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Nguyen, Cuong Q.
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A design of experiments study of procedure for assembling bascule bridge fulcrum
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by Cuong Q. Nguyen.
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[Tampa, Fla] :
b University of South Florida,
2006.
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Thesis (M.A.)University of South Florida, 2006.
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Includes bibliographical references.
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Text (Electronic thesis) in PDF format.
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System requirements: World Wide Web browser and PDF reader.
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Title from PDF of title page.
Document formatted into pages; contains 93 pages.
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Adviser: Autar K. Kaw, Ph.D.
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ABSTRACT: A bascule bridge is a type of movable bridge that can be opened or closed to facilitate the movement of waterborne traffic such as ships and yachts. TrunnionHubGirder (THG) assembly plays a role as a fulcrum in the bascule bridge. To make the fulcrum, the trunnion is shrinkfitted into the hub, and then the trunnionhub assembly is shrinkfitted into the girder. Hundreds of thousands of dollars could be lost due to failures during this step. Crack formations in the hubs of various Florida bascule bridges during assembly led the Florida Department of Transportation to commission a project with USF professors to investigate.Finite elements method (ANSYS package) is employed to model the THG assembly procedure and solve for the critical crack length and critical stress in this transient thermal structural problem.Design of experiments (DOE) is used with different cooling processes and the geometrical dimensions of the THG assembly to find the sensitivity of these parameters on the outputs.The influence of the hub outer diameter and the radial interference (between the trunnion and hub) is at different levels on the critical crack length and the stress ratio as it is dependent on fulcrum geometry. If we include four staged cooling methods as follow Type 1: liquid nitrogen Type 2: dryice/ alcohol bath followed by liquid nitrogen Type 3: refrigerated air chamber followed by liquid nitrogen Type 4: refrigerated air chamber followed by dryice/alcohol bath and then by liquid nitrogenthe cooling type factor contributes the most to both critical crack length (up to 79%) and the stress ratio (up to 84%) in the TH assembling procedures in all three considered bascule bridges. The staged cooling procedure type 2, which is, immersing the TH assembly into the dry ice medium, followed by immersing the TH set into the liquid nitrogen, give larger critical crack length (up to 400%) and stress ratio (up to 87%) compared to the case that used only liquid nitrogen.
653
Mixed level factorial design of experiment.
Shrunk fitting.
Staged cooling.
ANSYS modeling.
Radial interference.
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Dissertations, Academic
z USF
x Mechanical Engineering
Masters.
773
t USF Electronic Theses and Dissertations.
4 0 856
u http://digital.lib.usf.edu/?e14.1705
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A Design of Experiments Study of Procedure for Assembling Bascule Bridge Fulcrum by Cuong Q. Nguyen A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in Mechanical Engineering Department of Mechanical Engineering College of Engineering University of South Florida Major Professor: Autar K. Kaw, Ph.D. Daniel Hess, Ph.D. Frank Pyrtle, III, Ph.D. Date of Approval: July 11, 2006 Keywords: Mixed Level Factorial Design of Experiment, Shrunk Fitting, Staged Cooling, ANSYS Modeling, Radial Interference Copyright 2006, Cuong Q. Nguyen
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DEDICATION This thesis is dedicated to the memory of my grandmothers and grandfathers, who took care me with all their love. To my parents who have supported and believed in me. To my professors Drs. Autar K. Kaw and Tong T. Nguyen, who mentally instructed me and guided me to my academic achievements.
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ACKNOWLEDGMENTS It gives me immense pleasure and great prid e to present my thesis report titled, A Design of Experiments Study of Procedure for Assembling Bascule Bridge Fulcrum. I express my thanks and gratitude to Dr. Autar K. Kaw, whose guidance and direction helped me tremendously to complete this work. It is my privilege that I had the chance to work with the 2004 Florida Professor of the Year. His expertise in the field of solid mechanics including thermal stresses is the best I have ever witnessed and is definitely admirable. I am deeply indebted to him fo r the financial support, and for the academic and computer resources he provided. I am al so thankful for his patience and for his generosity. I also thank Dr. Frank Pyrtle, III, and Dr. Da niel Hess for their contributions in helping me finalize this thesis an d for being on my committee. I want to thank Mr. Jai P. Paul who helped me with the ANSYS code and I wish to thank Mr. Son H. Ho, a Ph.D. candidate at the U SF, for helping me in verifying the data.
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i TABLE OF CONTENTS LIST OF TABLES.............................................................................................................iii LIST OF FIGURES...........................................................................................................iv ABSTRACT......................................................................................................................v ii CHAPTER 1 INTRODUCTION.........................................................................................1 1.1 Bascule Bridges and THG Assembly.................................................................1 1.2 Previous Works and Objectives of the Thesis...................................................6 CHAPTER 2 LITERATURE...............................................................................................9 2.1 Introduction........................................................................................................9 2.2 Geometrical Details...........................................................................................9 2.3 Analytical Details.............................................................................................13 2.3.1 Parameters used..........................................................................13 2.3.2 Governing equations and boundary conditions...........................16 2.4 Nonlinear Material Pr operties of Metal........................................................23 2.4.1 Youngs modulus........................................................................23 2.4.2 Coefficient of thermal expansion................................................24 2.4.3 Thermal conductivity..................................................................25 2.4.4 Density........................................................................................26 2.4.5 Specific heat................................................................................27 2.5 Nonlinear Material Proper ties of Cooling Media.........................................28 2.5.1 Convection heat transfer coefficient, h c and assumptions..........28 2.5.2 Convection to refrigerator air......................................................29 2.5.3 Convection to dry ice/ alcohol bath............................................30 2.5.4 Coefficient of convectio n of liquid nitrogen...............................31 CHAPTER 3 ANSYS MOD ELING AND ANALYSIS...................................................34 3.1 Introduction......................................................................................................34 3.2 Why Inspect Only 1/6 th of the TH Assembly?.................................................34 3.3 Assumption for ANSYS Sequentia l Coupled Field Approach........................34 3.4 Coupled Field Analysis....................................................................................35
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ii 3.5 Direct Versus Sequential Coupled Field Analysis...........................................35 3.6 The Finite Element Model...............................................................................36 3.7 ANSYS Element Selection..............................................................................37 3.7.1 ANSYS element library and classification.................................37 3.7.2 Selection of elements..................................................................38 3.7.3 Selected element characteristics.................................................40 3.8 The Procedure of the ANSYS Modeling and Analysis...................................43 CHAPTER 4 DESIGN OF EXPERIMENT AND SENSITIVITY ANALYSIS..............47 4.1 Introduction......................................................................................................47 4.2 Screening the Variations..................................................................................48 4.3 2 k Factorial Design and Mixed Le vels Design of Experiment.......................50 4.3.1 2 k factorial design.......................................................................50 4.3.2 Mixed level factorial design........................................................54 4.4 General Factorial Design.................................................................................57 CHAPTER 5 RESULTS AND CONCLUSIONS.............................................................61 5.1 Specifications for Three Inspected Bascule Bridges......................................61 5.1.1 Bascule bridges general geometrical dimension.........................61 5.1.2 Radial interferences calculations................................................62 5.1.3 Hub outer diameter calculations.................................................63 5.2 Fracture Toughness and Yield Strength..........................................................65 5.2.1 The fracture toughness, K I ..........................................................66 5.2.2 The stress ratio............................................................................67 5.3 Typical Results of the Critical Points.............................................................68 5.4 Sensitivity Analysis of 2 Mixedlevel Factors................................................75 5.4.1 Specification of geometries of the hub and the trunnion............75 5.4.2 Collected data and se nsitivity analysis.......................................75 5.5 Sensitivity Analysis of 3 Mixedlevel Factors................................................82 5.5.1 Specification of levels of each factor..........................................82 5.5.2 Collected data and se nsitivity analysis.......................................83 REFERENCES..................................................................................................................92
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iii LIST OF TABLES Table 2.1 The TH geometry parameters summary.....................................................12 Table 2.2 Summary of parameters used for TH assembly analysis...........................13 Table 4.1 Notations for experiment combinations.....................................................52 Table 4.2 Manipulation for 3level factor to apply 2 k factorial design......................54 Table 4.3 Manipulation for 4level factor to apply 2 k factorial design......................54 Table 4.4 2, 3, 4 levels fact orial design of experiment..............................................55 Table 4.5 The ANOVA table for general factorial design.........................................58 Table 4.6 Sum of square of factors............................................................................59 Table 5.1 Geometrical dimensions of three Bascule bridges.....................................61 Table 5.2 Radial interference s of three Bascule bridges ............................................63 Table 5.3 Specifications of three levels of hub outer diameters................................64 Table 5.4 Yield strength, Y s as a function of temperature (Greenberg and Clark, Jr., 1969)....................................................................................................67 Table 5.5 Fracture toughness, K I as a function of temperature..................................68 Table 5.6 Specifications of two mixed level factors..................................................75 Table 5.7 Collected data for sensitivity analysis ........................................................76 Table 5.8 The contributions of factors on the OMCCL and the OMSR .....................76 Table 5.9 Specifications of three mixed level factors................................................82 Table 5.10 Collected data for sensitivity analysis........................................................83 Table 5.11 The contributions of factors on OMCCL and OMSR .................................84
