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Gary_Styger_M_Phil_Dissertation

  1. 1. AN INVESTIGATION OF THE EFFECT OF THE MANUFACTURING PROCESS ON THE PERFORMANCE OF CONVEYOR PULLEYS by GARY STYGER a dissertation submitted in partial fulfilment of the requirements for the degree of MAGISTER PHILOSOPHIAE In MECHANICAL ENGINEERING in the FACULTY OF ENGINEERING AND THE BUILT ENVIRONMENT at the UNIVERSITY OF JOHANNESBURG Supervisor: Professor R F Laubscher
  2. 2. An investigation of the effect of the manufacturing process on the performance of conveyor pulleys - G Styger 2 ABSTRACT Pulleys are critical items in belt conveyors. Their primary role is to drive large mining conveyor systems, facilitating the transportation of ore over extensive distances, both in South Africa and abroad. The effect of the manufacturing process (with specific emphasis on the induced residual stresses) on the fatigue performance of conveyor pulleys is herein investigated and reported. A pre-selected pulley was chosen based on size, suitable for experimental work as well as practical specifications. The static and fatigue performance of the pulley were investigated both with the current design criteria as well as Finite Element Analysis, with comparisons drawn. The material data for the Finite Element Models was obtained experimentally with tensile tests of the SANS 1431 350 WA plate. The magnitude of the residual stresses were obtained experimentally by using the incremental hole-drilling technique for non-uniform residual stresses. The method was verified by comparison with the Finite Element Analysis results for the non-linear material analysis of the roll-bending of the shell. The fatigue analysis revealed that the stress ranges of interest for the pulley were below the non-propagating stress range, and hence theoretically infinite fatigue life would be possible under constant amplitude conditions. The operational fatigue life required for the pulley would be possible, when considering the latest S-N curve for "very high cycle fatigue". The stress intensity factors for the weld details were also below the threshold value and hence crack growth should not occur, upon crack initiation. A new design criteria was proposed for the fatigue analysis considering either fatigue assessment standards or fracture mechanics for the assessment of the butt-welds. This investigation showed that the manufacturing-induced residual stresses may play a significant role in the fatigue life of a pulley. The fatigue strength of a machined stress - relieved joint is higher if the stress range is partly compressive. The fatigue strength of a machined as-welded joint is higher than estimated by the fatigue classifications. This is due to residual stress relaxation that occurs at the weld toe because of yielding and hence a subsequent reduction and redistribution of the residual stresses. This reduction in the mean stress level, with a stress range that is partly compressive, would mean an increase in the fatigue strength of the joint.
  3. 3. An investigation of the effect of the manufacturing process on the performance of conveyor pulleys - G Styger 3 This would in conclusion result in similar fatigue strengths for a stress-relieved and an as- welded joint. This would additionally depend on the extent of the reduction of the residual stress in the as-welded joint. Recommendations were suggested for further experimental and numerical work for both the T-bottom and Turbine-type pulleys.
  4. 4. An investigation of the effect of the manufacturing process on the performance of conveyor pulleys - G Styger 4 ACKNOWLEDGEMENTS I would like to thank the following people for their contribution to this dissertation: Professor R.F. Laubscher for his valued inputs and comments as well as continued support during this process. Mr. R. Shuma for the assistance with purchasing raw materials that were required for the experimental work. Mr. W. Dott for preparing the tensile samples. Mr. M. Mukhawana for the assistance with the tensile testing and residual stress measurement at the University of Johannesburg. Doug and Steve of CPM Engineering for the fabricated samples and information for the pre- selected pulley. Mr. W. Rall and Professor D.A. Hattingh from the Nelson Mandela Metropolitan University, whose support and assistance during the incremental hole-drilling was invaluable. I very much appreciated their insight. The support from Mr. P.E. Marsden and Dr. A. Kolahi of LUSAS was highly appreciated during the preparations of the FE models. My wife Dalene, whom has been a continuing support and understanding partner during this dissertation. My mother and father whose support and understanding was greatly appreciated.
  5. 5. An investigation of the effect of the manufacturing process on the performance of conveyor pulleys - G Styger 5 TABLE OF CONTENTS ABSTRACT _________________________________________________ 2 ACKNOWLEDGEMENTS_______________________________________ 4 1. INTRODUCTION__________________________________________ 25 1.1 Introduction ................................................................................................... 25 1.2 Aims of Investigation ..................................................................................... 27 1.3 Scope of Investigation ................................................................................... 28 1.4 Document layout ............................................................................................ 29 2. LITERATURE REVIEW ____________________________________ 30 2.1 Introduction ................................................................................................... 30 2.2 Conveyor pulley design ................................................................................. 30 2.2.1 Historic perspective ________________________________________________30 2.2.2 Conveyor pulley configurations _____________________________________31 2.2.3 Design procedure according to King _________________________________42 2.2.4 Design procedure according to Perry ________________________________48 2.3 Experimental investigation of conveyor pulleys ............................................ 59 2.3.1 Stress condition in belt conveyor drums______________________________59 2.3.2 Comparison of Experimental results and FEM for test pulley ____________62 2.4 Conveyor pulley standards ............................................................................ 64 2.4.1 Sizing of the pulley from SANS 1669-1:2005___________________________64 2.4.2 Design and manufacture of the pulleys according to Anglo American ____66 2.4.3 Anglo American Standard Pulleys ___________________________________68 2.5 Conveyor pulley failures................................................................................ 70 2.5.1 Conveyor pulley failures in South Africa ______________________________70 2.5.2 Failure analysis of conveyor pulleys _________________________________72 2.5.3 Case Study 1 : Fatigue failures of welded conveyor drums______________73 2.5.4 Case Study 2 : Fatigue in the shell of a conveyor drum _________________74 2.5.5 Case Study 3: Fracture analysis of a collapsed heavy-duty pulley in a long- distance continuous conveyor application __________________________________75 2.6 Conveyor pulley manufacture ........................................................................ 76
  6. 6. An investigation of the effect of the manufacturing process on the performance of conveyor pulleys - G Styger 6 2.6.1 The pulley shaft ___________________________________________________76 2.6.2 The locking element________________________________________________76 2.6.3 The end-disk ______________________________________________________77 2.6.4 The pulley shell____________________________________________________78 2.6.5 The pulley_________________________________________________________80 2.6.6 Assemble of the pulley _____________________________________________83 2.7 Residual stresses........................................................................................... 84 2.8 Residual stress due to butt-welds.................................................................. 86 2.8.1 Typical transverse and longitudinal residual stress in a butt-weld _______86 2.8.2 Estimated residual stress levels according to BS 7910 _________________86 2.8.3 Residual stresses in butt-welds of conveyor pulleys ___________________88 2.9 Reduction of residual stress due to thermal stress-relieving......................... 88 2.9.1 Stress relief procedures for welded low-carbon mild steel ______________88 2.10 Residual stress measurement methods......................................................... 89 2.10.1 Mechanical stress measurement methods ____________________________89 2.10.2 Residual stress measurements by diffraction _________________________91 2.11 Incremental Hole-Drilling for Non-Uniform Residual Stresses....................... 93 2.11.1 Incremental Strain Method __________________________________________93 2.11.2 Average Stress Method _____________________________________________94 2.11.3 Power Series Method _______________________________________________94 2.11.4 Integral Method ____________________________________________________95 2.11.5 Comparison of the methods_________________________________________95 2.11.6 Discussion of the non-uniform residual stress methods available _______96 2.12 Fatigue assessment in welds......................................................................... 96 2.12.1 Constant amplitude stress-life method _______________________________98 2.12.2 Constant amplitude strain-life method_______________________________107 2.12.3 Constant Amplitude Crack Growth __________________________________115 2.12.4 Nominal fatigue assessment according to BS 7608 ___________________120 2.13 Fatigue assessment according to classification standards ..........................122 2.13.1 Consideration of the mean stress influence in classification standards _122 2.13.2 Residual stress relaxation consideration in classification standards ____124
  7. 7. An investigation of the effect of the manufacturing process on the performance of conveyor pulleys - G Styger 7 2.13.3 Residual stress relaxation considerations according to IIW-1823-07 ____126 2.13.4 "Very high cycle fatigue" for constant amplitude loading ______________128 2.13.5 Modern conveyor pulley assessment________________________________128 2.14 Stress-relieving of conveyor pulleys.............................................................130 2.15 Finite Element Method ..................................................................................130 2.15.1 Introducing the Finite Element Method ______________________________130 2.15.2 Brief history of the Finite Element Method ___________________________131 2.15.3 Finite Element Method Formulation _________________________________132 2.15.4 Finite Element Method Process _____________________________________133 2.15.5 Description of the LUSAS Finite Elements ___________________________133 2.15.6 Use of the Finite Element Method in the assessment of pulleys ________139 2.15.7 Checking procedure for the Finite Element Method ___________________150 3. EXPERIMENTAL WORK __________________________________ 152 3.1 Introduction ..................................................................................................152 3.2 Tensile testing of SANS 1431 GR 350 WA plate ............................................152 3.2.1 Experimental procedure (Tensile Tests) _____________________________152 3.2.2 Discussion of the tensile test results ________________________________157 3.3 Residual stress measurement.......................................................................157 3.3.1 Residual stress measurement method_______________________________157 3.3.2 Preparation of the cylindrical sample (residual stress measurement) ___162 3.3.3 Heat input of the welds ____________________________________________166 3.3.4 Thermal stress-relieving of sample__________________________________168 3.3.5 Experimental procedure (residual stress measurement) _______________169 3.3.6 Uncertainty analysis (residual stress measurement) __________________175 3.3.7 Results of the residual stress measurement _________________________179 3.4 Material study of the cylindrical samples......................................................181 3.4.1 Experimental procedure (Material Study) ____________________________181 3.4.2 Results of the material study _______________________________________182 4. NUMERICAL AND ANALYTICAL WORK _____________________ 188 4.1 Introduction ..................................................................................................188 4.2 The Finite Element Models............................................................................188
