Rectangular Silos; Interaction of Structure and Stored Bulk Solid

7/21/2019 Rectangular Silos; Interaction of Structure and Stored Bulk Solid

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Rectangular Silos; Interaction ofStructure and Stored Bulk Solid

A thesis submitted forthe degree of Doctor of Philosophy

byRichard J. Goodey

Department of Mechanical Engineering

Brunel UniversitySeptember 2002


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Rectangular Silos; Interaction ofStructure and Stored Bulk Solid

A thesis submitted forthe degree of Doctor of Philosophy

byRichard J. Goodey

Department of Mechanical Engineering

Brunel UniversitySeptember 2002


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AbstractThemain aim of this research s directedtowardsthe studyof thin-walled rectangularplanform silos with a view to maximising their structural efficiency. In thin plates ofthe type making up the wall, membrane action may increase the load carryingcapability andcurrent design guides make no accountof this. Designingrectangularsilos with this in mind can lead to significant structural savings.The core of the research involves using the finite element method to study thepatternsof pressureexerted by the weight of a granularbulk solid on the walls of thesilo structure. The stored granular solid must use an elastic-plastic material law inorder to account or large deformations that canoccur in a thin-walled structure. Theneedfor this type of constitutive law led to the investigation of bulk solid propertiesand showsthat parameters hat have previously beenused to categorisebulk solidsmay not be sufficient to describe all aspects of their behaviour. The finite elementmodel createduses material constitutive laws that can be found in a number ofpackages. The required granular material parameterscan be determinedfrom anumber of simple tests. This approach aims to enable engineers o routinely usesimilar modelswhen designing silos.The results obtained from the finite element model exhibited some anomalies hathad been observed n previous work. These were mainly apparent n the form oflocalisedpressurepeaks near the baseof the model. Theseeffects were investigatedand possiblemechanisms hat leadto them wereproposed.The results from the finite element model were comparedto previous experimentalwork and existing theories. The model was then usedto conduct parametricsurveyson squareand rectangular planform silos and the distribution of pressureacross hewall comparedo previous predictive models.Finally, a scale hin-walled metal silo was constructedand pressuremeasurementsnfilling with pea gravel made. Theseare compared o predictions madeby the finiteelement model.

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AcknowledgementsThere are of course a number of people to whom I would like to express mygratitude. First and foremost, would like to thank my supervisor,Mr Chris Brownfor providing me with guidance (and funding) throughout the duration of myresearch. I would also ike to thankhim for giving methe opportunity of studying fora Ph.D. in the first place.Secondly, I must thank ProfessorMichael Rotter of Edinburgh University for hisassistancewith papersdrawn from this work. I must also mentionProfessor JurgenNielsen for his guidanceat the onset of this project. Thanks also to my secondsupervisors,ProfessorLes HenshallandProfessorLuiz Wrobel.This researchwould not have beenpossible without the assistance f the techniciansand staff in the department. Many thanks go to Keith Withers, Bob Webb, JohnLangdon,andBrian Dear.Therehasalso beena goodnumber of reprobateswho I havecrossedpaths with andhavemadethe whole researchexperiencemore enjoyable. My thanksto Dr RichardTorrens, Dr Anthony Morgan, Dr Chang-JiangWang and Dave Simpsonto mentionbut a few.Thanksmust go to my parents or their supportandhard cash over the large numberof years I have managed to stay in full-time education.

Finally, thanks to my girlfriend Lanawho hassupportedme for severalyears and hasonly spenta small amount of thattime telling meto geta ob.

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Table of ContentsAbstract iAcknowledgements iiNotation ixList of figures xiList of tables xvi1. Introduction 1

1.1 Thesissummary 22. Background to the problem 4

2.1 Definitions 52.1.1Planform 62.1.2Height to width ratio 62.1.3 Constructionmaterials 62.1.4 Flow type 6

2.2 Silos studied 83. Stressesand loads in the ensiled material and the silo structure 103.1 Silo loads 10

3.2 Stressstate n the stored solid 113.2.1 Interactionbetween silo and contents 12

3.3 Calculations or wall pressures 123.3.1Theoryof Janssen 133.3.2Theoryof Reimbert andReimbert 163.3.3 Theoryfor squat silos 20

3.4 Assumptionsof theories 213.5 Evaluationof pressurecoefficient -k 223.6 Dischargepressures 253.7 Other oadingconsiderations 25

3.7.1 Wind loading 263.7.2Seismicoading 263.7.3Thermaloading 26

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3.8 Application of theories o rectangularplanform silos 273.9 Silo response 27

3.9.1Load supportingactions n silosstructures 273.10 Membraneactions n rectangularplates 283.10.1Generalequationsor rectangularplates 30

3.10.1.1 Small deflection heory 313.10.1.2 Largedeflection heory 313.10.1.3Comparisonof smalland arge deflection

theory 313.10.2Generalnumerical echniquesor plateproblems 32

3.11 Structural considerations 323.12 Rectangularbin design 343.13 Summary 34

4. Numerical methods for the prediction of silo wall pressures 364.1 Introduction 364.2 Available numericaltechniques 36

4.2.1 Finite differencemethod 374.2.2Finite elementmethod 374.2.3Boundaryelementmethod 384.2.4 Discreteelementmethod 38

4.3 Choiceof numerical technique or use n the currentproject 394.4 Previous application of the finite element method to silo

problems 394.4.1 Filling pressures 394.4.2 Discharge pressures 41

4.5Summary 425. The finite element method applied to silo problems 44

5.1 Analysis type 445.1.1 Sourcesof non-linearity 465.1.2 Obtaining a solution in non-linear inite element

analysis 46

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5.1.3 Non-linearity in silo problems 475.1.3.1Largedeformations 475.1.3.2Contactanalysis 47

5.2 Constitutivemodels o describematerialbehaviour 485.2.1 Propertiesof granular bulk solids 495.2.2 Determininggranularbulk solids propertiesfor finite

elementanalysis 515.3 Material constitutive aws for use in the current work 55

5.3.1Elastic laws 565.3.2 Non-elastic aws 565.3.3 Elastic-plastic aws 56

5.4 Constitutivelawsavailablein ABAQUS 575.4.1Linearelastic law 585.4.2Porouselastic law 585.4.3Mohr-Coulomb law 60

5.4.3.1Plasticpotential andflow rule 625.4.3.2Mohr-Coulomb criteria in three-dimensional

stressspace 645.4.4 Drucker-Prageraw 64

5.5 Summary 666. Calibration and validation of material constitutive laws 67

6.1 Introduction 676.2 Axisymmetric model - geometry and boundary conditions 676.3 Material models 696.4 Elasticmaterialmodels 70

6.4.1Linearelasticaw 706.4.1.1Poissons atio asthe controlling factor in the

linear elastic model 726.4.2Porouselasticmodel 76

6.5 Elastic/plasticmodels 83

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6.5.1Application of plasticity criteriato the axisymmetricmodel 83

6.6 Three-dimensionalmodels 856.6.1 Furthervalidation 866.6.2 Comparisonwith Janssen ressure istribution 896.6.3 Comparisonwith experimentalsanddata 916.6.4 Modelling of peagravel 946.6.5 Comparison o Lahlouhet al's (1995)experimental

data 956.6.6 Best fit of finite element esults o experimentaldata 97

6.7 One-dimensionalconsolidation estsonthe two granularbulksolids 97

6.8 Conclusionsand choice of values or PE-DPmodel for LeightonBuzzardsand and pea gravel 102

7. The effect of the boundary condition at the baseof the axisymmetricbin 1047.1 Flat-bottomedaxisymmetricbin 104

7.1.1 Frictionlessbasecondition 1057.1.2 Frictional base condition 1077.1.3 Baseand wall not connected 109

7.2 Horizontal and vertical stressn the storedsolid 1117.3 Strain in the stored material 1147.4 Summary of flat-bottomed model 1167.5 Hopper basecondition 117

7.5.1 Pressuresn the hopper 1227.6 Conclusion 125

8. Further investigation of the geometry of Lahlouh et at (1995) 1278.1 Introduction 1278.2 Patternsof vertical stress n thestoredmaterial 127

8.2.1 Averageand local valuesof k 1308.3 Predictive law for wall normal pressures 133

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8.3.1 Comparison o experimentalresults in sand 1348.3.2 Comparison o experimental esults in gravel 138

8.4 Comparisonwith finite element esults 1428.5 Summary 1448.6 Finite elementpredictions or an idealisedwheat 145

8.6.1 Comparison o Janssen istribution 1458.6.2 Distribution of pressureacross he wall 1468.6.3 Patternsof vertical stress n the solid 1478.6.4 Averageand ocal values of k in the solid 1478.6.5 Comparisonwith predictive law 149

8.7 Summary 1509. Parametric study of a square planform silo 152

9.1 Introduction 1529.2Typeof stored material 153

9.2.1 Relativestiffnessbetweenwall and storedsolid 1559.2.2Deformationsn thewall 158

9.3 Planform 1599.3.1Squareplanform silo with 20mm thick wall 1599.3.2 Stiffness of ensiled material 164

9.4 Wall thickness 1669.5 Relativestiffness of silo wall andensiledmaterial 172

9.5.1 Effect of large deformations in the wall 1739.6 Variation of height of silo 1749.7 Summary 176

10.Experiment 17810.1Silogeometry ndconstruction 178

10.1.1Adhesivetests 18010.2 nstrumentation 181

10.2.1Deflections f thewall 18210.2.2Strain n the wall 18310.2.3Wall normalpressures 183

