus Pieper s results give values that are approximately.6 times larger than predicted by McLean OOMS AND ROBERTS Ooms and Roberts (985) conducted tests on a flat bottomed acrylic model silo, 3.8m tall and.m in diameter, fitted with an anti-dynamic tube and filled with wheat. Their model and tube arrangement has been shown in figure 4.5. Unlike the Reimbert s tube, theirs did not extend the full height of the silo and was open at the top, with port holes only at the base of the tube. The purpose of their tests was to determine the effectiveness of this tube arrangement in controlling flow patterns and wall pressures, before installing tubes in full scale silos. After installation of the tube, the model silo emptied in two stages as shown in figure 4.5, thus operating as two short silos in series. They report that with no tube installed, the flow pressures at the effective transition were over three times the static measured value. However, the flow pressures were nearly equal to the measured static pressures after the tube was installed. Ooms and Roberts state that the minimum height of the tube is determined by the angle, such that no effective transition is formed between the material and the silo wall during the first stage of discharge. Therefore, from the geometry of the model silo and tube: Stage Effective transition Stage h max Dead material h min (a) (b) Figure 4.5: Ooms and Roberts Tube: (a) Discharge Sequence; (b) Tube geometry

143 ANTI-DYNAMIC TUBE THEORY: LITERATURE SURVEY 4.5 h min = H D/(Tan) (4.3) Similarly, the maximum height of the tube is determined such that no effective transition intersects the wall during the second stage of discharge. Therefor h max =D/(Tan) (D/) Tan (4.4) The angle () of the effective transition to the vertical is given by Jenike et al (973c) and Arnold et al (989) to be a function of the internal friction angle of the material, whereas Hasra and Bazur (98) give as a function of both the wall and internal friction angles. Ooms and Roberts derived an expression for the static vertical pressure in a silo with a tube installed, by considering the equilibrium of a horizontal element. z o v t w dz z The forces acting on a horizontal slice of thickness dz in the silo at the level of the anti-dynamic tube have been shown in figure 4.6. The silo and tube diameters are given by D and d respectively. At the level of z=z o the vertical pressure, zo is equal to the v +( v /z)dz static pressure given by the Janssen equation (.) Figure 4.6: Forces acting on a horizontal slice v / B Bzo Bz e e / B zo (4.5) and B=4(DK o o +dk i i )/(D -d ) (4.6) In their derivation, Ooms and Roberts assumed the stress ratios on the inside and outside of the tube to be different. In equation 4.5, K o is the stress ratio on the outside of the tube and K i is the stress ratio on the inside of the tube. Assuming fully mobilised flow along the outside of the tube wall, Ooms and Roberts have given the vertical drag as a direct function of the internal friction angle and stress ratio, as shown in equation 4.7. BH e t B H t Fv Ki i d zo (4.7) B B However, as there is no direct relationship between the vertical drag force on the

144 4.6 DES NORTJE: PhD THESIS: THE ANTI-DYNAMIC TUBE IN MASS FLOW SILOS outside of the tube and the internal properties of the tube, equation 4.7 has been re-written as shown in equation 4.8. Therefore, K o and o have been substituted for K i and i, and the drag down force becomes: BH e t B H t Fv Ko od zo (4.8) B B The effect of the tube diameter on the vertical pressure and the vertical drag down force as given in equation 4.5, has been shown in figure 4.7 (a) and (b) respectively. These graphs were calculated for the material propeties and silo geometry of the model used in the laboratory. The tube height used in the calculations is.6m at a depth of.6m below the top of the silo. By increasing the tube to silo diameter ratio from one quarter to a half, the vertical pressure as calculated by equation 4.5 decreases from 84% of the Janssen pressure to 6% of the Janssen pressure. Thus, the vertical static pressure in a silo, and hence the horizontal static pressures on the walls, can be reduced by introducing a larger tube to silo diameter ratio.. Depth below surface (m) Vertical pressure (kpa) (a) Janssen Figure 4.7: Effect of tube to silo diameter ratio of.5 and.5 on: (a) Vertical pressure ; and (b) Vertical drag on outside of tube Height of tube (m) Vertical drag (kn) (b).5 The function B in the vertical drag down force given by equation 4.6, was calculated assuming equal internal and external stress ratios, and equal internal and external wall friction angles. The vertical drag force increases with increasing height of the tube as expected, as this is directly related to the tube surface area. The graph in figure 4.7(b) shows the relation to be approximately linear at depths of.4m and greater.

145 ANTI-DYNAMIC TUBE THEORY: LITERATURE SURVEY 4.7 The vertical drag force on the inside of the tube during flow has been given in the report by Ooms and Roberts, as: F vi d 4 zo d e 4iK i 4 i Ki Ht / d d 4 H t (4.9) This expression contains only the properties relating to the inside of the tube, as would be expected. The same material and silo properties as used in the above calculations, was used in the calculation of the vertical drag inside the tube. For a tube of length.6m the vertical drag force is.7kn, which is 45% of the vertical drag on the outside of the tube. This implies that the drag should be increased by a factor of.5 when the material flows inside and outside the tube at the same time KAMINSKI AND ZUBRZYCKI Kaminski and Zubrzycki (985) conducted experiments on a concrete model silo.5m in diameter by 3.78m tall, fitted with anti-dynamic tubes and filled with wheat. They report that the wheat had a bulk density 8 t/m 3, which is too heavy for wheat and should probably be 8kN/m 3. In their report, they do not give the diameters of the tubes nor do they give an indication of the number of tubes installed. A sketch of their model arrangement is shown in figure 4.8. Figure 4.8: Kaminski and Zubrzycki model silo spring dynamometer steel rods d S support tube The purpose of their tests was to determine the vertical forces acting on perforated and non-perforated tubes during material flow. The tubes were suspended by steel rods from a supporting structure above the model silo. Spring dynamometers, with a maximum capacity of kn, were fixed to each steel rod to measure the loads on the tubes as shown in the diagram. The results of their tests showed that the drag on the non-perforated tubes increased by a factor of 6 times the filling vertical force, whereas for a perforated tube there was no increase in the force during discharge of the material. Kaminski and Zubrzykci state that the flexibility of the tube supports has a large influence on the measured value of the vertical force. They conducted experiments by varying the flexibility of the supports and then measuring the deflection of the bottom of the tube, d S, during material flow.the results of their tests have been

146 4.8 DES NORTJE: PhD THESIS: THE ANTI-DYNAMIC TUBE IN MASS FLOW SILOS reproduced in figure 4.9. The vertical axis in their graph shows the ratio of the measured vertical deflection, d S, to the hydraulic radius of the tube, r h. Thus, they define the flexibility parameter as the ratio of the deflection, d s, to the hydraulic radius of the tube, r h. This seems inappropiate as the flexibility can not be a factor of the tube geometry. The horizontal axis is the ratio of the measured vertical force, F V, to the calculated Reimbert value, F R. They state that an increase in the flexibility parameter beyond 5.6, does not give a further decrease in the measured vertical force on the tube. The lowest measured vertical force was 4% of the calculated Reimbert value. For an inflexible support (a value of 3.3x -3 ), the measured vertical force was 7% of the calculated Reimbert value. They propose correction factors to be applied to the calculated Reimbert value, as shown in the graph by the stepped solid line, which represents an envelope of their test results ds / rh (x -3 ) F V / F R Figure 4.9: Kaminski and Zubrzycki test results for varying support flexibility They state that their results were compared with measurements on full scale silos. However, there is no reference or description of the extent of the full scale tests. In their report they state that the optimal geometric parameters for an antidynamic tube were previously derived in a dissertation in 977. Unfortunately this reference is not available in English, and hence the recommendations from their report have been included in this thesis for completeness only. The parameters are as follows: ) The diameter of the tube should fall within the following range:.3 d t / H.64 (4.) where H is the hydraulic radius of the silo and d t is the tube diameter.

147 ANTI-DYNAMIC TUBE THEORY: LITERATURE SURVEY 4.9 In this relationship, the tube diameter is a linear function of the silo diameter. ) The total area of the tube perforations should fall within the following limits:.8 A. (4.) and A = A h /A t = A h / d t H t (4.) where A h is the total area of the perforations in the tube wall, and A t is the surface area of a solid tube. This parameter gives the tube perforations as a linear function of the tube surface area. 3) Flow characteristics of the tube perforations.5.8 (4.3) 3 and = A * d t /(dg * D) (4.4) where d G is the silo discharge gate diameter, and D is the silo diameter. For a cylindrical or square silo, H =D/4, equation (4.) can be re-written as:.3 D/4 d t.64 D/4 (4.5) Re-writing the second tube parameter given in equation (4.), in terms of the tube diameter gives:.8 d t L t A h. d t L t (4.6) Re-writing the third tube parameter gives the discharge gate diameter as a function of the area of the tube perforations: Ah dt.5 D dg Ah dt.8 D (4.7) The maximum and minimum limits as given by equations (4.5), (4.6) and (4.7) have been plotted in figure 4.. Therefore, for a tube diameter of m, the acceptable range for the total area of the tube perforations must fall within.5m upto.5m per meter length of tube, while the silo discharge gate must fall within.m up to.5m in diameter.

148 4. DES NORTJE: PhD THESIS: THE ANTI-DYNAMIC TUBE IN MASS FLOW SILOS 3 Tube 5 Silo diameter 5. Tube perforation area. Discharge gate.3.4 Figure 4.: Maximum and minimum limits for tube parameters 4..7 SCHWEDES AND SCHULZE Schwedes and Schulze (99) report on a discharge tube being succesfully installed in two cement clinker silos of 6m diameter and 7m tall, with a funnel flow hopper. The tube diameter was.8m, with a wall thickness of 5mm. The ratio of the tube diameter to the silo diameter was.. The vertical hole spacing was set at 3.5m c / c with four rectangular holes at each level. The holes were orientated vertically, with an open area of.6m (.6x.m). In their report the maximum vertical stress inside the tube of diameter d t was calculated from: v =d t /(4K) (4.8) This is the Janssen equation with the exponential term approximately equal to one. Using the above formula given by Schwedes and Schulze, the pressure on the tube during flow of the material, for the silo and tube model used in this research, would be: v = ( 6.4) / (4.4Tan9) = 4.7 kpa For the tube length of.6m and diameter of 4mm, this gives a total drag down force of: F v =4.7 *.6 * *.4 = 4.66 kn = kg This is. times larger than the value determined using the equation recommended by Ooms and Roberts.

149 THE ANTI-DYNAMIC TUBE: EXPERIMENTAL SET-UP EXPERIMENTAL SET-UP 4.. ANTI-DYNAMIC TUBE MODEL The model silo set up in the laboratory has been described in chapter 3.. of this thesis. The same cohesionless material as described in chapter 3.., was used for the experiments of the frictional drag on the anti-dynamic tube. Anti-dynamic tubes of.4m and.m in diameter, and.4mm wall thickness, were placed centrally in the model silo and suspended from a support at the top of the model silo, as shown in figure 4.. The angle of wall friction ( t ) between the tube and the material was measured in a standard shear box test and was found to be 9. strain gauged support beam BUCKET ELEVATOR inextensible chain anti-dynamic tube:.6m total length locating rods 8 Figure 4. : Model Silo and Tube The support frame was made up from rectangular hollow sections (RHS) of 65mm x 35mm, welded to two short struts with base plates which were clamped to the top flange of the model silo, as shown in figure 4.. The deflection of the support frame is directly related to the frictional drag acting on the side of the tube during flow of the material. The beam in the support frame was deliberately orientated across its weaker axis to ensure adequate deflection, as well as provide sufficient space for fixing of the strain gauges. The chain connecting the tube to the support

150 4. DES NORTJE: PhD THESIS: THE ANTI-DYNAMIC TUBE IN MASS FLOW SILOS beam was an inextensible chain, thus avoiding excessive vertical movement of the tube during material flow. The chain was located across the centre of the support beam. 65 x 35mm RHS average strain readings from top and bottom of beam Base plates clamped on to top flange of model silo inextensible chain threaded rod and eye cover over top of tube bent plate nut, top and bottom pop-rivets anti-dynamic tube Figure 4.: Support frame and tube connection The chain was connected to the top of the tube by a strip of plate which was bent to fit across the tube, as shown in figure 4.. Two holes were drilled through the side of the plate, which was then connected to the top of the tube. A hole was drilled through the centre of the strip to accommodate a threaded bolt and eye with two nuts, top and bottom of the bent strip. The chain was passed through the eye of the threaded bolt and fixed over the top of the support beam. The cover to the top of the tube was fixed around the threaded bolt to prevent material flowing down the inside of the tube. For three tests this cover was removed giving the drag down force for material flow on the inside as well as the outside of the tube. The model tube lengths varied from.8m to.6m and were made up from three sections of galvanized pipes which were riveted together. To ensure the tubes

151 THE ANTI-DYNAMIC TUBE: EXPERIMENTAL SET-UP 4.3 inextensible chain cover over top of pipe skirt around gap (not shown) disk inside the pipe strain gauged rods Figure 4.3a: Anti-Dynamic Tube Sections Anti-dynamic tube Disks bolted to side of tube Strain gauged 5mm diameter threaded rods Skirt around the gap to prevent entry of material (not shown) Figure 4.3b: Detail of Disks and Rods remained in a central position during filling of the silo, six locating rods, three near the base and three at mid height of the tube, were used. The locating rods were removed when the fill level reached the level of the rods. The three sections of the tube were connected by means of disks and 5mm threaded rods, placed inside the tubes, as shown in figures 4.3a and4.3b. These 5mm rods were strain gauged at three points around their circumference to give the average strain reading. The support beam at the top of the silo, and the 5mm rods were calibrated by hanging a series of weights at the bottom of the tube and recording the voltage output. The weights were added in kg increments up to a maximum of 6kg. The calibration constants of the strain gauges, with units of kg/v, were determined from the slopes of the data output which plotted as a straight lines as shown in the Appendix B.. To determine the rate of flow of the material during discharge, the multi-turn potential meters (mtpm) and mm plates, as described in chapter 3..7, were used. The filling process was interrupted to place the plates in the material and record their positions. The total time taken to empty the silo was also recorded and compared to the flow rate given by the multi turn potential meters. A photograph of the inside of the silo with the tube installed has been shown in figure 4.4. The photograph shows the group of experiments with the tube divided into three sections.