  8. 8. An investigation of the effect of the manufacturing process on the performance of conveyor pulleys - G Styger 8 4.2.1 Description of the geometry________________________________________189 4.2.2 Material Properties________________________________________________192 4.2.3 Analyses_________________________________________________________193 4.2.4 Load and boundary conditions _____________________________________194 4.2.5 Finite Element Mesh_______________________________________________200 4.2.6 Number of Fourier harmonic components required ___________________205 4.3 Final Analyses...............................................................................................206 4.3.1 Results for the pulley______________________________________________206 4.3.2 Results for the shell_______________________________________________209 4.4 Comparison of current design techniques to the numerical study ...............211 4.4.1 Discussion of the conveyor design comparison ______________________213 4.5 Comparison of the bending residual stresses in the shell ............................213 4.5.1 Discussion of the results of the bending residual stress comparison study 214 4.6 Comparison of the Welding Residual Stresses in the Shell ..........................215 4.6.1 Discussion of the results of the welding residual stress comparison study 215 4.7 Fatigue Performance of the Pulley................................................................216 4.7.1 Fatigue assessment results according to the classification standards __218 4.7.2 Discussion of the fatigue performance assessment of the pulley _______221 4.8 Fatigue Assessment according to fundamental theory.................................222 4.8.1 Discussion of the results __________________________________________226 5. INFLUENCE OF RESIDUAL STRESS IN PULLEY DESIGN _______ 228 6. CONCLUSIONS AND RECOMMENDATIONS__________________ 232 6.1 Overview .......................................................................................................232 6.2 Literature Review ..........................................................................................232 6.3 Experimental Work........................................................................................234 6.4 Numerical Work ............................................................................................235 6.5 New criteria for conveyor pulley design........................................................236 6.6 Overall Conclusion .......................................................................................236 6.7 Recommendations ........................................................................................237
  9. 9. An investigation of the effect of the manufacturing process on the performance of conveyor pulleys - G Styger 9 APPENDIX 1 : TENSILE RESULTS_____________________________ 245 APPENDIX 2 : RESIDUAL STRESS RESULTS____________________ 247 APPENDIX 3 : CERTIFICATES ________________________________ 259 APPENDIX 4 : MATERIAL CERTIFICATES ______________________ 261 APPENDIX 5 : HEAT TREATMENT CERTIFICATES _______________ 263 APPENDIX 6 : NON-DESTRUCTIVE INSPECTION SHEET __________ 265 APPENDIX 7 : RESIDUAL STRESS MEASUREING EQUIPMENT COMPLIANCE CERTIFICATE _________________________________ 266 APPENDIX 8 : CONTOUR PLOTS OF THE RESULTS OF THE ROLL- BENDING SIMULATION _____________________________________ 267 APPENDIX 9 : INPUT DATA FILE OF THE 3D FINITE ELEMENT MODEL OF THE PULLEY ___________________________________________ 277 APPENDIX 10 : INPUT DATA FILE OF THE 2D FOURIER FINITE ELEMENT MODEL OF THE PULLEY ___________________________ 280 APPENDIX 11 : INPUT DATA FILE OF THE 2D FINITE ELEMENT MODEL OF THE SHELL ____________________________________________ 291
  10. 10. An investigation of the effect of the manufacturing process on the performance of conveyor pulleys - G Styger 10 LIST OF FIGURES Figure 1.1 : Typical layout of a Conveyor System [1].................................................... 25 Figure 1.2 : The components of a Conveyor Pulley [1] ................................................. 26 Figure 2.1 : Typical pulley configurations used in South Africa [5] .............................. 32 Figure 2.2 : Boss-type pulley [10] ................................................................................. 32 Figure 2.3 : Turbine-type pulley [10] ............................................................................. 33 Figure 2.4 : L-Bottom type pulley [10] ........................................................................... 34 Figure 2.5 : T-Bottom type pulley [10] ........................................................................... 35 Figure 2.6 : Weld along the face of the shell for the T-Bottom type pulley [11] ............. 37 Figure 2.7 : Large heavy shell rim bending decay [6].................................................... 38 Figure 2.8 : Small thin shell rim bending decay [6] ....................................................... 38 Figure 2.9 : Inside fillet and full penetration weld of Turbine-type pulley [11]............... 39 Figure 2.10 : Locking element Type "A" [11]................................................................. 40 Figure 2.11 : Locking element Type "B" [11]................................................................. 41 Figure 2.12 : Locking element Type "C" [11]................................................................. 42 Figure 2.13 : Static and fatigue stress model [6] ........................................................... 43 Figure 2.14 : Belt loading on the pulley diameter [6]..................................................... 43 Figure 2.15 : Rim bending of the shell [6] ..................................................................... 45 Figure 2.16 : S-N Curve for steel [15] ............................................................................ 48 Figure 2.17 : Stress concentration factors for filleted shafts in bending [16] ............... 49 Figure 2.18 : Surface finish modification factors for steel [7] ....................................... 49 Figure 2.19 : Modified Goodman diagram [7] ................................................................ 50 Figure 2.20 : Support and load distribution assumption for the shell [3] ...................... 50 Figure 2.21 : Circumferential load distribution assumption shell [3] ............................ 51 Figure 2.22 : Calculated and measured axial stresses in the pulley shell [3] ................ 51 Figure 2.23 : Mean - 2 standard deviation S-N curve [17].............................................. 52 Figure 2.24 : Determination of radial and hoop stress through the end-disk [4]........... 56 Figure 2.25 : Effect of disk shape [3]............................................................................. 58 Figure 2.26 : Effect of hub diameter/pulley diameter [3] ............................................... 59 Figure 2.27 : Diagram of drum with attached pick-ups [18]........................................... 60 Figure 2.28 : Analytical and experimental results in the middle of the shell (A-A) [18] . 61 Figure 2.29 : Analytical and experimental results in the extreme section of the shell (B- B) [18] ........................................................................................................................... 61 Figure 2.30 : Analytical and experimental results in the external contour section of the end-disk (C-C) [18] ....................................................................................................... 61 Figure 2.31 : Analytical and experimental results in the internal contour section of the end-disk (C-C) [18]........................................................................................................ 62 Figure 2.32 : Stresses measured on the test pulley [9] ................................................. 62 Figure 2.33 : Finite element analysis - shaft axial range stress [9] ............................... 63
  11. 11. An investigation of the effect of the manufacturing process on the performance of conveyor pulleys - G Styger 11 Figure 2.34 : Finite element analysis - Disk radial stress range [9]............................... 63 Figure 2.35 : Stresses measured on the test pulley during start-up [9] ........................ 64 Figure 2.36 : Dimensions of the conveyor pulley [20]................................................... 64 Figure 2.37 : Dimensions and tolerances of the turbine-type pulley [25] ...................... 69 Figure 2.38 : Full penetration fillet welds of the end-disk-to-shell interface of the turbine-type pulley [25] ................................................................................................. 69 Figure 2.39 : Dimensions and tolerances of the T-bottom type pulley [25] ................... 70 Figure 2.40 : Ground full penetration double butt-welds of the end-disk-to-shell interface of the T-bottom type pulley [25] ..................................................................... 70 Figure 2.41 : Position of position of failure at the weld root of a single -sided welded turbine-type pulley [6]................................................................................................... 71 Figure 2.42 : Shaft fatigue failure at locking element device shoulder [9]..................... 72 Figure 2.43 : Finish criteria of a shell circumferential weld in a T-bottom type pulley [9] ...................................................................................................................................... 72 Figure 2.44 : Shell fatigue failure at disk/shell circumferential weld [9]........................ 73 Figure 2.45 : Possible design of the shoulder in the end-disk [27] ............................... 74 Figure 2.46 : A fatigue region indicating beach marks and stress raisers [28] ............. 75 Figure 2.47 : Manufacture of a locking element ............................................................ 77 Figure 2.48 : T-bottom and turbine type end-disks ....................................................... 77 Figure 2.49 : Final machined end-disks ........................................................................ 78 Figure 2.50 : Roll-bending of the plate to produce the pulley shell............................... 78 Figure 2.51 : Sub-merged arc welding of the seam of the shell .................................... 79 Figure 2.52 : Ends of the shell being joined with sub -merged arc welding................... 79 Figure 2.53 : Inner weld of a T-bottom type pulley........................................................ 80 Figure 2.54 : Machining of the outer weld to return to sound weld............................... 80 Figure 2.55 : Dye penetrate testing of the machined weld before the final outer diameter circumferential weld runs are performed...................................................................... 81 Figure 2.56 : Completed outer diameter circumferential weld....................................... 81 Figure 2.57 : Machining of the outer diameter of the shell ............................................ 82 Figure 2.58 : Stress-relieved pulley............................................................................... 82 Figure 2.59 : Lagging on the pulley............................................................................... 83 Figure 2.60 : Balance weights welded to the outer radius of the end-disk.................... 83 Figure 2.61 : Complete pulley with Plummer block bearings mounted in position ....... 84 Figure 2.62 : Different cases of macro and micro residual stresses [31] ...................... 85 Figure 2.63 : Graph of the length scale indicating the different types of residual stresses verses grain size [31] ..................................................................................... 85 Figure 2.64 : Longitudinal and transverse residual stresses in a single-butt weld [33] 86 Figure 2.65 : Typical residual stress distribution in welded joints [32] ......................... 87 Figure 2.66 : Residual stress distribution measured from a shot-peened Ni alloy using hole-drilling method [31]............................................................................................... 90