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10.2.3.1 Wall pressurecells 18310.2.3.2Free ield cells 184

10.3 Finite element predictionsof the experimental ig 18510.4 Experimental results 187

10.4.1 Pressures cross he centreline 18810.4.2Comparisonwith predictive law 18810.4.3 Strain in the silo wall 190

10.5 Summary 19111. Rectangular planform silos 192

11.1Limitations to this study 19211.2 Current guidance available 19311.3Rigid walled silo 19611.4Variation of planform ratio 19711.5Variation of wall thickness 20311.6Summary 205

12. Conclusions and further work 20612.1Summaryof main conclusions 20712.2Further work 209

References 210Publications 223Appendix A- Derivation of the Janssen formulae 224Appendix B- Convergence test on the three-dimensional model 226Appendix C- Sample finite element input file 228Appendix D- Example contour plot output from ABAQUS 236Appendix E- Best fit of finite element predictions to the experimental

sanddata 238Appendix F- Investigation of wall pressure cells 240

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Notation(A dot abovea symbol ndicatesa rate(eg. 4. = directional strainrate))A Crosssectionalareaof silo/Characteristicabscissaa Lengthof long wallb Lengthof shortwallC Circumferenceof binCo Overpressure ischargecoefficientc cohesionD Diameterof bin/Plate lexural rigiditye Voids ratioG Plasticpotential/ShearmodulusH Heightof siloh,, Heightof surcharge oneIl First invariantof stressensorJ2 Secondnvariantof stress ensorJe Elasticvolumechangek Ratio of horizontalto vertical stressn solidL Lengthof silo wallPa Initial stressPelt Elastictensile imitPh Horizontal wall pressureP. Mean wall pressurePn Normal pressuren hopperPt Tractiveforce in hopperP, Vertical pressuren solidP. Tractiveforcedownbin wallp Averagepressurestressq Frictional force/DeviatorstressR Hydraulicradiusof binr Radiusof bin/Radiusof plate

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t Thickness of bin wallU Perimeter of rectangularbinv Volume of solidw Deflection of plateY Yield functiony Depth from surfaceof filla Redistribution parametera' Hopper wall angle Drucker-Prager internal angle of frictiony Bulk density/ShearstrainE Young's modulusES Young's modulus of storedsolidE, Young's modulus of silo wallcj Directional strainx Gradient of reloading lines1% Logarithmic bulk modulus/Gradientof initial loading line Coefficient of frictionv Poisson's ratioa Stressa, Initial yield stressas Axial stressar Radial stressa;1 Directional stressT ShearstressT; Directional shearstress

Internal angle of frictionyr Dilation angle

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List of figuresFigure 2.1 - Comparisonof ahydrostaticdistributionand a Janssen istribution in a deep

silo 4Figure 2.2- Figureused o determine heflow type in a silo (from ENV 1991-4 (1995)) 7Figure 2.3 - Planformshapes f silos 8Figure 3.1- Loadsexertedonthesilo structure rom theensiled material 10Figure 3.2 -Notation used or silo geometry 13Figure 33 -Equilibrium consideration or Janssenheory 14Figure3.4-A typicalJanssenistributionor wall normalpressures 16Figure 3.5- Comparisonof Janssen ndReimbert heories n a similarly sized circular bin 18Figure 3.6 - Normal pressures n the long andshort wall of a rectangularbin accordingtoReimbertandReimbert(1976) 20Figure 3.7- Janssen istributionsof wall normal pressureusingdifferent values of k 24Figure 3.8 -A rectangularsilo with externally applied stiffening 28Figure 3.9- Comparisonof largeand smalldeflection theory for a circular plate 30Figure 3.10- Structural eaturesof cylindrical silos 33Figure 5.1-A simple inearelastic oad-displacementresponse 44Figure 5.2-A simple non-linearelastic oad-displacementresponse 45Figure 5.3-A non-linearsystemwith a non-uniquesolution 45Figure 5.4- Coulombmodelfor friction 48Figure 5.5-A Jenikeshearcell 52Figure 5.6- Yield loci of granularmaterialdetermined rom theJenikeshear cell 52Figure 5.7- Typical results rom an oedometerest 53Figure 5.8- Apparatus or triaxial test 54Figure 5.9- Triaxial testresults or wheat material (Ooi, 1990) 55Figure 5.10- Volumetric response f a granularsolid in compression 59Figure 5.11- Distortion of a block of granularmaterialby the application of a force 60Figure 5.12- ThecompleteMohr-Coulombcriterion 61Figure 5.13- Variation of a) shear orceandb) volumetricstrain with shear strain in a

granularmaterialAtkinsonandBransby,1978) 63Figure 5.14- The Mohr-Coulombcriterion in principal stressspace 64Figure 5.15- The Drucker-Prager riterion in principal stressspace 65Figure 5.16- Thecoincidenceof theDrucker-PragerandMohr-Coulomb criteria (Chen,

1994) 66Figure 6.1- Restraintat thebaseof the finite elementmodel of the bin 68Figure 6.2- The finite elementmeshof an axisymmetricsilo 69

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Figure 6.3 - Wall normal pressureascalculated rom linear elastic constitutive law 72Figure 6.4 - Comparisonsbetween finite elementandJanssen redictions for a rangeof

materials 74Figure 6.5 - Theeffect of changingthe elastic stiffness n theaxisymmetricbin 75Figure 6.6 - Wall normal pressureascalculated rom the porous elasticconstitutive aw 77Figure 6.7- Theeffect of changingthe initial stress n theporous elasticmodel 78Figure 6.8 - Simplified figure to show compressionof solid in a typical silo 79Figure 6.9 - Finite element resultsasdeterminedby the recalibratedporous elasticmodel 81Figure 6.10- The effect of using different initial conditions in the porous elastic model 82Figure6.11- Theeffect onwall normalpressuresf usingelasto-plastic onstitutiveawsin an axisymmetricmodel 85Figure 6.12- Schematicshowing the

layout of the experimental ig usedby Lahlouhet al(1995) 87

Figure 6.13- Finite element model of Lahlouh et als (1995) geometry 88Figure 6.14- Averagewall normal pressuresas predictedby different constitutive aws 89Figure 6.15- Pressures cross he wall as predictedby linear elastic law 90Figure 6.16- Integratedexperimental (Lahlouh et at, 1995)andfinite elementwall normal

pressuresomparedo Janssenistributionor sand 92Figure 6.17- Finite element predictions of wall normal pressureacross hewall compared

to experimental dataof Lahlouh et al (1995) 93Figure 6.18- Comparisonbetween Janssen ndFEA results or pea gravel 95Figure 6.19- Integratedexperimental andfinite element wall normal pressures ompared

to Janssendistribution for gravel 96Figure 6.20- Finite element predictions of wall normal pressureacross hewall 97Figure 6.21- Schematicshowing the one-dimensionalconsolidation test cell 99Figure 6.22- One-dimensionalconsolidation of sand 99Figure 6.23- One-dimensionalconsolidation testson Leighton Buzzard sand 100Figure 6.24- One-dimensionalconsolidationtestson peagravel 101Figure 6.25- One-dimensionalconsolidationtestson wheat 102Figure 7.1- Theeffect of Poisson's ratio on the depth overwhich endeffectsare present 105Figure 7.2 - Pressures ear the baseusingdifferent constitutive laws 106Figure 7.3- Effect of doubling the meshdensityon wall normal pressures 106Figure 7.4- Theeffect of a frictional base n an axisymmetricbin 108Figure 7.5- Theeffect of adopting different values of 109Figure 7.6- Wall normal pressurepredictionsin the baseandwall disconnectedmodel 110Figure 7.7- Thedistribution of horizontal andvertical stressat two levels in the

axisymmetric silo model 111

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Figure 7.8- Effective coefficient of wall friction in thesandmodel 112Figure 7.9 - Wall normal pressuresrom themodel with spring-supported ase 113Figure 7.10- Vertical straincontourplot in axisymmetricmodel 114Figure 7.11- Radial strain contourplot in axisymmetricmodel 114Figure 7.12- Mechanism o account or observedend-effect 115Figure 7.13- Effect of smallangledhopperonthewall normal pressureprediction 116Figure 7.14- The wall normal pressure bove he transitionwith a45 concentrichopper 118Figure 7.15- Wall normalpressure ear hebaseof themodel with a hopper 119Figure 7.16- Comparisonof analysiswith LE andPEDPconstitutive aws 120Figure 7.17- Theeffectof alteringheangleof thehopperonpressuresbovehe

transition 121Figure 7.18-

The effect on wall pressures bove hetransition of a rigid hopperwall 122Figure 7.19-Normal pressuren hoppersof varying angles 123Figure 7.20- Deformation of thehopperwall 123Figure 7.21- Effect on hopperwall pressurepredictionsof varying wall stiffness 124Figure 7.22- Effective coefficientof friction in hoppersof different angle 125Figure 8.1- Proposedarchingmechanism f Rotteret al (2002) in rectangularsilo 128Figure 8.2 - Patternsof verticalstressat 1mand2mbelow the surfaceof thesolid 129Figure 83 - Local and average aluesof k in sand 131Figure 8.4- Local and average aluesof k in gravel 132Figure 8.5 - Wall pressure distributions from sand experiments and predictive law 135Figure 8.6- Best fit meanpressuresn sand 136Figure 8.7 - The variation of the parameter a with depth in sand in the geometry of