152 4.4 DES NORTJE: PhD THESIS: THE ANTI-DYNAMIC TUBE IN MASS FLOW SILOS Figure 4.4: View of the Anti-Dynamic tube installed in the steel model silo. 4.3 EXPERIMENTAL RESULTS A total of 4 tests were conducted with the anti-dynamic tube installed in the model silo. The tube lengths, diameters and the diameter of the silo discharge gate were varied. A matrix of tube tests performed is shown in table 4.. The first group of tests, Group A, were done with a silo discharge gate diameter of 6mm, while the second group of tests, Group B, had a discharge gate diameter of 4mm. Number of tests performed Tube diameter (4mm) Tube diameter (mm) Silo Solid T ube Solid Flow down discharge tube split in 3 tube inside & gate 6 mm sections 8 mm outside of diameter long long solid tube Group A: 6 mm Group B: 4 mm 3 Table 4.: Matrix of tube tests Three tests were performed with a tube of 6 mm long, followed by another three tests with the tube separated into three sections as shown in figure 4.3a. These six tests had a silo discharge gate diameter of 6mm and a tube diameter of 4mm.

153 THE ANTI-DYNAMIC TUBE: EXPERIMENTAL RESULTS 4.5 The opening size of the silo discharge gate was then varied from 6mm to 4mm by placing a wooden disk in the base of the hopper, and another three tests performed with the tube split into three sections. The tube diameter was then changed to mm and the length reduced to 8 mm. Two tests were then performed with a silo discharge gate diameter of 6mm. Finally, the last three tests were performed with material flowing down the inside and outside of the tube, with a silo discharge gate of 6mm in diameter. These variables on the tube were considered to be of greatest influence on the magnitude of the drag down force. The complete set of tube test results has been given in Appendix E. In nearly all the tests there was an initial peak force which only lasted a matter of seconds, followed by a second peak force of a longer duration. The maximum value of the initial peak varied considerably between tests, as can be seen in the two typical test results shown in figure 4.5 and 4.6 below. In figure 4.5, the initial peak value was lower than the second peak, where as in figure 4.6, the initial peak value is greater than the second peak value. The second peak represents established flow in the cylindrical section of the silo. Once the level of material drops below the top of the tube, there is a non-linear decrease in the drag force, which can be seen in figures 4.5 and 4.6. In figure 4.6, the drag force on the top and middle portion of the tube is decreasing, while on the bottom portion of the tube there is a constant drag force acting, until the material level drops below the bottom portion of the tube. Drag Down Force (kg) Time (seconds) Figure 4.5: Test number :Drag down force on solid tube.6m long

154 4.6 DES NORTJE: PhD THESIS: THE ANTI-DYNAMIC TUBE IN MASS FLOW SILOS Drag Down Force (kg) Time (seconds) Figure 4.6: Test number 6: Drag Down Force on tube split into three sections The first test performed to determine the drag down force, was test number. As can be seen from the output of this test in Appendix E, the pop rivets fixing the plate to the tube broke approximately 6 seconds after the discharge gate was opened. The tube moved down the silo with the flowing material and blocked the silo outlet. This test had to be abandoned and the silo top emptied before testing could resume. The plate was re-fixed to the top of the tube using larger diameter pop rivets, 5mm, to ensure this did not occur again. Silo Discharge Gate Diameter (6 mm) T est T ube Initial Shear Constant Shear No. length Drag Stress drag Stress (kg) (kpa) (kg) (kpa) Tube Diameter (4 mm) Tube Diameter ( mm) Tube Diameter ( mm) Flow down inside and outside of tube Table 4.: Results from Group A Tests The results from Group A Tests for the solid tubes, have been given in table 4. The shear stress on the tube has been determined from the drag down force divided by the surface area of the tube. The column of constant drag in the table refers to established flow in the silo, which occurs after the switch pressure has

155 THE ANTI-DYNAMIC TUBE: EXPERIMENTAL RESULTS 4.7 travelled up the silo. In some graphs of the output, the established flow is clearly visible, as in test numbers 3, 4, 6, 7 and 8. Except for test number 3, these tests correspond to the tube being split into three sections. In test number 3, the section of constant flow is very short due to the level of material going below the top of the tube, as is the case for the remaining tests using a solid tube. The results for the Group A tests with the tube split into three sections have been given in table 4.3 below. For this group of tests, there was an initial peak drag force followed by a second peak representing established flow in the cylindrical section of the silo. Silo Discharge gate Diameter (6mm) Test Tube Initial Drag Force (kg) Shear Established Flow Shear No. length Top Middle Bottom Stress Drag Force (kg) Stress (kpa) Top Middle Bottom (kpa) Tube Diameter (4 mm) Table 4.3: Group A Test Results for the tube split into three sections Drag Force (kn) 3 Gate = 6 mm: Initial peak Tube Surface Area (m ) The test results for Group A tests have been plotted in graphs as shown in figures 4.7 and 4.8. These test results have been divided into two subgroups showing the initial peak drag value, shown in figure 4.7, and the peak drag value for established flow, shown in figure 4.8. Figure 4.7: Group A Tests: Initial Peak Drag Force

156 4.8 DES NORTJE: PhD THESIS: THE ANTI-DYNAMIC TUBE IN MASS FLOW SILOS The drag force on the vertical axis, has been given in units of kilonewtons, while Drag Force (kn) Peak Drag Values for Established Flow Tube Surface Area (m ) Figure 4.8: Group A Tests: Established Flow Drag Force the tube surface area on the horizontal axis has been given in units of meters squared. From figure 4.7 and 4.8 it can be seen that there was a wide scatter of results, with the maximum initial drag on the tube reaching approximately.6 kn while the maximum drag value for established flow reached approximately.8 kn. The results from the Group B tests have been given in table 4.4. For these tests the tube was split into three sections and the discharge gate diameter was changed to 4mm diameter. The data output showed no initial peak value in the drag force on the tube when the discharge gate was opened, as can be seen in the graphs given in Appendix E. The values given in table 4.4 represent the maximum values of the drag force on the tube. For each section of the tube, the shear stress has been calculated by dividing the drag force by the surface area of the tube. The drag for each section Silo Discharge Gate Diameter (4mm) Tube Diameter (4 mm) Test Tube Drag Force (kg) Shear No. length Top Middle Bottom Stress (kpa) Table 4.4: Group B Test results has been determined by subtracting the value of the drag force for the lower sections. Thus for test number 7, the drag force acting on this section is equivalent to 85.9kg subtract 46.3kg, which gives 39.6kg. This value was then divided by the surface area over the length of.84m.

157 THE ANTI-DYNAMIC TUBE: EXPERIMENTAL RESULTS 4.9 Drag Force (kn).7 Peak Drag Values for 4mm Gate Tube Surface Area (m ) The results for Group B tests have been shown in figure 4.9 for established flow only, as there was no initial peak drag for these tests. From figure 4.9 it can be seen that the maximum drag force reached a value of approximately.64 kn. Figure 4.9: Group B Tests: Established Flow Drag Force The time taken to completely empty the silo was determined using a stop watch. This time gives the average flow rate of the material, compared to the results from the multi turn potential meters, which give the variation of the flow rate during material flow. These average flow rates are given in units of meters per second, which represents the rate of travel of the top surface of material. Therefore, the total height of the silo was divided by the time measured on the stop watch. While it is accepted that this is not an accurate measurement of the flow rate of a silo, it gives an indication of the variability of the flow rate for varying sizes of discharge gate diameter. The results are given in table 4.5 and 4.6 below. Silo Gate Diameter 6 mm Test Time from Flow Number stop watc h Rate (seconds) (m/s) Average Silo gate diameter 4 mm Test Time from Flow number stop watch Rate (seconds) (m/s) Average Table 4.6: Average Flow Rates of silo with gate diameter of 4mm Table 4.5: Average Flow Rates of silo with gate diameter of 6mm

158 4. DES NORTJE: PhD THESIS: THE ANTI-DYNAMIC TUBE IN MASS FLOW SILOS From table 4.5, it can be seen that the average time taken to empty the silo was one minute and thirty four seconds. For the total silo height of 3. meters, this gives an average flow rate of.396 m/s. This is faster than the rate given by the multi turn potential meters, since the material undergoes free fall in the hopper section of the silo. The average flow rates for the silo with a discharge gate diameter of 4mm are given in table 4.6. The discharge gate of 6mm has a plan area of. m while the gate of diameter 4mm has a plan area of.539 m. The gate of diameter 4mm is approximately 3.4 percent smaller than for the gate of diameter 6mm. The comparison between the flow rates shows that the 4mm gate gives an average flow rate of 4.5 times slower than the 6mm gate. The average value of the initial shear stress from table 4. and 4.3 for Group A tests is.5 kpa. This frictional stress corresponds to a silo with a gate diameter of 6mm, which has a plan area of.m. The average shear stress for established flow is.5 kpa for the same silo discharge gate. These values exclude the results for material flow down the inside as well as the outside of the tube. The average value of the shear stress for established flow for Group B tests, is.95 kpa, taken from table 4.4. This corresponds to a silo gate diameter of 4mm which has a plan area of.539m. These results have been shown in graphical form in figure 4.. Shear Stress (kpa) Exponential Trend Line Power Trend Line Linear Trend Line Silo Gate Plan Area (m ) Figure 4.: Shear Stress vs Discharge Gate Area

159 THE ANTI-DYNAMIC TUBE: EXPERIMENTAL RESULTS 4. Only three trend lines could be fitted through the two data points. These are an exponential curve, a power curve and a linear trend line. The equation for each of the trend lines are: Exponential Curve: =.39 e 7.x Power Curve: = 4x 6 A Linear Trend Line: =.85 A where is the shear stress acting on the walls of the tube, and A is the plan area of the gate. To determine which of these lines would be most appropriate, the drag force on a full scale silo which is five times larger than the model silo has been calculated. The full scale silo would have a diameter of 4.8m and a height of 6m, while the discharge gate would be.8m in diameter. The corresponding tube size would be.7m in diameter and 3.5m tall. For a discharge gate of.8m in diameter, the plan area is.53m. This would give the following shear stresses: Exponential Curve: = 6.897x 43 kpa. Power Curve: = 3.48x 6 kpa. Linear trend Line: = 5,3 kpa. The full scale anti-dynamic tube would have a surface area of approximately 8.7m. Thus the corresponding total friction force acting on the full scale tube, as determined from each of the trend lines, would give.979x 45 kn, 93.x 6 kn, and 47 kn, respectively. Clearly the calculated value from the exponential curve is far too large and therefore not practical. The value determined from the power curve is 9.5x 6 tonnes of force, which is equally impossible for the scope of the assumed model. When the silo discharge gate is opened the material suddenly changes from a stationary state to a state of constant motion under the

160 4. DES NORTJE: PhD THESIS: THE ANTI-DYNAMIC TUBE IN MASS FLOW SILOS acceleration of gravity. Therefore, the impact force as the material starts moving can be calculated as follows: F = mg = 6 (4.8 6/ /4) = 455. kn (4.9) If the value of the friction force as determined from the linear trend line was greater than the calculated impact force, it would imply that the material had an acceleration greater than gravity. Since the value of the friction force calculated from the linear trend line, 47 kn, is less than the force calculated for impact, as given in equation 4.9, the linear trend line can be accepted as a reasonable curve applied to the two data points, as shown in figure 4.. Therefore, it is proposed in this thesis that an equation for estimating the effect of the speed of the flowing material on the magnitude of the friction stress acting on the sides of a tube is as follows: =.85 A (4.) where is the shear on the walls of the tube and A is the plan area of the discharge gate. This equation is an empirical equation and it is not proposed that this is an acceptable expression for all silo and tube arrangements. This expression is limited to the scope of this thesis only.