  12. 12. An investigation of the effect of the manufacturing process on the performance of conveyor pulleys - G Styger 12 Figure 2.67 : Hoop, axial and shear stress through thickness [36] ............................... 90 Figure 2.68 : Contour method and neutron diffraction residual stress measurement comparison of the normal residual stress distribution in a welded plate [38] .............. 91 Figure 2.69 : Comparison of the calculation methods [39]............................................ 96 Figure 2.70 : Nominal stress approach for fatigue assessment of welds [41] ............... 97 Figure 2.71 : Structural stress approach for a weld [41] ............................................... 97 Figure 2.72 : Local strain approach for welds [41] ........................................................ 98 Figure 2.73 : Crack growth approach for welds [41] ..................................................... 98 Figure 2.74 : Log S-N curve fitted to data [41]............................................................... 99 Figure 2.75 : Correlation of fatigue strength and tensile strength of a material [42] ....100 Figure 2.76 : Determination of the S-N curve of a material [41]....................................101 Figure 2.77 : Effects of surface finish on the fatigue limit of steel [43] ........................101 Figure 2.78 : Surface finishes compared to fatigue limit and ultimate tensile strength [44]...............................................................................................................................102 Figure 2.79 : Stress concentration factors for aluminium alloys [45]...........................104 Figure 2.80 : Notch sensitivity for different material hardness [46] .............................104 Figure 2.81 : The affect of the fatigue notch factor on the fatigue life of a component [41]...............................................................................................................................105 Figure 2.82 : Goodman diagram [41]............................................................................106 Figure 2.83 : Axial strain range in a notch subjected to alternating stress [41] ...........107 Figure 2.84 : Cyclic stress-strain hysteresis loop [41] .................................................108 Figure 2.85 : Strain-life curve [41]................................................................................108 Figure 2.86 : Determination of the strain-life curve [41] ...............................................109 Figure 2.87 : Cyclic stress-strain curve [41].................................................................109 Figure 2.88 : Affect of surface finish on the strain-life curve [41] ................................110 Figure 2.89 : Fatigue notch factor as compared to weld toe radius [41] ......................111 Figure 2.90 : Local elastic-plastic stresses determined from nominal stresses [41]....112 Figure 2.91 : Stable cyclic stress-strain hysteresis loop [41].......................................113 Figure 2.92 : The affect of mean stress on the cyclic stress-strain hysteresis loop [41] .....................................................................................................................................114 Figure 2.93 : The influence of residual stress on the cyclic stress-strain hysteresis loop [41]...............................................................................................................................115 Figure 2.94 : Stress field around a crack [41]...............................................................116 Figure 2.95 : Geometry of cracks depending on position and loading [41] ..................116 Figure 2.96 : Sigmoidal da/dN - ΔK curve [41] .............................................................118 Figure 2.97 : Typical semi-elliptical shape for a surface crack [41]..............................119 Figure 2.98 : Edge and center cracks [41] ....................................................................119 Figure 2.99 : Weld classifications of two fillet welds [23].............................................120 Figure 2.100 : Mean-Line Weld classifications [23] ......................................................121
  13. 13. An investigation of the effect of the manufacturing process on the performance of conveyor pulleys - G Styger 13 Figure 2.101 : Scatter in the experimental results of fatigue tests of welded members [23]...............................................................................................................................122 Figure 2.102 : Influence of the residual stress on a stress cycle of an as-welded joint [33]...............................................................................................................................123 Figure 2.103 : Tensile mean stress due to the applied loading in a stress-relieved joint [33]...............................................................................................................................123 Figure 2.104 : Reduction of the stress range due to stress-relieving [49]....................124 Figure 2.105 : Reduction factor of the stress cycle for stress-relieving [50]................124 Figure 2.106 : Enhancement factor f(R) for residual stress level versus stress ratio [53] .....................................................................................................................................127 Figure 2.107 : Constant amplitude S-N curves for Steel for "very high cycle fatigue" [53] .....................................................................................................................................128 Figure 2.108 : Stress ranges assessed at the weld toes for the T-bottom type pulley as per BS 7608 [24]...........................................................................................................128 Figure 2.109 : Stress ranges assessed at the weld toes for the Turbine-type pulley as per BS 7608 [24]...........................................................................................................129 Figure 2.110 : Stress cycle below the endurance limit of a joint [50] ...........................129 Figure 2.111 : Finite Element Model of a structure [55] ................................................131 Figure 2.112 : Nodal configuration for Standard 2D Isoparametric element [55]..........132 Figure 2.113 : 3D beam element degrees of freedom [55] ............................................133 Figure 2.114 : 2D plane strain problem and finite element mesh [55] ..........................134 Figure 2.115 : Axisymmetric problem and finite element mesh [55] ............................134 Figure 2.116 : Stress output of the Axisymmetric element [55]....................................135 Figure 2.117 : Fourier element definition [55] ..............................................................135 Figure 2.118 : 3D element library [55] ..........................................................................138 Figure 2.119 : 3D mesh of component [55] ..................................................................139 Figure 2.120 : Stress outputs for the 3D element [55] ..................................................139 Figure 2.121 : King's analytical procedure compared with finite element analysis for Turbine and T-bottom type pulleys [6] .........................................................................140 Figure 2.122 : Cross-section of a pulley assembly [9] .................................................141 Figure 2.123 : Quadrant section of the finite element mesh [9]....................................142 Figure 2.124 : Effect of number of Fourier terms on belt (radial) loading approximation [9].................................................................................................................................143 Figure 2.125 : Effect of number of Fourier terms on torque (shear) loading approximation [9].........................................................................................................143 Figure 2.126 : Maximum von mises stress plot [9].......................................................144 Figure 2.127 : Quadrant section deformed shapes at different angles of wrap [9] .......144 Figure 2.128 : Comparison of the Finite Element Analysis and PSTRESS results [9] ..145 Figure 2.129 : Geometry and mesh of Option 1 and 2 respectively [27] .......................146
  14. 14. An investigation of the effect of the manufacturing process on the performance of conveyor pulleys - G Styger 14 Figure 2.130 : Quarter symmetry 3D mesh depicting loads and boundary conditions [27]...............................................................................................................................147 Figure 2.131 : Von mises contour plot of the quarter symmetry model [27] ................147 Figure 2.132 : Stresses vs angular position for the hot spot in option 1 (continuous duty : 250 kN) [27]........................................................................................................148 Figure 2.133 : Stresses vs angular position for the hot spot in option 2 (continuous duty : 250 kN) [27]........................................................................................................148 Figure 2.134 : Geometry of the half model for the finite element analysis [24] ............149 Figure 2.135 : Typical stress distribution in a T-bottom type pulley [24] .....................149 Figure 3.1 : Dimensions of the tensile specimen .........................................................154 Figure 3.2 : Machined tensile specimens.....................................................................154 Figure 3.3 : Instron 1195 universal testing machine ....................................................155 Figure 3.4 : Extensometer............................................................................................155 Figure 3.5 : Drilling through the centre of the strain gauge rosette [66] ......................158 Figure 3.6 : MTS-3000 RESTAN hole-drilling system [67] ............................................159 Figure 3.7 : Relieved stress at P along with the variation of the principal strains a distance away from the centre of the hole [68] ............................................................159 Figure 3.8 : Typical strain gauge rosette arrangement used for determining residual stress [64]....................................................................................................................160 Figure 3.9 : Biaxial uniform residual stress in a component [64] .................................160 Figure 3.10 : Biaxial non-uniform residual stress in a component [64] ........................160 Figure 3.11 : Dimensions of the weld preparation detail for the plate (all dimensions are in mm)..........................................................................................................................162 Figure 3.12 : Weld run-off plates at each end of roll-bent plate for longitudinal seam welds...........................................................................................................................163 Figure 3.13 : Sub-merged arc welding of the longitudinal seam ..................................163 Figure 3.14 : Back-gouging of the outer diameter weld ...............................................164 Figure 3.15 : Inner and outer diameter longitudinal seam welds ground -flush ............164 Figure 3.16 : Re-rolling of the cylindrical samples.......................................................165 Figure 3.17 : Circumferential weld runs .......................................................................165 Figure 3.18 : Back-gouging of the outer diameter weld ...............................................166 Figure 3.19 : Completed fabrication of the cylindrical shell sample ............................166 Figure 3.20 : Diagram of the RESTAN equipment experimental set-up [69].................169 Figure 3.21 : Measurement positions on the inside and outside diameter...................170 Figure 3.22 : Polished surface for application of the strain gauge...............................171 Figure 3.23 : EA-062RE-120 residual stress strain gauge rosette (Type A rosette) ......171 Figure 3.24 : Positioning of the strain gauge ...............................................................171 Figure 3.25 : RESTAN system secured in position over the strain gauge ....................172 Figure 3.26 : Method of optimising the degree of the polynomial [69] .........................173 Figure 3.27 : A typical polynomial fitted strain vs depth graph ....................................173