Lahlouh et al (1995) 137Figure 8.8- Predicteddistributionresulting rom best it valueof a compared o original

experimentaldata or sand Lahlouh et a!, 1995) 138Figure 8.9- Wall pressuredistributions rom gravelexperimentsand predictive law 139Figure 8.10- Best fit valuesof pmagainstexperimentaldataandJanssen istribution 140Figure 8.11- The variationof theparametera with depth in gravel in thegeometryof

Lahlouh etal (1995) 140Figure8.12- Predicted istributionesultingrombest it valueof a comparedo original

experimentaldata or gravel (Lahlouhet al, 1995) 141Figure 8.13- Comparisonof a determined rom finite elementmethod and experiment in

sand 142Figure 8.14- Comparisonof a determined rom finite elementmethod and experiment ingravel 143Figure 8.15- ComparisonbetweenFEA results andpredictive law of Rotter et al (2002) 144

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Figure 8.16- Comparison o Janssendistribution in wheat 145Figure 8.17- Wall normal pressuredistribution across he wall in wheat 146Figure 8.18- Patternof vertical stress n wheat lm from the surfaceof the solid 147Figure 8.19- Local andaveragevalues of k in wheat 148Figure 8.20-a determined rom finite elementmethod in wheat 149Figure 8.21- Comparisonof pressuresacross he wall from finite elementmodel and

predictive law in wheat 150Figure 9.1 - Wall normal pressuresn deepsquareplanform bin 154Figure 9.2 - Variation of a with depth in different materials 155Figure 9.3 - Normal deformations(w/t) downthe centreline of the bin wall 158Figure 9.4 - Wall normal pressuresor sand n squareplanform silos 159Figure 9.5 - Wall normal pressuresor gravel in squareplanform silos 160Figure 9.6 - Wall normal pressures or wheat n squareplanform silos 160Figure 9.7- Comparisonof Janssenand FEA prediction in 3m squareplanform bin with

100mm hick wall 161Figure 9.8- Coefficient a for sand n the squareplanform bins 162Figure 9.9- Coefficient a for gravel in the squareplanform bins 162Figure 9.10- Coefficient a for wheat in the squareplanform bins 163Figure 9.11- Distribution of pressure above he transition 164Figure 9.12- Relationshipbetweenwall length and elasticstiffness of storedmaterial 165Figure 9.13- Relationshipbetweenwall length anda 166Figure 9.14- Averagenormal wall pressuresdown the depthof a 1.5msquareplanform

bin with varying wall thickness(sand fill) 167Figure 9.15- Averagenormal wall pressuresdown the depthof a 1.5msquareplanform

bin with varying wall thickness(gravel fill) 168Figure 9.16- Averagenormal wall pressuresdown the depthof a 1.5msquareplanform

bin with varying wall thickness(wheat fill) 168Figure 9.17- Variation of a with depth in bins of varying wall thickness(sand) 169Figure 9.18- Variation of a with depth in bins of varying wall thickness(gravel) 169Figure 9.19- Variation of a with depth in bins of varying wall thickness(wheat) 170Figure 9.20- Variation of a at 6m depth with wall thickness 171Figure 9.21- Comparisonof EJLIFt3 with a in sand, gravel and wheat 172Figure 9.22- Effect of large deformations onthe value of a 173Figure 9.23- Comparisonof results obtained in al Omdeepbin and5m deep bin (sand

fill) 175Figure9.24- Averagewall pressuresn a2msquare lanformsilo of differentheight(wheat ill) 176

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Figure 10.1- Overall view of the experimental ilo structure 179Figure 10.2- Internal view of the filler box showingdischargeholes 179Figure 103 - The chute and conveyor or emptying heexperimentalsilo rig 180Figure 10.4- Wall normal deflectionscalculated rom FEA underdifferent loading

conditions 182Figure 10.5- Principle of wall pressure ell 184Figure 10.6- Free field cell of the type designed y Askegaard 1978) 185Figure 10.7- Finite elementandJanssen redictionsof themeanpressuren the

experimental ig 186Figure 10.8- Finiteelement rediction f pressurecrosshewall in theexperimentalig 187Figure 10.9- Pressuresn themodel silo from free field cells andfinite elementmodel 188Figure 10.10

-Predictive law of Rotter et al (2002) fitted to experimental esults 189

Figure 10.11- Horizontal and vertical straindown hecentrelineof thebin wall 190Figure 11.1-Notation for rectangularplanform silos 192Figure 11.2- Ratio of pressureson long side o short side(after Reimbert) 194Figure 113 - Wall normal pressurepredictionsn a rectangularplanform silo 195Figure 11.4- Long and short wall pressuresn a2:1planfomi ratio bin compared o

predictionsromENV 1991-4 1995) 196Figure 11.5- FEA predictions comparedo thepredictionsof ReimbertandReimbert

(1976) 197Figure 11.6- Wall normal pressuren a 1.1:1planformratio silo (sand ill) 198Figure 11.7- Wall normal pressuren a 1.3:1planformratio silo (sand ill) 199Figure 11.8- Wall normal pressuren a 1.S:1planform ratio silo (sand ill) 199Figure 11.9- Wall normal pressuren a 2: 1planform ratio silo (sand ill) 200Figure 11.10- Deformation of thewall at the top of thesilo 201Figure 11.11- Wall normal pressuren a 1.1:1planformratio silo (wheat fill) 201Figure 11.12- Wall normal pressuren a 1.3:1planform ratio silo (wheat fill) 202Figure 11.13 - Wall normal pressure in a 1.5:1 planform ratio silo (wheat fill) 202Figure 11.14- Wall normal pressuren a2: 1planformratio silo (wheat ill) 203Figure 11.15- Redistribution of wall normal stressn a rectangularbin of planform ratio

1.5:1 204

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List of tablesTable 3.1 - Stressesn a rectangularplate as calculatedfrom largeand small deflection

theories 31Table 3.2 - Finite elementsolution to the large and small deflection rectangularplate 32Table 5.1 - Somepropertiesof granularbulk solids 51Table 6.1- Valuesof bulk density or Leighton Buzzard sand rom Lahlouh et al (1995) 71Table 6.2- Propertiesof various granular bulk solids as given by Rotter (2001) 73Table 6.3- Propertiesof granularbulk solids for use n the linear elastic model 73Table6.4- Valuesusedo calibrateheporouselasticconstitutiveaw 76Table6.5- Calculationf.% rom nitial and inal densities 80Table 6.6- Wall normal pressuresin kPa) in the experimentalsilo from free field cells insand Lahlouh et al, 1995) 92Table 6.7- Propertiesof pea gravel measuredby Lahlouh et al (1995) 94Table 6.8- Results rom consolidationtestson sand and gravel 101Table 6.9- Final propertiesused or the materials in the finite element model 103Table 9.1- Equivalentelasticstiffness of the three materials 157Table 9.2- Equivalentelasticstiffness n the squareplanform bins 164Table 9.3- Equivalentelasticstiffnesswith varying wall thickness 170Table 10.1- Results rom shear estsof different adhesives 181Table 10.2- Valuesof a and p. determined from the predictive law of Rotter et al (2002) 189Table 11.1- Definitionsaccording o EN1991-4 (Note: d,,--b for rectangular silos) 193Table 11.2- Lengthof walls in rectangular planform silo models 198

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Chapter 1- IntroductionIn the modem industrial environment therearemany situationswhere granularbulksolids need o be stored. Typical granularbulk solids includecorn,plastic pellets fora production line or coal to feed furnaces. Thestructuresused o hold thesematerialsareknown as silos, although they areoccasionally eferred o asbunkers (particularlywhen referring to the ground supported,squattype typically used to store coal inpower stations). Silos areusually constructedrom steel or reinforcedconcreteand itis apparent that much researcheffort should be directedtoward the study of thesetypes of structuresas over 1000 structural failures occur in silos in North Americaalone eachyear (Jenkynand Goodwill, 1987).Silos are most commonly found in circular planform dueto the structuralefficiencyof this type (Trahair, 1985). Circular planform structuressupportthe loadsfrom theensiled material mainly by membrane actions. They are axisymmetric, potentiallysimplifying the designprocess(Rotter, 1985a)but this can lead to a weak structurewere any bending actions to arise (Rotter, 1985b). Circular planform silos must beaccurately constructedbecausethey are sensitive o any imperfections of the shellsurface. One of the alternative silo structural forms is a rectangular planformstructure, but this has largely been ignored because t is difficult to assess heresponse to the loads acting on the structure. In this type of structure the main loadcarrying action is bending but if the wall deforms sufficiently there may be somemembrane action induced. This membrane action will always be dominated by thebending action but can significantly increase the ultimate strength of the plate if it istaken into consideration. Advantageously however, rectangular silos are simplyconstructed from flat plate which requires little preparatory work and could thereforeprovide a cost effective alternative to circular planform silos if design codes could besuitably adapted to better predict the behaviour of this type of structure.