161 THE ANTI-DYNAMIC TUBE: MATHEMATICAL MODEL MATHEMATICAL MODEL A mathematical model of the friction force on an insert tube has been developed by considering the equilibrium of vertical forces acting on an elemental slice of material as shown in figure 4.. The figure represents an element of material of thickness dz, from a silo fitted with an anti-dynamic tube. The lengths of the element sides are S, S, and S 3 where: S = Rd S = rd S 3 = (R- r) (4.) The top and bottom surface areas of the element are equal, and are given by: A (S +S ) S 3 / d (R -r )/ (4.) The volume of the element is given by: V = Adz = d (R -r )dz/ (4.3) A top A silo S R S 3 v dz S r s A tube A bottom t v z v dz d Figure 4.: Forces acting on an elemental slice of material in a silo fitted with an anti-dynamic tube The areas of the element in contact with the silo wall and tube wall are given respectively by: A silo = S dz = Rd dz and A tube = S dz = r d dz (4.4) By considering downward forces on the element as positive, equilibrium of the vertical forces acting on the elemental slice gives: v s silo t tube v (4.5) z v A V A A dza Noting that = K v Tan w, and substituting equations 4., 4.3, and 4.4 in equation 4.5 gives the following differential equation for the vertical pressure acting on the element :

162 4.4 DES NORTJE: PhD THESIS: THE ANTI-DYNAMIC TUBE IN MASS FLOW SILOS dv dz G v (4.6) where : G = [ (K s Tan sw R + K t Tan tw r)]/[ (R -r )] (4.7) This is the same expression as derived by Ooms and Roberts (985). To solve equation 4.6, multiply throughout by a function (z) and add and subtract (z) v from the left hand side of the equation. Re-arranging the terms gives: z) (z) (z) G(z) (z) ( v v v (4.8) Noting that the first term of equation (4.8) is the derivative of the product of (z) and v, and if the second term equals zero then: z z v (4.9) From the second term in equation (4.8), v, therefor let: G = (z)/(z), and Ln(z) G z Gz C (4.3) Gz Therefore, z e Ae Substitute equation 4.3 into 4.9 and solving gives: v G Ce Gz (4.3) To find the constant of integration: at z =, v =, and C= -/G, Then equation 4.3 becomes: v e G Gz (4.3) O Neil (995) gives this general method for solving linear first order differential equations. Substitute equation (4.3) into the expression for t to give: G z t Kt v Tant Kt Tant e (4.33) G The total friction force acting on the tube can be found by multiplying equation 4.33 by the circumference of the tube and integrating over the length of the tube: h G h FF r dz K Tan r h e C t t t (4.34) G G From the boundary conditions at h=, F F =, and therefore the constant of integration can be found:

163 THE ANTI-DYNAMIC TUBE: MATHEMATICAL MODEL 4.5 C (4.35) G Substituting equation 4.35 in equation 4.34 gives the expression for the total friction force acting on the tube. FF Kt G G h Tan t r e (4.36) where G is given in equation TUBE PARAMETERS From the first geometric parameter defined by Kaminski and Zubrzycki 985, the diameter of the tube divided by the hydraulic radius of the model silo used in this research gives a value of.5833 which lies within their recommended optimum range. Therefore, this model tube and silo is considered a good geometric arrangement. As mentioned previously in this paper, the second and third parameters have no meaning for a tube with zero perforations and have not been considered. As can be seen from equation 4.36 the friction force on the tube is directly related to the stress ratio within the material at the tube wall, the material density and the tube radius. Substituting for =6kN/m 3, t =9, s =,r=4mm, R=96mm, h=.6m and using various values of the stress ratio, the variation of the friction force has been shown in figure 4.. The stress ratios used in figure 4. have been discussed on the following page. From this figure it can be seen that the correct value of stress ratio must be used to determine the friction force. Friction Force F F Length of tube (m) : Active Stress ratio : At Rest Stress ratio 3: Minimum given in AS : Blight s Upper limit Experimental results Figure 4.: Graph of Friction force for various stress ratios

164 4.6 DES NORTJE: PhD THESIS: THE ANTI-DYNAMIC TUBE IN MASS FLOW SILOS The active stress ratio is given by: ( Sin ) /( Sin ) =.7, for the tube K a and silo wall. Using this value in equation 4.36 gives the first curve in figure 4.. The stress ratio for the at rest condition is given by ( Sin ) =.99 for the tube and silo wall, which gives the second curve in figure 4. K o The stress ratio given in the Australian Code [] is : Sin Sin Cos K (4.37) 4 Cos where is the coefficient of wall friction and is the material friction angle For the silo and tube model this gives: K t =.768 and K s =.79. The minimum value recommended by AS 3774 is.35, which is given in figure 4. by the third curve. The fourth curve of the friction force in figure 4. has been calculated using the upper limit recommendation of the stress ratio for granular materials from Blight (993). This upper limit is given by: K = +Sin =.77 Substituting the values for the model into equation 4.36 and using the Blight upper limit stress ratio gives a total friction force of 93. kg. Substituting a stress ratio of.77 into the expression given by Ooms and Roberts, equation 4.7, the total friction force acting on the tube is: FF.77 Tan9.4 z e F F.894 kn kg In the above expression, the values of the wall friction and stress ratio inside the tube have been taken equal to the outside of the tube, and the v has been assumed zero. In both equations 4.9 and 4.36 for the friction force, the variables G and B have the same numerical value because of the assumption that the internal tube wall has the same surface as the external tube wall VARIABLE VERTICAL PRESSURE ACROSS A SLICE From the literature survey of the pressure measurements in the silo, the vertical pressure in the centre of the silo is approximately 5% to 5% greater than the average vertical pressure. In this thesis an average value of 3% is recommended. Suzuki et al (985) give an approximation of the vertical pressure distribution

165 THE ANTI-DYNAMIC TUBE: MATHEMATICAL MODEL 4.7 across a horizontal element in the silo. This distribution has been given in chapter 3... and has been shown schematically in figure 4.3 Silo center line.6 This pressure distribution has been adapted to a silo fitted with an anti-dynamic tube. Since the vertical pressure near the silo wall has been shown to be less than average, this has been assumed at the wall of the tube. Similarly, since the vertical pressure at the centre is greater than average, it has been assumed in this thesis that the vertical pressure at the midspan between the tube and silo wall is greater than average. This results in the pressure distribution shown in figure /3 R 3/4 R Silo wall Figure 4.3: Vertical Pressure distribution according to Suzuki et al. Average vertical stress Silo center line Tube wall. to.3 Silo wall F F F 3 Average vertical stress.9 to.7 /4 / 3/4.9 to.7 Figure 4.4: Assumed vertical pressure distribution in a silo fitted with an anti-dynamic tube The shape was chosen such that the area of the curve below the average pressure line equals the area above the average pressure line. Two distributions were considered where the maximum and the minimum varied from % above and below the average line to 3% above and below the average pressure line. The equations for the two distributions are : v.cosx for the % variation (4.38)

166 4.8 DES NORTJE: PhD THESIS: THE ANTI-DYNAMIC TUBE IN MASS FLOW SILOS and v.3 Cosx for the 3% variation (4.39) Thus the element has been divided into four segments as shown in figure 4.5. The vertical forces acting over the element segments, F, F and F 3, have been shown in figure 4.4. These forces were determined by integrating the expression for the vertical pressure distribution across the element and multiplying by the average length of the segment. R S S 6 S 3 S 5 dz S 4 S r 3 / 4 / / 4 d Figure 4.5: Segments of the elemental slice The lengths of the segments S 4, S 5 and S 6 are given by the following: S 4 = (r+(r-r)/4) d = (3r+R) d /4 S 6 = (r+(r-r)3/4) d = (r+3r) d /4 (4.4a,b) For the distribution of a % variation above and below the average pressure line, the vertical force, F is given by: S S4 F v.v Cos x dx (4.4) where (S +S 4 )/ is the average length of the segment and is given by: (S +S 4 )/ = (7r+R) d /4 (4.4) Integrating equation 4.4 from to.5(r-r) and substituting equation 4.4 in equation 4.4 gives: F v 4. v 7r R 4 7r v d 6.7r v 8 R v 6.R v 8

167 THE ANTI-DYNAMIC TUBE: MATHEMATICAL MODEL 4.9 which can be simplified to give: 7r v R v. F d (4.43) 8 8 Similarly the expression for the vertical force acting over the middle segment is given by: F S v.v Cos x dx 5 (4.44) and the length of the segment S 5 is given by S 5 = (r+r) d / (4.45) Integrating equation 4.44 and substituting equation 4.45 gives the expression for F, after simplification, as follows: r v R v. F d (4.46) And the expression for the vertical force, F 3, acting over the third segment of the element is given by: S S F3 (4.47) 6 v.v Cos x dx where (S +S 6 )/ is the average length of the segment and is given by: (S +S 6 )/ = (r+7r) d /4 (4.48) Again integrating equation 4.47 and substituting equation 4.48 into 4.47, after simplification, gives the following expression for the force F 3 : r v 7 R v. F3 d (4.49) 8 8 Therefore, there are three forces acting upwards on the element from the material below. These three forces are: ( F / z)dz, ( F / z)dz, and ( F / z)dz (4.5a,b,c) These three forces have an expression which can be found in a similar manner to the definition of the vertical forces acting downwards. The final expression for the vertical forces acting upwards on the element are: 7r R v. F v dz d (4.5) 8 8 z

168 4.3 DES NORTJE: PhD THESIS: THE ANTI-DYNAMIC TUBE IN MASS FLOW SILOS d. dz z R r F v v (4.5) d. dz z 8 7 R 8 r F v v 3 (4.53) The terms containing σ v in the expressions of the vertical forces acting downwards cancel out with the σ v terms in the expressions of the vertical forces acting upwards on the element. Therefore, only terms containing ( σ v / z)dz will appear in the expression for vertical equilibrium of the forces acting on the element, which is as follows: d dz z 6 R 7r dz r d Tan dz R d Tan dz d r R v t H s H d dz z R r. dz d z 4 R r dz d z 8.R.7r v v v dz d z 8.7 R.r dz d z 6 7 R r v v (4.54) Dividing equation 4.54 throughout by dzdθ and cancelling terms gives the following expression: v t t s s v r R r K R K z R r. 3 (4.55) which can be written as: J C z v v (4.56) where R r. 3 r K R K C t t s s (4.57) and. 3 r R R r. 3 r R J (4.58) The expression for C in equation 4.57 differs from the expression for G given in equation 4.7 by the term: π/((3π/-.)(r-r)), which is inversely proportional to the difference between the silo radius and the tube radius.

169 THE ANTI-DYNAMIC TUBE: MATHEMATICAL MODEL 4.3 The expression for J in equation 4.58 differs from the right hand term in equation 4.6 by the term: π(r-r)/(3π/-.), which is linearly related to the difference between the silo and tube radius. Solving equation 4.56 by the same method as given for equation 4.9 gives the solution for σ v as: J Cz v e (4.59) C where J and C have been defined in equations 4.57 and If the silo wall and the tube wall have the same coefficient of friction and assuming K s =K t =K, then the expression for J/C becomes: J C ( R r K (R r) ) (R r) K (4.6) The total drag force on the tube can be determined from: h F r dz (4.6) F t The shear stress on the walls of the tube is determined from: t = h Tan t = K t v t (4.63) Therefore, the friction force is given by: h F F rk t t v dz (4.64) Substituting the expression for v from equation 4.59 into the equation 4.64, and integrating over the height of the tube gives the following expression for the friction force acting on the tube walls: h Ch FF rk t tj e (4.65) C where h is the total height of the tube, K t is the stress ratio at the wall of the tube, t is the coefficient of friction of the tube wall, r is the radius of the tube and C and J are constants defined in equations 4.57 and 4.58.