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iv LIST OF FIGURES Figure 1.1 Bascule bridge open for road traffic (the Tower Bridge, London, UK)......2 Figure 1.2 Bascule bridge open for water traffic (the Tower Bridge, London, UK)....2 Figure 1.3 The THG assembly get installe d on the leaf of the bascule bridge.............3 Figure 1.4 The trunnion is located onto the trunnion frame (the Gateshead Millennium Bridge, Gateshead, UK)...........................................................3 Figure 1.5 Two different assembly procedures (Besterfield, Kaw, and Crane, 2001)............................................................................................................5 Figure 2.1 The THG assembly (Besterfield, Kaw, and Crane, 2001)...........................9 Figure 2.2 The TH assembly.......................................................................................10 Figure 2.3 The side view of the TH assembly.............................................................11 Figure 2.4 The top view of the TH assembly..............................................................12 Figure 2.5 Coordinate for the hub and trunnion (front view)......................................18 Figure 2.6 Coordinate for the TH set (side view)........................................................19 Figure 2.7 Youngs modulus of steel as a functi on of temperature (Collier, 2004)..........................................................................................................24 Figure 2.8 Coefficient of thermal expansion of steel as a function of temperature (Collier, 2004)........................................................................25 Figure 2.9 Thermal conductivity of steel as a function of temperature (Collier, 2004)..........................................................................................................26 Figure 2.10 Density of steel as a f unction of temperature (Collier, 2004)....................27 Figure 2.11 Specific heat of steel as a function of te mperature (Collier, 2004)...........28 Figure 2.12 Convective heat transfer coefficient of refrigerator ai r as a functions of temperature (Collier, 2004)...................................................................30
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v Figure 2.13 Convective heat transfer coe fficient of dry ice as a function of temperature (AspenTech, 2004).................................................................31 Figure 2.14 Convective heat transfer coefficient of liquid nitrogen as a functions of temperature (Brentari and Smith, 1964)................................................32 Figure 2.15 Heat flux versus temperature di fference for liquid nitrogen (Barron, 1999)..........................................................................................................33 Figure 3.1 SOLID453D structural solid, and SOLID703D thermal solid (ANSYS release 10.0 documentation).......................................................41 Figure 3.2 Trunnionhub assembly with SOLID45 and SOLID70 elements..............41 Figure 3.3 CONTACT174 overlaying the trunnion outer diameter surface...............42 Figure 3.4 TARGET170 overlaying the hub inner diameter surface..........................43 Figure 3.5 Division of the parts and meshed model TH ..............................................44 Figure 3.6 ANSYS flow chart ....................................................................................46 Figure 4.1 General model of a process or system (Montgomery, 2001).....................48 Figure 5.1 Temperature dependence of fr acture toughness and yield strength of ASTM E24 steel casting (Gree nberg and Clark, Jr., 1969)......................65 Figure 5.2 Critical crack length...................................................................................66 Figure 5.3 Temperature versus time of the node ( r = 9.0000 = 90.000 o z = 19.000 )......................................................................................................69 Figure 5.4 Hoop stress versus time of the node ( r = 9.0000 = 90.000 o z = 19.000 )......................................................................................................70 Figure 5.5 CCL versus time of the node ( r = 9.0000 = 90.000 o z = 19.000 ).......71 Figure 5.6 Temperature versus time of the node ( r = 26.000 = 120.00 o z = 33.667 )......................................................................................................72 Figure 5.7 VonMises stress versus time of the node ( r = 26.000 = 120.00 o z = 33.667 )......................................................................................................73 Figure 5.8 SR versus time of the node ( r = 26.000 = 120.00 o z = 33.667 )...........74 Figure 5.9 The effects of the treatments on the OMCCL and OMSR ..........................77
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vi Figure 5.10 The variations of the OMCCL versus the changing of selected factors.....79 Figure 5.11 The variations of the OMSR versus the changing of selected factors........81 Figure 5.12 The effects of the interested factors in the OMCCL and OMSR ................85 Figure 5.13 The variations of the OMCCL versus the changing of selected factors.....87 Figure 5.14 The variations of the OMSR versus the changing of selected factors........89
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vii A DESIGN OF EXPERIMENTS STUDY OF PROCEDURE FOR ASSEMBLING BASCU LE BRIDGE FULCRUM Cuong Q. Nguyen ABSTRACT A bascule bridge is a type of movable bridge that can be opened or closed to facilitate the movement of waterborne traffic such as ships and yachts. TrunnionHubGirder ( THG) assembly plays a role as a fulcrum in the bascule bridge. To make the fulcrum, the trunnion is shrinkfitted into th e hub, and then the trunnionh ub assembly is shrinkfitted into the girder. Hundred s of thousands of dollars could be lost due to failures during this step. Crack formations in the hubs of various Florida bascule bridges during assembly led the Florida Department of Transportation to commission a project with USF professors to investigate. Finite elements method (ANSYS package) is employed to model the THG assembly procedure and solve for the cr itical crack length and critical stress in this transient thermal structural problem. Design of experime nts (DOE) is used with different cooling processes and the geometrical dimensions of the THG assembly to find the sensitivity of these parameters on the outputs. The influence of the hub outer diam eter and the radial interference (between the trunnion and hub) is at differe nt levels on the critical crack length and the stress ratio as it is dependent on fulcrum geometry. If we include four staged cooling methods as follow o Type 1: liquid nitrogen o Type 2: dryice/ alcohol ba th followed by liquid nitrogen o Type 3: refrigerated air chamber followed by liquid nitrogen
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viii o Type 4: refrigerated air chamber followed by dryice/alcohol bath and then by liquid nitrogen the cooling type factor contributes the most to both cr itical crack length (up to 79%) and the stress ratio (up to 84%) in th e TH assembling procedures in all three considered bascule bridges. The staged cooling procedure type 2, which is, immersing the TH assembly into the dr y ice medium, followed by immersing the TH set into the liquid nitrogen, give larger critical crack length (up to 400%) and stress ratio (up to 87%) compared to the case that used only liquid nitrogen.
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1 CHAPTER 1 INTRODUCTION 1.1 Bascule Bridges and TGH Assembly A bascule bridge is a type of movable bridge that can be opened or closed to facilitate the movement of waterborne traffic such as ships and yachts (Figures 1.1 and 1.2). Bascule is the French word for seesaw It belongs to the firstclass lever, where the fulcrum is located between the effort and the resistance. However, the bascule bridges belong to the secondclass levers depending on how the load is designated. The bascule bridge opens like a lever on a fulcru m. The fulcrum that is fit into the girder of the bridge is made of a trunnion shaft a ttached to the leaf girder via a hub, and supported on bearings to permit rotation of the leaf. The trunnion, hub and girder when fitted together are referred to as a trunnionhubgirder ( THG) assembly. The THG assembly forms the pivotal element of the bascule mechanism. To open and close the girder (that is, the leaf) of the basc ule bridge, power is supplied to the THG assembly by means of a curved rack and pinion gear at the bottom of the girder or by hydraulic systems.
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Figure 1.1 Bascule bridge open for road traffic (the Tower Bridge, London, UK) Figure 1.2 Bascule bridge open for water traffic (the Tower Bridge, London, UK) The bascule bridge opens like a first order level on a fulcrum (Figure 1.3 and 1.4). The fulcrum that is fit into the girder of the bridge is made of a trunnion and a hub. This trunnion, hub and girder when fitted together are referred to as a trunnionhubgirder (THG) assembly. 2
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Figure 1.3 The THG assembly get installed on the leaf of the bascule bridge Girder Trunnion Hub Figure 1.4 The trunnion is located onto the trunnion frame (the Gateshead Millennium Bridge, Gateshead, UK) The THG assembly is generally made by interference fits between the trunnion and hub, and between the hub and girder. Typical interference fits used in the THG assemblies for Florida bascules bridges are FN2 and FN3 fits which are US Standard Fits. According to Shigley and Mishke (1986): FN2 designation is, Mediumdrive fits that are suitable for ordinary steel parts or for shrink fits on light sections. They are about the tightest fits that can be used with highgrade castiron external members. 3
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4 FN3 designation is, Heavy drive fits th at are suitable for heavier steel parts or for shrink fits in medium sections. In general, since there is ex istence of interference between two cylinders A and B, to insert cylinder A into cylinder B one n eeds to shrink down the cylinder A by cooling media, or to warm up the cylinder B by heat so urce. As the result, there are two different THG assembly procedures (AP) in the practic al field, named AP1 and AP2 (Figure 1.5). AP1 involves the following steps: 1. The trunnion is shrunk by c ooling in liquid nitrogen. 2. This shrunk trunnion is th en inserted into the hub and allowed to warm up to ambient temperature to develop interference fit on the trunnionhub interface. 3. The resulting trunnionhub assembly is shrunk by cooling in liquid nitrogen. 4. This shrunk trunnionhub assembly is then inserted into the girder and allowed to warm up to ambient temper ature to develop interference fit on the hubgirder interface. AP2 involves the following steps: 1. The hub is shrunk in the liquid nitrogen. 2. This shrunk hub is then inserted into the gi rder and allowed to warm up to ambient temperature to develop interference fit on the hubgirder interface. 3. The trunnion is shrunk by c ooling in liquid nitrogen. 4. This shrunk nitrogen is the insert ed into the hubgirder assembly and allowed to warm up to ambient temperat ure to develop interference fit on the trunnionhub interface.