- Of most importanceto structural designersarethe (horizontal andvertical) pressuresacting on the silo wall. There are several heories hat are commonly used o predictpressures Janssen,1895; Reimbert andReimbert, 1976). The resulting pressuresare

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then usually multiplied by some factor of safety in order to account for any excesspressurehat may occur during filling or discharge. Once theseareascertained t isthen a matter of using appropriate methodology to design the silo for strength,stiffnessand stability. Most of these theories assume he silo (and its contents)hassymmetry of rotation but it has beenshown that even in a cylindrical silo this is notthe case(Nielsen, 1983). It is also obvious that rectangular structuresdo not haverotational symmetry and therefore the available theories have to be adaptedto suit.This adaptation s unsatisfactorybecause t makes further assumptionswhich mayhavea deleteriouseffect uponthe economyof any rectangularsilo design. Structuresproduced n this way tend to have an excess of stiffness; they usethick plates and alargenumberof externaland internal stiffeners. This increaseshe weight of the siloandhence he cost. Thesedesignscould be improved by allowing a certain amountof deformation to occur in the wall of the silo which in turn stiffens the plate bymembraneactions.In previous work (Jarrett, 1991; Lahlouh et al, 1995) the emphasis was onexperimental determination of the effect of wall flexibility on measured pressures.This project aims to extend the knowledge in the area of rectangular planform steelsilosin the following ways:

" By using finite elementmodels to predict wall pressures n flexible walledsilos,

" By determining the effect of a very flexible, thin silo wall on wall pressures,

" By comparing experimental results with predictive theories, and

" By assessmentnddesignof instrumentation for usein scalemodel silos.1.1Thesis summaryThe work presented in this thesis consists of twelve chapters of which thisintroduction s the first.

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Chapter 2 gives somebackground nto the problembeingstudied aswell asdefiningsome of the terms and conceptsusedthroughout he remainderof the work. Chapter3 describessomeof the methods currently used o assesshe loadson, andresponsesof silo structures. Chapter 4 introduces various numerical methods that can beapplied to problems of the type studied n this work. It goeson to describe he choiceof the finite element method usedfor the remainderof thestudy. Chapter 5 describesthe finite elementmethodwith respect o silo problemsoutlining solution techniquesand modelling methodsused.Chapter 6 describes the initial investigation and validation of the finite elementmodel. This includes determination of parametersused n various constitutive lawsthat can describegranular materials of the type stored n silos. This is achievedby acombination of interpretation of experimentaldata,comparisonwith current theoryand comparisonwith previous experimental work (Lahlouhet al, 1995). Chapter7 isa study of someof the modelling anomaliesexperiencedn Chapter6. This is withregard to observedendeffectsnearthe baseof thebin.Chapter 8 introduces a two parameter predictive law conceived by Rotter et al(2002). This law aims to describe the experimentally observed non-uniformdistribution of wall pressure at a given depth in the ensiled material. The law iscompared to experimental and finite element results. Chapter 9 describes aparametric survey of a square silo. It is shown that the previously introducedpredictive law is a good fit to finite element results for a number of geometricallydifferent bins. Chapter 10 describes an experiment performed on a thin-walledsquare planform silo and makes comparisons with the predictive law and the finiteelement model. Chapter 11 presents a parametric study of a rectangular silo andshows that the actions in structures of this type are very different from the actions ina square planform silo.Finally, Chapter 12presentsconclusionsand suggestionsor further work.

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Chapter 2- Background to the problemThe structural designer of silos is concerned with the horizontal and verticalpressures acting on the wall of the silo, because these pressures determine themembrane and bending actions that govern design. The response of the silo can bedifficult to predict due to the large number of different possible failure modes,interaction between the stored solid and the structure and sensitivity to geometricimperfections.

The wall pressures in silos were originally assumed to be equivalent to hydrostatic(fluid) pressures. Early experiments by Roberts (1882; 1884) showed that this wasnot the case as some of the weight of the stored material is carried by the wall due tofriction at the interface between the materials. Janssen (1895) confirmed this andpublished his still widely used theory that accounts for this phenomenon. Soon after,Airy (1897) published a second theory to compute wall pressures but this is not insuch widespread use as Janssen's. A typical Janssendistribution for normal pressuredown the depth of a silo wall is shown in figure 2.1, along with the hydrostaticdistribution that was previously assumedto be correct.

0Janssen distribution2 Hydrostatic distribWon

4

100 20 40 60 80 100 120 140 160 180

Horizontal wall pressure (kPa)

Figure 2.1 - Comparison of a hydrostatic distribution and a Janssendistribution in a deepsilo

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It is obviousthat the pressurepredictedby the Janssen heory is substantially lowerthan the hydrostaticdistribution. The reasonfor this observedphenomenon s thatgranular materials have shear strength (i.e. they can support some of their ownweight) andfriction exists at the interfaceof the wall and the material. This leadstothe developmentof an equilibrium between he weight of the solid and the supportingwall friction. The observation of this property has led to the proposal of manytheories in the field of soil mechanics, several of which are specifically for thepredictionof pressuresn silos. Howevereach of thesetheoriescontains assumptionsand imitations thatmaymakethemunsuitable or particular typesof problems.Theories for silo pressures include the Janssen formula (1895), those developed byReimbert and Reimbert (1976) from empirical data and earth pressure theories suchas Coulomb's (1776). These theories consider the wall to be rigid and the pressure tobe invariant at any given depth but it has been shown by Jarrett et al (1995) and morerecently emphasised by Rotter et al (2002), that this is not the case for rectangularplanform silos. This research showed that when the wall is flexible an arch can formin the granular solid over the deforming wall leading to lower pressures in the middleof the wall and higher pressures near rigid, supporting boundaries. This is in contrastto Janssen's original postulation that a rectangular planform silo might experiencehigher pressuresat the mid-side of the wall and lower pressures in the comers.2.1 DefinitionsThere are several terms used in this thesis that require definition. The word silodescribesthe entire structure. The vertical walled section of this is usually referred toas the bin or box, while the angled section at the base is referred to as the hopper.Silos are further classified according to their planform, height to width ratio, thematerial they are constructed from and the type of flow they exhibit. The definitionsgiven are consistent with the current ENV 1991-4 (1995).

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2.1.1 PlanformCurrently, silos most commonly occur in either circular or rectangularplanform.Other planforms are occasionally usedand some were investigatedby Reimbert andReimbert (1976). Circular planform silos are efficient structures hat carry most oftheir loads by membrane(hoop) actions. As this form of silo is the most common,the majority of researchactivity is not unreasonablydirectedtowards their study.Rectangular planform structures may however, need to be used for a number ofpractical reasonsand if the efficiency of this type of structurecouldbeimprovedthenit may provide a viable alternativeto circular silos.2.1.2 Height to width ratioSilos can also be classified asdeepor shallow andan approximateguideto this factorwould be that a shallow silo has a height not exceedingone and a half times thediameter (or shortest side length). A deep bin obviously has a ratio greater than this(ENV 1991-4,1995).2.1.3 Construction materialsSilos are usually constructed from either steel or reinforced concrete. Concretestructures are more usually used in larger applications such as the storage of cementclinker, coal and grain. Concrete structures are obviously designed with permanencyin mind whereas steel structures can be easily constructed, moved, recycled etc.Recently, large concrete designs have been replaced by batches of squat steel silos.The design and testing of concrete structures has been well reported (Eibl, 1998) andwill not be covered in this project, although some reference will be made to very stiffstructures which could be considered to be like concrete in behaviour.

2.1.4 Flow typeConsideration must be given to the type of flow experiencedn a silo. The type offlow leads to different pressure regimes within the silo contents and thereforedifferent pressures on the silo wall (Nielsen, 1998). Flow types can be generally

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classified asmass low and funnel flow (sometimesknown as core or internal flow).In mass low, the ensiledmaterialmoves down the silo as one during discharge. Infunnel flow, the material tends to flow down a central flow channel while the outermaterial remains stationary. Whether funnel flow or mass flow occurs in a silo isgovernedby a numberof factorsamongstwhich thematerial properties andthe angleof the hopperwalls are most important. A graphicaldesign method for determiningwhich type of flow will occur is employedby most codes and a typical diagram isshownin figure 2.2 (ENV 1991-4,1995).

I

g 60 asp9a 50 %f Ft and Floht%` 50g 40 \ssllowortn waowmay orrwldn f ` c 40

j

FuioeFFkwS 30 fheses 30li 2D %% 1 4-02D3:

10

[Mm Flow %I, %.

