Thesis Report                         OnANALYSIS AND IMPROVEMENT OF JACKING SYSTEMS               FOR JACK-UP RIG         ...
SUMMARYAs the need for energy increases globally, explorers have went out deeperand deeper into the ocean for oil and gas ...
AcknowledgementThe author wishes to express his sincere gratitude to his guide, A/P Mr. H.P Lee, for hisvaluable guidance,...
TABLE OF CO TE TS    Title                                                                   Page umber     Summary .........
2.6.2 Stress Reduction by Use of Fillets ...............123. Problem Definition .............................................
List of Figures1. Different working depths of offshore units.2. Photograph of an offshore Jack-up in operation.3. Top plan...
19. The rack’s top and bottom is constrained in the x and y directions.20. The rack’s side is constrained in the x directi...
40. Plastic Strain Variation in Rack and Pinion when fillet radius, r = 7.2mm.41. Plastic Strain Variation in Rack and Pin...
List of Tables1. Results of mesh element size and corresponding highest stress.2. Table of property of steel3. Table of Re...
1. IntroductionThe continuously rising oil price has been driving oil companies to put increasingly moreefforts in explori...
retract and extended during the transition. The upwards and downwards linear motion ofthe lifting mechanism- the rack and ...
A Jack-up rig is an offshore structure composed of a hull, legs and a lifting system thatallows it to be towed to a site, ...
current and wave conditions. In the afloat transit mode, the jack-up would also have agreater draught due to the weight of...
legs to be able to match the strength of the cylindrical legs, one must carefully considerits flexural and axial strength ...
is towed by several towboats to the location. In some other cases, the jack-up unit isbrought up onto the deck of another ...
into the water. The consequences are kept to a minimal as the hull is near the water, thusreducing the wave impact.For the...
Fig 7: Diagram of a Jack-up Rig under transition2.5 Lifting Mechanism – Rack and PinionAll Jack-ups have mechanisms for li...
Fig 8: Photograph of a Rack and Pinion systemOne point to take note is that the cross-sectional thickness of the pinion is...
Fig 9: Case Histories classified according to causes of failures2.6.1   Failure of Jack-up Rigs by FatigueIn high cycle fa...
The factor that determines fatigue failure is mean stress. As the mean stress decreases, thefatigue life increases. The de...
Fig 10: Causes of Jack-up Rigs During period of 1979 to 19882.6.2   Stress Reduction by Use of FilletsGears develop high s...
3. Problem DefinitionThe hull is supported by the Jack-up’s lifting mechanism; the consequences would becatastrophic if th...
the stress around the fillet radius. However with larger fillet radius, the area of contact ofthe rack with the pinion is ...
(a)                                        (b)    (c)                                        (d)Fig 12: Diagram of the cha...
Graph of Stress against Mesh Size                            8.00E+08                            7.50E+08          Stress,...
The FEM model is made up of all solid tetrahedral elements.                                          17
Fig 14: Different views of rack model                 18
Fig 15: Different views of pinion model                  19
20
Fig 16: Different views of Rack and Pinion Configuration               Fig 17: Comparison of Model with and without Fillet...
4.3.1 DisplacementFirstly the pinion is given the boundary condition that the inner bore is constrained in allthe 3 direct...
Fig 19: The rack’s top and bottom is constrained in the x and y directions                 Fig 20: The rack’s side is cons...
Fig 21: Bottom elements are given pressure4.3.3 ContactThe surface of the rack has slave nodes and the surface of the pini...
Fig 23: Overview of contact surface of the model4.4 MaterialAll the models are given the property of steel. The properties...
5. ResultsThe results of the simulations can be summarized into the following: Stress variationagainst fillet radius, plas...
Fig 25: Stress Variation in Rack and Pinion when fillet radius, r = 2.5mmFig 26: Stress Variation in Rack and Pinion when ...
Fig 27: Stress Variation in Rack and Pinion when fillet radius, r = 6.25mmFig 28: Stress Variation in Rack and Pinion when...
Fig 29: Stress Variation in Rack and Pinion when fillet radius, r = 7.2mmFig 30: Stress Variation in Rack and Pinion when ...
Fig 31: Stress Variation in Rack and Pinion when fillet radius, r = 8.125mmFig 32: Stress Variation in Rack and Pinion whe...
Fig 33: Stress Variation in Rack and Pinion when fillet radius, r = 10mm                Fillet radius r, mm               ...
Stress Against Fillet Radius                         7.6                         7.5   Stress, σ (Pa) x108                ...
Fig 36: Plastic Strain Variation in Rack and Pinion when fillet radius, r = 2.5mmFig 37: Plastic Strain Variation in Rack ...
Fig 38: Plastic Strain Variation in Rack and Pinion when fillet radius, r = 6.25mmFig 39: Plastic Strain Variation in Rack...
Fig 40: Plastic Strain Variation in Rack and Pinion when fillet radius, r = 7.2mmFig 41: Plastic Strain Variation in Rack ...
Fig 42: Plastic Strain Variation in Rack and Pinion when fillet radius, r = 8.125mmFig 43: Plastic Strain Variation in Rac...