170 4.3 DES NORTJE: PhD THESIS: THE ANTI-DYNAMIC TUBE IN MASS FLOW SILOS Thus the expression for the friction force assuming a vertical pressure distribution across the element, as given by equation 4.38, has been determined. Substituting in the values for the model, where =6kN/m 3, t =Tan9, s =Tan, and assuming the Blight (993) upper limit on the stress ratio of K=+sin, is applicable gives the total friction force on the tube of 3.43kg. The second expression for the vertical pressure distribution, as given by equation 4.39, can be solved in a similar manner. For this pressure distribution the final expression for the friction force is as given in equation However the constants J and C are as follows: J C R r (4.66) 3 R r 4 Ks sr K t t r (4.67) 3 R r 4 Substituting the model constants for the silo and tube walls into equations 4.66 and 4.67, and solving equation 4.65 gives a total friction force acting on the tube of 37.6kg. These equations give values of the friction force in accordance with the values measured in the laboratory, refer table 4.3.

171 BUCKLING OF THIN CYLINDRICAL SHELLS: CLASSICAL THEORY 5. CHAPTER 5 INTRODUCTION TO CYLINDRICAL THIN SHELL BUCKLING THEORY Shell structures are defined as thin by their radius to thickness ratio, r/t. The definition of small and large in terms of r/t is very ambiguous in most texts. The minimum limit for the radius to thickness ratio is approximately, 5 to whereas the maximum limit is solely dependent on the practical aspects of constructing such a thin shell. As the r/t ratio gets larger, so the walls of the shell deviate from their initial curved shape. Thus there are imperfections in the shape of the shell. These imperfections greatly influence the maximum load carrying capacity of the shell. The early theories dealing with shell analysis assume a perfectly curved wall and therefore the load capacities are much higher than the results obtained from experimental research. These imperfections are most pronounced in the axial load capacity of the shell. The middle surface of an element of the cylindrical shell wall, shown in figure 5., is synonymous to the neutral axis of a beam. The shell element has a radius r, a wall thickness t and sides of length dx and rd. The x, y, and z-axes are as shown in figure 5., where the x-axis is parallel to the length of the shell, the y axis is tangential to the shell wall and the z-axis is positive towards the centre of the shell. The corresponding displacements along the x, y and z axes are u, v, and w respectively. dx z d x rd t/ middle surface dz -t/ Figure 5.: Element of a cylindrical shell wall y

172 5. DES NORTJE: PhD THESIS: THE ANTI-DYNAMIC TUBE IN MASS FLOW SILOS 5. UNIFORM EXTERNAL LATERAL PRESSURE In the consideration of a thin cylindrical shell subjected to a uniform external lateral pressure, the simple equation, as given in appendix F, for a uniformly compressed circular ring can be used as the theoretical buckling load, if the ratio of length to shell diameter is larger than fifty. For shorter cylinders, Timoshenko and Gere (963) use the general equations for the deformation of a cylindrical shell, to determine the critical buckling load due to a uniform external pressure. This allows the effects of end restraints to be taken into account, which introduces local bending for shells with rigid end supports. The analysis of a thin shell requires the determination of the equilibrium of forces acting on a small element of the shell wall. Figure 5. shows a cylinder subjected to a uniform external lateral pressure, q, around it s circumference and along it s length. The internal membrane forces acting on the deformed shell element have been shown in Figure 5.3. q x z y d Figure 5.: External lateral pressure on a Cylindrical shell z N y dx x Qx Q x x dx N yx N xy r N x Nx N x x dx Nxy Nxy x dx Nyx Nyx Q x d y rd Ny Ny d Figure 5.3: Enlarged Deformed Element showing the internal forces

173 BUCKLING OF THIN CYLINDRICAL SHELLS: CLASSICAL THEORY 5.3 The general equations of equilibrium of membrane forces for deformation of a cylinder due to uniform external lateral pressure are given below: In the x-direction: Nx r x N yx N y v x w x (5.) In the y-direction: Ny Nxy r x Qy (5.) In the z-direction: Qx r x Q y N y v r w r qr (5.3) In equations 5., 5. and 5.3, Timoshenko and Gere assumed that the resultant forces, except N y, are small. They neglected all terms containing products of these resultants with the derivatives of the displacements u, v and w. Figure 5.4 shows the internal moments acting on the sides of the deformed element. d M y z dx y M yx x M xy r M xy M x M x M yx rd M y Figure 5.4: Deformed Element showing the internal moments

174 5.4 DES NORTJE: PhD THESIS: THE ANTI-DYNAMIC TUBE IN MASS FLOW SILOS Equilibrium of moments with respect to the x, y, and z axes given in figure 5.4 are as given below. Again Timoshenko and Gere assumed that the bending and twisting moments are small and therefore neglected the products of these moments with the derivatives of the displacements u, v, and w. M xy r x M y rq y Q y M y r M xy x (5.4) M yx M x r x r Q x M xy Qx r M x (5.5) x where Q x and Q y are the shear forces acting perpendicular to the element sides as shown in figure 5.3. Substituting for Q y and Q x from equations 5.4 and 5.5 into equations 5., 5., and 5.3 gives the following: In the x-direction: N x r x N yx v N y x w x (5.6) In the y-direction: N y N xy r x M y r M xy x (5.7) In the z-direction: M x r x M y r v N y r w qr r (5.8) Timoshenko and Gere give the following expressions for N x, N xy and N y : N x t / dz x t / Et u x v r w r (5.9) N xy t / t / dz xy Et u r v x (5.) N y =-qr (5.)

175 BUCKLING OF THIN CYLINDRICAL SHELLS: CLASSICAL THEORY 5.5 Substituting equation 5. and the derivatives of equations 5.9 and 5. and into equation 5.6 for the forces in the axial direction, and multiplying this result throughout by r(- )/Et gives the following: In the x-direction: x w x v r x v r u x w r x u r (5.) where =qr(- )/Et (5.3) The equation for equilibrium in the x-direction, given by Timoshenko and Gere (963) has been repeated below: x w x v r x v r u x w r x u r (5.4) The difference between the two equations (5. & 5.4) lies in the second term in brackets, which comes from the substitution for N xy. Thus there is a printing error in equation 5.4 given in Timoshenko and Gere. Working back from this equation, and integrating this second term in brackets with respect to, and multiplying by Et/r(- ) gives the following substitution for N xy : x v u r Et N xy (5.5) Unless =, which for steel is not true, equation 5.5 does not equal equation 5., and therefore equation 5.4 is incorrect. Only small deflections from the uniformly compressed form of equilibrium have been considered, and therefore N y differs by a small amount from the value of qr: N y = -qr+n y (5.6) where N y is the small change in the value of N y due to the deformation of the shell. Also taking into account stretching of the middle surface during buckling gives the following expressions for N y and the external pressure q: N y =N y (+ ) and q=q(+ )(+ ) (5.7a,b)

176 5.6 DES NORTJE: PhD THESIS: THE ANTI-DYNAMIC TUBE IN MASS FLOW SILOS where the axial and circumferential strains are given by: =u/x and =(v/-w)/r (5.8a,b) The bending moments in the x and y directions are given by: x x w r v r x w D M (5.9) y x w x w r v r D M (5.) The twisting moment is given by: x w v r D M xy (5.) Substituting the derivatives of equations 5., 5.6, 5. and 5. into equation 5.7 for the forces in the y-direction gives the following: In the y-direction:... w v x v r x u r x v r x w r w v (5.) where =t /(r ) (5.3) Similarly substituting the derivatives of equations 5.6, 5.9, and 5. into equation 5.8 for the forces in the z-direction gives the following: In the z-direction:... w x w r x v r v w v x u r w w x w r... (5.4) The displacements in the x, y, and z-directions, for the buckled shape, are given in Timoshenko and Gere(963) by the following equations, respectively : u = Asin(n)sin(x/L), v = Bcos(n)cos(x/L), w = Csin(n)cos(x/L) (5.5a,b,c)

177 BUCKLING OF THIN CYLINDRICAL SHELLS: CLASSICAL THEORY 5.7 where n is the number of half waves around the circumference and A, B and C are arbitrary constants. The derivatives of these displacement equations with respect to x is: u x A L Sin n x Cos L and x u L u (5.6a,b) v x B Cos n L x Sin L and x v L v (5.7a,b) w x C Sin n L x Sin L and w x L w and Similarly the derivatives of equations 5.5a,b,c with respect to is: 4 x w 4 L 4 4 w (5.8a,b,c) u x AnCos(n)Sin L and u n u (5.9a,b) v BnSin n x Cos L and v n v (5.3a,b) w x CnCos(n)Cos L and w n w and 4 w 4 n 4 w (5.3,a,b,c) Substituting the derivatives given in equations 5.6 to 5.3 into equations 5., 5. and 5.4, after considerable manipulation, gives the following three simultaneous equations with three unknown constants A, B, and C, in Timoshenko and Gere as follows: In the x-direction: A n B n n C (5.3) In the y-direction: A n B n n 3 C n n n (5.33) In the z-direction: 4 4 n C n n n 3 B n n A (5.34) where qr and Et t r and r (5.35a,b,c) L

178 5.8 DES NORTJE: PhD THESIS: THE ANTI-DYNAMIC TUBE IN MASS FLOW SILOS These three equations can be written in matrix form as follows: C B A n n n n n n n n n n (5.36) Buckling of the cylinder is only possible if the constants A, B, and C in equations 5.3, 5.33 and 5.34, are not equal to zero. Therefore, the determinant of the above matrix, equation 5.36, must be equal to zero. The equation for the critical lateral pressure can be simplified by further neglecting small terms in the determinant to give the following final expression for the critical lateral pressure acting on the cylinder: cr r L n n r L n n n r t r Et q (5.37) The expression as given by equation 5.37 has been plotted in graphical form in figure 5.5, for a steel shell with a Young s Modulus of x 3 Mpa, a Poisson s ratio of.3 and a length to diameter ratio of 4. A group of curves has been plotted for n= to n=8 and the ratio of thickness to radius was varied from.5 to.5. For a thickness to radius ratio of.35, the shell buckles at a pressure of.66 MPa with 8 half waves around the circumference, as shown by point A in figure5.4. However, a similar shell will buckle at a higher pressure with only 6 half waves, as shown by point B, and with 4 half waves as shown by point C in figure 5.5. Therefore, as the lateral pressure acting on a shell increases, so the number of half waves into which the shell buckles decreases.

179 BUCKLING OF THIN CYLINDRICAL SHELLS: CLASSICAL THEORY n= n=8 n=6 External lateral pressure, q cr (MPa) C B A n= t/r Figure 5.5: Critical External Lateral Pressure acting on a Thin Cylindrical Shell Figure 5.6 shows the pressure on a cylinder as given by equation 5.37 for a steel cylinder with the number of half waves around the circumference kept constant at n=3. A group of curves has been plotted for length to diameter ratios of 5, 6 and 5. As can be seen from figure 5.6, the critical pressure on the shell decreases as the length to diameter ratio increases, for a given thickness to radius ratio. Figure 5.7 shows the critical lateral pressure on the same cylinder, however the number of half waves around the circumference has been increased to 8. As can be seen for any given thickness to radius ratio of the shell, there is very little change in the value of the critical pressure for a cylinder of length 5 times the diameter compared to a cylinder of length 5 times the diameter. Therefore, the statement made by Timoshenko and Gere that the equation for a uniformly compressed circular ring can be used to calculate the critical pressure for a cylinder if the length to diameter ratio is large, is only partially true. Figures 5.6 and 5.7 show that as the number of half waves around the circumference increases, so the critical pressure tends towards the same value, regardless of the cylinder s length.

180 5. DES NORTJE: PhD THESIS: THE ANTI-DYNAMIC TUBE IN MASS FLOW SILOS.4. Critical Lateral Pressure, q cr, (Mpa) Figure 5.6: Critical Lateral pressure for n=3 L=5D L=5D t/r L=6D Critical Lateral Pressure, q cr, (Mpa) L=5D L=5D t/r Figure 5.7: Critical Lateral pressure for n=8 5. SYMMETRICAL BUCKLING DUE TO A UNIFORM AXIAL PRESSURE The stiffness of a shell around it s circumference (the membrane stiffness), is much larger than the bending stiffness of the shell, ie along its axis. Therefore, a great deal of membrane energy can be absorbed without deforming too much. However to absorb the same amount of strain energy in bending, the shell has to deform much more. When the shell is loaded such that most of it s strain energy is in the form of membrane compression, the shell will fail dramatically in buckling if this stored up membrane energy is converted to bending energy. Two types of buckling exist, non-linear collapse and bifurcation buckling. Non-linear collapse is predicted by means of a non-linear stress analysis, while the onset of bifurcation buckling is predicted by means of an eigenvalue analysis. An idealised load-deflection curve has been given in figure 5.8 for an axially compressed cylinder, showing the general shape of the various types of buckling paths. For all paths of buckling, as the load approaches the maximum load, the load-deflection curve has a nearly zero slope. If the load is maintained, snap through buckling occurs resulting in dramatic and instantaneous failure.