  15. 15. An investigation of the effect of the manufacturing process on the performance of conveyor pulleys - G Styger 15 Figure 3.28 : A typical stress vs depth graph...............................................................174 Figure 3.29 : Stress-relieved and as-welded cylindrical samples.................................181 Figure 3.30 : Samples prior to sectioning ....................................................................181 Figure 3.31 : Macro showing the cross-section through the weld for the stress-relieved condition......................................................................................................................182 Figure 3.32 : Macro showing the cross-section through the weld for the as-welded condition......................................................................................................................182 Figure 3.33 : Hardness profile across the weld bead for both conditions ....................183 Figure 3.34 : Micrograph showing the structure of the parent material for the stress- relieved condition ........................................................................................................184 Figure 3.35 : Micrograph showing the structure of the parent material for the non stress-relieved condition .............................................................................................184 Figure 3.36 : Micrograph showing the structure of the heat-affected zone for the stress- relieved condition ........................................................................................................185 Figure 3.37 : Micrograph showing the structure of the heat-affected zone for the non stress-relieved condition .............................................................................................185 Figure 3.38 : Micrograph showing the structure of the weldment for the stress-relieved condition......................................................................................................................186 Figure 3.39 : Micrograph showing the structure of the weldment for the non stress- relieved condition ........................................................................................................186 Figure 4.1 : Pulley assembly ........................................................................................190 Figure 4.2 : Exploded view of the pulley assembly ......................................................190 Figure 4.3 : Cross-section of the FE prepared pulley geometry (half symmetry model) (all dimensions are in mm)...........................................................................................191 Figure 4.4 : Dimensions of the end disk (all dimensions are in mm)............................191 Figure 4.5 : Half model of the pulley geometry in the FE system.................................191 Figure 4.6 : Quarter model of the 3D pulley .................................................................192 Figure 4.7 : Half model of the FE shell geometry .........................................................192 Figure 4.8 : Uniaxial yield stress vs total strain defined in LUSAS @ [55] .....................194 Figure 4.9 : Self-weight applied to the FE model..........................................................195 Figure 4.10 : Belt pressure distribution over the circumference of a driven pulley [79] .....................................................................................................................................195 Figure 4.11 : Graph of the Belt pressure vs circumferential position for the mesh sensitivity analysis ......................................................................................................196 Figure 4.12 : Graph of the Belt pressure vs circumferential position for the final Fourier analysis .......................................................................................................................197 Figure 4.13 : Various functions used to apply the belt pressure over the belt width [3] .....................................................................................................................................197 Figure 4.14 : Belt pressure applied to the half model of the pulley..............................197
  16. 16. An investigation of the effect of the manufacturing process on the performance of conveyor pulleys - G Styger 16 Figure 4.15 : Locking element interface pressure applied to the model for the end disk and shaft......................................................................................................................198 Figure 4.16 : Built-in boundary conditions for the mesh sensitivity and harmonic component assessment...............................................................................................198 Figure 4.17 : Simply-supported boundary condition of the final analysis of the pulley .....................................................................................................................................199 Figure 4.18 : Quarter symmetry model for the 3D analysis ..........................................199 Figure 4.19 : Graph of loading for the non-linear analysis...........................................199 Figure 4.20 : Loading of the shell.................................................................................200 Figure 4.21 : Symmetry and simply-supported boundary conditions of the shell........200 Figure 4.22 : Typical mesh of 2D analysis of the pulley...............................................201 Figure 4.23 : Typical mesh of the 2D plane strain analysis of the shell .......................201 Figure 4.24 : Contour plot of the maximum vertical displacement for the 3D analysis 202 Figure 4.25 : Contour plot of the maximum radial displacement of the pulley shell ....207 Figure 4.26 : Contour plot of the Von Mises stress in the pulley shell .........................207 Figure 4.27 : Contour plot of the axial stress in the pulley shell ..................................208 Figure 4.28 : Radial stress due to the assemble of the pulley......................................209 Figure 4.29 : Hoop stress due to the assembly of the pulley .......................................209 Figure 4.30 : Maximum bending stress through the thickness ....................................210 Figure 4.31 : Residual bending stress through the thickness......................................211 Figure 4.32 : Constant amplitude bending stress range ..............................................217 Figure 4.33 : Weld measurement position of the pulley...............................................217 Figure 4.34 : Weld measurement position of the pulley...............................................217 Figure 4.35 : True Stress-Strain Curve for NSR2..........................................................225 Figure 5.1 : Suggested modified design procedure for conveyor pulleys....................231
  17. 17. An investigation of the effect of the manufacturing process on the performance of conveyor pulleys - G Styger 17 LIST OF TABLES Table 2.1 : Basic outer diameter of the pulley [20]........................................................ 65 Table 2.2 : Preferred shaft and bearing journal diameters pulley [20].......................... 65 Table 2.3 : Surface finish of the shaft [29]..................................................................... 76 Table 2.4 : Surface finish of the locking element [29] ................................................... 77 Table 2.5 : Surface finish of the end-disk [29]............................................................... 77 Table 2.6 : Surface finish of the shell [29] ..................................................................... 79 Table 2.7 : Parametric ranges for recommended residual stress distributions [32] ...... 86 Table 2.8 : Summary of the measurement technique and their attributes [31] .............. 92 Table 2.9 : Determination of surface factor based on tensile strength of the material [41]...............................................................................................................................102 Table 2.10 : Load factor for type of loading condition [41]...........................................103 Table 2.11 : Evolution cases for typical welded components for reduction in residual stress [52]....................................................................................................................126 Table 2.12 : Weld classification according to BS 7608 [23] ..........................................129 Table 2.13 : Manufacturing specifications of both options [27] ...................................146 Table 3.1 : SANS 1431 GR 350 WA Tensile specimen information ...............................154 Table 3.2 : SANS 1431 GR 350 WA Mechanical properties for the specimens .............156 Table 3.3 : SANS 1431 GR 350 WA Average Mechanical Properties for the Specimens .....................................................................................................................................157 Table 3.4 : Placement of the principal angle β [64] ......................................................162 Table 3.5 : Longitudinal seam weld of shell 1 (Inner Diameter)....................................167 Table 3.6 : Longitudinal seam weld of shell 1 (Outer Diameter)...................................167 Table 3.7 : Longitudinal seam weld of shell 2 (Inner Diameter)....................................167 Table 3.8 : Longitudinal seam weld of shell 2 (Outer Diameter)...................................167 Table 3.9 : Circumferential weld of the shell (Inner Diameter)......................................168 Table 3.10 : Circumferential weld of the shell (Outer Diameter)...................................168 Table 3.11 : Holding temperature for stress-relieving..................................................168 Table 3.12 : Contributions of uncertainty in residual stress measurement [70]: ..........176 Table 3.13 : ASTM E837-08 Non-Uniform maximum stress results nearest to the surface for the as-welded and stress-relieved conditions........................................................179 Table 4.1 : Overall dimensions of the pulley................................................................190 Table 4.2 : SANS 1431 GR 350 WA Average mechanical properties for the specimens .....................................................................................................................................193 Table 4.3 : Uniaxial stress vs total strain for the 350 WA plate ....................................194 Table 4.4 : Circumferential belt pressure for the mesh sensitivity analysis.................196 Table 4.5 : Circumferential belt pressure for the final Fourier analysis .......................196 Table 4.6 : Types of elements used for the analyses [55] ............................................200 Table 4.7 : Results of mesh sensitivity assessment of the 3D Linear Analysis............202
  18. 18. An investigation of the effect of the manufacturing process on the performance of conveyor pulleys - G Styger 18 Table 4.8 : Results of mesh sensitivity assessment of the axisymmetric analysis ......203 Table 4.9 : Results of the mesh sensitivity assessment of the Fourier analysis..........204 Table 4.10 : Results of the mesh sensitivity assessment of the Non-Linear Analysis .204 Table 4.11 : The results of the harmonic component assessment ...............................205 Table 4.12 : Results of the final analysis......................................................................206 Table 4.13 : Results of the assembly analysis .............................................................208 Table 4.14 : Results after loading for the bending analysis .........................................210 Table 4.15 : Results after unloading for the bending analysis .....................................210 Table 4.16 : Comparison of King's design calculation to the FEA for the static results .....................................................................................................................................212 Table 4.17 : Comparison of the King's design calculation to the FEA for the fatigue results..........................................................................................................................212 Table 4.18 : Comparison of Perry's design calculation to the FEA for the displacement results..........................................................................................................................212 Table 4.19 : Comparison of Perry's design calculation to the FEA for the static results .....................................................................................................................................212 Table 4.20 : Comparison of the Perry's design calculation to the FEA for the fatigue results..........................................................................................................................213 Table 4.21 : Comparison of estimated residual stress .................................................214 Table 4.22 : Comparison of measured and estimated residual stress for the as-welded condition......................................................................................................................215 Table 4.23 : Comparison of measured and estimated residual stress for the stress- relieved condition ........................................................................................................215 Table 4.24 : (BS EN 1993-1-9) - Fatigue ratio in the hoop direction for the as-welded condition......................................................................................................................218 Table 4.25 : (BS EN 1993-1-9) - Fatigue ratio in the axial direction for the as-welded condition......................................................................................................................218 Table 4.26 : (BS 7608) - Fatigue ratio in the hoop direction for the as-welded condition .....................................................................................................................................218 Table 4.27 : (BS 7608) - Fatigue ratio in the axial direction for the as-welded condition .....................................................................................................................................218 Table 4.28 : (RP-C203) - Fatigue ratio in the hoop direction for the as-welded condition .....................................................................................................................................218 Table 4.29 : (RP-C203) - Fatigue ratio in the axial direction for the as-welded condition .....................................................................................................................................219 Table 4.30 : (IIW-1823-07) - Fatigue ratio in the hoop direction for the as-welded condition......................................................................................................................219 Table 4.31 : (IIW-1823-07) - Fatigue ratio in the axial direction for the as-welded condition......................................................................................................................219