Fig 44: Plastic Strain Variation in Rack and Pinion when fillet radius, r = 10mm               Fillet radius r, mm        ...
Plastic Strain Against Fillet Radius                             8                             7     Plastic Strain x10 -2...
Fig 46: Displacement Variation when fillet radius, r= 0mmFig 47: Displacement Variation when fillet radius, r= 2.5mm      ...
Fig 48: Displacement Variation when fillet radius, r= 5mmFig 49: Displacement Variation when fillet radius, r= 6.25mm     ...
Fig 50: Displacement Variation when fillet radius, r= 6.875mm Fig 51: Displacement Variation when fillet radius, r= 7.2mm ...
Fig 52: Displacement Variation when fillet radius, r= 7.5mmFig 53: Displacement Variation when fillet radius, r= 8.125mm  ...
Fig 54: Displacement Variation when fillet radius, r= 8.75mmFig 55: Displacement Variation when fillet radius, r= 10mm    ...
Fillet radius r, mm                 Displacement x10-3 , mm                                            0.000              ...
without the fillets. The plastic strains are also the lowest when the fillet radius is6.875mm. The maximum displacements a...
8. References  1) Bennett and Associates, L.L.C and Offshore Technology Development Inc (July     1, 2005). “Jack up units...
10) Litvin, F.L (1996), “Application of Finite Element Analysis for Determination of   Load Share, Real Contact Ratio, Pre...
18) Shuyan Ji, Daizhong Su and Jiansheng Li (2006), “Gear Design Optimisation   with a variable penalty function.” Proceed...
Upcoming SlideShare
Loading in …5
×

Download

1,372 views

Published on

Published in: Technology, Business
0 Comments
1 Like
Statistics
Notes
  • Be the first to comment

No Downloads
Views
Total views
1,372
On SlideShare
0
From Embeds
0
Number of Embeds
168
Actions
Shares
0
Downloads
97
Comments
0
Likes
1
Embeds 0
No embeds

No notes for slide

Download

  1. 1. Thesis Report OnANALYSIS AND IMPROVEMENT OF JACKING SYSTEMS FOR JACK-UP RIG Submitted by NG JUN JIE U048851X Department of Mechanical Engineering In partial fulfillment of the requirements for the Degree of Bachelor of Engineering National University of Singapore Session 2007/2008 1
  2. 2. SUMMARYAs the need for energy increases globally, explorers have went out deeperand deeper into the ocean for oil and gas which forms the world’s mainenergy source. Jack-up rigs, which is one of the offshore structures that isused in today’s extraction of oil from the seabed, is being analyzed.This paper attempts to provide an insight to improving the fatigue life of thelifting mechanism which comprises of the rack and pinion of a jack-up rigby reducing the mean stress.A practical model of a rack and pinion in a jack-up rig is modeled usingCAD and simulations are run on the model using finite element methodprograms. The strength of the structure would be evaluated according to theanalysis results.A study of the optimum fillet radius for the contact stress between the rackand pinion is proposed which helps in reducing the fatigue failures by cyclicloading of jack-ups. i
  3. 3. AcknowledgementThe author wishes to express his sincere gratitude to his guide, A/P Mr. H.P Lee, for hisvaluable guidance, proper advice and constant encouragement during the course or hiswork on this project.The author also feel very much obliged to his co-supervisor, Dr. X.M Tan, ResearchScientist at Institute of High Performance Computing for his encouragement andinspiration for execution of the project work.The author is deeply indebted to his parents for their inspiration and ever encouragingmoral support, which enabled him to pursue his studies.The author is also very thankful to the entire faculty and staff members of MechanicalEngineering Department for their direct–indirect help and cooperation. ii
  4. 4. TABLE OF CO TE TS Title Page umber Summary ................................................................................ i Acknowledgement ................................................................. ii Table of Contents ................................................................. iii List of Figures ........................................................................v List of Tables ...................................................................... viii 1. Introduction .......................................................................1 1.1 Thesis Outline ...........................................................2 2. Literature Review ..............................................................2 2.1 Jack-up Rigs .............................................................2 2.2 Types of Jack-up Units .............................................3 2.3 Types of Leggings.....................................................4 2.4 Modes of Operations in a Jack-up .............................5 2.4.1 Afloat Transit Mode .....................................6 2.4.2 Preload Mode ...............................................7 2.4.3 Elevated Mode .............................................8 2.5 Lifting Mechanism – Rack and Pinion ......................8 2.6 Failures of Jack-up Rigs ............................................9 2.6.1 Failure of Jack-up Rigs by Fatigue .............10 iii