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Figure 1.5 Two different assembly procedures (Besterfield, Kaw, and Crane, 2001) During either of these assembly procedures, the trunnion, hub and girder develop both structural stresses and thermal stress. The structural stresses arise due to interference fits between the trunnionhub, while the thermal stress develop when the trunnion, the hub, or the trunnionhub assembly is immersed in liquid nitrogen or when the cold trunnion is inserted into the hub. Transient Stress term will be use in this study to mean stresses during the assembly procedure and the term Steady State Stress will be used to mean the stresses in the trunnion and hub at the end of the assembly procedure. One good assumption can be made that the structural stresses is depended on the structural stresses but not vice versa. 5
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6 1.2 Previous Works and Objectives of the Thesis Pourmohamadian and Sabbaghian (1985) incorpor ated the varying of material properties to model the transient stresses in a solid cylinder under an axisymmetric load. Therefore, it can not be applied to the THG assembly which has complex geometries, nonsymmetric loading and thermoelastic contact. Then, Noda (1985) studied the thermoelastic contact between two standard cylinders. Again, his work cannot be applied to nonstandard geometries like in the THG assembly. In 1987, Noda also modeled a transient thermoelastic contact problem with a position dependent heat transfer coefficient and transi ent thermoelastic stresses in a short length cylinder. It still does not address the issues of temperature dependent material properties and complex geometries of the THG. Parts of the THG assembly are subjected to thermal shock when they are cooled down before shrink fitting. Oliveira and Wu (1987) inspected the fracture toughness for hollow cylinders subjected to stress gradient arising due to thermal shock, but the results covered a wide range of cylinder geometries. Denninger (2000) created software tools that gave the user the opportunity to examine the torque generated in actual THG bascule bridge designs, the e ffect of the interference and fit specifications on the stresses in the THG assembly, and the bolt patterns. But, the steady stresses in the THG assembly are well below the ultimate tensile strength and yield strength of the material. So, are the transien t stresses more than the allowable stresses? A parametric finite element model was de signed by Ratnam (2000). This study developed to find both the transient and steady state st resses occurred during the assembly process. ANSYS was used for the finite element an alysis to determine those stresses. In AP1, a combination of high hoop stress and low temperature result in smaller values of critical crack length that possi bly lead to crack development and growth.
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7 In AP2, stresses due to interference never o ccur together with the thermal stresses during the cooling process, resulting in la rger values of critical crack length, thereby reducing the probability of crack development and growth. There is a need of a genera l guideline for assembling the THG fulcrum, since his work also showed that it has to be analyzed separately for every bascule bridge. Nichani (2001) validated the theoretical valu es of stresses of two procedures assembly (AP1 and AP2) from previous two studies (Denninger, 2000 and Ratnam, 2000). Two identical sets of trunnionhubgirder were a ssembled, one using assembly procedure AP1 and the other using AP2. The results came out that confirm the statement that the procedure AP2 is safer than the procedure AP1 in the term of lower hoop stresses, VonMises stresses and larger critical crack length ( CCL ). But in real practice, the result may be different since each bascule bridge ha s different geometrical dimensions. Further more, in both procedures AP1 and AP2, the THG assembly experiences thermal shock which reduces the CCL tremendously, so a staged cooli ng assembly procedure needs to be investigated. Berlin (2004) gives an innovative assembly procedure to install the trunnionhub assembly into the girder by heating up the girder to create a clearance. Since the placement of the coil in warming the girder is critically based on the geometry of the girder, the work has a limitation of genera l application for all bascule bridges. In the same year, Collier (2004) devel oped a finite difference model of a long compounded cylinder with axisymmetric response with temperature dependent properties. The study showed that the resistan ce to failure was increased by as much as 50% when the compounded cylinder is cooled first in a refrigerated air chamber and followed by immersion in liquid nitrogen. Bu t, we still need to analyze the threedimensional complex geometry of the THG.
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8 Paul (2005) changed the geometrical dimensions of the inside diameter of the trunnion, the outside diameter of the hub and interfer ence between trunnion and hub according to design of experiments standards to find the sensitivity of these parameters on critical stresses and critical crack lengths during the assembly. But the results were limited to cooling in liquid nitrogen a nd to one bascule bridge. In this thesis, based on what has been done in the previous works, we investigate the influences of main geometry parameters of the trunnion, hub, and di fferent cooling stage combinations. Three different bascule brid ges which are different in geometrical dimensions have been considered to verify the validity of this design of experiment study.
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CHAPTER 2 LITERATURE 2.1 Introduction The chapter introduces detail about the geometry of the THG assembly (Figure 2.1) and the AP1 assembly procedures will be explored. Discussion of equations of equilibrium, the straindisplacement equations and the stressstrain equations for the trunnionhubgirder assembly are discussed in this chapter. Boundary conditions for the trunnionhub assembly will also be introduced and explained clearly. The nonlinear material properties of metal that is used to make the trunnion and the hub will be included in this section. 2.2 Geometrical Details Figure 2.1 The THG assembly (Besterfield, Kaw, and Crane, 2001) 9
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Due to the purpose of this thesis, we only focus on the geometry of the trunnion and hub assembly only, and as shown in Figures 2.2, 2.3 and 2.4 Figure 2.2 The TH assembly 10
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WF L F L L H L T Figure 2.3 The side view of the TH assembly 11
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12 Figure 2.4 The top view of the TH assembly The geometric specifications of the trunnion, hub and girder are tabulated in Table 2.1. Table 2.1 The TH geometry parameters summary Dimensional Terms Description LT Total length of the trunnion L Extension of the trunnion on the gusset side (length to hub on the trunnion on the gusset side) LH Total length of the hub RTI Inner radius of the trunnion RTO Outer radius of the trunnion (inner radius of the hub) RHO Outer radius of the hub RFO Outer radius of the hub flange 2 RTO 2RFO 2 TG2 R T I 2 RHO
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Table 2.1 (Continued) WF Width of hub flange TG Gusset thickness LF Distance to hub flange IN Radial interference between T and H 2.3 Analytical Details The equations of equilibrium, the straindisplacement equations and the stressstrain equations for the trunnionhubgirder assembly are discussed in this section. 2.3.1 Parameters used To develop these equations, the following symbols are used, where i = 1 and 2 represents the trunnion, hub, respectively. Summary of parameters are given in Table 2.2. Table 2.2 Summary of parameters used for TH assembly analysis Terms Description Stresses: iriizieir Radial stress Hoop stress Axial stress VonMises stress Shear stress in r plane Strains: iriiz Radial strain Hoop strain Axial strain 13
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14 iuiu Table 2.2 (Continued) Displacements: r izu Radial displacement Hoop displacement Axial displacement Shear strains: ir iz izr Shear strain in the rz plane Shear strain in the plane Shear strain in the z r plane Temperature dependent parameters: Ti TTGiThcThc)(T Ki C p (T) TEi (T) Temperature dependent Poissons ratio Temperature dependent thermal conductivity Temperature dependent shear modulus Temperature dependent heat transfer coefficient of cooling medium Temperature dependent heat transfer coefficient of cooling medium Temperature dependent material density Temperature dependent material specific heat Temperature dependent Youngs modulus Temperature dependent thermal expansion coefficient