I.`m 10 Lb=Flow0 ` L 1090 80 70 60 50 40 30 90 80 70 60 50 40 30Angleof 'i tionof 2opper waUct Angle of iridme ion of hopperwadC[

Figure 2.2 - Figure used o determine the flow type in a silo (from ENV 1991-4 (1995))The figure shows that mass flow will more likely occur in smooth walled silos withsteep hoppers and those with rough walls or shallow hoppers will more likelyexperience funnel flow. The type of flow experienced in the silo is an importantstarting point in the structural design process. Mass flow may be deemed necessaryin food processesbecausethey operate on a first in/first out principle, thus avoidingstagnant zones associated with funnel flow. This stagnant food material could spoilin a low volume production situation. Conversely, space may be limited whichprecludes the steephopper angle required for mass flow hoppers and therefore othersolutions for assisted discharge would have to be considered such as flow agitators.Another disadvantage is that mass flow silos wear heavily on the wall due to the

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abrasive effect of the granular bulk solid sliding down the wall and this must befactored nto the design.2.2 Silos studiedThis thesis is concerned with metal, rectangular planform, thin-walled silos. Metalsilos appear in many different plan-forms: circular, rectangular, square andhexagonal. They can be tall or squat, ground supported or elevated on columns.Somedifferent forms of silo are shown in figure 2.3.

abcFigure 2.3 - Planform shapesof silos

This variation of form allows the designer to accommodatemany different types ofloading but the interaction betweenthe storedbulk solid andthe silo wall makes heassessment f the wall pressuresdifficult. Figure 2.3(a) showsa diagramof a typicalmetal,cylindrical silo. It canbe seen hat the basicstructural form of this type of silois a thin, axisymmetric shell of revolution making the stressn the structure elativelyuniform around the circumference. This makes the analysis of the structureslightlyeasier or simple, un-stiffened cases. However, silos are regularly stiffened n some

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way and when considering a rectangularplanform silo their natural shapecontainsstress aiserssuch as corners hatmake heprediction of stressmore difficult.Figure 2.3(b) shows he basic ayout of a rectangular plan-form structure of the typethat is to be consideredn this project. Eachform of silo has ts own advantagesanddisadvantages. Rectangularplanform silos can be easily constructed from platematerial, meaningthere s less abricationwork required prior to construction on site.However, rectangularplanform silos are not as efficient structuresas circular onesbecausehey do not takeadvantage f membranestressessection 3.10). Rectangularstructures may also make more use of available ground-space (because theytessellate),which maybe mportant n applications where spaces limited.It is noted that the silos studied n this project are different from the type shown infigure 2.3(c), which is known as a trough bunker. Theseareusually squat and long(the ratio of long wall to short wall is large and they are often treated as infinitelyextendingparallel walls) and exhibit plane flow (essentially2-D flow, the stressstateis not a function of theco-ordinateperpendicular o theplaneof flow).

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Chapter 3- Stresses and loads in the ensiled material and the silostructure

3.1 Silo loadsSilos are subjected o a number of load casesof various load conditions. Of mostimportancefor designpurposes are the loads imposedon the structureby the weightof the granularbulk solid when the silo is filled and emptied. These oadsconsist ofpressurecomponentsand frictional tractive forcesand areshownin figure 3.1.

--- -

Ph- I

PwJ

Pv

UPfl\PtFigure 3.1 - Loads exerted on the silo structure from the ensiled material

Wall pressures anbe influenced by a number of factors including the silo geometry,wall friction andtype of stored material. There are also a numberof other oads hatmay need to be considered in the design process such as wind loading, thermalloadingandotherenvironmental loads.This thesis is mainly concerned with the determination of the filling pressuresn thesilo structure and as such most consideration will be given to these oads and theinteractionbetween he ensiled material and the silo structure.

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3.2 Stressstate in the stored solidGranular materials at rest in a silo can be said to exist in an elastic state which islimited by the Rankine active and passive pressure coefficients (Gaylord andGaylord, 1984). Thesedefine thepoint of active failure (causedby expansion of thematerial) and passive failure (causedby compressionof the material). However,throughout the granular material this state can differ which can affect the internalstressesexperienced by an ensiled material which in turn affects the pressuresexperiencedon the wall.This behaviour of stored materials is known to affect the pressures exerted on thewalls of a silo structure (Nielsen, 1983). Therefore it follows that the overall stressstate that the stored material assumes nfluences the wall pressures. There are threecategories of wall pressure that are usually considered by designers; filling, static anddischarge. The assumption that the filling and storage pressures are approximatelyequal is widely accepted (although some types of granular materials may cause thepressure regimes to alter over a period of time), and so consideration is normally onlygiven to two stress states. The problem is often further simplified by basing theevaluation of discharge pressures on the static/filling pressures which are multipliedby some pre-determined factor that accounts for the apparent excess pressure. Thisfactor is usually determined from codes or design guides (ENV 1991-4,1995; DIN1055,1987). These flow load multipliers are generally derived by consideration ofchanges in the stress field upon the onset of discharge (Nanninga, 1956; Walters,1973a,b) and were originally obtained empirically. This process requires thedesigner to perform only one set of calculations upon which the final design will bebased even though there are three different phases of loading. This is before anyconsideration of any other external loadings. This would appear to be a considerableshortcoming given the variability of experimentally observed static pressures(Nielsen, 1979; Nielsen and Anderson, 1982; Harden et al, 1984) and dischargepressures (Nielsen, 1998).

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The stress state (and hence the wall pressures) is also greatly affected by theeccentricity of the fill and discharge processes. During filling it is unlikely thatmaterial will be deposited rom directly abovethe bin in the centre. It is more likelythat material will be travelling up a conveyer and will be deposited into the bintowards one side, possibly impacting on the opposite wall in the case of a slendersilo. This will havea large effect on the stateof stress hat exists in the solid at theend of the filling process and this has subsequentlybeen shown to affect the wallpressuresZhonget al, 2001). Similarly, it is possiblethat the material will be drawnfrom the silo via an eccentric hopper and this has been shown to producehighlyasymmetricpressuresn cylindrical bins (Rotter, 1986).3.2.1 Interaction betweensilo and contentsThepressureexertedon the silo wall is dependenton the interactionbetween he walland the stored bulk solid. This is especially so when the wall is flexible in asystematicway, as in a rectangular planform silo. Little work is currently available,although several researchgroups have shown that the flexibility of the wall of acylindrical silo can have an effect on the wall pressures(Ooi and Rotter, 1990;Mahmoud and Abdel-Sayed, 1981). Others have performed studies of the silo-material interaction (Emanuel et al, 1983; Ibrahim and Dickenson, 1983). Most ofthesestudieshaveused a two-dimensional finite elementmodel. In order to improvethe study of this phenomenona three-dimensionalmodel would be the most suitablebut until recentlythis would have been impractical due to the constraints of computersystems.Jarrett et al (1995), Lahlouh et al (1995) and Rotter et al (2002) have presentedexperimental esultsfor squareplanform steel silos showing that the flexibility of thewall andthe type of granularmaterial used affects the measuredpressures.3.3 Calculations for wall pressuresThereare a number of theories available for the prediction of the wall pressures ndthesehavebeenreviewedextensivelyelsewhereArnold et al, 1980; Gaylordand

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Gaylord, 1984). Most of these theoriesare applicable to cylindrical silos becausethesestructurescanbe reduced o two dimensionsby the assumptionof axisymmetry.As it is known that the wall deformationandhence he wall pressuren a rectangularsilo varies at constant depth then a three dimensionalanalysis is required (Ibrahimand Dickenson, 1983). However, astherearenot any specific formulae for this typeof calculation, the cylindrical theoriesare usually applied to rectangular planformstructures, although these are sometimesmodified slightly to take account ofparameterssuch as the ratio of the lengthof the side walls. Only the most commonlyused theories will be assessedhere, along with their assumptions and suitabilitytowards the current problem. Figure 3.2 shows he notation usedfor the geometry ofthe silo in the forthcoming sections.

DFigure 3.2 - Notation used or silo geometry

3.3.1 Theory of JanssenPressuresn deep silos of circular planform are usually calculatedby the theory ofJanssen 1895). This is derived by considering he equilibrium of a horizontal sliceof material in the silo as shownin figure3.3.

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Summationand integration of these forces yields the following equation for theverticalpressuren the solid:

Pv= (1-e-r Y/R)yk

(3.1)

Wherey= bulk densityof the storedmaterial, R= A/C = hydraulic radius of the silo,= the coefficient of wall friction, k= the ratio of horizontal to vertical stress andy= the depth of the solid above the section, which is assumed o be level. Theoutlinederivationof this formula canbe seen n Appendix A.

The Janssen ormula makes a number of assumptionsthat have an effect on theusefulness f the model." The vertical stresses re zero at the free surface" The coefficient of friction between the material and the wall is constantand

friction ismobilisedhroughout" The averageatio of horizontal to vertical stress s constant" The storedmaterial is isotropic and uniform in weight

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r 'gurr- -7.2 - Jyuiuua Iuua cuuDlucI auvu av uau"cu ruvul y


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Theseassumptions imit the cases o which theJanssenormulashouldbeappliedbutit is still used as the basis for a large number of designcodes(ENV 1991-4,1995;DIN 1055,1987).Thepressurecoefficient, k, relates he horizontalto vertical stressn the solid:

k= Ph (3.2)-PVAnd hencehehorizontalwall pressures givenbythe ormula:

(3.3)

The usefulness of the Janssen theory is therefore dependant on the method used toassessthis coefficient, k, and this is discussed further in section 3.5. The use of thehydraulic radius of a bin (rather than the circumference or the diameter) allows thedesigner to account for non-circular cross-sections, but the resulting solution is stilltwo dimensional and therefore gives no indication about the pressure distributionacross a non-circular bin at any given depth. It is therefore commonly assumedto beuniform. A typical Janssen distribution for wall normal pressures (Ph) is shown infigure 3.4.