  19. 19. An investigation of the effect of the manufacturing process on the performance of conveyor pulleys - G Styger 19 Table 4.32 : (BS EN 1993-1-9) - Fatigue ratio in the hoop direction for the stress-relieved condition......................................................................................................................219 Table 4.33 : (BS EN 1993-1-9) - Fatigue ratio in the axial direction for the stress-relieved condition......................................................................................................................220 Table 4.34 : (BS 7608) - Fatigue ratio in the hoop direction for the stress-relieved condition......................................................................................................................220 Table 4.35 : (BS 7608) - Fatigue ratio in the axial direction for the stress-relieved condition......................................................................................................................220 Table 4.36 : (RP-C203) - Fatigue ratio in the hoop direction for the stress-relieved condition......................................................................................................................220 Table 4.37 : (RP-C203) - Fatigue ratio in the axial direction for the stress-relieved condition......................................................................................................................221 Table 4.38 : (IIW-1823-07) - Fatigue ratio in the hoop direction for the stress-relieved condition......................................................................................................................221 Table 4.39 : (IIW-1823-07) - Fatigue ratio in the axial direction for the stress-relieved condition......................................................................................................................221 Table 4.40 : Fatigue properties for the assessment .....................................................224 Table 4.41 : Fatigue outputs for the as-welded condition of the pulley........................224 Table 4.42 : Fatigue outputs for the stress-relieved condition of the pulley ................225 Table 4.43 : Fatigue outputs for the as-welded condition of the pulley (Residual Stress = Yield Strength) ..........................................................................................................226 Table 4.44 : Allowable stress range for "very high cycle" constant amplitude fatigue of the pulley, according to IIW-1823-07............................................................................226
  20. 20. An investigation of the effect of the manufacturing process on the performance of conveyor pulleys - G Styger 20 LIST OF SYMBOLS a Asymmetric term am Fourier coefficient for the cosine function α Angle of wrap (radians) b Inner diameter of the end-disk (mm) bm Fourier coefficient for the sin function B Bearing center (mm) a, b and u Constants depending on the shape of the end-disc C.V. Coefficient of Variance d Shaft diameter under the end-disk (mm) D Pulley diameter (mm) Di Inside diameter of the shell (mm) ds Bearing shaft diameter (mm) Ds Stepped-up shaft diameter (mm) E Young's modulus (MPa) F Functions of the poisson's ratio F Vector of known forces (N) H Non-dimensional depth from surface h Contact length/width of the locking element (mm) h1 Width of the hub (mm)
  21. 21. An investigation of the effect of the manufacturing process on the performance of conveyor pulleys - G Styger 21 hr Width of the hub at the radius of interest (mm) Hb Rim section force (N) Is Second Moment of Area of shaft (mm 4 ) I Current (Amp) k Constant for the geometry of the shell k distance from fillet to disc centre (mm) K Global stiffness matrix K5 End-disk stiffness constant K6 Shaft stiffness constant K7 End-disk stiffness constant l Belt width (mm) L Spacing between end-disks (mm) m Fourier component number m The total number of harmonics in the series from n = 0, or n = 1 to n = m Mb Rim bending section moment (N.mm) M Shaft bending moment (N.mm) Md End-disk bending moment (N.mm) n Harmonic component number Pb Belt pressure (MPa) Ph Locking element hub pressure (MPa)
  22. 22. An investigation of the effect of the manufacturing process on the performance of conveyor pulleys - G Styger 22 Pu Locking element pressure on the shaft (MPa) PL Pressure loss in contact due to the belt forces (MPa) P1 Locking element pressure on the inner radius of the end-disk (MPa) Pm Mean pressure (MPa) Q Heat input per unit length (kJ/mm) ra Radius of the drilled hole (mm) rb Rim bending radial deflection due to belt pressure (mm) r1 Radial expansion due to the locking element pressure (mm) R Ratio of end-disk diameters rt Radius of the shell (mm) Red Ratio of the hub diameter/pulley diameter r1 Inner radius of the end-disk (mm) rr Radius of interest at the hub portion of the end-disk (mm) rm Mean radius of the strain gauge circle (mm) s Symmetric term s Standard Deviation s Velocity (mm/s) σb Hoop stress due to belt pressure (MPa) σ1 Hoop stress at the inside diameter of the shell (MPa) σab Axial rim stress (MPa)
  23. 23. An investigation of the effect of the manufacturing process on the performance of conveyor pulleys - G Styger 23 σhb Hoop rim stress (MPa) σd Radial direct stress (MPa) σa Axial bending stress (MPa) σa max. Maximum axial stress (MPa) σh max. Maximum hoop stress (MPa) σXG Axial stress (MPa) σYG Hoop stress (MPa) σR Radial stress (MPa) σH Hoop stress (MPa) σf Flow stress for a plane strain elastic-perfectly plastic isotropic material (MPa) σY Yield or 0.2% Off-Set Strength of the material (MPa) σX Bending stress on the top and bottom surface (MPa) σ‟X Bending residual stress on the top and bottom surface (MPa) T1 Tight belt tension (N) T2 Slack belt tension (N) T Instantaneous belt tension (N) Te Equivalent belt tension (N) t Shell thickness (mm) ted Thickness of the end-disk (mm)
  24. 24. An investigation of the effect of the manufacturing process on the performance of conveyor pulleys - G Styger 24 Tx Tension distribution along the belt width (N) τXυ Shear stress (MPa) tLE Contact length/width of the locking element (mm) θ Active drive arc (radians) θ Angular position (degrees) θs Deflection angle (minutes) ν Poisson's ratio V Volt (V) W Belt width (mm) ̅ Mean X Functions of the poisson's ratio XF Functions of the poisson's ratio X Vector of unknown field quantities such as displacement (mm) z Hole depth (mm) Z Shell constant (mm), Depth from surface (mm) λ Constant for the shape of the shell
  25. 25. An investigation of the effect of the manufacturing process on the performance of conveyor pulleys - G Styger 25 1. INTRODUCTION 1.1 Introduction The mining and industrial sectors process raw material into final products to be used by various industries. The distances between the raw material delivery and the subsequent process undertaken are often large, and require an automated means of transportation of the material. Conveyor systems facilitate the automation of the moving of goods, removing the requirement for expensive labour and allowing it to be done safely. Materials handling and packaging industries are the main users of this effective transportation system. Different configurations of conveyor systems exist, such as:  Chain conveyor  Screw conveyor  Belt conveyor The typical layout of a belt conveyor system configuration, as used by the mining industry is shown in Figure 1.1 (below). The main components, the head and tail pulleys, are an integral part of the system, and are the focus of this dissertation. Figure 1.1 : Typical layout of a Conveyor System [1] The head pulley is often referred to as the drive-end pulley, as the motor and gearbox are mounted on this end, to drive the system. The tail end pulley is positioned on the non-drive end of the system. The snub pulley is appropriately positioned to achieve the required angle of wrap for the head pulley. The bend pulleys guide the belt towards the take-up pulley. The
  26. 26. An investigation of the effect of the manufacturing process on the performance of conveyor pulleys - G Styger 26 take-up pulley supports a weight, which provides the initial tension for the system. Idlers are positioned on each side of the belt between the head and tail pulleys. A typical conveyor pulley assembly is presented in Figure 1.2 (below): Figure 1.2 : The components of a Conveyor Pulley [1] The pulley consists of the following components:  The shell is constructed from rolled plate, with the ends of the plate welded together.  The lagging is a rubber layer with cut grooves, to assist with traction and grip between the pulley and conveyor belt.  The end plate or end-disk is machined from forged or rolled plate.  The shaft is machined from low-carbon steel round bar.  The locking element is machined from high-strength steel and is used to induce a “press-fit” between the shaft and the pulley, once assembled. The manufacturing process of the pulley is typically as follows:
  27. 27. An investigation of the effect of the manufacturing process on the performance of conveyor pulleys - G Styger 27  An end-disk is circumferentially welded into each end of the shell. Two different end plate configurations currently exist, and these will be discussed later.  The end-user may specify that the pulley is stress-relieved before the final assembly.  The rubber lagging is applied to the shell of the pulley.  The locking elements are placed in each end plate, with the shaft in position. The locking elements are then bolted up to the required torque, thereby inducing the appropriate contact pressure, between the shaft and pulley. The contact pressure is induced due to the cone angle of the locking element faces, which ensures that the inner and outer rings, contract and expand respectively.  The completed pulley assembly is then placed into the bearing assemblies, which are in turn bolted to the conveyor system structure. The design of the pulleys is based on an empirical approach that considers calculations separately for each component, and in turn their effect on each other. Static and fatigue loading are considered in the evaluation of the design, by the choice of appropriate design applied service loads. The current design philosophy for conveyor pulleys does not, however, consider the effect of the manufacturing process as part of the design philosophy. The design process allows for the inclusion of the assembly and in-service induced loads only. Because of the nature of the manufacturing techniques employed, residual stresses may also be additionally introduced and could therefore affect the in-service performance. Currently these are ignored for smaller-sized pulleys. Larger pulleys are sent for stress relieving. The current design philosophy therefore does not specifically cater for residual stresses introduced by the manufacturing. It's perceived effect is however widely recognised and addressed by appropriate stress relieving. The nature and effectiveness of the stress relieving is however not well documented. 1.2 Aims of Investigation The main aim of this investigation is, therefore to assess the effect of the manufacturing process (with specific emphasis on the introduced residual stresses) on the structural integrity of a conveyor pulley, in-service. This implies the following: 1. Investigating the current conveyor pulley design philosophy and manufacturing technique, with specific emphasis on the introduction of residual stresses.