181 BUCKLING OF THIN CYLINDRICAL SHELLS: LITERATURE SURVEY 5. P L A Limit load of a perfect shell P C B Bifurcation C P S E D Post-Buckling Curve F Load P Limit of an imperfect shell Deflection Figure 5.8: General P- curve for a non-linear analysis In figure 5.8, the cylinder deforms approximately axi-symmetrically along the line OA, until the limit load P L is reached at point A. At point A, snap through buckling occurs and the load-deflection curve follows the line AC. The path OAC is called fundamental or primary buckling and is associated with axi-symmetric buckling. The bifurcation buckling point, B, lies between O and A, as shown in figure 5.8. At this buckling load, the deformations begin to grow in a pattern which is different from the pre-buckled pattern. Failure of this new deflection mode occurs if the postbifurcation load-deflection curve has a negative slope and the applied load is independent of the deformation amplitude. The post bifurcation buckling path BD is called the secondary or post buckling path and along this path the cylinder buckles non-symmetrically. In both load paths of collapse and bifurcation buckling, the maximum limit occurs at loads for which some or all of the material is stressed beyond its elastic limit point. In the case of real structures which contain imperfections there is no such thing as true bifurcation buckling. The loading path of a real shell follows the curve OEF, with the failure corresponding to snap through at point E. The collapse load,p s, at point E involves significant non-symmetric displacements.

182 5. DES NORTJE: PhD THESIS: THE ANTI-DYNAMIC TUBE IN MASS FLOW SILOS Although true bifurcation buckling is fictitious, a bifurcation buckling analysis is valid as it gives a good approximation of the actual failure load and mode. The difference between the bifurcation load P C of the perfect shell, and the actual collapse load P S of the imperfect shell, depends on the magnitude of the initial imperfections. A plot of P C /P S versus deflection characterises the sensitivity of the maximum load, P S, to initial geometric imperfections. This corresponds to the term of an imperfection sensitive shell. Figure 5.9 shows a plot of test data from Brush and Almroth of the normalised buckling stress, Pc/Ps, for various r/t ratios of cylindrical shells. The curve shows that the greater the r/t ratio (ie smaller wall thicknesses) the lower is the Pc/Ps ratio. Therefore, shells with very thin walls are highly sensitive to initial imperfections in the walls. Normalised buckling stress..8.6 Theory A design recommendation Practical range Radius/Thickness Figure 5.9: Test data for cylinders subjected to axial compression. Taken from Brush and Almroth

183 BUCKLING OF THIN CYLINDRICAL SHELLS: LITERATURE SURVEY BUCKLING DUE TO A UNIFORM AXIAL PRESSURE ( SPECIAL CASE ) N cr /unit circumference x z Figure 5.: Axial buckling of a thin cylindrical shell y Timoshenko and Gere (963) give a theoretical method for determining the critical axial buckling load acting on a thin shell by considering the potential energy of the shell during deformation. The shell in figure 5. shows a typical thin shell subjected to a uniform axial pressure, N cr per unit circumference length, which Timoshenko and Gere assume remains constant during buckling. The shell has a radius r, a wall thickness t, and a length of L. They define the middle surface of the shell as the surface which is located in the centre of the wall thickness. The associated displacements for the axes system, x,y,z, shown in figure 5. is u,v and w, respectively. In this theory they have used energy methods to determine the critical axial force by equating the strain energy to the work done by the external forces. The strain energy of the shell during buckling consists of the strain due to axial compression as well as strain of the middle surface in the circumferential direction and bending of the middle surface. The strain energy term is made up of bending energy and the strain energy due to stretching of the middle surface. U = U C +U B (5.38) U C is the circumferential strain energy and is given by: L E h U dy dx (5.39) C a c a c ( ) where a is the axial strain and c is the circumferential strain. The circumferential strain energy of the middle surface has been given in Timoshenko and Gere, using the following expressions for the axial and circumferential strains. These strains have been shown schematically in figure 5..

184 5.4 DES NORTJE: PhD THESIS: THE ANTI-DYNAMIC TUBE IN MASS FLOW SILOS N cr o is the axial strain before buckling and is given by: o = N cr /Et (5.4) o a a is the axial strain and is given by: a = o -w/r (5.4) c c is the circumferential strain and is given by c =w/r - o (5.4) Figure 5.: Schematic diagram of strains due to axial pressure The function for the axial displacements has been given by Timoshenko and Gere as: w= A Sin(mx/L) (5.43) Substituting equation 5.4, 5.4, 5.4 and 5.43 into equation 5.39 gives the following expression for the change in circumferential strain energy: U c L Et Et L A Sin m x o r L o A Sin mx A Sin mx r d dx o r L A Sin mx r d dx r L o r L (5.44) In equation 5.38, U B is the bending energy term and is given in Timoshenko and Gere as: L dy dx U D B x (5.45) where D is the flexural rigidity, (the derivation of D is given in Appendix F): 3 E t D (5.46) In the expression for the change in bending strain energy given by equation 5.45 x is the curvature of the shell along its length and is given by: x = w/x (5.47)

185 BUCKLING OF THIN CYLINDRICAL SHELLS: LITERATURE SURVEY 5.5 Substituting the second derivative of the displacement function given by equation 5.43 into equation 5.47, gives the curvature of the shell along it s axis as: x = A(mx/L) Sin(mx/L) (5.48) Substituting equation 5.48 into 5.45 gives the following expression for the change in bending strain energy: 4 l D m mx UB A r Sin dx (5.49) l l Substituting equation 5.49 and 546 into equation 5.38 gives the final expression for the increase in strain energy in the shell during buckling by Timoshenko and Gere as: L 4 4 mx A EtL A m U te A Sin dx r L D (5.5) o L r 4 L The work done by the compressive forces, W, is equal to force times distance. The distance through which the axial force travels is given by: X = ( a - )+L (5.5) where L is the shortening due to bending effects. In general terms L is determined as given in figure 5., which shows the wall of the shell buckling in the vertical plane. The distance through which the axial force travels is given by L=L -L, where L is the original length, L is the length after buckling, z is the radial axis, and dz is the change in radius due to buckling. dx L dz By small angles : dz/dx L Shell wall L L L dz Cos dx dx L L z dx... dx (5.5) Figure 5.: Shell wall during buckling L L L L z dx

186 5.6 DES NORTJE: PhD THESIS: THE ANTI-DYNAMIC TUBE IN MASS FLOW SILOS Therefore, the work done is given by: W = N cr ( a - c ) + N cr (L) (5.53) The radial displacements in the z-direction are given in equation 5.4. Substituting equation 5.4, 5.4 and 5.4 into 5.53, and multiplying around the circumference of the shell gives: L L A mx Am mx W r N Sin dx Cos dx cr (5.54) r L L L Timoshenko and Gere equate the strain energy in the shell to the work done by the axial compressive force, to give the following expression for the critical axial stress acting on the shell: N cr B E D B E cr D (5.55) ts t D r B t r B where B= m/l, S is the shell circumference and m is the number of half waves along the length of the shell. The shell in figure 5. has been shown with 5 half waves along it s length. Since D, E, S, r and t are all constants, the minimum axial force required to cause the shell to buckle is given by: d cr ts N cr min db EtS BDS 3 r B (5.56) The expression for the critical load given by equation 5.55, has been plotted in figure 5.3 for a steel shell with a Youngs modulus of x 3 MPa, a Poisons ratio of.3 and a wall thickness of.8mm. The graphs in figure 5.3 have been plotted as a function of the number of half waves along the length of the shell. Four curves have been shown for a cylinder with a length to diameter (r) ratio varying from to 8, in increments of. Figure 5.3 shows that the minimum axial load required to cause buckling remains constant, which in this example is kN. For a constant shell length at this minimum buckling load, the shell diameter increases as the number of half waves along the shell decreases, as shown by points A and B in figure 5.3. Alternatively, for the same number of half waves along the length of the shell, the axial load increases as the diameter of the shell decreases, as shown by points C and D.

187 BUCKLING OF THIN CYLINDRICAL SHELLS: LITERATURE SURVEY L/r=4 L/r= 7 D C N cr (kn) 5 45 Min = kn A Length/number of half waves, L/m (mm) Figure 5.3: Variation of the Critical Axial Load as a function of the Cylinder Length to Number of Half Waves ratio B Since the number of lobes around the circumference of the shell is zero, this analysis given by Timoshenko and Gere, represents a special case of buckling of a thin cylindrical shell. Figure 5.4 graphically demonstrates this special case of a cylinder buckling due to an axial pressure, as given by equation 5.56 with the number of longitudinal half waves increasing from m= to m=36. m= m=3 m=9 m=36 Figure 5.4: Special case of axial buckling with number of half waves increasing from to 36.

188 5.8 DES NORTJE: PhD THESIS: THE ANTI-DYNAMIC TUBE IN MASS FLOW SILOS From equation 5.56 the length of waves into which the shell buckles can be determined as follows: d cr ts db BD t E r d db B BD t E 3 r B (5.57) Therefore, Et m (5.58) r D L B 4 Solving for L/m gives: L m r t 4 (5.59) For steel =.3, therefore equation 5.59 can be approximated by: L / m.7 r t (5.6) Equation 5.6 can be re-arranged to give the number of half waves, m, into which a shell of known length, radius and thickness will buckle, as follows: m = L / (.7 rt) (5.6) The number of half waves as a function of the t/r ratio as given by equation 5.6 has been plotted in figure 5.5 for shell lengths varying from r to 3r. The graph shows that as the length of shell increases so does the number of half waves into which it will buckle. It can also be seen that thicker shells buckle into fewer half waves than do shells with a smaller t/r ratio, ie thinner shells. 9 Number of half waves, m Shell thickness to radius ratio, t/r L=3r L=r L=r Figure 5.5: Number of half waves as a function of t/r

189 BUCKLING OF THIN CYLINDRICAL SHELLS: LITERATURE SURVEY GENERAL CASE OF BUCKLING DUE TO AN AXIAL PRESSURE The analysis given in section 5.. gives the result for the special case of buckling when the deformed shape only has half waves along it s longitudinal axis. There are cases when the cylinders will buckle with m half waves in the longitudinal direction as well as n half waves around it s circumference. Timoshenko and Gere have approached the general case of axial buckling in the same way as the analysis for buckling due to a uniform external lateral pressure. Referring to figure 5.3 given in section 5., the resultant internal forces due to an axial pressure give the following three equations of equilibrium in the x, y and z- directions. Assuming that all the forces except Nx are small, and neglecting the products of these forces with the derivatives of the displacements u,v and w: In the x-direction: N x r x N yx (5.6) In the y-direction: N y N xy r x rn x v x Q y (5.63) In the z-direction: Q x r x Q y rn x w x N y (5.64) Similarly, referring to figure 5.4, neglecting products of moments with the derivatives of the displacements, the equations of moment equilibrium are the same as given in section 5.. These equations have been repeated below: In the x-direction M xy r x M y rq y Q y M y r M xy x (5.4 repeated) In the y-direction M yx M x r x r Q x M xy Qx r M x x (5.5 repeated) In the z-direction: M yx r(n N ) M (5.65) xy yx yx

190 5. DES NORTJE: PhD THESIS: THE ANTI-DYNAMIC TUBE IN MASS FLOW SILOS Substituting for Q x and Q y from equations 5.5 and 5.4, respectively, into equations 5.63 and 5.64 gives the following: Equilibrium of forces in the x-direction remains unchanged: N x N r yx x (5.57 repeated) In the y-direction: x M M r x v rn x N r N xy y x xy y (5.66) In the z-direction: N x w rn M r x M r y x y x (5.67) The definitions of N x, N xy, M x, M y and M xy are as given by equations 5.9, 5., 5.9, 5. and 5. respectively. Substituting for N x and N xy into equation 5.57 gives: In the x-direction: u r x w r x v r x u (5.68) Timoshenko and Gere give the following definition for N y : x u r w v r Et dz N / t / t y y (5.69) Substituting for N y from equation 5.69, and for M y and M xy into equation 5.66 gives the following: In the y-direction:... w v x v r x u r x v r x v r x w r w v (5.7)

191 BUCKLING OF THIN CYLINDRICAL SHELLS: LITERATURE SURVEY 5. Note that equation 5.7 differs from equation 5. only by the last term: ie equation 5.7 has the extra term of r v/x. Finally substituting for M x, M y, N x and N y into equation 5.68 gives the following equation for the equilibrium of forces: In the z-direction: x w r w r x w r x v r v r (5.7) The expressions for the displacements have been given in equations 5.5a,b,c in section 5.. Substituting the derivatives of these displacements into the above three equations, gives the following three simultaneous equations, in matrix form: C B A n n n n n n.5 n n n (5.7) where, as before, =t /(r ) and = mr/l The solution to equation 5.7 can be found by setting the determinant equal to zero and solving for. Timoshenko and Gere have ignored small quantities of higher order containing terms in and. The solution for then becomes: G F Et N x (5.73) where F and G are as follows:... n 3 n F n n 3 n 7 n... (5.74).... n n n G n... (5.75)

192 5. DES NORTJE: PhD THESIS: THE ANTI-DYNAMIC TUBE IN MASS FLOW SILOS Both the numerator and denominator to equation 5.73 are unitless. Therefore, the total axial force carried by a shell is given by: N r Et F (5.76) x G TOTAL Timoshenko and Gere state that since is a large number, equation 5.76 can be approximated as follows: 3 n N Et (5.77) x r TOTAL n Both equations 5.76 and equation 5.77 for the total critical axial load have been plotted in figure 5.6, as a function of the longitudinal half waves, m. These curves were calculated for a shell with a Young s Modulus of x 3 MPa, a Poisson s ratio of.3, a radius of 3mm, a thickness of.6mm, (t/r=.), a length of 3 times the radius (L=9mm) and the number of half waves around the circumference n=4, ie n= kn Total Axial Load Capacity, N xtot (kn) Equation kn Equation Number of Longitudinal half waves, m. Figure 5.6: Total axial load as a function of the number of longitudinal half waves.