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024

0 810

gE 12 - Janssen distributionw 14V 16

1820 02468 10 12 14 16

Wall normal pressure (kPa)

Figure 3.4 -A typical Janssen distribution for wall normal pressures

It can be seen that at great depth the pressures tend to an asymptotic value. Thisasymptotic value can be calculated by reducing the exponential component ofequation 3.1 to zero making the maximum horizontal and vertical pressures equal toequations 3.4 and 3.5 respectively.

Pn=yR (3.4)

pv _ yR (3.5),uk

3.3.2 Theory of Reimbert and Reimbert

Another method of prediction of wall pressure is that of Reimbert and Reimbert(1976) which is an empirically derived method resulting from a large number ofexperiments. This theory allows for the differences in planform that can occur insilos and also the value of k to change with respect to the depth below the surface offill. It is based on the calculation of what is termed the "characteristic abscissa". Fora cylindrical silo the horizontal pressureat a given depth s:

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(P1 1J12 (3.6)

Where P. is themaximum lateralthrust given by:

, _ yD (3.7)

It may be noted that this value is comparableto the maximum value given by theJanssen quation equation3.4).

And A is the"characteristicabscissa"given by:A=Dk (3.8)4uk 3D= diameterof silo andh, = heightof cone of surcharge.Thevalueof k taken n the original formulaewas the Rankineactive ratio (seebelow,equation3.17). It may also be notedthat this theory takes accountof any surchargeon the material's surface which the Janssenequations do not (they are modified toaccommodatean equivalent surface). This is a major deficiency in the Janssentheory as the surfaceboundary condition is clearly incorrect. A conical surchargemust leadto a finite value of vertical stresswhere the solid first makes contactwiththe wall. However, he wall normal pressureat this point must be zero andthereforethe value of k must also be zero. This discrepancyhas no real effect in a deepsilobecausehe surcharges small relative to the overall depth but in a squat silo thesurchargecanhavea large effect and different methods of treating this problem areshown n section3.3.3.The following graph shows the horizontal pressure distribution down the wall of acylindrical deep silo as calculatedby the Janssenand the Reimbert and Reimbertformulaeassumingahorizontal surface figure 3.5).

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0

-5

10Et 15 - Janssen distribution

- - Reimbart and Reimbart

2002468 10

Horizontal wall pressure (kPa)Figure 3.5 - Comparison of Janssen and Reimbert theories in a similarly sized circular bin

The value of k taken here was that of the Rankine active ratio. The choice of thisratio is discussed in further detail in section 3.5. The graph clearly shows that bothmethods tend towards an asymptote at a great depth and the maximum value isidentical.The Reimbert and Reimbert method also gives specific formulae for the calculationof the pressure in a rectangular bin. This is based on the calculation of the relevantPmax nd A for the long and the short walls using the same basis as for the cylindricalsilo (i. e. the equilibrium requirement is maintained).If the width of the short wall is denoted as b and the long wall as a then the pressureon the short wall can be calculated from:

Ph =P .b 1-y +1

(3.9)A,

Where Pma.bis given by:

yb (3.10)Pmx.n =4p

And Al given by:

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b he (3.11)A' rpk 3

For the long wall the formula is similar:Ph=P.. a 1- y

(3.12)AZ

Where Pm. e s given by:

7h1 (3.13)P. a =4p

With:

b, -2ba - b2 (3.14)

a

And A2 given by:(3.15)

'cock 3

Figure 3.6 shows the horizontal pressures on the walls of a deep rectangular binwhere the ratio of long wall to short wall is 2: 1.

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o23

'`t S I6

7 -+- Short waH8 -a- Long wall9

10 02468 10 12 14Wall normal pressure (kPa)

Figure 3.6 - Normal pressures on the long and short wall of a rectangular bin according toReimbert and Reimbert (1976)

It is clear that this theory predicts that the shorter wall will experience a much lowerpressure than the

longer one. This arises from the granular bulk solid spanningacross the shorter distance between the long walls and hence more load is transferredto the long walls.3.3.3 Theories for squat silosThe above theories are most suited to deep bins but when the height to diameter ratioof a silo becomes less than about 1.5 then the silo can be referred to as squat althoughH/D values of between 1 and 1.5 are defined as intermediate by the Eurocode (ENV1991-4,1995). There is even some debate about this definition with other limitingvalues being proposed (Fischer, 1966), for example:

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height of the surcharge one nto the equationbut Janssen heory assumesa level filland must thereforebemodified to accountfor the surcharge. If Janssenheory is tobe used for squat silos then the origin of the first contact is moved to createanequivalent surface. The height of this surface s calculated by consideringthe heightof a level fill giventhe volume of solid encompassed y the fill andthe surcharge. Itis thereforedependanton the silo geometryand the angle of the surcharge. Thismethod eads o non-zeropressures t the first wall contact.To bettercalculate he pressuresn squat silos, designerscan use an earth pressuretheory such as those of Coulomb (1776) or Rankine (1857). However, there aredifficulties with using these. Rankine's theory assumes that the stored materialextends nfinitely andthereforeshouldnot beapplied to circular or rectangular silos.Neither does t takeaccount of the wall friction and thus tends to underestimate hewall pressures.TheCoulombtheoryalso fails to take account of the wall friction andassumesa horizontal fill surface which can lead to errors in silos of this type. TheCoulomb methodhas therefore been adaptedby some researchers n order to takeaccount of these factors (Mayniel, 1808; Muller-Breslau, 1906). Muller-Breslau(1906) modified the Coulomb theory to account for wall friction and a slopingbackfill (this theory is essentiallyfor earth pressurebehind a retaining wall but thesloping backfill canbe equated o the surchargen a silo). However, the volume ofsurchargeon the silo is limited by its size and hence incorporation of the slopingbackfill tends o over-estimatehe pressureson the wall and a more accurateanswermay beobtained rom this theoryby assuminghe surface s a level fill.It is apparenthat morework needs o be directed owards squat structures n ordertoprovide designerswith satisfactory guidelines for the assessmentof loads. This isespecially mportantasbatteriesof squat silos are becoming more common in placeof largerconcretestructures.3.4 Assumptions of theoriesThe abovedescribed heories make a number of assumptions of which somehavebeen mentioned. There are a great number of implied assumptions that can affect the

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accuracyand usefulness of the theories and the most important of thesearenow setout.

1. Wall friction is assumedo be constantdown the depthof the bin. This maynot always be the case as relative movement between he wall and solid canaffect the value of .

2. The stored material is incompressible,homogeneousand isotropic. Granularmaterials exhibit very complicated behaviour which is discussedurther insection 5.2.1. Granular materials contain voids which make themcompressibleand they are sensitiveto the method of filling in the silo whichcan affect the stressstateproducing a very anisotropic ill (thong et al, 2001).However, Munch-Anderson et al (1992) and Ooi et al (1990) haveshown hatthe average pressureon a level in a circular silo is well represented y theJanssendistribution.

3. Discharge pressures are simplified by using an over pressure factor dependenton the type of flow exhibited in the silo. This is a gross over-simplification ofthe mechanisms that are occurring in a discharging silo; these have beenshown to be very complex and produce wall normal pressuresthat are abovethe estimations made from over-pressure factors, and in most cases are notconstant at a given depth in a cylindrical silo (Nielsen, 1983). The onset ofdischarge also has a large effect on the behaviour of the stored material whichfurther emphasises the limitations described in Point 2 above. The modellingof discharge is outside the scope of this thesis.

3.5 Evaluation of the average pressure coefficient -kAs mentioned,the evaluation of k affects the predictions of theabove heories or thewall pressures n a silo. This thesis will discuss the averagevalue of k (the ratiodeterminedby consideration of the silo contents asa whole) andthe local valueof k(theratio determined at a point of inspection). It is usually assumedhat the averageratio is constant throughout the contentsof the silo but local valuesof k can differ

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from the assumed alue dueto the influence of factorssuch as wall deformation andwall friction. Methods that allow for the variation of the average value of k havebeen developed(ReimbertandReimbert, 1976)but the overall result is often onethatdoesnot differ too greatlyfrom using a singlevalueof k. Using the single value of khas been consideredsufficiently accurate or most researchers Jaky, 1948; Pieperand Wenzel, 1965;Walker, 1966)but there has beenmuch discussion of what valuek should be. Rankine (1857)shows hat therearetheoreticalboundaries for the valueof k known as the active and passive pressure atios. Thesetwo extremes can bedemonstratedby the considerationof a mass of cohesionlesssoil behind a retainingwall and are:

" The active pressure atio that results from the movement of the wall awayfrom the granularmaterial,and

" The passivepressureatio that results from the movement of the wall towardsthe granularmaterial.

The active pressure is the minimum value and occurs just before failure of thegranular material as the wall moves away. The passive pressure is the maximumvalue and occurs just before the granular material compressively fails. As calculatedby Rankine thesetwo coefficients are given by equations 3.17 and 3.18.Active: k_ 1- sino (3.17)1+sinq'Passive: k =1 + sino (3.18)1-sinq$Where 4 is the internalangleof friction of the granularmaterial.The active pressure atio could be experienced n a flexible walled silo as there isscopefor the material to fail if the wall deforms. This ratio has been usedby somedesigners. It is unlikely that the passivepressure atio would be reached in a siloproblem.