  28. 28. An investigation of the effect of the manufacturing process on the performance of conveyor pulleys - G Styger 28 2. Identifying possible sources of manufacturing-induced residual stresses, and quantifying and assessing their severity. 3. Investigating the effect of these residual stresses on pulley performance, in order to develop a guideline of how it may be included in a conveyor pulley design. 1.3 Scope of Investigation To achieve the stated aims above the scope of the investigation are then summarised in the following steps: 1. Complete a detailed literature review of pulley manufacture and design to ascertain the current state of the technology. 1 2. Review the manufacturing techniques employed to determine their effect on the introduction of residual stresses. 3. Experimentally determine the residual stress state of the critical components of a pre- selected T-Bottom type pulley. 4. Evaluate the effectiveness of the stress-relieving process by duplicating the experimental investigation mentioned above, for a stress-relieved T-Bottom type pulley. 5. Evaluate the effect of the experimentally determined residual stresses by appropriate finite element analysis of a pulley subjected to in-service loads. 6. Reach a conclusion regarding the relevance of the manufacturing-induced residual stresses and the use of stress-relieving to reduce their effect. Recommend possible inclusion into the design process. 1 The literature review is extensive and may seem excessively long. It is how ever seen as one of significant contribution of this dissertation as it summarises a complex design issue of an important mechanical engineering product in a compact and easily accessible (UJ Open Library) document.
  29. 29. An investigation of the effect of the manufacturing process on the performance of conveyor pulleys - G Styger 29 1.4 Document layout Chapter 2 presents a literature review, to ascertain the current state of the technology relating to conveyor pulley design. It also introduces the reader to the manufacturing techniques employed and the effect thereof. A brief introduction of the Finite Element Method, is also presented. Chapter 3 describes the experimental work conducted for a pre-selected T-Bottom type pulley to determine the levels of the residual stresses induced due to the manufacturing processes employed. The findings of the tensile tests for determining the material properties and residual stress measurements, are reported, for both the stress-relieved and non stress- relived states. Chapter 4 describes a numerical and analytical evaluation of the effect of residual stress on a pre-selected T-Bottom type pulley. A comparison is made between the current design process and the results of the finite element analysis. The performance of the pulley is estimated based on high cycle, low stress fatigue life, and is compared with relevant fatigue standards. Chapter 5 discusses the influence of the residual stresses and their possible implementation in the current empirical and numerical design processes as described in a flow chart. Chapter 6 presents the conclusions drawn on the relevance of the residual stresses and the use of stress-relieving to reduce them. Recommendations are also made regarding the possible inclusion of this effect within the current empirical and numerical design processes.
  30. 30. An investigation of the effect of the manufacturing process on the performance of conveyor pulleys - G Styger 30 2. LITERATURE REVIEW 2.1 Introduction This chapter is a literature review of the current state of the technology regarding conveyor pulley design. The manufacturing techniques employed and the effect thereof are also introduced. The chapter concludes with a brief introduction to the Finite Element Method. 2.2 Conveyor pulley design The history of the design of conveyor pulleys over the past 80 years, both in Europe and in South Africa is discussed. Different pulley configurations are then outlined, along with their advantages from a functional and manufacturing point of view. The keyless connections used in pulley design are detailed as well as the implications of the various configurations. The design procedures by King and Perry as used in South Africa are discussed in detail. Current conveyor pulleys standards that are used in industry, are reviewed. Prominent conveyor pulley failures experienced in industry, and the affect on the design of pulleys, are also discussed. 2.2.1 Historic perspective The origins of systematic conveyor pulley design can be traced back to the 1930s and 1940s when classical theory of plates and shells, as developed by Timoshenko et al. [2], was utilised. The research that followed in the 1960s and 1970s in Germany, then set the benchmark for the fundamental understanding of pulley behaviour, which is still used today. Lange [3] was the first to formalise the representation of the triaxial stress state in the pulley shells and end-disks as a Fourier series expansion. Schmoltzi [4] investigated the contact stress field of the keyless locking element connections (which became popular at the time) used between the shaft and pulley. These workers also conducted extensive strain gauging to verify their work. Their work will be described in further detail in a subsequent section. The application of the work of Lange and Schmoltzi in South African industry was discussed by King [5] in 1983. He concluded that the reasons for the slow uptake of these practices into local design offices related to their highly technical and mathematical nature. He also subsequently compared local manufacturing practices with those employed in Europe. He identified the following differences:
  31. 31. An investigation of the effect of the manufacturing process on the performance of conveyor pulleys - G Styger 31  The European shaft and end-disk materials were of significantly higher tensile strength than used in local construction.  European pulley diameters were larger (1 000 to 1 750 mm), while the pulley widths were similar to local practices of the time.  The European manufacturing methods aimed at low mass. The local methods of focused less on mass due to lack of standardisation and the undemanding conditions under which local pulleys operated. King [6] developed a design procedure for the pulleys that could be used in a drawing office. His work considered both steady-state static and fatigue conditions of the pulley, and is still extensively used by the local industry. Perry [7] used fundamental fatigue theory and the works of Lange and Schmoltzi to develop a design procedure for conveyor pulleys. These design procedures will be detailed in a subsequent section. Qiu et al. [8] developed a new pulley stress analysis system based on the modified matrix method. The system allowed for simpler analyses than was possible with Finite Element Analysis (FEA) at the time. The system could also correctly determine the complex stress state at the shell-to-end-disk interface which had not previously been possible. The integrated analysis of the pulley and shaft as in FEA, was also possible. Sethi et al. [9] described the Conveyor Dynamics, Inc. (CDI) design criteria of pulleys, with a test case investigated empirically, as well as with FEA and experimental work. Recently, FEA has been used more frequently in conveyor pulley design due to its widespread use and ability to model the complex interaction of the shell and shaft under the bending conditions resulting from belt tensions. 2.2.2 Conveyor pulley configurations The pulley construction configurations have developed based on the critical areas of connection between the components, namely the end-disk to shell and the end-disk to the shaft. The end-disks are typically welded inside the shell or along the face of the shell. The end-disk is connected to the shaft via a keyed connection or shrink-fit, or by the currently used keyless connection utilising locking elements to induce an interference-type fit between the shell and the shaft. King [5] reviewed typical pulley construction configurations used in South Africa (Figure 2.1, below). Shaded areas indicate the stress levels, within the pulley. A darker shading indicates a higher stress level.
  32. 32. An investigation of the effect of the manufacturing process on the performance of conveyor pulleys - G Styger 32 Figure 2.1 : Typical pulley configurations used in South Africa [5] The most commonly used pulley configurations along with their advantages and disadvantages, are now discussed. 2.2.2.1 Boss-type pulley This pulley (see Figure 2.2) is specifically suited for light to medium duty applications. The pulleys incorporated plates fillet welded to mild steel bosses fitted to the shaft with an interference-fit. Drive pulleys used parallel keys if the torque requirements were high. Figure 2.2 : Boss-type pulley [10] The typical dimensions for this type of pulley are:  Diameters of 200 to 1 000 mm.  Belt widths of 500 to 1 200 mm.  Shaft diameters of 40 to 150 mm.
  33. 33. An investigation of the effect of the manufacturing process on the performance of conveyor pulleys - G Styger 33 The advantages of this type of pulley construction are:  Low cost.  Maintenance-free.  Shaft fixed for life of the pulley.  Tolerates higher deflection. The main disadvantage of this type of pulley construction is:  Shaft is not removable. 2.2.2.2 Turbine-type pulley This pulley (Figure 2.3) is suited to medium duty applications, with the option of a removable shaft. The end-disk is designed to allow flexure near the shell, by virtue of a reduced thickness. This reduces stress levels in the weld at the end-disk-to-shell interface and the locking element. The end-disk is then thicker at the hub, in order to distribute the induced pressure of the locking element into the end-disk and then into the shell. The locking elements must be sized correctly to ensure that the transmittable torque is not exceeded during operation. Figure 2.3 : Turbine-type pulley [10] The typical dimensions for this type of pulley are:  Diameters of 200 to 1 250 mm.  Belt widths of 500 to 2 100 mm.  Shaft diameters of 50 to 260 mm.
  34. 34. An investigation of the effect of the manufacturing process on the performance of conveyor pulleys - G Styger 34 The advantages of this type of pulley construction are:  Cost effective.  Removable shaft.  Solid end-disk - no welds in the shaft area. The disadvantages of this type of pulley construction are:  Locking element failure if overloaded.  Tolerates less deflection than the boss-type pulley.  Locking element bolts protrude past end-disk face. 2.2.2.3 L-Bottom type pulley The stresses in the weld of the end-disk-to-shell interface is reduced in this type of pulley, by being moved along the face of the shell. This type of pulley is normally used when shaft diameters are greater than or equal to 200 mm, and the pulleys are non-drive. The pulleys can be utilised for the drive-end, as long as the turbine-type narrow locking elements torque capacity is not exceeded. Wide bearing centers can only be used for this type of pulley. Stress-relieving of the pulley shell is recommended. Figure 2.4 : L-Bottom type pulley [10] The typical dimensions for this type of pulley are:  Diameters of 200 to 1 250 mm.  Belt widths of 500 to 2 100 mm.  Shaft diameters of 50 to 300 mm.