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15 TKIC Table 2.2 (Continued) The critical fracture toughness of the material Terms in use: r, z Cylindrical coordinates is used T T(r,z,) is the temperature distribution [ o F] T w T(r,z,) is the wall temperature[ o F] T c The cooling medium temperature [ o F] t Inspected time [s] 0E Elastic constant at absolute zero S A constant eT Einstein characteristic temperature Volume coefficient of thermal expansion 3T Lattice contribution Normal electronic specific heat T Electronic contribution a Crack length [in] ef Edge effect factor RDIV Radial divisions on trunnion, hub, flange CDIV Circumferential division of trunnion, hub, and flange LTDIV Division along the length of the trunnion LHDIV Division along the length of the hub GDIV Division on the triangular side of the gusset GWDIV Division on the thickness side of the gusset LFDIV Division on the width of the flange
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Table 2.2 (Continued) g Acceleration due to gravity D The cylinder diameter [in] The kinematics viscosity The absolute viscosity k The coefficient of thermal conductivity of cooling agent Gr The Grashoff number Pr The Prandtl number Nu The Nusselt number 2.3.2 Governing equations and boundary conditions In this thesis, we assume that the TH assembly already exists at the room temperature after the cooled trunnion is inserted into the hub. The most critical part of the assembly, from previous studies, was identified as the trunnionhub interface immediately after it is immersed in liquid nitrogen for cooling, and before sliding into the girder. Hence, the main interests of this thesis are the structural conditions after the trunnionhub contact, the thermal conditions during cooling of the trunnionhub and the structural conditions immediately after the cooling, and before sliding it into the girder. The main goal of this thesis is to change the geometrical parameters of the trunnion and the hub, and to find the sensitivity of these parameters on critical stresses and critical crack lengths during this stage of the assembly. The trunnionhub assembly is immersed in a cooling medium at temperature until it approaches steady state at time. cT ct Structural elasticity equations and structural boundary conditions: (since our problem involve two disciplines field, structure and thermal) 16
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Equilibrium Equations The equations of equilibrium are given by Equations 2.1, 2.2 and 2.3 01zrrrirziiririr (2.1) 021 rzrririziir (2.2) 01 rrrzirzizirziz (2.3) StressStrain Equations The stressstrain equations are given by TTiziiiriirodTTTTE)()()(1 (2.4) TTiziriiiidTTTTE0)()()(1 (2.5) TTiriiiziizdTTTTE0)()()(1 (2.6) TGiirir (2.7) TGiizrizr (2.8) TGiiziz (2.9) StrainDisplacement Equations The normal strains and displacements are related by the following equations ruirir (2.10) 17
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rururirii1 (2.11) zuiziz (2.12) The shear strains and displacements relations are given as the follow ruruuriiirr1 (2.13) ruzuizirrz (2.14) zuuriizz1 (2.15) Structural Boundary Condition Figures 2.5 and 2.6 with geometrical dimension abbreviations will help to understand the structural boundary conditions. 18 Figure 2.5 Coordinate for the hub and trunnion (front view) 2or 2ir1or1ir3or3ir
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z 2el3el3sl2sl1sl1el Figure 2.6 Coordinate of the TH set (side view) 19
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The structural boundary conditions on the inside radius of the trunnion, are 1irr 0),,,(11tzrir 20 , 11eslzl ctt 0 (2.16) 0,,,11tzrir 20 , 11eslzl ctt 0 (2.17) At the outer radius of the trunnion, where there is no contact 1orr 0),,,(11tzror 20 , (2.18) 21sslzl 12eelzl ctt0 0,,,11tzror 20 ,, (2.19) 11eelzl 12eelzl ctt0 At the surface in contact at the trunnion outer radius, 1orr tzrtzriror,,,,,,2211 20 ,, (2.20) 22eslzl ctt0 tzrtzriror,,,,,,2211 20 ,, (2.21) 22eslzl ctt0 tzrutzruiror,,,,,,2211 20 ,, (2.22) 22eslzl ctt0 tzrutzruio,,,,,,2211 20 ,, (2.23) 22eslzl ctt0 At the outer radius of the hub, 2orr 0,,,22tzror 20 ,, (2.24) 22eslzl ctt0 0,,,22tzror 20 ,, (2.25) 22eslzl ctt0 At the right edge of the hub at, the hub is constrained to avoid rigid body motion by the following conditions: 2slz 0,,,22tlrusz 22oirrr 20 (2.26) ctt0 0,,,22tlrsrz 22oirrr 20 (2.27) ctt0 0,,,22tlrsz 22oirrr 20 (2.28) ctt0 20
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Thermal governing equations and thermal boundary conditions: Now, when we have the shrink fit stresses solved from the above elasticity equations and structural boundary conditions, we now immerge the TH assembly into the cool medium which has a temperature of T c and want to investigate for time t c At and for from ziik (1993) we have: initialTTt:0, ctt0 Thermal Governing Equations: tTTKTCTzTTrrTrrrp)()()(1122222 (2.29) Thermal Boundary Condition: Trunnion experienced convection on its inner radius, is: 1irr 1irr ,, 11eslzl ctt0 ))(()(1ccTTTh r TTK (2.30) At the outer radius of the trunnion,, there are noncontact and contact surfaces. 1orr At the inside noncontact surface, 1irr , 11eslzl ctt 0 ))(()(1ccTTTh r TTK (2.31) At the contact surface, 1orr ,, 22eslzl ctt 0 r TTK r TTK )()(21 (2.32) At the outer radius of the hub, 2orr 21
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2orr ,, 22eslzl ctt0 ccTTThrTTK2 (2.33) At the outside noncontact surfaces, o ,,, 1orr 21sslzl 12eelzl ctt 0 ))(()(1ccTTTh r TTK (2.34) o ,, 2orr 32sslzl ctt 0 ccTTThrTTK 2 (2.35) o ,, 3orr 33eslzl ctt 0 ccTTThrTTK 2 (2.36) o ,, 2orr 23eelzl ctt 0 except the areas that contact to 6 gussets ccTTThrTTK 2 (2.37) o For all three sides of each gusset: ccTTThrTTK 2 (2.38) At the endsurfaces of trunnion o coisrrrlz111, ctt 0 ))(()(1ccTTThzTTK (2.39) o coiettrrrlz0,,111 ))(()(1ccTTThzTTK (2.40) At the endsurfaces of hub o coisttrrrlz0,,222 22
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))(()(2ccTTThzTTK (2.41) o coiettrrrlz0,,222 ))(()(2ccTTThzTTK (2.42) At the endsurfaces of the flange o coisttrrrlz0,,333 ))(()(2ccTTThzTTK (2.43) o except the areas that contact to the gussets coiettrrrlz0,,333 ))(()(2ccTTThzTTK (2.44) 2.4 Nonlinear Material Properties of Metal The nonlinear material properties for a typical steel, Fe2.25 Ni (ASTM A203A) are plotted in the next several pages. Though nonlinear material properties in general are explored, particular emphasis is given to properties at low temperatures. 2.4.1 Youngs modulus The elastic modulus of all metals increases monotonically with increase in temperature. The elastic modulus can be fitted into a semiempirical relationship TE 10TTTeeSEE (2.45) The Youngs modulus remains stable with change in temperature, that is, the variation is not very large as shown in Figure 2.7, and hence is assumed to remain constant throughout this analysis. 23
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Figure 2.7 Youngs modulus of steel as a function of temperature (Collier, 2004) 051015202530354003002001000100Temperature Young's Modulus (Mpsi) )(Fo 2.4.2 Coefficient of thermal expansion The coefficient of thermal expansion at different temperatures is determined principally by thermodynamic relationships with refinements accounting for lattice vibration and electronic factors. The electronic component of coefficient of thermal expansion becomes significant at low temperatures in cubic transition metals like iron (Reed, 1983). The coefficient of thermal expansion increases with increase in temperature by a factor of three from o F to 80 o F as shown in Figure 2.8. 24
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Figure 2.8 Coefficient of thermal expansion of steel as a function of temperature (Collier, 2004) 00.0000010.0000020.0000030.0000040.0000050.0000060.0000074003002001000100Temperature Coefficient ofThermal Expansion )(Fo)//(Finino 25 2.4.3 Thermal conductivity The coefficient of thermal conductivity increases with an increase in temperature by a factor of two from 321 o F to 80 o F (Figure 2.9). Thermal conduction takes place via electrons, which is limited by lattice imperfections and phonons. In alloys, the defect scattering effect T is more significant than the phonon scattering effect 2T (Reed, 1983).