  5. 5. 2.6.2 Stress Reduction by Use of Fillets ...............123. Problem Definition ..........................................................13 3.1 Implementation .......................................................144. Numerical Investigation...................................................14 4.1Mesh sensitivity .......................................................14 4.1.1 Results ........................................................15 4.2 FEM model .............................................................16 4.3 Boundary Conditions ..............................................21 4.3.1 Displacement ..............................................22 4.3.2 Pressure.......................................................23 4.3.3 Contact........................................................24 4.4 Material ..................................................................255. Results .............................................................................26 5.1 Stress against Fillet Radius .....................................26 5.2 Plastic Strain against Fillet Radius ...........................32 5.3 Displacement against Fillet Radius ..........................386. Conclusion.......................................................................447. Recommendations ...........................................................458. References .......................................................................46 iv
  6. 6. List of Figures1. Different working depths of offshore units.2. Photograph of an offshore Jack-up in operation.3. Top plane diagrams of (a) 4-legged jack-ups (b) 3-legged jack-ups.4. Photographs of (a) cylindrical legs (b) truss legs.5. Photograph of (a) Jack-up with legs retracted. (b) Jack-up under tow (c) Jack-up on a loading vessel, accompanied by supporting towboats.6. Photograph of a Jack-up Rig under preload conditions.7. Diagram of a Jack-up Rig under transition.8. Photograph of a Rack and Pinion system.9. Case Histories classified according to causes of failures.10. Causes of Jack-up Rigs During period of 1979 to 1988.11. Schematic diagram of contact of pinion and rack with varying fillet radius.12. Diagram of the changes in mesh sensitivity (a) Mesh = 0.015 (b) Mesh =0.075 (c) Mesh =0.025 (d) Mesh =0.00125.13. Table and Graph of Stress against Mesh Size.14. Different views of rack model.15. Different views of pinion model.16. Different views of Rack and Pinion Configuration.17. Comparison of Model with and without Fillets.18. The pinion is constrained in all the three directions. v
  7. 7. 19. The rack’s top and bottom is constrained in the x and y directions.20. The rack’s side is constrained in the x direction.21. Bottom elements are given pressure.22. (a) Slave nodes of the rack (b) Master nodes of the pinion.23. Overview of contact surface of the model.24. Stress Variation in Rack and Pinion when fillet radius, r = 0mm.25. Stress Variation in Rack and Pinion when fillet radius, r = 2.5mm.26. Stress Variation in Rack and Pinion when fillet radius, r = 5mm.27. Stress Variation in Rack and Pinion when fillet radius, r = 6.25mm.28. Stress Variation in Rack and Pinion when fillet radius, r = 6.875mm.29. Stress Variation in Rack and Pinion when fillet radius, r = 7.2mm.30. Stress Variation in Rack and Pinion when fillet radius, r = 7.5mm.31. Stress Variation in Rack and Pinion when fillet radius, r = 8.125mm.32. Stress Variation in Rack and Pinion when fillet radius, r = 8.75mm.33. Stress Variation in Rack and Pinion when fillet radius, r = 10mm.34. Graph of Stress Against Fillet Radius.35. Plastic Strain Variation in Rack and Pinion when fillet radius, r = 0mm.36. Plastic Strain Variation in Rack and Pinion when fillet radius, r = 2.5mm.37. Plastic Strain Variation in Rack and Pinion when fillet radius, r = 5mm.38. Plastic Strain Variation in Rack and Pinion when fillet radius, r = 6.25mm.39. Plastic Strain Variation in Rack and Pinion when fillet radius, r = 6.875mm. vi
  8. 8. 40. Plastic Strain Variation in Rack and Pinion when fillet radius, r = 7.2mm.41. Plastic Strain Variation in Rack and Pinion when fillet radius, r = 7.5mm.42. Plastic Strain Variation in Rack and Pinion when fillet radius, r = 8.125mm.43. Plastic Strain Variation in Rack and Pinion when fillet radius, r = 8.75mm.44. Plastic Strain Variation in Rack and Pinion when fillet radius, r = 10mm.45. Graph of Plastic Strain against Fillet Radius.46. Displacement Variation when fillet radius, r= 0mm.47. Displacement Variation when fillet radius, r= 2.5mm.48. Displacement Variation when fillet radius, r= 5mm.49. Displacement Variation when fillet radius, r= 6.25mm.50. Displacement Variation when fillet radius, r= 6.875mm.51. Displacement Variation when fillet radius, r= 7.2mm.52. Displacement Variation when fillet radius, r= 7.5mm.53. Displacement Variation when fillet radius, r= 8.125mm.54. Displacement Variation when fillet radius, r= 8.75mm.55. Displacement Variation when fillet radius, r= 10mm.56. Graph of Displacement Against Fillet Radius. vii
  9. 9. List of Tables1. Results of mesh element size and corresponding highest stress.2. Table of property of steel3. Table of Results of Fillet radius and Stress4. Table of Results of Fillet radius and Strain5. Table of Results of Fillet radius and Displacement viii