193 BUCKLING OF THIN CYLINDRICAL SHELLS: LITERATURE SURVEY 5.3 From figure 5.5 it can be seen that the approximate equation 5.77 predicts a higher axial load capacity than given by the full solution, equation 5.76 for m=9. The difference for this particular cylinder is.5% (=( )/535). From figure 5.5 it was shown that shells with smaller thickness to radius ratios buckle into a larger number of half waves than do shells with larger ratios. Furthermore as the length of the shell increases, so does the number of longitudinal half waves. For the shell under discussion the approximate number of half waves into which the shell will buckle, at the minimum axial load capacity has been calculated from equation 5.6 and found to be 39 waves. Thus the area of the graph in figure 5.6 applicable to this shell lies beyond the graph where the two equations coincide. As the number of half waves around the circumference increases, so the two curves approach each other. The case of n= results in the largest difference between the approximate curve and the exact curve. For n=3 this difference is reduces to 9.7% and for n=4 the difference is only 5%. Therefore, in figure 5.6, it is not possible to be working within the range of fewer longitudinal half waves, m, where there is a noticeable difference between the two curves. Therefore, the approximation made by Timoshenko and Gere is reasonable. 5.3 COMBINED AXIAL AND UNIFORM EXTERNAL LATERAL PRESSURE The problem of a thin cylindrical shell subjected to a combination of an axial and uniform external lateral pressure is approached in the same way as before. Equilibrium equations for the internal forces and moments acting in the x, y and z- directions due to the applied loads are first assembled. It is to be expected that these equations are a combination of the equilibrium equations for the case of lateral pressure and the case of axial loads. They can be written as follows: In the x-direction (this is the same as the equation for lateral pressure only): Nx r x N yx N y v x w x (5. repeated)

194 5.4 DES NORTJE: PhD THESIS: THE ANTI-DYNAMIC TUBE IN MASS FLOW SILOS In the y-direction (this is the same as for axial pressure only) Q x v rn x N r N y x xy y (5.63 repeated) In the z-direction (this is a combination of both cases) qr x w rn r w r v N Q x Q r x y y x (5.78) Similarly the equations of internal moment equilibrium are as follows: In the x-direction (this is the same as the case of lateral pressure): x M M r Q rq M x M r xy y y y y xy (5.4 repeated) In the y-direction (this is the same as the case of lateral pressure): x M M r Qx r Q x M r M x xy x x yx (5.5 repeated) Substituting equations 5.4 and 5.5 into equations 5., 5.63 and 5.78 gives the following: In the x-direction: x w x v N N x N r y yx x (5.6 repeated) In the y-direction: x M M r x v rn x N r N xy y x xy y (5.66 repeated) In the z-direction: qr x w rn w r v r N M r x M r x y y x (5.77)

195 BUCKLING OF THIN CYLINDRICAL SHELLS: LITERATURE SURVEY 5.5 The definitions for N x, N y, N xy, M x, M y, and M xy have been given previously in equations 5.9, 5.69, 5., 5.9, 5. and 5. respectively. Substituting these equations into equations 5.6, 5.66, and 5.77 gives the following: In the x-direction: x w x v r x v r u x w r x u r q (5. repeated) In the y-direction:... w v x v r x u r x v r x v r x w r w v... n (5.7 repeated) In the z-direction:... w x w r x v r v w v x u r n q 4 x w r w w x w r... (5.78) where q is due to the external lateral pressure and is given by: q =qr(-v )/(Et) (5.79) and n is due to the axial pressure and is given by: n =N x (-v )/(Et) (5.8) and is as before: =t /r Using the definitions for the displacements as follows: u=a Sin(n) Cos(mx/L) v=b Cos(n) Sin(mx/L) (5.8a,b,c) w=c Sin(n) Sin(mx/L)

196 5.6 DES NORTJE: PhD THESIS: THE ANTI-DYNAMIC TUBE IN MASS FLOW SILOS Substituting the derivatives of the displacements given in equations 5.8,a,b,c in equations 5., 5.7 and 5.78 gives the matrix given in equation 5.8. Note that in the x-direction the term Sin(n)Cos(mx/L) cancels out; in the y- direction the term Cos(n)Sin(mx/L) cancels out and in the z-direction the term Sin(n)Sin(mx/L) cancels out. C B A M and =mx/l as before (5.8) where [M]= n q n q q r n r n r r n n r r r n n n.5 r rn r rn n r (5.83) Furthermore, Timoshenko and Gere give the solution to the matrix in equation 5.8 by setting the determinant equal to zero. This results in the following solution: H + R = K q + P n (5.84) where: H=(- ) 4 (5.85) R=( +n ) 4 - ( n + (4-) n 4 + n 6 ) + (-) n + n 4 (5.86) K = n ( +n ) (3 n +n 4 ) (5.87) P = ( +n ) + n (5.88)

197 BUCKLING OF THIN CYLINDRICAL SHELLS: LITERATURE SURVEY 5.7 Equation 5.84 shows that the lateral pressure and the axial pressure are linearly related. This equation has been plotted in figure 5.7 for a cylinder with a Young s modulus of x3mpa, a Poisson s ratio of.3, a wall thickness of.6mm, a radius of 5mm (t/r=.4), a length of 75mm (=5r) and 4 half waves around the circumference ie n=. Three curves have been plotted for the number of longitudinal half waves, increasing from m=5 to m=35 in increments of. Figure 5.7 shows that the axial load capacity for a cylinder buckling into 35 longitudinal waves is less than the load capacity for the cylinder to buckle in 5 longitudinal waves. Furthermore from the difference in the slope of the three lines in figure 5.7, it can be seen that the external lateral pressure has a greater influence on the axial pressure for a lower value of m than a higher value. 7 Total Applied Axial Load, N x (kn) m=5 m=35 m= External Lateral Pressure, q cr (MPa) Figure 5.7: Total Axial Load, (N x ), as a function of External Lateral Pressure, (q cr ). Figure 5.8 shows the interaction curve, as given by equation 5.84, for a thin cylindrical steel shell with a length to radius ratio of 4 and a thickness to radius ratio of.5. The number of half waves along the length and around the

198 5.8 DES NORTJE: PhD THESIS: THE ANTI-DYNAMIC TUBE IN MASS FLOW SILOS circumference have been varied to give the lowest possible failure combination of the axial and lateral pressures. 8 m=7,n=4 m=3,n=3 6 m=3,n=4 Total axial load (kn) 4 m=3,n=5 m=3,n= Lateral Compressive Pressure (MPa) Figure 5.8: Interaction Graph for a Thin Shell with L=4r.

199 CYLINDRICAL THIN SHELL EXPERIMENTS: LITERATURE SURVEY 6. CHAPTER SIX CYLINDRICAL THIN SHELL EXPERIMENTS 6. LITERATURE SURVEY 6.. TENNYSON Tennyson(968) conducted tests on thin cylindrical shells with cut outs. The shells were made from photo-elastic plastic which were spun cast by pouring liquid epoxy plastic into a cylindrical mould and rotated at high angular speed until the epoxy plastic had cured. Tennyson conducted tests on 6 cylinders, each with only one circular cut out located at mid height of the cylinder. The shells were fitted with end r r c Figure 6.: Diagram of thin shell with cutout. plates to provide clamped constraints. A diagram drawn to scale of a shell with the largest cut out has been given in figure 6.. From a photograph of the shells given in Tennyson, the shells appear to have a height to diameter ratio of. As can be seen, the holes in Tennyson s tests were small compared to the radius of the cylinder. The shells were tested axially till elastic buckling occurred. Each cylinder was loaded axially before the holes were cut out, to determine the reference buckling load. The results from Tennyson s tests have been given in table 6. along with the dimensions of the cylinders and holes. The radius of the cylinders is r, while the radius of the cut out is r c. In Table 6., the reference failure load for cylinders without holes is N, and the failure load for cylinders with holes is N x. The ratio of shell thickness to radius (t/r) has been given in the third column of the table, with the inverse of this ratio, ie r/t, in brackets. The last column of the table gives the ratio of the buckling load of the cylinders with cut outs to that of the reference buckling load for no cut outs. A graph of the results from Tennyson has been given in figure 6. for the ratio of N x /N o to the ratio of hole radius to cylinder radius. As Tennyson points out there is a unique correlation between the test data which appears to be independent of the cylinder thickness to radius ratio, t/r.

200 6. DES NORTJE: PhD THESIS: THE ANTI-DYNAMIC TUBE IN MASS FLOW SILOS r (mm) t (mm) t/r (r/t) r c (mm) r c /r N x (kn) N x /N (33).34 (9).67 (6) Table 6.: Test data from tests by Tennyson (968) As can be seen from the graph the larger the cut out the lower is the buckling load of the cylinder. Tennyson gives a trendline through the data which gives an apparent minimum for the ratio of N x /N o. This does not seem to be a reasonable conclusion as it is doubtful whether a hole to shell radius of.5 gives the same buckling load as a ratio of., and therefore the trend line has been omitted in figure N x /N o r c /r Figure 6.: Results from Tennyson(968)

201 CYLINDRICAL THIN SHELL EXPERIMENTS: LITERATURE SURVEY 6.3 Tennyson states that the reason for the drastic reduction of buckling load for a cylinder with a cut out is due to the growing imperfection in the shape of the cylinder in the region of the cut out. 6.. ALMROTH AND HOLMES Almroth and Holmes (97) conducted tests on thin walled aluminium cylinders, four with unreinforced cutouts and five with reinforced cutouts. In all cases two rectangular cutouts were made in the cylinder at mid height and 8 apart on the circumference. Before the rectangular holes were cut, the cylinders were subjected to axial pressure to determine a reference buckling load N o. Table 6. gives the test data from Almroth and Holmes, converted to SI units. The length of the rectangular cutouts has been given in Almroth and Holmes as a 3 or 45 arc in plan, as shown in figure 6.3 drawn to scale. The segment length of the arc, S, has been calculated and added to table 6.. Their cylinders were 9 inches high with an outer diameter of.75 inches, and a varying wall thickness. The average radius of each cylinder, r av, has been given to the middle surface of the cylinder. S r Cut outs r S Plan Elevation Figure 6.3: Diagram of cylinders with cut outs tested by Almroth and Holmes The first four rows of data in table 6. are for cylinders with unreinforced cut outs while the remaining seven rows of data are for cylinderrs with reinforced cut outs. From table 6. it can be seen that the cut outs were all of approximately the same dimensions, with only one being smaller than the rest. Consequently little information can be gained from plotting their results as a function of the hole dimensions. However, the wall thicknesses of the cylinders tested do vary, and therefore the ratio of the buckling load to the reference buckling load, N x /N o, has

202 6.4 DES NORTJE: PhD THESIS: THE ANTI-DYNAMIC TUBE IN MASS FLOW SILOS been plotted against the ratio of the wall thickness to the radius, t/r, for each cylinder. The graph of their test results has been shown in figure 6.4. The data points marked with a dot are for unreinforced cut outs, while the data points marked with a star are for reinforced cut outs. r av (mm) t (mm) A (mm ) S (mm) A c (mm ) t/r N x (kn) N x /N o Table 6.: Test data from Almroth and Holmes (97) Ratio of buckling load to reference buckling load, N x /N o Wall thickness to cylinder radius,t/r. Reinforced cut outs Unreinforced cut outs Figure 6.4: Test data from Almroth and Holmes (97) The data from Almroth and Holmes shows that cut outs which are reinforced around the edges fail at approximately 68% to 88% of the reference buckling load, whereas unreinforced cut outs fail at approximately 48% to 68% of the reference buckling load.