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Figure 2.9 Thermal conductivity of steel as a function of temperature (Collier, 2004) 0.00020.000250.00030.000350.00040.000450.00050.000554003002001000100Temperature Thermal Conductivity )(Fo)///(FinsBTUo 2.4.4 Density For the range of temperatures of interest to our study the density remains nearly constant, as shown in Figure 2.10. 26
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Figure 2.10 Density of steel as a function of temperature (Collier, 2004) 00.10.20.30.44003002001000100Temperature Density )(Fo)/(3inlb 2.4.5 Specific heat Lattice vibrations and electronic effects affect the specific heat of a material. The contribution of two effects can be shown by Equation 2.46. TTC3 (2.46) Note that specific heat decreases by a factor of five over the temperature range in question, as shown in Figure 2.11. 27
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Figure 2.11 Specific heat of steel as a function of temperature (Collier, 2004) 00.020.040.060.080.10.124003002001000100Temperature SpecificHeat )(Fo)//(FlbBTUo 2.5 Nonlinear Material Properties of Cooling Media 2.5.1 Convective heat transfer coefficient, h c and assumptions The assumptions in the calculation of the convective heat transfer coefficient for cooling media are as follow: the geometry of the assemblies is assumed to be cylindrical. We need to calculate the Grashoff number, Prandtl number and the Nusselt number before we can obtain the convection coefficient, h. According to Incropera, and DeWitt (1996), we have: 33)( DTTgGrcw (2.47) and the Pr then is calculated 28
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kcpPr (2.48) as the result, the Nu is calculated as 227816961Pr492.01Pr387.0825.0GrNu (2.49) finally, the convective heat transfer coefficient for the cooling media is obtained as DkNuhc (2.50) Note: The value for the hydraulic diameter, D for the TH assembly is the trunnion outer diameter. Turbulent flow is assumed. The convection coefficient is assumed to be dependent on the wall temperature and the bulk or ambient temperature, and D. 2.5.2 Convection to refrigerator air There is another cooling stage option that we can apply to cool down the assembly by using the refrigerated air to cool the assembly first and before next cooling stage. For the convective heat transfer coefficient of fridge air as a function of the wall temperature is given in Figure 2.12. 29
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Figure 2.12 Convective heat transfer coefficient of refrigerator air as a function of temperature (Collier, 2004) 00.0000020.0000040.0000060.000008402002040608Wall TemperatureConvective Heat Transfer Coefficient 0 )(Fo)///(2inFsBTUo 2.5.3 Convection to dry ice/ alcohol bath Dry ice is frozen carbon dioxide a normal part of our earth's atmosphere. It is also the same gas commonly added to water to make soda water. Dry ice is particularly useful for freezing, and keeping things frozen because of its very cold temperature (108F). Dry ice is widely used because it is simple to freeze and easy to handle using insulated gloves. Dry ice changes directly from a solid to a gas sublimationin normal atmospheric conditions without going through a wet liquid stage. The convective heat transfer coefficient versus temperature is given in Figure 2.13. 30
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Figure 2.13 Convective heat transfer coefficient of dry ice as a function of temperature (AspenTech, 2004) 00.000010.000020.000030.00004120100806040200Wall TemperatureConvective Heat Transfer Coefficient )(Fo)///(2inFsBTUo 2.5.4 Coefficient of convection of liquid nitrogen The convective heat transfer coefficient of liquid nitrogen is dependent on many factors, such as, surface finish, size of the object and shape of the object, to name a few. Based on the previous discussion, the convective heat transfer coefficient of liquid nitrogen is shown in Figure 2.14 (Brentari and Smith, 1964). This data was chosen because it very closely matches the surface finish, and object sizes and shapes used for trunnions and hubs. Note that the convective heat transfer coefficient of liquid nitrogen is evaluated at the wall temperature. 31
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Figure 2.14 Convective heat transfer coefficient of liquid nitrogen as a function of temperature (Brentari and Smith, 1964) 32 The phenomenon of convection to liquid nitrogen is quite complex and involves multiphase heat transfer (Figure 2.15). Whenever an object at ambient temperature (say, 80 o F) comes into contact with liquid nitrogen, film boiling occurs until the temperature of the object reaches approximately o F. This phenomenon of film boiling occurs when there is a large temperature difference between the cooling surface and the boiling fluid. At the point when film boiling stops, the minimum heat flux occurs and the phenomenon of transition boiling occurs until the temperature of the object reaches o F. At the point when transition boiling stops, the maximum heat flux occurs and the phenomenon of nucleate boiling occurs until the temperature of the object reaches the temperature of liquid nitrogen. Nucleate boiling occurs when small bubbles are formed at various 0.480.520.560.64003002001000100Temperature Convective Heat TransferCoefficient )(Fo)///(FinhrBTUo
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nucleation sites on the cooling surface. When nucleate boiling st arts the object cools very rapidly. 1000 10000 100000 1 10 100 1000Change in Temperature, Twall Tsaturation (oF)Heat Flux, q/A (BTU/hrft2) Transition Boiling Film Boiling N ucleate Boiling Maximum Minimum heat flux Flux Figure 2.15 Heat flux versus temperature di fference for liquid nitrogen (Barron, 1999) 33
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34 CHAPTER 3 ANSYS MODELING AND ANALYSIS 3.1 Introduction This chapter introduc es the selection of method that is used to approach this thermal structural problem. Also, it is going to give a literature survey about appropriate element types that are used to bu ild different part in the TH assembly. Finally, the ANSYS code which is a batch of ANSYS comm ands is explained in detail. 3.2 Why Inspect Only 1/6 th of the TH Assembly? Since the TH assembly has an axisymmetric shape, for the sake of computational time spent on running ANSYS program (according to design of experiment, we are going to run 36 different cases for each bridge, and th ere are three different bridges will be inspected). Then, just 1/6 th of the TH set will be modeled and analyzed. 3.3 Assumption for ANSYS Seque ntial Coupled Field Approach The assumption in this approach is that the structural results are dependent upon the thermal results but not viceversa. This is a fair assumption as the effect of strains on the thermal analysis is negligible. Also, the sta ndard inaccuracies associated with any finite element model due to mesh density, time incr ements, number of substeps, etc. are present in this model. The material prope rties of the trunnion hub assembly and the cooling medium are temperature dependent an d are evaluated at specified temperature increments. The properties in between or out side the extremes of these values are interpolated and extra polated, respectively.
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35 3.4 Coupled Field Analysis A coupledfield analysis is one that consists of the interactions between two or more disciplines or fields of engi neering. For example, a piezoe lectric analysis, handles the interaction between the stru ctural and electric fields: it solves for the applied displacements due to voltage distribution, or vice versa. Thermalstress analysis, thermalelectric analysis, fluidstructure analysis, magneticthermal analys is, magnetostructural analysis and microelectromechanical system s (MEMS) are other examples of coupledfield analysis. This study involves the coupling of the therma l and structural fields. ANSYS features two types of Coupled Field anal ysis: Direct and Sequential. 3.5 Direct Versus Sequential Coupled Field Analysis Direct Coupled Field Analysis The direct method often consists of just one analysis that uses a coupledfield element type (for example, SOLID5, PLANE13, or SOLID98) containing all necessary degrees of freedom. Coupling is handled by calculating el ement matrices or element load vectors that contain all necessary terms, simultaneousl y. This method is used when the responses of the two phenomena are dependent upon each other, and is computationally more intensive. Sequential Coupled Field Analys is (Indirect Coupled Field) The sequential method involves two or more sequential analyses in which, the results of one analysis are used as the loads of the follo wing analysis, each belonging to a different field. This method is used where there is oneway interaction between the two fields.
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36 There are two types of sequential coupled field analysis: sequentially coupled physics and sequential weak coupling. Sequentially coupled physics analysis: Th e results from the first analysis are applied as loads for the second analysis. The load is transferred external to the analysis, and they must explicitly be transferred using the physics environment. An example of this type of analysis is a sequential thermalstress analysis where nodal temperatures from the thermal analysis are applied as body force loads in the subsequent stress analysis. Sequential weak coupling analysis: Th e solution for the fluid and solid analysis occurs sequentially, and the load transfer between the fluid and the solid region occurs internally across a similar or dissimilar mesh interface. An example of this type of analysis is a fluidstructure interaction analysis requiring transfer of fluid forces and heat flux from the fluid to the structure and displacements and temperature from the structure to the fluid. 3.6 The Finite Element Model This thesis concentrates on the procedure that is one of the steps in AP1 in which the trunnionhub assembly is cooled in the cooli ng medium. Prior to this step, the assembly has interference stresses from the 2 nd step, in which the shrunk trunnion is inserted into the hub to form an interference fit. Hence, it becomes imperative to have the interference stresses present in the assembly befo re subjecting it to cooling media. To incorporate this, the interference valu es are calculated base d on the trunnion outer diameter or the hub inner diameter, using FN2 fit specifications. These values are then added to the diameters and the geometry is constructed in ANSYS. A structural analysis to determine the interference stresses is done by allowing the interference fit to take place. The problem is solved with no additi onal displacement constraints or external forces. The trunnion is constrained within the hub due to its geometry. Stresses are
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37 generated due to the general misfit between the target (hub) and the contact (trunnion) surfaces. The trunnionhub assembly is hence obtained w ith the interference stresses and this assembly is now subjected to cooling in a li quid nitrogen bath. This is done in ANSYS by subjecting the exposed areas of the assembly to convection in a cooling medium whose properties are the same as that of liquid nitroge n. The result of this thermal analysis is the temperature distribution in the trunnionhub assembly. The te mperature distribution thus obtained is applied as the load to the subseque nt structural analysis, to obtain the thermalstresses in the trunnionhub assembly. It is important to understa nd that the stresses obtained after this analysis is the combination of the stresses due to the interference between the trunnion and the hub (interference stresses), and the stresses due to the temperature gradient (thermal stresses). 3.7 ANSYS Element Selection The elements for the finite element model are chosen from the ANSYS element library (ANSYS Element Reference Manual, Rel ease 10.0, 2005), which consists of various elements to represent the different p hysical materials used in real life. 3.7.1 ANSYS element library and classification Element library is grouped based on the foll owing characteristics to make element type selection easier. Totally, there are three different groups as follows. Two dimensional versus three dimensional models : ANSYS models may be either twodimensional or th reedimensional depending upon the element types used. Axisymmetric models are considered to be twodimensional. Element characteristic shape : in general, there are four different shapes those are:
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38 o Point o Line o Area o Volume Degrees of freedom and discipline : The degrees of freedom of the element determine the discipline for which the element is applicable: structural, thermal, fluid, electric, magnetic, or coupledfield. The element type should be chosen such that the degrees of freedom are sufficient to characterize the model's response. 3.7.2 Selection of elements The elements used in this model are chosen based on all of the characteristics described in the previous section, including the diffe rent physical analyses the model undergoes. The method of selection of the elements is briefly described in this section. The geometry of the trunnionhub assembly is 3dimensional and has volume. Therefore, the elements used for the finite element model are chosen only from among the solid elements of the element library. The firs t analysis that th e trunnionhub assembly undergoes is a structural analysis which is done to include the interference stresses that develop at the previous ste p, caused when the trunnion is sh rink fit into the hub. Since it is a structural analysis, a structural solid element (SOLID45) is chosen. The interference between the tr unnion and the hub is simulated with the help of special elements called Contact Elements. ANSYS supports both rigidtoflexible and flexibletoflexible surfacetosurface contact el ements. These contact elements use a target surface and a contact surface to form a contact pair. The target and associated contact surfaces are identified via a shared real cons tant set. These surfacetosurface elements are wellsuited for applications such as interf erence fit assembly contact or entry contact,
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39 forging, and deepdrawing problems. Since, the trunnion and the hub are expected to undergo deformation; the contact is identif ied as flexibletoflexible contact. In problems involving cont act between two boundaries, one of the boundaries is conventionally established as the target surface, and the other as the contact surface. Contact elements are constrained against penetr ating the target surface. However, target elements can penetrate through the contact surf ace. For flexibletoflexible contact, the choice of which surface is designated contact or target can cause a different amount of penetration and thus affect the solution accu racy. Many guidelines are presented in the ANSYS Structural Analysis Guide, Re lease 10.0, which can be followed when designating the surfaces. The most releva nt guideline for this model reads, If one surface is markedly larger than the other surface, such as in the inst ance where one surface surrounds the other surface, the larger surface should be the target surface . Using the above guideline, the hub is designated as the target surfac e and the trunnion is designated as the contact surface. TARGE170 is used to model the target surface with CONTA174 as the contact surface, since the co ntact pair is 3dimensional. They behave as structural contact having structural degrees of freedom in the first analysis. The interference fit trunnionhub assembly, the n, undergoes a thermal analysis when it is cooled in liquid nitrogen. A thermal solid el ement is required for this analysis. However, it is not required to select another elemen t from the ANSYS element library as ANSYS automatically changes the structural elemen t to its corresponding thermal element when the element type is changed from structural to thermal. In this case, ANSYS changes SOLID45 to its corresponding thermal element SOLID70. However, the contact elements cannot be changed as they do not have any ot her elements associated with them. Hence, their degrees of freedom are changed to make them behave as thermal contact. The final analysis the trunnionhub assembly unde rgoes is a structural analysis where the total stress, that is, the combination of inte rference stresses and the stresses due to the
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40 temperature gradient (thermal stresses) is obtai ned. Since this is a structural analysis, the elements are changed back to structural elements, as they were in the first analysis. The thermal element SOLID70 is changed back to SOLID45 by ANSYS when the element is changed from thermal to structural. The contact elements are changed back to structural contact by changing their degrees of freedom. In summary, four elements are used in this thesis: SOLID45, SOLID70, TARGE170, and CONTAC174. 3.7.3 Selected element characteristics SOLID45: the structural solid element used for the structural analyses is SOLID45 (Figures 3.1 and 3.2). It is generally used for the threedimensional modeling of solid structures. The element is defined by eight nodes having three degrees of freedom at each node: translations in the nodal x y and z directions. SOLID70: the thermal solid element used for the thermal analysis is SOLID70. It has a threedimensional thermal conduction capabi lity. The element has eight nodes with a single degree of freedom, temperature, at each node. The element is applicable to a threedimensional, steadystate or transient therma l analysis. The following figure shows an 8node (I,J,K,L,M,N,O,P) hexahedral solid element with 6 surfaces. It represents both, SOLID45 and SOLID70, since they have the same geometry, node locations, and coordinate system.