  10. 10. 1. IntroductionThe continuously rising oil price has been driving oil companies to put increasingly moreefforts in exploring and producing oil from the sea. The demand for oil has pushedexplorers to venture deeper and deeper into the ocean. There are now a lot of facilitiesused for the extraction of oil from the seabed. They vary from Jack-ups to drill ships.They are used in different environments and they are used for different sea depths. Figure1 illustrates further on the different working depths of the mobile offshore units. Fig 1: Different working depths of offshore unitsJack-up rigs are capable of working in sea depths up to 400ft (121m). Forsemisubmersible rigs, they usually work up to a depth of 3280ft (1000m) [9]. Lastly,drillships are usually used in very deep waters to extract oil from the seabed.Jack-up rigs are mobile and they would be towed from one place to another after theoriginal oil site has either low production and cannot meet the demand or it is no longercommercially profiteering to produce from that well. The legs of the jack-up would 1
  11. 11. retract and extended during the transition. The upwards and downwards linear motion ofthe lifting mechanism- the rack and pinion, would be experiencing stress repeatedly. It isthen proposed through the chamfering of the rack edges, to reduce the maximum contactstress and thus increasing its fatigue life.The purpose of this project is to conduct a parametric study of the relationship betweenhighest contact stress and the fillet radius of the rack edge.1.1. Thesis OutlineChapter 2 provides an introduction to the jack-up rig and some of the causes of failuresthat offshore mobile units undergo. The chapter also goes through some of the paststatistics of jack-up rigs failures and its causes. Chapter 3 presents the problem of theprevailing jack-up rigs and the numerical approach to solve for the optimum solution.Chapter 4 shows the lifting mechanism which the rack and pinion is being developed intoa CAD model. It also presents the boundary conditions used in the simulation of realenvironmental loads as well as the material properties. Chapter 5 presents the findings ofthe simulations. Chapter 6 gives the conclusion of the study done. Lists ofrecommendations are made in Chapter 7.2. Literature Review2.1 Jack-ups RigsJack-up rigs are used in the exploration of oil since the 1950’s. They have been used forexploration drilling, tender assisted drilling, production, accommodation, and work ormaintenance platforms. [1] 2
  12. 12. A Jack-up rig is an offshore structure composed of a hull, legs and a lifting system thatallows it to be towed to a site, lower its legs into the seabed and elevate its hull to providea stable work deck capable of withstanding the environmental loads.Jack-up rigs are used because they can be towed to another oil well after the wellproduction no longer can produce the required demand. A fixed platform would besimilar to a jack-up rig just that it is only built at that particular worksite and cannot bemoved. Jack-up rigs are thus more expensive to build than a fixed platform. (b) (a) Fig 2: Photograph of an offshore Jack-up in operation2.2 Types of Jack-up UnitsThere are typically 3-legged and 4-legged jack-ups in the world today. However themajority of the jack-ups that are produced are 3 legged. [1]The advantages of 4-legged jack-ups are that they have more work space as they requireno preload tankage and they are usually stiffer in the elevated mode because of the extraleg. It is also because of the extra leg that the jack-up would experience additional wind, 3
  13. 13. current and wave conditions. In the afloat transit mode, the jack-up would also have agreater draught due to the weight of the additional leg.The 3-legged jack-ups on the other hand weighs lesser for a given hull size and can carrymore load. They also eliminate the construction of an additional leg, thus reducing thenumber of lifting mechanism (racks, pinions, etc). This helps to reduce the power andmaintenance requirements. However unlike 4-legged jack-ups they require preload tanksonboard which take up usable space. (a) (b) Fig 3: Top plane diagrams of (a) 4-legged jack-ups (b) 3-legged jack-ups2.3 Types of LeggingsThere are two main types of leggings: cylindrical and trussed.Cylindrical Legs are made up of hollow steel tubes. They may be fitted with rack andpinions or holes in the shell to allow jacking up or down of the hull. They belong to anolder class of jack-ups rigs and they are used in water depths less than 300 ft. The mainadvantage of using cylindrical legs is that it is smaller in cross-section and takes up lessdeck space.The newer jack-up units are equipped with truss legs as they are lighter and use lessmaterial whilst providing the same resistance to the environmental loads. For the truss 4
  14. 14. legs to be able to match the strength of the cylindrical legs, one must carefully considerits flexural and axial strength of the trusses’ chords and braces. (a) (b) Fig 4: Photographs of (a) cylindrical legs (b) truss legs2.4 Modes of Operations of a Jack-upThere are basically 3 types of modes a jack-up would experience when transiting from anwork site to another. They are the afloat transit mode, preload mode and the elevatedmode.2.4.1 Afloat Transit ModeWhen the hull of a jack-up is lowered from its elevated mode, the legs are retracted andthey leave the seabed. It is not necessary to full retract the legs as long as they haveenough clearance from the seabed. This would increase stability of the jack-up andreduces the risk of wind overturning. The jack-up floats on the sea on its own hull and it 5