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Values of k for use in silo problems have been suggested including Jenike et al's(1973) suggestion of 0.4 (no matter what the material is) and Jaky's (1948) ratio forearth pressure at rest:

k=1-sinn (3.19)This is used mainly for silos with rough walls and has been adopted (with slightmodification) by the modem ENV 1991-4 (1995) which gives the value of k as:k =1.1(1- sin) (3.20)Figure 3.7 shows the effect of using the different values of k on the normal wallpressure distribution in a deep cylindrical silo when using the Janssen ormula.

Figure 3.7 - Janssen distributions of wall normal pressure using different values of k

The two limiting cases can clearly be seen and are quite different from the majorityof the predictions which tend to have similar values of k.

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3.6 Discharge pressuresThis thesis s concernedwith the prediction of the filling pressuresbut it is necessaryto have some understanding of the methods currently used to predict dischargepressuresn designcodes. Most current codes(ENV 1991-4,1995; DIN 1055,1987)handle the prediction of the discharge pressures in a silo by calculation of a flow loadmultiplier and then applying this to the filling pressures. This method would appearsimplistic given the number of variables that have been identified that can affect thesymmetry f wall pressures venn a cylindricalbin.

The Eurocode (ENV 1991-4,1995) gives guidelines for discharge rom concentricand eccentrichoppers. In the caseof the concentrichopper the normal wall pressuremultiplier is basedupon the characteristicsof the material.Phd CoPh (3.21

Co=1.35for4_ 0 (3.23)In the caseof the eccentric discharge the procedure is more complicatedand firstinvolves the calculation of the flow channeleccentricity and geometry. This impliesthat the method is for funnel flow silos and no mention is made of eccentricmassflow hoppers. Knowledge of the flow channel then allows calculation of the wallpressuren the flow zone and also in the staticzone.3.7 Other loading considerationsAs well as the loads imposed by the ensiled material there are a number of otherfactorsthat must be consideredin the designprocess. Loads are mposedon the silostructurefrom external factors that include wind loads, seismic loads, and thermalloads causedby expansion and contraction of either the structure or the storedmaterial.

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3.7.1 Wind loadingAny immovable structure experiences ctionsfrom the wind. Many statistical toolsare available for the predictionof possible wind speedsn a given geographical areaandthesecan be used o assesswind loads. The shapeof the silo and the material itis constructed from affect this loading, a tall, rough square silo will experience ahigher wind load than a squat, smooth cylindrical bin. Considerationmust also begiven to the location of the silo with respectto other structures. This can includeother silos if a multi-cellular installation is being considered. The design of the roofcan also have considerableeffect on the passageof wind round the silo and if thedesigndoesnot call for a roof at all thenthepossibleeffect of the wind acting on theinternal surfacesof the bin must be examined. Wind loading can be critical forempty silos so considerationmustbegiven to whetherthe filling/emptying cycle willinvolve periodswhere hesilo may remain emptyfor some ime.3.7.2 Seismic loading

This type of load is not really applicable to designs intended for use in the UK orother areas of the world where seismic activity is low. However, there are a largenumber of regions where seismic activity is an important factor and some codes giveguidance on designing for this type of load (ENV 1998-4,1999). The response of thestructure will be a function of the material it is constructed from, the type offoundation and the structure's natural periods of vibration.

3.7.3 Thermal loadingIf the overall temperatureof the silo's immediate environment increasesthen thestructure will expand. If the relative magnitude of the expansion of the structurecompared o the ensiled material is large enough hen the material stored inside mayassumea new position. However,on contraction he materialwill experience a largepassive pressurewhich in turn leads to compaction and stiffening of the storedmaterial. As well as the high stressesn the wall of the structure there may bedischarge problems associatedwith this compaction. Some researchers have

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investigated his effect (Zhang et al, 1986; 1989). This phenomenoncould haveasignificant effect in areaswhere extremetemperaturecycles areapparent.3.8 Application of theories to rectangular planform silosAll of the previously described theories are most applicable to circular planformsilos. Some codes recommend ways in which the theory can be adjusted to takeaccount of the different shape (ENV 1991-4,1995; DIN 1055) and some have beenadapted to make them more applicable (Reimbert and Reimbert, 1976). Janssen'soriginal paper (1895) postulated that the pressure at the mid-side of a rectangular silowould be higher than that at the comer. For a flexible walled silo this has beenshown experimentally to be incorrect and the pressure at the mid-side can beconsiderably lower than the pressure at the comer (Jarrett et al, 1995). The Reimbertand Reimbert (1976) formulae would appear to be most useful as they give values forthe wall pressure on the long and the short wall. None of the theories however giveany sort of information about the variation of the pressure across the wall at anygiven depth. This is where structural savings can be made because if the pressureisshown to be lower in the middle of the wall then the bending moment is lower andhence the required strength is reduced. There is also the strengthening effect oftaking into consideration the membrane stiffness of the plate.3.9 Silo response3.9.1 Load supporting actions in silo structuresThis sectionof the thesisdiscusseshow the alternative forms of silo carry their loadsandhow this canhavean effect on silo design. Cylindrical silos support most of theirloadsusing in-plane(membrane) forces althoughsomebendingmay occur in order omaintain compatibility of the boundary (Rotter, 1985a). Therefore, in most casesmembrane theory is used for the design of cylindrical shells. Rotter (1985b)discussesseveralcaseswhere the effects of bending stressmay be of importance,such as under repeatedcyclic loadings, where large bending stressescould causeafatigue failure.

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Rectangular structures however are usually designed so that the primary loadsupporting action is that of bending stresses (Brown, 1998). Structures are oftendesigned with externally applied stiffening which divides the more flexible platematerial into a number of panels which are treated and analysed separately. Thistends to lead to an inefficient structure being designed with an excess of strength.Figure 3.8 shows a typical rectangular silo with a large amount of external stiffening.

However, as will be shown later, in an unstiffened structure using thin plates,membrane stress can be induced when the plate undergoes large deflections.Therefore, if a design could be formulated that permitted large deflections then someof the load would be carried by this membrane stress, creating a more efficientstructure. This would have particular benefit for smaller, unstiffened rectangularstructures.3.10 Membrane actions in rectangular platesWhen considering plates of the sort that a rectangular planform silo may beconstructed from, the main supporting action is that of bending actions. However,with thin plates, when the deflection at the midpoint is large compared to thethickness of the plate, the middle surface becomes strained, resulting in in-plane

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Figure 3.8 -A rectangular silo with externally applied stiffening


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tensile stress evels that stiffen the plate by a considerableamount. This stress iscalled diaphragm, direct or membrane stress. Ugural (1981) defines a largedeflection in a plate as one wherethe deflection is greater han the thickness of theplate but goes on to show that significant differences between large and smalldeflection solution methodsoccurwhen hedeflectionexceedshalf of the thickness.A plate undergoing large deflections hasnon-linear elastic load-deflection and load-stressrelationships which are not accounted or by small-deflection bending theory(the usual approach o platebendingproblems). Thereforeamodified theory must beemployed which accounts for the large deflections. This is usually achieved bydetermining the bending stressas per the original theory and then adding to this themembrane stress. As an exampleconsideration s given to a circular plate that isfully fixed about its edge(Ugural, 1981). The bending(small deflection) solution ofthis problem is given by equation3.24.

r4wm. =D or6464DP1= r4 Wmax (3.24)

The bending andmembrane largedeflection) solution is given by equation3.25 (forderivation seeUgural (1981)).

364DCw 1+8 Et (2Lmaxpl -- r3 r3 (1- v) rr)(3.25)

Figure 3.9 shows deflections in the range 0


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4-3.5-

SmalldeflectiontheoryLarge deflection theory

2.5- 1c 2w' 1.5 "1

0.5-0

0 0.25 0.5 0.75 1 1.25 1.5W. )t

3.5-- -Smalldeflectiontheory

Large deflection theory2 5- 1. 00OF2- '101.5 "

1 '0.5-

0

Figure 3.9 - Comparison of large and small deflection theory for a circular plate3.10.1 General equations for rectangular platesVon Karman (1910) introducedgoverning differential equationsfor large deflectionsof thin plates. These take the form of coupled, non-linear partial differentialequations and where realistic problems are concerned obtaining a solution is acomplex and time-consuming task. Some approximate solutions of simple shapesunder uniform loading have been determined (Timoshenko and Woinowsky-Krieger,1959; Roark and Young, 1975) but it is only since the advent of numerical techniquesthat the general problem has been treated satisfactorily (Zienkiewicz and Taylor,1989).Solutions for rectangular plates are therefore normally obtained by experimental ornumerical techniques. However, some examples are available from literature (e.g.Bares(1979) for small deflectionsand Levy (1942) for large deflections) and thesehavebeencollatedby RoarkandYoung (1975) for engineeringdesignpurposes.The effect of calculating the stresses n rectangular plates using the two differenttheoriescanbe easily shown using an example. Consider a squareplate, fully fixedand loadedwith a uniform load of 2 kPa. It is madefrom steelwith E=210 GPaandv=0.3 andhasdimensionsof lm squareand thickness3mm.