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Figure 3.1 SOLID453D structural solid, and SOLID703D thermal solid (ANSYS release 10.0 documentation) Figure 3.2 Trunnionhub assembly with SOLID45 and SOLID70 elements CONTA174: the contact surface for the trunnion at the trunnionhub interface is modeled using CONTA174 (Figure 3.3). This type of element is used to represent contact and 41
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sliding between 3D target surfaces and a deformable surface (trunnion), defined by this element. This element is located on the surfaces of 3D solid or shell surfaces. It has the same geometric characteristics as the solid or shell element face with which it is connected. It can be used in almost every discipline of engineering as it can support any degree of freedom when the corresponding KEYOPT is changed. Contact occurs when the element surface penetrates one of the target segment elements on a specified target surface. The following figure shows the element CONTA174 overlaying the outside diameter surface of the trunnion at the trunnionhub interface. Figure 3.3 CONTA174 overlaying the trunnion outer diameter surface TARGE170: the target surface for the hub at the trunnionhub contact is modeled using TARGE170 (Figure 3.4). This type of element is used to represent various 3D target surfaces for the associated contact elements. The contact elements themselves overlay the solid elements describing the boundary of a deformable body (trunnion), and are potentially in contact with the target surface (hub), defined by TARGE170. This target 42
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surface is discretized by a set of target segment elements (TARGE170) and is paired with its associated contact surface via a shared real constant set. Any translational or rotational displacement, temperature, and voltage can be imposed on the target segment element. Forces and moments can also be imposed on target elements. Figure 3.4 shows the target element TARGE170 overlaying the inside diameter surface of the hub at the trunnionhub interface. Figure 3.4 TARGE170 overlaying the hub inner diameter surface 3.8 The Procedure of the ANSYS Modeling and Analysis The procedure of the ANSYS code is explained carefully as follows. START THE ANSYS PROGRAM. THE STRUCTURAL ANALYSIS 43 o Define trunnion and hub geometrical parameters.
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o Define element type, material of TH assembly, and properties of cooling media. o Build TH and then perform controlledmesh by the following parameters (Figure 3.5): RDIV: Radial divisions on trunnion, hub, flange CDIV: Circumferential division of trunnion, hub, and flange LTDIV: Division along the length of the trunnion LHDIV: Division along the length of the hub GDIV: Division on the triangular side of the gusset GWDIV: Division on the thickness side of the gusset LFDIV: Division on the width of the flange Figure 3.5 Division of the parts and meshed model TH o Create contact pair Generate the target surface Generate the contact surface o Apply boundary conditions, symmetric displacement conditions o Solve for the interference stresses of the structural problem of TH assembly 44
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45 THE TRANSIENT THERMAL ANALYSIS: o Return to PREP7 and modify the database to prepare for thermal analysis: Switch element types, from structural type into thermal Switch structural contact to thermal contact o Specify thermal boundary conditions: Specify areas that come into play in convection and those ones which are not. Specify nodes that come into play in convection and those ones which are not. o Solve the transient thermal problem for TI seconds. Create a table of four columns are r z , and temperature, respectively from 1 st second to TI th second. FINAL: STRUCTURAL AND THERMAL ANALYSIS AT EVERY SECOND o Return to PREP7 and modify the data base to prepare for final analysis (combine both results from stru ctural and thermal analysis): Load results from initial interference stresses Switch element types from thermal type into structural Switch thermal contact to structural contact Apply symmetric displacement boundary condition For every n th second from 1 to TI load the thermal temperatures of that second as a body force, then so lve as a the structural problem. o Write the result [ r z , temperature, Hoop stress, and VonMise stress] to an EXCEL file for each second for later use. STOP ANSYS.
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The ANSYS flow chart program is also developed as followed. Start Read: RTI, RTO, LT . ., RDIV, CDIV, LTDIV . ., E, , C p TI, h Define: SOLID45, T o T c Generate:1/6 th TH assembly, TARGE170, CONTA174 Apply: symmetric displacement conditions SOLVE: structural interference stresses Structure Thermal (SOLID45 to SOLID70, contact) Apply: thermal BCs, loads SOLVE: transient thermal stresses for TI second(s) Write: r , z, T Thermal Structure (SOLID70 to SOLID45, contact) Apply: structural load (interference stresses), sym displacement conditions Apply: thermal load (thermal stresses) SOLVE: the total stresses Write: r ,, z, T, e to Excel no 46 I TI y es Stop Figure 3.6 ANSYS flow chart
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47 CHAPTER 4 DESIGN OF EXPERIMENT AND SENSITIVITY ANALYSIS 4.1 Introduction Experimental design, which deals with collect ion of data, was deve loped at about the same time as the analysis of variance, ma inly by R. A. Fisher (C. L. Chiang, 2003). Scientific research depends on the quality of data collected. A good design is as essential to a successful study as a proper method of analysis. Corresponding to an observation in statistical analysis, there is an experimental unit in the design. An experimental unit then is one on which a treatment is applied and an observation is made. The need for the experiment design is because of the variati on among experimental units. The objective of an experimental design is to control the e xperimental variation by proper assignment of treatments to experimental units. The met hod of assigning treatments to experimental units is the design of the experiment. Generally, experiment al variations can be classified into two categories: systematic and ra ndom variations or factors (Figure 4.1). Systematic factors: is associated with a specific factor, or factors in the study, usually can be controlled or subject to a st atistical test. Random factors: generally cannot be controlled and some may not even be detected before an experiment is pe rformed. It is principally due to the innate difference among experimental units.