  15. 15. is towed by several towboats to the location. In some other cases, the jack-up unit isbrought up onto the deck of another vessel. In this case, the legs of the jack-up unit mustbe retracted to the maximum before loading onto the loading vessel. (a) (b) (c) Fig 5: Photograph of (a) Jack-up with legs retracted. (b) Jack-up under tow (c) Jack-up on a loading vessel, accompanied by supporting towboats.2.4.2 Preload ModeThe jack-up unit has to be preloaded to simulate operating conditions. In this mode, thehull is jacked up slowly to a height no more than 5 feet above the sea level. By pumpingin seawater from the surroundings to the onboard preload tanks, the hull carries extraweight apart from its own weight. In this mode, there are chances that a leg shift or soilfailure might occur. If that happens, the jack-up would lose its balance, dropping its hull 6
  16. 16. into the water. The consequences are kept to a minimal as the hull is near the water, thusreducing the wave impact.For the 4-legged jack-up rigs, there is little or no preload water. It is usually done bypreloading 2 of its diagonally opposite legs by the weight of the hull itself. After settlingoccurs, the 2 legs are lifted slightly to bring the other 2 legs to its preload period. Afterthe 4 legs are settled, the hull is then brought up to its operating height. Fig 6: Photograph of a Jack-up Rig under preload conditions2.4.3 Elevated ModeOnce the hull is in the operating height, the brakes are set and its weight lies fully on thestrength of the legs. The jack-up rig is ready to begin operations. Figure 7 illustrates thevarious modes of operation from arriving at the oil site to operational mode. 7
  17. 17. Fig 7: Diagram of a Jack-up Rig under transition2.5 Lifting Mechanism – Rack and PinionAll Jack-ups have mechanisms for lifting and lowering the hull. Majority of Jack-ups areequipped with a Rack and Pinion system for continuous jacking operations. The powersources used for jacking include both electric and hydraulic. Figure 8 shows a typicalrack and pinion system found on a jack-up rig. 8
  18. 18. Fig 8: Photograph of a Rack and Pinion systemOne point to take note is that the cross-sectional thickness of the pinion is usually largerthan the thickness of the rack. This is to prevent slipping of the rack off the pinion.2.6 Failure of Jack-up RigsAccording to a report by MSL Engineering Ltd prepared for Health and Safety Executivein 2004, 53% of the failures that jack-up rigs experiences is due to punch through of thelegs. Other causes include uneven seabed, volcanic activities, unexpected penetration ofthe legs, sliding of mat foundation and mudslide. 9
  19. 19. Fig 9: Case Histories classified according to causes of failures2.6.1 Failure of Jack-up Rigs by FatigueIn high cycle fatigue situations, materials performance is normally characterized by theS-N curve. The graph depicts of a cyclical stress(S) against cycles to failures (N). Failuredue to repeated loading is called fatigue.Fatigue failures are often caused by the degradation of metal surface. A rough surfacefinish, a scratch or oxidation will provide an initial crack. Cracks will propagate aftercyclical loading and eventually lead to fatigue failure. 10
  20. 20. The factor that determines fatigue failure is mean stress. As the mean stress decreases, thefatigue life increases. The defects on the contact surface will cause a decrease in the lifeof a material. As such, sharp corners which stresses concentrate on will probably be thefirst where cracks will occur and propagate.In another paper by B. P. M. Sharpies, W. T. Bennett, Jr and J. C. Trickey, it spoke of thefailures that jack-up rigs experience due to a certain factor during the period 1979 to 1988.Fatigue was one of the factors with 13 accidents out of the 226 accidents that occurredduring this period. 11
  21. 21. Fig 10: Causes of Jack-up Rigs During period of 1979 to 19882.6.2 Stress Reduction by Use of FilletsGears develop high stress concentration at the gear tooth root stress and the contact point.It is usually at these areas where there is a higher chance of fatigue failure. In normaloperations the contact point shifts along the profile of the tooth and a surface fatiguefailure is likely. In the normal jack up operation, the rack and pinion is held in place afterthe hull is at its operational height. In this case, the contact point between the rack andpinion will stay in one place for an extended period of time. This is when the relieffeatures are needed to reduce the highest stress at the contact point. Fatigue life will beincreased if fillets are introduced at the contact point.Vasilios [21] introduces another method to find the minimum fillet stress using BEM andfurther verification has been done using 2D photoelasticity. Math [20] proposes anapproach to determine the geometry of the spur gear tooth fillet. Equations have been setup for the tangency of the involute curve and root fillet. 12