203 CYLINDRICAL THIN SHELL EXPERIMENTS: LITERATURE SURVEY 6.5 In some instances, reinforcing the cut out edges can almost double the axial load capacity of a cylinder with perforations. From figure 6.4 it can be seen that the buckling load of cylinders with cut outs, reinforced or unreinforced, depends on the ratio of wall thickness to cylinder radius, which is in contrast with the test results from Tennyson. One possible explanation is the difference in the shape of the cut outs and the different materials of the cylinders. Tennyson used plastic cylinders whereas Almrtoh and Holmes used aluminium cylinders STARNES JR Starnes Jr (97) conducted tests on thin walled cylindrical shells made of Mylar polyester film and made of copper, with a single circular hole at mid height of the cylinder. Mylar was chosen by this researcher for it s ability to buckle elastically as well as not being sensitive to handling, thereby allowing the same shell to be tested several times with an increasing hole size. These shells were made from rectangular sheets of Mylar which were lap jointed with a flexible adhesive. Starnes Jr does not state in the report where the location of the hole was relative to the lap joint, and it is assumed in this thesis that the hole was placed diametrically opposite to the joint. The copper shells were manufactured by an electroforming process. A fly cutter was used to cut the hole in the shell and subsequent surface measurements showed no apparent bending around the hole due to cutting. The Mylar shells were 3.mm in diameter and 54mm long, while the copper cylinders had a length and diameter of 3.mm. The results from Starnes Jr tests on copper shells have been given in table 6.3 and the results from the Mylar shells have been given in table 6.4 in SI units. Starnes Jr does not give the value of N x and N o for the copper shells however, the ratios of the stress S x due to the applied load to the reference buckling stress S o have been given. r c (mm) r (mm) r c /r S x /S o Table 6.3: Test data for copper shells from Starnes Jr (97)

204 6.6 DES NORTJE: PhD THESIS: THE ANTI-DYNAMIC TUBE IN MASS FLOW SILOS r c (mm) r c /r N x (kn) N x /N o Table 6.4: Test data for Mylar shells from Starnes Jr (97) A graph of the test results on the copper shells have been given in figure 6.5 with the vertical axis in terms of S x /S o plotted as a function of the ratio of the cut out radius to the cylinder radius. The results from the Mylar shells have been given in figure 6.6 with the vertical axis in terms of N x /N o also plotted as a function of the ratio of the cut out radius to the cylinder radius, r c /r. As can be seen from figure 6.5 and 6.6, the results lie on a similar shaped curve to the results from Tennyson. Ratio of buckling stress to reference buckling stress, N x /N o Ratio of cut out radius to cylinder radius, r c /r Figure 6.5: Test results for copper shells from Starnes Jr (97)

205 CYLINDRICAL THIN SHELL EXPERIMENTS: LITERATURE SURVEY 6.7 Ratio of buckling load to reference buckling load, N x /N o Ratio of cut out radius to cylinder radius, r c /r Figure 6.5: Test results for Mylar shells from Starnes Jr (97) 6..4 SCUTELLA L. Scutella (998) conducted eighteen tests on thin walled cylindrical steel shells with multiple perforations around the circumference and along the length of the shell. The cylinders had a wall thickness of.8mm, a radius of 5mm and a length of 6mm. The cylinders were made from cold formed steel sheets which were prepunched and then rolled and butt welded to form a cylinder. The cylinders were fitted with end caps which were clamped on, giving constrained edge conditions. The cylinders were then subjected to a uniform axial compression in a displacement controlled test. A diagram of the layout of the holes on the flat sheet prior to rolling has been given in figure 6.7. The ratio of the hole area to the cylinder surface area (here after called cut out area) was varied by increasing the hole sizes from an 8mm diameter hole to a mm diameter hole. The same ratio of cut out area was also achieved by increasing the centre to centre spacing of the holes. The cut out area has been calculated by the ratio of one hole area to the area of one diamond square. These concepts have been diagrammatically shown in figure 6.8 (a) and (b).

206 6.8 DES NORTJE: PhD THESIS: THE ANTI-DYNAMIC TUBE IN MASS FLOW SILOS 45 6mm 94mm Figure 6.7: Hole layout for flat sheet rolled into cylinders Cut out area =5.3% Cut out area =6.5% Cut out area =36.6% Cut out area =65.9% Figure 6.8 (a): Increasing the cut out area by increasing the hole sizes and maintaining the centre to centre spacing of the holes. Cut out diameter =8mm Cut out diameter =3mm Cut out diameter =9mm Figure 6.8 (b): Maintaining the cut out area at 5.3% by increasing the hole sizes and increasing the centre to centre spacing of the holes.

207 CYLINDRICAL THIN SHELL EXPERIMENTS: LITERATURE SURVEY 6.9 Three sets of tests were done on a cylinder with no holes to determine a reference buckling load, N o, which was found to give an average value of 3.46kN. Table 6.5 gives the average results of the data for three tests of each cut out area and hole arrangement, by Scutella. From the data for a cut out area of.5% there appears to be little difference between a hole size of 8mm and 3mm, however when the spacing and hole size was increased to 9mm there appears to be a drop in the buckling capacity of the cylinder by approximately 37%. r c (mm) A c (mm ) A c /A N x (kn) N x /N o Table 6.5: Test data from Scutella (998) A graph of the test results from Suctella has been shown in figure 6.9. As can be seen from figure 6.9, the data points tends to lie on a curve similar to those from Tennyson and Starnes Jr. There is a very rapid decrease in the buckling capacity of a cylinder with multiple perforations as the ratio of cut out area is increased from.5 to approximately.5. Ratio of buckling load to reference buckling load, N x /N o Ratio of cut out area to cylinder surface area, A c /A Figure 6.9: Test results from Scutella (998)

208 6. DES NORTJE: PhD THESIS: THE ANTI-DYNAMIC TUBE IN MASS FLOW SILOS Figure 6. shows the data points for a cylinder with a cut out area of 5% with an increasing hole size and an increasing centre to centre hole spacing. Since the data points appear to give a linear trend, a line has been drawn through these points with the intercept on the y-axis set to which represents the data point for a cylinder with no cut outs. Ratio of buckling load to reference buckling load, Nx/No Centre to centre hole spacing (mm) Figure 6.: Test results from Scutella (998) for a cylinder with a cut out area of.5% From the results by Scutella, given in figure 6.9 and 6., the buckling capacity of a cylinder with multiple perforations appears to be a function of not only the cutout area but also the centre to centre hole spacing. The trendline through the data points in figure 6. has the following equation: N x /N o = -.4 s where s is the centre to centre spacing of the holes. 6. DISCUSSION In both the papers by Tennyson(97) and Starnes Jr (97), reference is made to a parameter which the researchers suggest governs the buckling capacity of the cylinder with cut outs. In Tennyson this parameter is as follows: r 4 c (6.) 8r t

209 CYLINDRICAL THIN SHELL EXPERIMENTS: LITERATURE SURVEY 6. where is Poissons ratio, r c is the cut out radius, r is the cylinder radius and t is the cylinder thickness. Since 8=, equation 6. can be reduced to the following expression: r r c c K K (6.) r t r t where K =.5 [(-v )] /4 (6.3) A similar parameter is found in Starnes Jr as follows: r 4 c (6.4) r t which can be reduced as follows: r c K (6.5) r t where K is as before and all other symbols are as defined previously. The only difference between equation 6. and 6.5 is the value of in the denominator of equation 6.. Multiplying the numerator and denominator in both equations by gives the following: r c K for Tennyson s expression (6.6) r t r c K for Starnes Jr s expression (6.7) r t Both equations 6.6 and 6.7 are expressions in terms of a constant multiplied by the square root of the ratio of the cut out area to the area of cylinder material in plan. However, in equation 6.7 the ratio has only half the cylinder material in the denominator. Both these expression suggest that the buckling capacity is a function of how much material has been removed from the cylinder by the cut out.

210 6. DES NORTJE: PhD THESIS: THE ANTI-DYNAMIC TUBE IN MASS FLOW SILOS If both Tennysons data and Starnes Jr s data are plotted as a function of the ratio of the cut out area to the cylinder surface area ( viz (r c /rl), the results plot as shown in figure 6.. Tennyson(968) Starnes Jr (97) N x /N o. Nx/No =.584(Ac/A) -.85 N x /N o. Nx/No =.574(Ac/A) Ratio of cutout area to cylinder surface area, Ac/A Ratio of cutout area to cylinder surface area, Ac/A Figure 6.: Test results from Tennyson (968) and Starnes Jr (97) plotted as a function of the cut out area to the cylinder surface area. For both sets of data a power series trendline has been plotted through the test results. As can be seen there is a very good agreement between the two sets of test results when using the ratio of cut out area to the cylinder surface area rather than the ratio of cut out area to the cylinder plan area. This similarity between the results may be due to the fact that both Tennyson and Starnes Jr conducted their tests on plastic cylinders which could be manufactured to a very high standard. In a similar manner plotting the test results of Almroth and Holmes (97) and Scutella(998), as a function of the ratio of the cut out area to the cylinder surface area, produces the graphs shown in figure 6.. As can be seen these curves also result in a power series plot, however the agreement between the results is not very good. While both tests were conducted on steel cylinders, the difference between the results may be due to a different degree of wall imperfection in each case.

211 PERFORATED CYLINDRICAL SHELLS: EXPERIMENTAL SET-UP Scutella (998) Nx/No =.3(Ac/A) Almroth and Holmes (97).4 N x /N o. Nx/No =.56(Ac/A) N x /N o Ac/A Ac/A Figure 6.: Test results from Almroth and Holmes(97) and Scutella(998) The buckling capacity of a shell with perforations is therefore a function of the amount of material that has been removed from the walls of the shell, and is not a function of the plan area of the cylinder as initially suggested by Tennyson and Starnes Jr. Furthermore, from figure 6. it can be seen that the reduction in buckling carrying capacity of a cylinder with multiple perforations is also a function of the centre to centre spacing of the cut outs. From the test results given in figure 6. and 6., it is can be seen that the reduction in buckling capacity of a cylinder with perforations, either multiple or single, subjected to a uniform axial pressure plots as a power series given by equation 6.8 as follows: N x /N o = C (A c /A) -B (6.8) where N x is the buckling capacity of a shell with perforations N o is the reference buckling load of a shell without perforations C, and B are empirically determined constants A c is the total area of the cut outs and A is the surface area of the cylinder.