  35. 35. An investigation of the effect of the manufacturing process on the performance of conveyor pulleys - G Styger 35 The advantages of this type of pulley construction are:  Removable shaft.  Solid end-disk - no welds in the shaft area.  Shell weld is in a low bending stress area. The disadvantages of this type of pulley construction are:  Locking element failure if overloaded.  Tolerates less deflection than boss type.  Difficult to handle, as it has no lip on the shell.  Locking element bolts protrude past end-disk face. 2.2.2.4 T-Bottom type pulley The T-bottom pulley (Figure 2.5) also uses the principle of an off-set face welded end-disk to the shell as in the L-bottom type pulley. The pulley is typically used when the shaft diameters are greater than or equal to 200 mm. The wider end-disk is suitable in drive-end applications when the turbine-type locking element torque transmission capacity has been exceeded, and the wider locking element must be used. This pulley is also suitable for heavy-duty applications for non-drive pulleys. This type of pulley can only be used on wide bearing centers, and stress-relieving of the pulley shell is recommended. Figure 2.5 : T-Bottom type pulley [10]
  36. 36. An investigation of the effect of the manufacturing process on the performance of conveyor pulleys - G Styger 36 The typical dimensions for this type of pulley are:  Diameters of 200 to 1 250 mm.  Belt widths of 500 to 2 100 mm.  Shaft diameters of 100 to 400 mm. The advantages of this type of pulley construction are:  Heavy duty.  Removable shaft.  Solid end-disk - no welds in shaft area.  Shell weld is in low-bending stress area. The disadvantages of this type of pulley construction are:  Expensive.  Tolerate less deflection than boss type.  Locking element bolts protrude past end-disk face. 2.2.2.5 Summary of the behaviour of the commonly used end-disk types Summary of the behaviour of T-bottom end-disk type:  As previously mentioned, the critical concern in the design of a pulley shell is the configuration of the end-disk. This is particularly with regard to the degree of flexibility at the end-disk-to-shell interface, and the correct sizing of the diameters of the hub portion of the end-disk required, due to the locking element pressure induced.  The T-bottom pulley type is best suited to medium and heavy duty applications, as the main circumferential weld is located away from the high stress zone, caused by bending and the end-disk locking expansion (see Figure 2.6). The weld‟s location away from the end-disk also make it easier for a high quality butt-weld to be achieved.
  37. 37. An investigation of the effect of the manufacturing process on the performance of conveyor pulleys - G Styger 37 Figure 2.6 : Weld along the face of the shell for the T-Bottom type pulley [11]  The large radii used in the T-bottom type end-disk reduces the stress concentrations and allows the stresses to redistribute more uniformly (see Figure 2.5).  The end-disk is profiled to ensure the stresses throughout are constant, the stress concentrations are minimised, and the thicknesses are not excessive This allows appropriate flexibility for the critical interfaces.  The T-bottom type of end-disk is expensive to manufacture. This is due to the amount of machining required to achieve the T-bottom profiled shape of the section. It is therefore only recommended for heavy-duty applications, where possible.  This configuration positions the weld away from the highest radial and axial stresses. However, there is still a reasonable level of stress that should not be ignored. Figures Figure 2.7 and Figure 2.8 (below) show the dissipation of the stresses along the length of the shell, and related to shell thickness. Complete decay is only achieved for a few thicknesses.
  38. 38. An investigation of the effect of the manufacturing process on the performance of conveyor pulleys - G Styger 38 Figure 2.7 : Large heavy shell rim bending decay [6] Figure 2.8 : Small thin shell rim bending decay [6] Summary of the behaviour of Turbine end-disk type:  Beneficial for use in light to medium-duty. It has similar characteristics to the T-bottom type end-disk, except that the top of the end-disk is welded to the inside diameter of the shell and it uses a different locking element (see Figure 2.9) .  The weld is positioned in a high stress zone. However, because this type of pulley is subjected to lighter loads, as long as the connection is performed with a full penetration weld with no lack of fusion and a inner fillet, for reducing the possibility of cracking through the throat of the weld, then this design is far more affordable than the T-bottom type of end-disk.
  39. 39. An investigation of the effect of the manufacturing process on the performance of conveyor pulleys - G Styger 39 Figure 2.9 : Inside fillet and full penetration weld of Turbine-type pulley [11]  The limited machining required for this type of end-disk, and the implementation of suitable welding practices renders this configuration the most suitable for a wide range of applications. 2.2.2.6 Types of locking elements Over the last 30 years, the keyless connection of the end-disk to shaft, made by virtue of locking elements, has become the most popular connection method. The locking element works based on a similar concept to that of a cone clutch with the addition of the inner and outer rings being split to eliminate shearing loads on the bolts. The inner ring (inner segment) contracts while the outer ring (outer segment) expands, upon tightening of the bolts. The locking elements facilitate the removal of the shaft. Additional advantageous characteristics are:  The torque applied to bolts induces a expansion of the outer ring and contraction of the inner ring thus inducing controlled pressure on the hub portion of the end-disk and shaft respectively. This allows determination of fairly accurate sizing of the hub portion of the end-disk, with thick-walled cylinder theory. This eliminates unnecessary end-disk material.  The torque transmission capacity of the locking element is determined by the torque applied to the bolts thus eliminating the need for keys and keyways. Thus additional stress concentrations are avoided.  The locking element can withstand high axial thrust, virtually eliminating the possibility of the pulley moving axially on the shaft. If the pulley were to move on the shaft, the locking elements could be loosened and the pulley re-positioned and then re- tightened once in position. A similar event with a shrink-fit or key-way would cause the bore to be worn rendering the entire pulley redundant.
  40. 40. An investigation of the effect of the manufacturing process on the performance of conveyor pulleys - G Styger 40 The different locking element configurations are: Locking Element Type "A" Locking element type "A" was used by the South African market about 20 years ago and was the most common locking element used in conveyor pulleys at the time. In the unloaded condition, it had the best stress pattern of the three locking elements to be discussed. However, when load is applied to the shell and the shaft deflects, the stress pattern changes. In this configuration. because of the angle of the tapered segments, it is neither self-centering nor self-locking. Thus, at all times the securing bolts are in tension, and it becomes necessary to use a centralising ring as shown in Figure 2.10 (below), as well as having to limit the deflection of the shaft to ensure that the bolts do not exceed the elastic limit. The acceptable level of deflection for this type of locking element is about 1/2500 of the bearing centers or a slope of 0.05 degrees. Although this deflection restriction is not totally unfavourable, it often forced the designer to use a bigger shaft than usual, in order to limit deflection of the shaft. The most unfavourable characteristic of this type of locking element is that it is virtually impossible to control the mating surfaces of the segments. This can result in high point loads initially, and also make it necessary to re-torque the bolts after the initial running in period, after the segments have settled. Figure 2.10 : Locking element Type "A" [11] Locking Element Type "B" This configuration type (Figure 2.11, below) is used with medium duty pulleys, such as the turbine or flat bottom design. The reason for using this type of locking element and not type "A", is that it has the following advantages:
  41. 41. An investigation of the effect of the manufacturing process on the performance of conveyor pulleys - G Styger 41  Due to the angle of the tapers on the segments, these elements are both, self- centering and self-locking. Therefore they do not require a centralising ring behind the end-disk. Because of the elimination of this ring, it is always possible to withdraw the shaft from the drum, because the inevitable build-up of rust between the centralising ring and the shaft does not occur.  Due to the self-locking tapers, the bolts are not in tension to the same extent as type "A". Hence greater shaft deflections are possible in the order of 1/1800 to 1/2000 of the bearing centers or a slope of 0.06 degrees.  Unlike locking element type "A" it is not so critical that the locking element bolts are tightened in sequence. This is because of the tapers it is impossible to tighten this element unsymmetrically.  The major disadvantage of this type of locking element, is that it cannot transmit the same torque as type "A". However, on conventional pulleys this seldom causes a problem as the torque to be transmitted by the shaft is usually well within the capabilities of the chosen locking element. Figure 2.11 : Locking element Type "B" [11]
  42. 42. An investigation of the effect of the manufacturing process on the performance of conveyor pulleys - G Styger 42 Locking Element Type "C" This configuration (Figure 2.12, below) was specifically designed for conveyor pulleys. It has all the advantages of types "A" and "B", although with far lower surface pressures. At the same time being able to transmit between 2 to 3 times the torque of type "A" or "B". Figure 2.12 : Locking element Type "C" [11] Summary of end-disk considerations for locking elements:  The hub portion of the end-disk must be sized correctly, in order to avoid cracking due to excessive pressure induced by the locking element.  The hub diameter should not be too large to accommodate the locking element pressure, as this could limit the degree of flexibility of the end-disk at the shell connection. In addition, this would cause cracking of the weld, particularly with the turbine type of end-disk. 2.2.3 Design procedure according to King This procedure was developed for use at drawing offices, and is based on the superposition of the individual effects of the pulley shell under the static and fatigue loadings in service [6]. The calculations are typically preformed for the "unworn" and "worn" condition of the shell, based on reduction of the shell thickness due to wear caused by the conveyor belt. Hub
  43. 43. An investigation of the effect of the manufacturing process on the performance of conveyor pulleys - G Styger 43 The static and fatigue stress model proposed by King is shown in the Figure 2.13 (below): Figure 2.13 : Static and fatigue stress model [6] 2.2.3.1 Belt pressure - fatigue loading The belt pressure on the surface can be shown from first principles to behave as shown in Figure 2.14 (below). This is where pressure is solely a function of curvature and instantaneous belt tension (T). Figure 2.14 : Belt loading on the pulley diameter [6]