  22. 22. 3. Problem DefinitionThe hull is supported by the Jack-up’s lifting mechanism; the consequences would becatastrophic if the rack and pinion were to fail by fatigue. In recurring instances ofjacking up and down the legs due to its mobile nature, the rack and pinion experiencesrepeated contact stress at sharp corners, especially at the edges of the rack. The cyclicnature of loading and unloading the hull causes the rack to fail possibly by fatigue.Therefore, it is paramount that the rack and pinion has relief features in areas where thehighest stress occurs. Presently there are no studies on this aspect of study. This is a newarea of study to look into. It could have possible tremendous positive impacts if this wereto be put into practice in the offshore industry. The relief fillet causes a lower distributedstress at the edges. The chamfering of the edges of the rack at the contact surface givesthe rack a longer fatigue life. It is then proposed to investigate the relationship betweenthe fillet radius and its corresponding stress level at the point of contact to find anoptimum fillet radius for the rack. Highest Highest stress stress Fig 11: Schematic diagram of contact of pinion and rack with varying fillet radiusThe highest stress is found at the sharp corners of the rack. The rationale of thesimulation is that by chamfering the edges, the highest stress is reduced by distributing 13
  23. 23. the stress around the fillet radius. However with larger fillet radius, the area of contact ofthe rack with the pinion is reduced. There is a point where the highest stress will start toincrease with larger fillet radius. The aim of this simulation is to find out the optimumfillet radius.3.1. ImplementationSoftware used: -Modeling using Solidworks 2005, transferring of coordinates to MSC Patran 2005r3 andanalysis is done using Abaqus Version 6.41.4. umerical Investigation4.1 Mesh SensitivityA study of varying element size on the rack surface has been done for convergencepurposes. This study is done on the purpose that the computational result does not deviatetoo much from the actual stress. The convergence test is done on a single model. Mesh size (mm) HighestStress (Pa) 0.00125 7.55E+08 0.0025 7.53E+08 0.0075 7.36E+08 0.015 7.10E+08 Table 1: Results of mesh element size and corresponding highest stress 14
  24. 24. (a) (b) (c) (d)Fig 12: Diagram of the changes in mesh sensitivity (a) Mesh = 0.015 (b) Mesh =0.075 (c) Mesh =0.025 (d) Mesh =0.001254.1.1 ResultsThe results of convergence are shown in the form of a table and graph shown below. 15
  25. 25. Graph of Stress against Mesh Size 8.00E+08 7.50E+08 Stress, σ (Pa)) 7.00E+08 6.50E+08 6.00E+08 5.50E+08 5.00E+08 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 Mesh Size (mm) Fig 13: Table and Graph of Stress against Mesh SizeThe graph above shows the variation of element size and its corresponding stress. Itshows that even after the element size has been halved to 0.00125, the stress deviate fromthe previous stress by less than 1%. Thus it would be more computationally expensive touse element size of 0.00125 and below. It would be fairly accurate to use element size of0.0025 in area of analysis.4.2 FEM modelThe model is based on a practical Rack and Pinion that is in a commercial Jack-up. Theoriginal Rack and Pinion has been cut into half since it is symmetrical about its centralplane.The rack is modeled and meshed as shown in Figure 15. Similarly the pinion can befound in Figure 16. The final rack and pinion configuration is shown in Figure 17.Finally, a contrast is shown between a rack with and without fillets in Figure 18. 16
  26. 26. The FEM model is made up of all solid tetrahedral elements. 17
  27. 27. Fig 14: Different views of rack model 18
  28. 28. Fig 15: Different views of pinion model 19
  29. 29. 20
  30. 30. Fig 16: Different views of Rack and Pinion Configuration Fig 17: Comparison of Model with and without Fillets4.3 Boundary ConditionsThe model has boundary conditions like displacement, pressure and contact. 21
  31. 31. 4.3.1 DisplacementFirstly the pinion is given the boundary condition that the inner bore is constrained in allthe 3 directions. Figure 19 shows the boundary condition given to the pinion. Fig 18: The pinion is constrained in all the three directions 22
  32. 32. Fig 19: The rack’s top and bottom is constrained in the x and y directions Fig 20: The rack’s side is constrained in the x direction4.3.2 PressureThe rack is given an upwards pressure of 1 kipf/sq in (6.89 MPa) and is applied at thebottom surfaces of bottom elements. 23
  33. 33. Fig 21: Bottom elements are given pressure4.3.3 ContactThe surface of the rack has slave nodes and the surface of the pinion has master nodes indefining the contact. A friction coefficient of 0.1 is being applied here. (a) (b) Fig 22: (a) Slave nodes of the rack (b) Master nodes of the pinion 24
  34. 34. Fig 23: Overview of contact surface of the model4.4 MaterialAll the models are given the property of steel. The properties of steel are presented in aform of graph below. Properties of Steel Elastic Modulus 2e11 Pa Poisson Ratio 0.3 Density 7850 kg/m3 Yield Stress 7.24e8 Pa Table 2: Table of property of steel 25