212 6.4 DES NORTJE: PhD THESIS: THE ANTI-DYNAMIC TUBE IN MASS FLOW SILOS 6.3 PERFORATED CYLINDER EXPERIMENTS 6.3. EXPERIMENTAL SET-UP Following on from the experiments done by L Scutella, tests were done in the laboratory on thin shells with multiple perforations subjected to a combined uniform axial and lateral pressure. The shells were formed from thin cold-formed steel sheets with a thickness of.8mm. The holes were pre-punched in the sheets prior to rolling into a cylindrical shell. Figure 6.4 and 6.5 shows photographs of the punching of the cut outs in the sheets and the rolled sheet once punching was complete. The setting out of the holes started from the centre of the sheet to minimise the effects of accumulated measurement errors. The centre to centre hole spacing was kept constant at 57.5mm and the diameters of the cut outs were 5mm and 76mm. This gave a cut out open area of 6.5% and 36.6% respectively. To ensure that no holes were located on the seam, the setting out was such that the centreline of the sheet aligned with the three-quarter line of the hole group. The dimensions of the sheet and hole centre to centre spacing have been drawn to scale in figure 6.6. Figure 6.4 : Photo of cut out punching process Figure 6.5 : Photo of punched sheet rolled into cylindrical shape prior to welding the seam

213 PERFORATED CYLINDRICAL SHELLS: EXPERIMENTAL SET-UP 6.5 Figure 6.6a shows the sheet with the 5mm diameter hole and figure 6.6b shows the sheet with the 76mm diameter holes. These figures give a good visual appreciation of the difference between cut out open area of the two groups of cylinders investigated. 6mm mm 57.5mm 94mm Figure 6.6a:.8mm thick sheet with 6.5% cut out open area (5mm diameter pre-punched holes) Figure 6.6b:.8mm thick sheet with 36.6% cut out open area (76mm diameter pre-punched holes). No holes were located on the shell seam

214 6.6 DES NORTJE: PhD THESIS: THE ANTI-DYNAMIC TUBE IN MASS FLOW SILOS After the holes were punched, the sheet was rolled into a cylindrical shape and the seam was butt welded. To weld the seam, the rolled sheet was clamped into the two end caps which had a diameter of 3mm. A diagram of the end clamps, which were also used during the tests to ensure constraint end conditions of the cylinder, is shown in figure 6.7. Loading sphere circular end caps circular clamps (not shown) hold cylinder in place Spaces between cross beams closed off with plates welded to the beams and cap Figure 6.7: Top View and Side view of cylinder end caps Side view Top view cylinder fitted over end caps 5mmx45mm deep cross beams for carrying axial load The circular clamps and end caps were ridged on the inside to ensure a good grip on the cylinder wall and thereby stop any possibility of slippage. The end caps were closed with a plate welded to the sides of the cross beams and the cap. To apply the axial pressure to the cylinder, one end cap was fitted with a solid half steel ball and notched for seating of the loading rod, as shown in figures 6.8a and b. The bottom end cap was also notched for seating on a small round stainless steel ball. Loading rod connected to Instron Loading ball Top of end cap Figure 6.8a : Loading ball and rod Figure 6.8b: Photo of Loading ball on end cap

215 PERFORATED CYLINDRICAL SHELLS: EXPERIMENTAL SET-UP 6.7 A dummy PVC tube of 5mm diameter (5mm smaller than the perforated cylinders) was placed inside the perforated cylinders. Polystyrene filler plugs were then fixed with double sided tape to the inner dummy tube through the cut outs. A plastic sheath was fitted over the perforated cylinder and clamped to the end caps with the cylinders, to ensure that the whole assembly was water tight. This assembly has been shown in diagrammatically figure 6.9 below. End cap Perforated cylinder Dummy PVC tube Polystyrene filler plugs plastic sheath End cap Figure 6.9: Perforated cylinder assembled with plastic sheath, dummy tube, plugs and end caps To ensure the dummy PVC tube did not carry any of the applied axial load, the top of the tube was notched by 3mm to match the alignment of the cross beams of the end caps. This allowed the end caps on the perforated cylinders to move the required vertical distance of 5mm as well as maintaining the dummy tube in place while lifting the assembly inside the loading bin. This has been shown diagrammatically in figure 6.. End cap and cross beams Cross beams fit inside notches Dummy PVC tube with notched top edge Figure 6.: Notched dummy PVC tube to accommodate end cap.

216 6.8 DES NORTJE: PhD THESIS: THE ANTI-DYNAMIC TUBE IN MASS FLOW SILOS To apply the lateral compressive pressure to the outside of the cylinder, the whole assembly as shown in figure 6.9 was placed inside a large container which had a removable top lid. This container, the loading bin, was fitted with a small stainless steel ball in the centre of its base, thereby ensuring the cylinder assembly could be placed centrally. The loading bin was first placed in the Instron machine and then the cylinder assembly was lifted into place so that the notched bottom end cap fitted the stainless steel ball at the base of the bin. The loading bin was then filled with water up to a short distance from the top, leaving an air gap. The lid of the loading bin was fitted with a rubber O-ring, and bolted to the main body of the container. Through the lid were two valves, to which an air supply and pressure gauge were fitted. Thus the top air gap inside the loading bin was pressurised from the compressed air supply, and the amount of pressure applied was read off the gauge. In this manner a constant lateral pressure was applied to the cylinder assembly during the testing process. A photograph of the loading bin and top valves has been shown in figure 6.. The lid also had an opening with a rubber O- ring through which the loading rod from the Instron machine was placed. Opening for loading rod Valve for compressed air supply Top lid bolted to container Valve for air pressure gauge Construction joint Valve for draining water after completed test Figure 6.: Photograph of container for cylinder assembly

217 PERFORATED CYLINDRICAL SHELLS: EXPERIMENTAL RESULTS EXPERIMENTAL RESULTS Once the perforated cylinder, dummy PVC tube, polystyrene fillers, plastic sheath and end caps had been assembled and placed in the loading bin, the perforated cylinders were subjected to axial displacement controlled tests in the Instron. For each test a constant lateral pressure was applied first and then the displacement controlled test started. For each group of cylinders, ie 5mm diameter cut outs and 76mm diameter cut outs, the lateral pressure was varied from kpa to 4kPa in increments of ten. The data from Scutella(998) has been used as the reference buckling capacity for perforated cylinders with only applied axial pressure. Table 6.6 gives a summary of the results from the combined axial and lateral pressure tests. The results from Scutella(998) have been included in Table 6.6 for the case of zero lateral pressure. The full set of test results have been plotted graphically in Appendix G. for the 6.5% cut out area and in Appendix G. for the 36.6% cut out area. Applied Lateral Pressure (kpa) Maximum Applied Total Axial Load (kn) 5mm Cutouts 76mm Cutouts 3.4 (Scutella) 3.4 (Scutella) Table 6.6: Test results for cylinders with multiple perforations As a reference point, the capacity of the perforated cylinders with a 6.5% and 36.6% cut out area due to a lateral pressure only was also determined. This was done using the assembled system in the loading bin and applying a lateral pressure in increments of 5kPa up to a maximum of 6kPa. Inside the perforated cylinders were placed four displacement transducers around the circumference, which measured the inwards movement of the cylinder walls. The applied lateral pressure was plotted as a function of the inward wall movement as shown in appendix G.3 for the 6.5% and 36.6% cut out areas. Figure shows a photograph of the cylinder with a 6.5% cut out area after failure due to a lateral pressure only.

218 6. DES NORTJE: PhD THESIS: THE ANTI-DYNAMIC TUBE IN MASS FLOW SILOS Figure 6.: Buckled perforated cylinder due to lateral pressure only (6.5% cut out area) The failure load for the cylinders with 6.5% and 36.6% cut out areas were 4kPa and 47kPa, respectively. The results from the perforated cylinder tests have been plotted in figure 6.3 for the perforated cylinders with 6.5% cut out area and in figure 6.4 for the cylinders with 36.6% cut out area Total Axial Load (kn) Applied Lateral Pressure (kpa) Total Axial Load (kn) Applied Lateral Pressure (kpa) Figure 6.3: Test results for combined pressure on perforated cylinders with 6.5% cut out area Figure 6.4: Test results for combined pressure on perforated cylinders with 36.6% cut out area

219 PERFORATED CYLINDRICAL SHELLS: EXPERIMENTAL RESULTS 6. From the theory of thin walled shells subjected to a combined axial and lateral pressure, given in chapter 5.3, it was shown that the relationship between the axial and lateral pressure is linear. Furthermore, from the finite element analyses given in section 6.4 of this chapter, it was shown that the relationship between the axial and lateral pressure for a cylinder with multiple perforations consists of two straight lines as shown in figure 6.3 and 6.4. On closer inspection of the test results for the 6.5% cut out area, the data point for an applied lateral pressure of kpa would appear to be too high. Therefore, ignoring this point, a best fit straight line can be drawn through the remaining data points for an applied lateral pressure up to 4kPa. The equation of this line is given as follows: N x6.5 = q (6.9) where N x6.5 is the total applied axial load in kn acting on a cylinder with cut out area of 6.5%, q is the applied lateral pressure in kpa and 3.4 is the total axial load capacity of a cylinder with 6.5% cut out area with zero lateral pressure. The second best fit line has been drawn through the data point for an applied lateral pressure of 4kPa and the reference pure lateral pressure, 47kPa. This line is given by the following equation: N x6.5 = q (6.) Similarly from the results for the cylinder with 36.6% cut out area, the data point for an applied lateral pressure of 3kPa appears too high and has therefore also been ignored. The best fit straight lines through the remaining points have been shown in figure 6.4. For the data points up to an applied lateral pressure of 4kPa, the equation of the straight line is as follows: N x36.6 = q (6.) where N x36.6 is the total applied axial load in kn acting on a cylinder with cut out area of 36.6%, q is the applied lateral pressure in kpa and 3.4 is the total axial load capacity of a cylinder with 36.6% cut out area with zero lateral pressure. The second straight line through the data point for an applied lateral pressure of 4kPa and the reference buckling pressure of 4kPa for an applied lateral pressure only is as follows:

220 6. DES NORTJE: PhD THESIS: THE ANTI-DYNAMIC TUBE IN MASS FLOW SILOS N x36.6 = q (6.) Using these equations, interaction curves can be drawn for the capacity of a cylinder with multiple perforations subjected to a combination of an applied lateral pressure and total applied axial load as shown in figure 6.5. The areas under the curves represent the load combinations which can safely be sustained by cylinders with a cut out area of 6.5 and 36.6% Total axial load N x, (kn) Applied lateral pressure q, (kpa) Safe combined loads for cut out area of 6.5% Safe combined loads for cut out area of 36.6% Figure 6.5: Interaction graphs for cylinders with multiple perforations (cut out areas of 6.5% and 36.6%) From figure 6.5 it can be seen that the influence of the lateral pressure is to reduce the applied axial load up to a point when the lateral pressure is close to the value of the cylinders lateral failure pressure. At this point the applied axial load is approximately half the value of the axial load for zero lateral pressure acting. Beyond this point the lateral pressure has a stronger influence on the axial load by rapidly reducing it s value to zero. From equation 6.9, the axial load which a cylinder with 6.5% cut out area can carry when subjected to a lateral pressure of 4kPa is as follows: N x6.5 = *4 = 6. kn Similarly for a cylinder with a cut out area of 36.6%, the total axial load which can be carried with a combined lateral pressure of 4kPa is as follows:

221 PERFORATED CYLINDRICAL SHELLS: EXPERIMENTAL RESULTS 6.3 N x36.6 = *4 = 6.6 kn Plotting the ratio of these values to the reference buckling loads (3.4 for the 6.5% cut out area and 3.4 kn for the 36.6% cut out area) as a function of the cut out area of the cylinder, has been shown in figure 6.6. From the work by Tennyson(968), Starnes Jr (97), Almroth and Holmes (97) and Scutella (998) it was shown that the curve for the axial load on cylinders with perforations has the shape of a power series. It has been assumed in this thesis that this same shape curve will be applicable to a cylinder with multiple perforations, which has a combination of an applied axial and lateral pressure. Therefore, plotting a power series trendline through the points in figure 6.6 gives the curve of the total axial load on a cylinder with multiple perforations and a constant lateral pressure of 4kPa as a function of the cut out area. An additional set of curves have also been plotted for a constant lateral pressure of kpa and 5kpa. Ratio of axial load to reference buckling, N x / N o Ratio of cut out area, A c /A Constant lateral pressure: 5kPa kpa 4kPa Figure 6.6: Axial load as a function of cut out area for a constant lateral pressure

222 6.4 DES NORTJE: PhD THESIS: THE ANTI-DYNAMIC TUBE IN MASS FLOW SILOS Table 6.7 gives the data points used to plot the curves in figure 6.6. The equations of the resulting power series trendlines have also been given in table 6.7 for each curve of constant lateral pressure. Lateral Pressure (kpa) 5 4 Cut out area Equation Number Calculated N x (kn) N x /N o Table 6.7 : Data points for curves in figure 6.5 Equation of power series trendline N x / N o =.357 q (-.947).74 q (-.94).65 q (-.84) Photographs of the buckled cylinders with 6.5% and 36.6% cut out area subjected to a combined total axial load and external lateral pressure have been shown in figures 6.7 and 6.8 respectively. Figure 6.7a: Perforated Cylinder with 6.5% cut out area subjected to a total axial load of 6.7kN and kpa lateral pressure Figure 6.7b: Perforated Cylinder with 6.5% cut out area subjected to a total axial load of 7.3kN and 3kPa lateral pressure

223 PERFORATED CYLINDRICAL SHELLS: EXPERIMENTAL RESULTS 6.5 Figure 6.7c: Perforated Cylinder with 6.5% cut out area subjected to a total axial load of 8.55kN and 4kPa lateral pressure Figure 6.8a: Perforated Cylinder with 36.6% cut out area subjected to a total axial load of.5kn and kpa lateral pressure Figure 6.8b: Perforated Cylinder with 36.6% cut out area subjected to a total axial load of 9.36kN and kpa lateral pressure