  44. 44. An investigation of the effect of the manufacturing process on the performance of conveyor pulleys - G Styger 44 Belt tension for a non-drive pulley: Eq. 2-1 Belt tension for a drive pulley: Where the belt tension is asymmetrical, an equivalent tension (Te) is found, and assumed to act evenly over the belt lap. The active drive arc [3] is determined for the highest probable coefficient of friction. . / Eq. 2-2 ( ) ( * Eq. 2-3 The equivalent belt tension (Te) is used instead of the instantaneous belt tension (T) to find the belt pressure and hoop stress for position "B" - the center of the shell. Belt pressure on the shell: Eq. 2-4 Hoop stress due to the belt pressure on the shell: Eq. 2-5 2.2.3.2 Shaft connection induced expansion - Static Loading The Lame's equation [12] for determination of the variation of the stresses through thickness of a thick cylinder, is used to determine the hoop stress induced at the shell inside diameter, or end-disk outside diameter. This equation can be used directly to approximate the stress variation in the turbine type of end-disk. A T-bottom type of end-disk requires an iterative approach, to accurately assess the variation of the stress through the thickness of the end- disk. This iterative method is discussed in the subsequent design procedure by Perry. ( ) ( ) Eq. 2-6
  45. 45. An investigation of the effect of the manufacturing process on the performance of conveyor pulleys - G Styger 45 2.2.3.3 Rim bending due to the belt pressure - Fatigue Loading In modern pulleys, this is often the dominant effect. It describes the stress caused when two adjacent sections of a plate deflect to different positions (Figure 2.15, below). Schorer [13] studied local rim-bending in large pipelines with hoop stiffeners. This behaviour was not previously assessed for pulleys. Figure 2.15 : Rim bending of the shell [6] The radial deflection due to the belt pressure is determined as follows: . / Eq. 2-7 For the rim-bending section moment: ( * Eq. 2-8 Eq. 2-9 For the rim section force: Eq. 2-10 For the axial rim stress: Eq. 2-11 Hoop rim stress:
  46. 46. An investigation of the effect of the manufacturing process on the performance of conveyor pulleys - G Styger 46 ( * Eq. 2-12 2.2.3.4 Rim bending due to the shaft connection induced expansion - Static Loading The radial expansion is determined using Lame's equation for thick cylinders, pertaining particularly to the turbine type of end-disk. T-bottom type of end-disks require an iterative process to determine the radial expansion. The radial expansion due to the locking element pressure is determined as follows: ( ) Eq. 2-13 The determination of Mb1, Hb1, σa1 and σh1 is performed based on the above formulae. 2.2.3.5 Shell bending due to the shaft - fatigue loading Hartenburg [14] found that when acting in isolation, bending closely followed conventional engineer's bending theory. For Shaft bending moment: ( ) Eq. 2-14 For Axial bending stress: ( ( ) ) Eq. 2-15 The above formulation can result in an overly-conservative estimation of the bending stress in the end-disk-to-shell interface. This is because it is assumed that the shell experiences the full bending moment of the shaft. King [5] notes that the end-disk bending moment is determined by the ratio of the inner and outer diameter of the end-disk and the thickness thereof. A relative stiffness of the shaft and shell are established, thus allowing the end-disk- to-shell bending moment to be reduced compared to the shaft bending moment. This is then re-entered into the above axial bending stress equation, in order to determine a less conservative bending stress in the end-disk-to-shell interface. For the end-disk-to-shell bending moment: Firstly, relative stiffness of the shaft and shell are established.
  47. 47. An investigation of the effect of the manufacturing process on the performance of conveyor pulleys - G Styger 47 For the shaft stiffness constant: ( ) Eq. 2-16 For the end-disk stiffness constant: Eq. 2-17 ( ( )) Eq. 2-18 A suitable end-disk thickness is selected, by setting the radial direct stress to a third of the allowable fatigue stress, and finding an estimated value for the end-disk thickness from the equation to follow. The end-disk stiffness is then determined, based on this estimated value. For radial direct stress: Eq. 2-19 For end-disk stiffness: Eq. 2-20 If the shell is then assumed infinitely stiff compared with the shaft and end-disk, which is reasonable because the stiffness is the fourth power of the diameter, as supported by Schmoltzi's experimental work [4]. The bending moment is then distributed in the end-disk and shaft in the proportion to stiffness, as follows: For the end-disk bending moment: Eq. 2-21 The axial bending stress for the end-disk-to-shell interface, is then calculated as follows: ( ( ) ) Eq. 2-22
  48. 48. An investigation of the effect of the manufacturing process on the performance of conveyor pulleys - G Styger 48 2.2.3.6 Total stress The total stress is determined as the superposition of the static and fatigue stress for both the axial and hoop components at position "A" and "B". The axial stress at position "A" is determined as follows: Eq. 2-23 The hoop stress at position "A" is determined as follows: Eq. 2-24 2.2.4 Design procedure according to Perry 2.2.4.1 Fatigue considerations Typically a pulley rotates 35 million times per year [7]. All major pulley components should therefore be designed for infinite fatigue life. Infinite fatigue life means that the actual stress must be less than the endurance limit of the steel for a parent material (Figure 2.16, below). Figure 2.16 : S-N Curve for steel [15] Actual stress implies that geometrical stress concentration factors, surface finish factors and size factors have been considered for the stress calculation. Figure 2.17 illustrates the effect of reduced bearing diameters on shaft stress and Figure 2.18 indicates the effect of surface finish.
  49. 49. An investigation of the effect of the manufacturing process on the performance of conveyor pulleys - G Styger 49 Figure 2.17 : Stress concentration factors for filleted shafts in bending [16] Figure 2.18 : Surface finish modification factors for steel [7] Fatigue failure is prevented by keeping the fatigue reserve factor (FRF) above 1.3. Figure 2.19 (below) illustrates the meaning of the term fatigue reserve factor.
  50. 50. An investigation of the effect of the manufacturing process on the performance of conveyor pulleys - G Styger 50 Figure 2.19 : Modified Goodman diagram [7] 2.2.4.2 Shell As previously stated the shell is more rigid, typically 10 times more rigid than the shaft and end-disk assembly. The shell is assumed to be a simply supported tube as shown in Figure 2.20 (below). The figure also shows the sinusoidal load distribution assumed along the length of the shell, for this design procedure. Figure 2.20 : Support and load distribution assumption for the shell [3] For the sinusoidal load distribution: ( * Eq. 2-25 For drive pulleys, the load distribution along the circumference is assumed to be linear, as shown in Figure 2.21 (below).
  51. 51. An investigation of the effect of the manufacturing process on the performance of conveyor pulleys - G Styger 51 Figure 2.21 : Circumferential load distribution assumption shell [3] Lange found that maximum axial and hoop stress ranges occur at the center of the shell and 3 stress reversals occur per revolution for a drive pulley with an angle of wrap of 180 degrees. This is seen in Figure 2.22 (below) where Lange compared calculated data versus measured data: Figure 2.22 : Calculated and measured axial stresses in the pulley shell [3] From these loading conditions the stresses at any point can be determined, and are as follows: Eq. 2-26 Eq. 2-27 * ,( ) ( ) ( ) ( )*( ) +- ,( ) ( )*( ) - + ,( ) ( ) - Eq. 2-28 In the longitudinal direction: [ ] ( * Eq. 2-29
  52. 52. An investigation of the effect of the manufacturing process on the performance of conveyor pulleys - G Styger 52 In the circumferential direction: [ ] ( * Eq. 2-30 Shear stress: [ ] ( * Eq. 2-31 Due to the many iterations required, these calculations should not be attempted without the use of a computer. The maximum values of these stresses are used to calculate the Fatigue Reserve Factor of the shell. BS 5400 [17] was used for the fatigue assessment of the circumferential and longitudinal welds in the shell. The code is used to assess if the calculated stress range is below the non-propagating stress range (endurance limit) at 1x10 7 cycles, for the particular class of weld. If this criterion is met then fatigue need not be considered as theoretically infinite fatigue life should be achieved as long as the weld is sound with no defects. The following classes of weld are used for the detail of concern according to BS 5400, under this design criterion:  Class C is used for the classification of the longitudinal weld of the shell. This is then compared with the corrected hoop stress range of the shell. A correction factor of 43/59, as observed by the experimental and numerical work conducted by Lange, is applied to the stress range before the fatigue assessment. This is based on the initial assumption that the shell is simply-supported. The mean minus two standard deviation S-N curve, is used for this assessment (Figure 2.23, below). Figure 2.23 : Mean - 2 standard deviation S-N curve [17]
  53. 53. An investigation of the effect of the manufacturing process on the performance of conveyor pulleys - G Styger 53  Class F is used for the classification of the circumferential full penetration T-weld for the end-disk-to-inside shell of the turbine type of end-disk. The axial and radial stress range is used for the fatigue assessment.  Class C is used for the classification of the circumferential full penetration butt-weld for the end-disk-to-shell of the T-Bottom type of end-disk. Lange determined by strain gauge measurement that the axial stress range perpendicular to the circumferential weld is 65% of the maximum axial stress range in the shell, for a pulley with an angle of wrap of 180 degrees. This reduction is used as the axial stress range at the weld in this design criterion. 2.2.4.3 Shaft Two criteria determine shaft size, namely deflection angle at locking element, and fatigue failure at the location of maximum stress. Deflection angle: The shaft and end-disk share the imposed bending moment. The portion carried by each is directly proportional to its rigidity. The locking element and hub part of the turbine-shaped end-disks are assumed to be inflexible, provided that the hub diameter is less than one half of the pulley diameter. The deflection angle is calculated (for turbine shaped end-disk pulleys) from: ( )( ) ( )( ) ( ) Eq. 2-32 ( )( ) Eq. 2-33 The maximum allowable shaft deflection angle at the locking element depends on the type of locking element and is specified by the manufacturers thereof. A typical figure is 5 minutes. Fatigue failure: For stepped shafts, the most probable location of failure is in the fillet or at the edge of the locking element. For straight shafts, it is at the edge of the locking element. ⁄ ( ⁄ ) Eq. 2-34

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