  35. 35. 5. ResultsThe results of the simulations can be summarized into the following: Stress variationagainst fillet radius, plastic strain against fillet radius and displacement against filletradius.5.1 Stress Variation against Fillet radiusThe following results are presented in the form of pictures shown below. Fig 24: Stress Variation in Rack and Pinion when fillet radius, r = 0mm 26
  36. 36. Fig 25: Stress Variation in Rack and Pinion when fillet radius, r = 2.5mmFig 26: Stress Variation in Rack and Pinion when fillet radius, r = 5mm 27
  37. 37. Fig 27: Stress Variation in Rack and Pinion when fillet radius, r = 6.25mmFig 28: Stress Variation in Rack and Pinion when fillet radius, r = 6.875mm 28
  38. 38. Fig 29: Stress Variation in Rack and Pinion when fillet radius, r = 7.2mmFig 30: Stress Variation in Rack and Pinion when fillet radius, r = 7.5mm 29
  39. 39. Fig 31: Stress Variation in Rack and Pinion when fillet radius, r = 8.125mmFig 32: Stress Variation in Rack and Pinion when fillet radius, r = 8.75mm 30
  40. 40. Fig 33: Stress Variation in Rack and Pinion when fillet radius, r = 10mm Fillet radius r, mm Stress x108 , Pa 0.000 7.53 2.500 7.20 5.000 7.10 6.250 6.98 6.875 6.95 7.200 6.99 7.500 7.01 8.125 7.06 8.750 7.11 10.000 7.19 Table 3: Table of Results of Fillet radius and StressThe highest stress from the diagrams is presented in a form of graph shown below. 31
  41. 41. Stress Against Fillet Radius 7.6 7.5 Stress, σ (Pa) x108 7.4 7.3 7.2 7.1 7 6.9 0 2 4 6 8 10 12 Fillet Radius, r (mm) Fig 34: Graph of Stress Against Fillet Radius5.2 Plastic Strain against Fillet RadiusThe following pictures depict the plastic strains that are found by the edges of the rack. Fig 35: Plastic Strain Variation in Rack and Pinion when fillet radius, r = 0mm 32
  42. 42. Fig 36: Plastic Strain Variation in Rack and Pinion when fillet radius, r = 2.5mmFig 37: Plastic Strain Variation in Rack and Pinion when fillet radius, r = 5mm 33
  43. 43. Fig 38: Plastic Strain Variation in Rack and Pinion when fillet radius, r = 6.25mmFig 39: Plastic Strain Variation in Rack and Pinion when fillet radius, r = 6.875mm 34
  44. 44. Fig 40: Plastic Strain Variation in Rack and Pinion when fillet radius, r = 7.2mmFig 41: Plastic Strain Variation in Rack and Pinion when fillet radius, r = 7.5mm 35
  45. 45. Fig 42: Plastic Strain Variation in Rack and Pinion when fillet radius, r = 8.125mmFig 43: Plastic Strain Variation in Rack and Pinion when fillet radius, r = 8.75mm 36
  46. 46. Fig 44: Plastic Strain Variation in Rack and Pinion when fillet radius, r = 10mm Fillet radius r, mm Plastic Strain x10-2 0.000 6.92 2.500 5.00 5.000 4.66 6.250 4.21 6.875 4.13 7.200 4.17 7.500 4.23 8.125 4.43 8.750 4.68 10.000 4.98 Table 4: Table of Results of Fillet radius and Strain 37
  47. 47. Plastic Strain Against Fillet Radius 8 7 Plastic Strain x10 -2 6 5 4 3 2 1 0 0 2 4 6 8 10 12 Fillet Radius, r (mm) Fig 45: Graph of Plastic Strain against Fillet Radius5.3 Displacement against Fillet RadiusThis simulation serves as a preliminary study of displacement. The following shows theresults of varying fillet radius against displacement. Figure 54 shows the graph of thedisplacement and fillet radius. 38
  48. 48. Fig 46: Displacement Variation when fillet radius, r= 0mmFig 47: Displacement Variation when fillet radius, r= 2.5mm 39
  49. 49. Fig 48: Displacement Variation when fillet radius, r= 5mmFig 49: Displacement Variation when fillet radius, r= 6.25mm 40
  50. 50. Fig 50: Displacement Variation when fillet radius, r= 6.875mm Fig 51: Displacement Variation when fillet radius, r= 7.2mm 41
  51. 51. Fig 52: Displacement Variation when fillet radius, r= 7.5mmFig 53: Displacement Variation when fillet radius, r= 8.125mm 42
  52. 52. Fig 54: Displacement Variation when fillet radius, r= 8.75mmFig 55: Displacement Variation when fillet radius, r= 10mm 43
  53. 53. Fillet radius r, mm Displacement x10-3 , mm 0.000 1.89 2.500 1.71 5.000 1.69 6.250 1.55 6.875 1.55 7.200 1.56 7.500 1.58 8.125 1.59 8.750 1.60 10.000 2.00 Table 5: Table of Results of Fillet radius and Displacement Displacement Against Fillet Radius 2.5 Displacement (mm) x10 -3 2 1.5 1 0.5 0 0 2 4 6 8 10 12 Fillet Radius, r (mm) Fig 56: Graph of Displacement Against Fillet Radius6. ConclusionFrom this study, we have found out an optimum fillet for the edges of the rack. It can beconcluded from the studies that a fillet radius of 6.875mm is the optimum radius. Itshows that with fillet radius of 6.875mm would yield the lowest stress of 6.95e8 Pa. Itmeans there is a reduction of about 7.7% in contact stress compared with the rack that is 44