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Systematical Factors 0. x p x 2 ... Inputs Output Process y Ther z 1 z q z 2 Random i zation F actors Figure 4.1 General model of a process or system (Montgomery, 2001) There are several types to design an experiment (Montgomery, 2001) Randomized complete block designs. Latin square designs. GraecoLatin square designs. Balanced incomplete block designs. Factorial designs: o General factorial design. o 2k factorial design. o Highlevel and mixedlevel factorial design. 4.2 Screening the Variations As we can see, there are many factors that may come into play on the critical crack length (CCL) and stress ratio (SR) in preparing the TH assembly, for example: length of the trunnion, length of the hub, thickness of the flange, thickness of the gussets, inner diameter of the trunnion, inner diameter of the hub, outer diameter of the hub, and 48
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49 cooling type combinations that are used to cool down the TH set before inserting into the girder. So, it is necessary to screen out those factors that seem affecting less on the output. This is usually done by experiences of the de signer(s), from the previous works, by conducting some pilot runs or by performi ng the fractional factorial design by which we can roughly estimate the percent of cont ribution of each fact ors. We keep those factors that contribute more th an a certain value, and then do the full experiment with them. Since, this thesis is a follow up work that has inherited all of the previous works of the USF professors and graduate students, a nd based on their work, it came to attention that the staged cooling type, the outer diam eter of the hub and the interference between the hub and the trunnion seem pl ay significant roles on th e output. There are three interested factors inspected in the thesis: tw o quantitative factors (radial interference and outer diameter of the hub) and one qualitative factor (cooling type). Radial interference factor: It is an unc ontrollable variance, but we can calculate the limitation of this parameter based on the geometry of the TH and the interference criteria fit (FN2 or FN3). S o, it is reasonable that we will detect the contribution of those limits (lower and upper values) of variation of the interference. Outer diameter of the hub: We want to ch eck the effect of the outer diameter of the hub. American Association of State Highway and Transp ortation Officials (AASHTO) standards call for a hub radi al thickness of 0.4 times the inner diameter, while currently a hub radial thickness of 0.1 to 0.2 times the inner diameter is used. Therefore, it is impera tive to study how the critical crack length and the stress ratio change when the AAS HTO standards are employed, and if it is better than the current pract ice. So, it comes up to the idea that we want to investigate the two limits of this ratio: minimum of 0.1 to a maximum of 0.4 and also the middle level to obtai n a clearly picture of how the CCL and SR vary over this range of hub radial thickness. Cooling type factor: Collier (2004) stated that the minimum cr itical crack length and stress ratio are increased by as mu ch as 200% when cooling first in refrigerated air followed by liquid nitrogen (comparison to the case that only
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using liquid nitrogen to cool down the TH assembly). So, we like to inspect four different cooling types as follow: o Type 1: using liquid nitrogen only (321o F). o Type 2: immersing the TH into the dry ice/ alcohol bath (108o F) then drop it into the liquid nitrogen (321o F). o Type 3: putting the TH into the refrigerated air chamber (32o F) and then immerge into the li quid nitrogen (321o F). o Type 4: the fourth type is the comb ination of all cooling media: let the TH cooling inside the fridge air chamber (32o F), then immerse it into the dry ice/ alcohol bath (108o F), and followed by dropping into the liquid nitrogen (321o F). To understand and appreciate the generalization characteristic of the gene ral factorial design method, it is good to do a quick review of the 2k factorial design, and mixed level factorial design. 4.3 2k Factorial Design and Mixed Le vels Design of Experiment 4.3.1 2k factorial design Now, we consider k interested factors which have 2 different levels of each, the high and low. A complete replicates of search a design requires observations and is called a factorial design. The model includes: k22......22 k2 (k) main factors. twofactor interaction factors. 2k 50
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three factor interaction factors. 3k One factor interaction factors. k There are total numbers of effects in a factorial design. For example, a factorial design has total number of effects. For example, in our study we involve three factors, so factorial design is appropriate for our case. So, let take a close look on factorial design. 1 2 k k232 1 233232 32 Factorial design Let A, B, C are three main factors involved in the experiment. The total number of observations or runs required is given by Each factor has two different levels indicated by 1 and +1. One leve l is indicated by 1 where as +1 represents other level. There are total number of effects in factorial design, they are 8231 23 3 main factors: A, B, C. 3 twofactor inte raction factors: AB, AC, BC. 1 threefactor interaction factor: ABC. If the factors involved are A, B, and C, then the 8 experiments are named as shown in the following Table 4.1. 51
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Table 4.1 Notations for experiment combinations Run A B C Labels A B C 1 (1) 0 0 0 2 + a 1 0 0 3 + b 0 1 0 4 + + ab 1 1 0 5 + c 0 0 1 6 + + ac 1 0 1 7 + + bc 0 1 1 8 + + + abc 1 1 1 The main effects of the factors A, B, and C, for n replicates, are found by using the Equation 4.1, 4.2, 4.3, respectively. ])1([41bcabccacbabanA (4.1) ])1([41accaabcbcabbnB (4.2) ])1([41abbaabcbcaccnC (4.3) The two factor interaction effects AB, AC, and BC, for n replicates, are found using the following formulae, respectively. ])1([41cacbcabcbaabnAB (4.4) ])1[(41abcbcaccabbanAC (4.5) ])1[(41abcbcaccabbanBC (4.6) The overall interaction effect ABC, for n replicates, is found using the Equation 4.7. 52
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)]1( [ 4 1 ababcacbcabc n ABC (4.7) In the above equations the quant ities in brackets are called Contrasts of the treatment combinations. The sums of squares for the e ffects are calculated as in Equation 4.8. n Contrast SS 82 (4.8) The total sum of squares is calculated by summing the squares of all the data values and subtracting from this number the square of the grand mean times the total number of data values. Mathematically, a i b j c k n l ijkl Tabcn y y SS111 2 1 2 (4.9) ,...2 ,1 ; ,2,...c 1 ,...2 ,1 ; ,...2 ,1 nl k bjai where and i, j, k are the three factors factor A, factor B, and factor C, respectively, and l is the number of n replicates. a i b j c k n l ijkly y1111 The property that the treatment sum of square s plus the error sum of squares equals the total sum of squares is utilized to compute th e error sum of squares. Hence, it is usually calculated by subtraction. ) (ABC BC AC AB CBA T ESSSSSSSSSSSSSSSSSS (4.10) The percentage contribution of each effect is then found by calculating the ratio of the respective sum of squares and the total sum of squares and multiplying by 100. Mathematically, the percenta ge contribution of effect A is calculated as 100 T ASS SS Once, the percentage contribution of each effect is found, the one with the highest value is said to have the most effect on the experiment. The p values are then found to confirm the magnitude of these effects. In general, smaller the p values, more significant are the effects. 53
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54 4.3.2 Mixed level factorial design A design in which some factors have two levels and some factors have three levels can be derived from the table of pl us and minus signs for the 2 k design. The general procedure is best illustrated by an example which is exactly related to our thesis. To apply the theory of 2 k factorial design for our thesis, we need to do manipulation by replace the 3level factor by two 2level factors as in Table 4.2. Table 4.2 Manipulation for 3 level factor to apply 2 k factorial design 2Level factor 3Level fa ctor Actual treatment Run # B C X X 1 1 1 x 1 Level 1 2 +1 1 x 2 Level 2 3 1 +1 x 2 Level 2 4 +1 +1 x 3 Level 3 Like in the same manner, we introduce two new 2level factors P and Q and make a convention like in Tabl e 4.3 and we have resu lts in the Table 4.4. Table 4.3 Manipulation for 4level factor to apply 2 k factorial design 2Level factor 4Level fa ctor Actual treatment Run # P Q Z Z 1 1 1 z 1 Level 1 2 +1 1 z 2 Level 2 3 1 +1 z 3 Level 3 4 +1 +1 z 4 Level 4