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3.10.1.1 Small deflection theoryFrom Roark and Young (1975) the formulas for the bendingstressanddeflection atthemid-point of a fully fixed squareplate underuniform loadaregiven as:

0.1386gb2 (3.26)t2

0.0138gb4 (3.27)maxw= Eta

whereq= load per unit area,b= width of plate,t= thickness,a= bendingstress,w=deflection andE= Young's modulus.3.10.1.2Large deflection theoryIn Roark and Young (1975) analytical results from various sourcesare tabulated.This table is given for v=0.316 so results would be slightly different from thosegiven by the small deflection theory. Large deflection solutions are expressed nterms of the coefficients w/t, gb4/Et4and ab2/Et2. Once the value of gb4/Et4 sdetermined then the table can be used to obtain values for the other coefficients.Interpolation betweenthe values in the table maybe necessaryo provide values ofstressanddeflection for a particular problem.3.10.1.3 Comparison of large and small deflection theory

Table 3.1 shows the maximum stress in the plate and the deflection at the middlepoint ascalculatedfrom the two theories.

Mid-point stress(MPa) Mid-point deflection (mm)Small deflection theory 30.8 4.87Largedeflection theory 23.8 3.12Table 3.1 - Stresses n a rectangular plate ascalculated from large and small deflection theories

It can be seen hat by accountingfor the membranestressof a plate undergoing argedeflections, lower values for stressand displacementare obtained. If membrane

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stresss accounted or it can eadto more efficient useof the strengthof a plate whendesigning structures. Designing the structure so that large deflections occur mayprove difficult and therefore accounting for membrane stiffness may have otherbenefits. More realistically, this method may make it possible to evaluate the truefactorof safety n the structure.3.10.2General numerical techniques for plate problemsAs previously mentionedsince the advent of computer basednumerical techniquesproblems n large deflection plate bending have becomeeasier o assess.The aboverectangularexample can be repeated using a simple finite element model and theresults or this areshown n table 3.2.Element behaviour a (MPa) y (mm)Bendingonly 31.98 4.94Bendingand membrane 24.64 3.16

Table 3.2 - Finite elementsolution to the large and small deflection rectangular plateIt is clear that the finite elementmethod agreeswell with both theories but of courseit is not limited to the small number of geometries and load cases hat are availablefrom literature,and it canbe used or many plate sizes or geometries.3.11 Structural considerationsTherearea largenumberof other structural components n a silo other than the mainshell (or plate) structure. All of thesecan have an effect on the structural responseofthe silo. Figure 3.8 showed a rectangular silo and figure 3.10 shows an elevatedcylindrical silo with a numberof commonly used structural features.

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Figure 3.10 - Structural features of cylindrical silos

These features include a ring beam which provides a junction between the parallelsection of the silo and the hopper. The overall support of the silo is by columnswhich are also attached to the ring beam. This ring beam is therefore subjected to acomplex loading from the weight of the bin structure and the membrane tensioncaused by the hopper. The ring beam also acts as a stiffener at this point of the silo.Comparable in this senseto the ring beam is the roof which as well as protecting thecontents from exposure to the elements also provides stiffening at the top of the bin.

Figure 3.8 shows a rectangular silo that features externally applied stiffening. Theuse of this type of stiffener results in a structure that can be considered to act like aseries of flexible panels supported between the stiffeners and there are a number ofmethods given for the analysis of this type of structure (Troitsky, 1980). Thismethod of design usually leads to an overly stiffened structure which does not takeadvantage of its full load carrying capability. This is obviously wasteful ofconstruction material but also underlines the lack of knowledge concerningrectangular structures. Although not shown in figure 3.8 another occasionally usedfeature of rectangular silos are internal ties across the comers. These are used toprevent spreading of the corners in the silo but must be carefully considered beforeuse as they can affect the flow of the material. There is little openly available

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material concerning the use of internal corner ties because t is commerciallysensitive but Khelil (1998) discussessome of the effects hat internal ties canhaveonwall pressuresandflow patterns.3.12 Rectangular bin designAs this thesis is concernedonly with design of rectangularsilos a full descriptionofthe design process for circular planform silos is unnecessary. There are a largenumber of codes, referencesand design guides that are devotedto this topic (DIN1055,1987; ENV 1991-4,1995; Rotter, 2001). It is worthwhile though to discusssome of the aspects of structural design specific to rectangular planform silos.Troitsky (1980) has produced a design guide specifically for rectangularsteel silostructures.Most silos of this type would be expected to contain a ring beam at the junctionbetween the hopper and the wall. The ring beam s subjected o a variety of loadsespecially if the structure is supported on discretecolumns. It must distribute theweight of the silo to the supporting columns. At the onsetof filling the ring beamwill be subjected to inward forces from the weight of the hopper. As fillingcontinues this force is offset by the horizontal pressureexertedon the walls of thesilo.The columns that support a circular silo usually terminate at the ring beam. Inrectangularbin designthey often are extended o the top of thebin in orderto providemore stiffness. There is often anotherring beamat thevery top of the silo to preventexcessive deflections at the free edge although this could be incorporated into theroof design.3.13 SummaryThe basis of silo design is knowledge of the internal pressures ausedby the weightof the material that is to be stored. A number of theories and methods fordetermining thesepressureshave beendeveloped. Somearespecific to a certain siloform (e.g. Muller-Breslau (1906) for squat silos) but generally most codes use one

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method no matter what the shape. The Eurocode (ENV 1991-5,1994) attempts toaddresshis problemby classifying silos according o their shape,height etc.The correct determinationof the wall pressuress also hindered by the assumptionsmade by the theories. Chiefly, the adoption of a constant value of k is a majorsimplification especially when considering rectangular planform silos which havebeenshownexperimentally o exhibit largevariations in the value of k throughout thestoredbulk solid (Rotteret al, 2002).When considering ectangularsilo design, bendingmoments are usually the basis. Ithasbeenshown hat by accounting or membraneactions that may arise in the platestructure, ower stressesand deflections for a given load are predicted. Membranestressarisesas a result of wall deformation but this deformation doesnot needto betoo large (compared o the thicknessof the plate) for structural savingsto be made.Rectangularplanform steelsilo designsareoften produced that use a large amountofexternal stiffening. This limits the plate deformation which in turn limits anymembraneactionsand removal of a large amount of this stiffening may result in astructurewith the same oad carrying capability but using a reducedamount of rawmaterial.

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Chapter 4- Numerical methods for the prediction of silo wallpressures

4.1 IntroductionDue to the many limitations of the analytical methods for determining silo wallpressures hat arise from the assumptionsdescribed n section3.4, researchers aveturned to numerical methods. The current work aims to investigate he interactionbetween he ensiled material and the silo structurewhenthe wall is flexible and largedeflections occur using a numerical model. There are a number of numericalmethods available to the structural designer, some of which may be suitable foranalysing silo problems. Initially, the use of these numerical methods and theirpotential applicability to the current problem is assessed eforeresultsare presentedfor analysisperformed using a finite element method.4.2 Available numerical techniquesWith the advent of affordable yet powerful computers,numerical techniqueshavebeendeveloped that take advantageof this increasedcomputationalpower. Systemsof difficult and time-consuming equations can now routinely be solved in a matter ofminutes. There are three main techniques used for structural analysis; the finitedifference method, finite element method and boundary element method. Thesemethods rely on some form of discretization (dividing the problem into units thathave known geometry and properties) of the problem (this could be a fluid flowdomain, structural component etc. and the resulting mesh (combined withknowledge of the boundary conditions) is used to constructa systemof equations odescribe the problem mathematically. These equationscan then either be solvedimplicitly or explicitly (depending on the type of problem) to producethe requiredresults (stresses,pressuresetc. . There also exists a fourth method, the discreteelementmethod, that can be usedfor the simulation of granularbulk solidsproblems.This method models individual particles and the interactionbetweenhem,and would

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appear o be ideally suited or problemsof the type studiedhere. Thereare howeverlimitations to eachmethod,describedbelow.4.2.1 Finite difference methodThis method is most commonly applied to problems that can be describedas two-dimensional, such as the temperaturedistribution across a thin plate or the velocityprofile of a fluid at a cross-sectionof a duct. The system s divided into a regularlyspacedgrid anda function to describe he required variable is derived at each of thepoints on the grid. Assumingknowledge of the function's value at the boundaries,then solution of the system of equationscan be performed manually or by using asimple computer program. This method is not commonly used in commercialstructural analysisanddesignbecausehe equations o be solved must be determinedfor each separateproblem which would be extremely time-consuming for a largeanalysis. The finite difference method is the most established numerical methoddiscussedhere but is consideredunsuitablefor the complex problem that is posedbythe threedimensionalsimulationof silo filling.4.2.2 Finite element methodThis technique is currently one of the most commonly used numerical techniques formechanical engineering design. It can be applied to many different types of problem,with the result that one package can be used to solve problems in a wide range offields (structural, magnetic flux, fluids, electrical). As in the finite difference methodthe problem is divided into a series of smaller elements connected by nodes. Thesenodes can be compared to the grid points in the finite difference method but they donot have to be regularly spaced and thus, much more geometrically complexproblems can be modelled using the finite element technique.

There are a large number of commercial finite element packages available; whichgenerally include a GUI (Graphical User Interface). This makes it possible for theengineer to create complex models that may contain many thousands of degreesoffreedom without having to have specialist programming knowledge. In the past,

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Rectangular Silos; Interaction of Structure and Stored Bulk Solid