  54. 54. without the fillets. The plastic strains are also the lowest when the fillet radius is6.875mm. The maximum displacements are near the applied pressure and they are foundat the places where there is the least deformation. At fillet radius of 6.875mm and6.25mm, the rack and pinion exhibits the least displacement.All of the results that are exhibited are static stresses produced from constant loads. Inactual operation, the rack and pinion experiences dynamic loads variations. However thedynamic loads will be small as the rotational speed of the pinion will be slow whenelevating the hull. For actual determination of the stresses, programs that are able tocalculate dynamic loads are suggested, for instance LS-DYNA.7. RecommendationsIn practical offshore industry, there is no record of filleting the edges of the rack.However from this study, it shows that by chamfering the edges of the rack, one is able toreduce the maximum contact stress, thus reducing mean stress. From there, the chances offatigue failure can be reduced too.One recommendation is that instead of chamfering special fillet radius of 6.875mm whichrequires precise machining, one can opt for common fillet radius like 6mm or even anodd number 7mm which is close to the optimum fillet radius to reduce chances of fatiguefailure.Another recommendation is that designers of jack-up rigs can indicate that in themanufacture of the rack; to save cost, chamfer only the operational range. That is to sayonly chamfer the range where the hull is in elevated mode. There is no need to chamferthe whole rack as at full operational height of the hull would experience more cycles tofailure with real environment loads in times of storms. 45
  55. 55. 8. References 1) Bennett and Associates, L.L.C and Offshore Technology Development Inc (July 1, 2005). “Jack up units: A technical primer for the offshore industry professional.” Retrieved 23 Sep 2007, from http://www.bbengr.com/jack_up_primer.pdf 2) B. P. M. Sharpies, W. T. Bennett, Jr and J. C. Trickey (1989), “Risk analysis of Jackup Rigs.” Marine Structures, Vol. 2, pp. 281-303. 3) Chien-Hsing Li, Hong-Shun Chiou, Chinghua Hung, Yun-Yuan Chang and Cheng-Chung Yen (2002), “Integration of Finite Element Analysis and Optimum Design on Gear Systems”, Finite Elements in Analysis and Design, Vol. 38, pp. 179-192. 4) D.P. Stewart and I.M.S. Finnie (2001), “Spudcan-Footprint Interaction During Jack-Up Workovers.”, Proceedings of the Eleventh International Offshore and Polar Engineering Conference. 5) Dudley, D.W (1954), “Practical Gear Design”, McGraw-Hill, New York. 6) Fredette L. and Brown M. (1997), “Gear Stress Reduction Using Internal Stress Relief Features”, Journal of Mechanical Design, Vol. 119, pp. 518-521. 7) Fumitaka Higuchi, Shuuichi Gofuku (2007), “Approximation of involute curves for CAD-system processing.” Engineering with Computers, Vol 23, pp.207-214. 8) John J. Coy Dennis P. Townsend and Erwin V. Zaretsky (1985). “Gearing.” NASA reference publication. 9) Keppel Fels (2008) Retrieved 23 Feb 2008, from http://www.keppelfels.com.sg/products 46
  56. 56. 10) Litvin, F.L (1996), “Application of Finite Element Analysis for Determination of Load Share, Real Contact Ratio, Precision of Motion, and Stress Analysis,” Journal of Mechanical Design, Transactions of the American Society of Mechanical Engineers, Vol. 118, No. 4, pp. 561–567.11) Litvin, F.L. (1994), Gear Geometry and Applied Theory, Prentice Hall, Englewood Cliffs, NJ.12) Litvin, F.L., and Hsiao, C.-L. (1993), “Computerized Simulation of Meshing and Contact of Enveloping Gear Tooth Surfaces,” Computer Methods in Applied Mechanics and Engineering, Vol. 102, pp. 337–366.13) Moriwaki, I., Fukuda, T., Watabe, Y., Saito, K (1993), “Global Local Finite Element Method (GLFEM) in Gear Tooth Stress Analysis”, Journal of Mechanical Design, Vol. 115, pp. 1008-1012.14) MSL Engineering Ltd (2004). “Guidelines for jack-up rigs with particular reference to foundation integrity.” Retrieved 1 Feb 2008, from www.hse.gov.uk/research/rrhtm/rr289.htm15) P. J. Mills, A. T. Dixon, H. Smallman & D. Smith (1997). “Some Aspects of the Safety of Jack-ups on Location and During Transit.” Marine Structures, Volume 10, Number 2, pp. 243-262.16) R. Gobithasan, R. Rofizah & M. A. Jamaludin (2005), “Straight line and circular arc methods for developing G1 and G2 involute curves.” Journal Teknologi, Vol. 43, pp. 55-66.17) S.Barone (2001), “Gear Geometric Design by B-Spline Curve Fitting and Sweep Surface Modelling.” Engineering with Computers, Vol 17, pp. 64-77. 47
  57. 57. 18) Shuyan Ji, Daizhong Su and Jiansheng Li (2006), “Gear Design Optimisation with a variable penalty function.” Proceedings of International Conference on Advanced Design and Manufacture.19) Tian Yong-tao, Li Cong-xin, Tong Wei and Wu Chang-hua (2003), “A finite- element-based study of the load distribution of a heavily loaded spur gear system with effects of transmission shafts and blanks.” Journal of Mechanical Design, Vol. 125, pp. 625-631.20) V.B Math and Satish Chand (2004), “An approach to the Determination of Spur Gear Tooth Root Fillet.” Journal of Mechanical Design, Vol. 136, pp. 336-340.21) Vasilios A. Spitas, Theodore N. Costopoulos and Christos A. Spitas (2006). “Optimum Gear Tooth Geometry For Minimum Fillet Stress Using BEM and Experimental Verification With Photoelasticity.” Journal of Mechanical Design, Vol. 128. pp. 1159-1164.22) Vijayarangan S. and Ganesan N. (1993), “Stress Analysis of Composite Spur Gear Using the Finite Element Approach”, Computers and Structures, Vol. 46, No. 5, pp. 869-875.23) Wikipedia- The free encyclopedia (2008). “Jack-up Barge.” Retrieved 1 Feb 2008, from http://en.wikipedia.org/wiki/Jack-up_barge24) Wildhaber, E., 1946c, “Tooth Contact,” American Machinist, Vol. 90, No. 12, pp. 110–114. 48

×