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55 Table 4.4 2, 3, 4 levels factoria l design of experiment Run # A B C X P Q Z AB AC AP AQ BC BP BQ CP 1 1 1 1 x 1 1 1 z 1 1 1 1 1 1 1 1 1 2 1 1 1 x 2 1 1 z 1 1 1 1 1 1 1 1 1 3 1 1 1 x 2 1 1 z 1 1 1 1 1 1 1 1 1 4 1 1 1 x 3 1 1 z 1 1 1 1 1 1 1 1 1 5 1 1 1 x 1 1 1 z 2 1 1 1 1 1 1 1 1 6 1 1 1 x 2 1 1 z 2 1 1 1 1 1 1 1 1 7 1 1 1 x 2 1 1 z 2 1 1 1 1 1 1 1 1 8 1 1 1 x 3 1 1 z 2 1 1 1 1 1 1 1 1 9 1 1 1 x 1 1 1 z 3 1 1 1 1 1 1 1 1 10 1 1 1 x 2 1 1 z 3 1 1 1 1 1 1 1 1 11 1 1 1 x 2 1 1 z 3 1 1 1 1 1 1 1 1 12 1 1 1 x 3 1 1 z 3 1 1 1 1 1 1 1 1 13 1 1 1 x 1 1 1 z 4 1 1 1 1 1 1 1 1 14 1 1 1 x 2 1 1 z 4 1 1 1 1 1 1 1 1 15 1 1 1 x 2 1 1 z 4 1 1 1 1 1 1 1 1 16 1 1 1 x 3 1 1 z 4 1 1 1 1 1 1 1 1 17 1 1 1 x 1 1 1 z 1 1 1 1 1 1 1 1 1 18 1 1 1 x 2 1 1 z 1 1 1 1 1 1 1 1 1 19 1 1 1 x 2 1 1 z 1 1 1 1 1 1 1 1 1 20 1 1 1 x 3 1 1 z 1 1 1 1 1 1 1 1 1 21 1 1 1 x 1 1 1 z 2 1 1 1 1 1 1 1 1 22 1 1 1 x 2 1 1 z 2 1 1 1 1 1 1 1 1 23 1 1 1 x 2 1 1 z 2 1 1 1 1 1 1 1 1 24 1 1 1 x 3 1 1 z 2 1 1 1 1 1 1 1 1 25 1 1 1 x 1 1 1 z 3 1 1 1 1 1 1 1 1 26 1 1 1 x 2 1 1 z 3 1 1 1 1 1 1 1 1 27 1 1 1 x 2 1 1 z 3 1 1 1 1 1 1 1 1 28 1 1 1 x 3 1 1 z 3 1 1 1 1 1 1 1 1 29 1 1 1 x 1 1 1 z 4 1 1 1 1 1 1 1 1 30 1 1 1 x 2 1 1 z 4 1 1 1 1 1 1 1 1 31 1 1 1 x 2 1 1 z 4 1 1 1 1 1 1 1 1 32 1 1 1 x 3 1 1 z 4 1 1 1 1 1 1 1 1
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56 Table 4.4 (Continued) Run # CQ PQ ABP ABQ ACP ACQ APQ BCP BCQ BPQ CPQ 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 1 1 1 3 1 1 1 1 1 1 1 1 1 1 1 4 1 1 1 1 1 1 1 1 1 1 1 5 1 1 1 1 1 1 1 1 1 1 1 6 1 1 1 1 1 1 1 1 1 1 1 7 1 1 1 1 1 1 1 1 1 1 1 8 1 1 1 1 1 1 1 1 1 1 1 9 1 1 1 1 1 1 1 1 1 1 1 10 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 12 1 1 1 1 1 1 1 1 1 1 1 13 1 1 1 1 1 1 1 1 1 1 1 14 1 1 1 1 1 1 1 1 1 1 1 15 1 1 1 1 1 1 1 1 1 1 1 16 1 1 1 1 1 1 1 1 1 1 1 17 1 1 1 1 1 1 1 1 1 1 1 18 1 1 1 1 1 1 1 1 1 1 1 19 1 1 1 1 1 1 1 1 1 1 1 20 1 1 1 1 1 1 1 1 1 1 1 21 1 1 1 1 1 1 1 1 1 1 1 22 1 1 1 1 1 1 1 1 1 1 1 23 1 1 1 1 1 1 1 1 1 1 1 24 1 1 1 1 1 1 1 1 1 1 1 25 1 1 1 1 1 1 1 1 1 1 1 26 1 1 1 1 1 1 1 1 1 1 1 27 1 1 1 1 1 1 1 1 1 1 1 28 1 1 1 1 1 1 1 1 1 1 1 29 1 1 1 1 1 1 1 1 1 1 1 30 1 1 1 1 1 1 1 1 1 1 1 31 1 1 1 1 1 1 1 1 1 1 1 32 1 1 1 1 1 1 1 1 1 1 1
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57 Table 4.4 (Continued) Run # ABCQ ABPQ ACPQ BCPQ ABCPQ 1 1 1 1 1 1 2 1 1 1 1 1 3 1 1 1 1 1 4 1 1 1 1 1 5 1 1 1 1 1 6 1 1 1 1 1 7 1 1 1 1 1 8 1 1 1 1 1 9 1 1 1 1 1 10 1 1 1 1 1 11 1 1 1 1 1 12 1 1 1 1 1 13 1 1 1 1 1 14 1 1 1 1 1 15 1 1 1 1 1 16 1 1 1 1 1 17 1 1 1 1 1 18 1 1 1 1 1 19 1 1 1 1 1 20 1 1 1 1 1 21 1 1 1 1 1 22 1 1 1 1 1 23 1 1 1 1 1 24 1 1 1 1 1 25 1 1 1 1 1 26 1 1 1 1 1 27 1 1 1 1 1 28 1 1 1 1 1 29 1 1 1 1 1 30 1 1 1 1 1 31 1 1 1 1 1 32 1 1 1 1 1 We then do analysis for the 2 k design ( k = 5 ), and use the same calculation method that has been introduced above. 4.4 General Factorial Design The 2 k factorial design is absolutely convenien t and very easy to apply on the problem that all the involved factors have two leve l of variation. Although there is a way to manipulate the problem which have mixed level f actors, but it is complicated. So, in this case of experiment, it is most efficient to analyze by using the general factorial design which can be used for all design of experiment s. By a factorial desi gn, we mean that in
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each complete trial or replication of the experiment all possible combinations of the levels of the factors are investigated. The following section will introduce the quick review of general factorial design. Take a general experiment that has a levels of factor A, b levels of factor B, c levels of factor C, and so on, arranged in a factorial experiment. In general, if there are n replicates of the complete experiment, there will be abc. .n total observations. If all factors in the experiment are fixed, we may easily formulate and test hypotheses about the main effects and interactions. For fixed effects model, test statistics for each main effect and interaction may be constructed by dividing the corresponding mean square for the effect or interaction by the mean square error. All of the F test will be uppertail, onetail tests. The number of degree of freedom for any main effect is the number of levels of the factors minus one, and the number of degrees of freedom for an interaction is the product of the number of degrees of freedom associated with individual components of the interaction. For example, consider the threefactor analysis of variance model. ijklijkjkikijkjiijkly )()()()( (4.11) ,...2 ,1 ; ,2,...c 1 ,...2 ,1 ; ,...2 ,1nlkbjai Assuming that A, B, C are fixed, the analysis of variance table is shown in the Table 4.5. The F test on main effects and interactions follow directly from the expected mean squares. Table 4.5 The ANOVA table for general factorial design Source of Variation Sum of Squares Degree of Freedom Mean Square Expected Mean Square F o A SS A a1 MS A 12 abcni EAMSMSF0 B SS B b1 MS B 122 bacnj EBMSMSF0 58
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Table 4.5 (Continued) C SSCc1 MSC 12 2 c abnk E CMS MS F 0 AB SSAB(a1 )( b1 ) MSAB )1)(1( )(2 2 ba cnij E ABMS MS F 0 AC SSAC(a1 )( c1 ) MSAC )1)(1( )(2 2 ca bnik E ACMS MS F 0 BC SSBC(b1 )( c1 ) MSBC )1)(1( )(2 2 cb anjk E BCMS MS F 0 ABC SSABC(a1 )( b1 )(c1 ) MSABC )1)(1)(1( )(2 2 cba nijk E ABCMS MS F 0 Error SSEabc(n1 ) MSE2 Total SSTabcn1 The total sum of square is found in the usual way as given in Table 4.6. Table 4.6 Sum of square of factors SSA abcn y y bcna i i 2 1 21 SSB abcn y y acnb j j 2 1 21 SSC abcn y y abna i k 2 1 21 SSAB BA a i b j ijSSSS abcn y y cn 2 11 21 SSAC CA a i c k kiSSSS abcn y y bn 2 11 21 SSBC C B b j c k jkSSSS abcn y y an 2 11 21 SSABC BC AC AB C BA a i b j c k ijkSSSSSSSSSSSS abcn y y n 2 111 21 59
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Table 4.6 (Continued) SSE abcn y y n SSa i b j c k ijk T 2 111 21 SST abcn y ya i b j c k n l ijkl 2 1111 2 60
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61 CHAPTER 5 RESULTS AND CONCLUSIONS 5.1 Specifications for Three Inspected Bascule Bridges 5.1.1 Bascule bridges general geometrical dimension In the effort of trying to inspect a wide range of dimensions of Florida bascule bridges, we examined three bridges with the dimensions are given as in Table 5.1. Table 5.1 Geometrical dimensions of three bascule bridges Bascule bridges Parameters Hallandale Christa McAuliffe 17 th Street Causeway 1 L 26.0 18.5 6.00 2 LF 7.00 4.25 4.25 3 LH 28.0 16.0 11.0 4 LT 80.0 53.5 23.0 5 RHO (Actual) 17.5 16.0 8.88 6 RFO 30.0 27.0 13.2 7 RTI 1.50 1.00 1.19 8 RTO 13.0 9.00 6.47 9 TG 2.00 1.50 1.25 10 WF 3.00 1.75 1.25
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5.1.2 Radial interferences calculations The radial interferences at the trunnionhub assembly is based on standard interference fitsFN2 or FN3. These standard interference fits dictate the limits of the dimensions of the TH parts as follows. If a cylinder B is fit into cylinder A, there is an upper and lower limit by which the nominal (outer or inner, respectively) diameter of each cylinder will varies. This limit, L, in thousand of an inch, is given by. 31CDL where D (nominal diameter) is in inches and the coefficient C, based on the type of fit. For example, a typical nominal trunnion outer diameter and hub inner diameter of the Hallandale Bascule bridge is 26. Using a FN2 fit, C values for the tolerance are the corresponding matrix 288.3717.2907.0000.0][C and the four limits are calculated as follows: ][0097407.00080491.00029870.00000000.010001)000.26(288.3717.2907.0000.0][31inL 62
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Hence, the outer diameter dime nsions of the trunnion would be and the inner diameter of the hub would be .These two pair s of extreme dimensions of the trunnion and hub diameters produce values of diam etrical interference ranging from 0.0053621 to 0.0097407 that is, the radial interferences range from 0.0026811 to 0.0048704 In the same manner, we have the summary of radial interferences for a ll three bascule bridges given as follows in Table 5.2. ][2600974070 00804910in ][2600268700 00000000in Table 5.2 Radial interferen ces of three bascule bridges Hallandale BB Christa McAuliffe BB 17th Street Causeway BB Inner Diameter Outer Diameter Inner Diameter Outer Diameter Inner Diameter Outer Diameter Trun_ nion 3.0000 00974070 0080491026 2.000000861700 0071206018 2.3800 00772000 00638000944.12 Hub 00268700 0000000026 35.00000237700 0000000018 32.00000213000 00000000944.12 17.760 Radial Interf_ rences 0048704.0,0026811.0 0043085.0,0023718.0 0038600.0,0021250.0 5.1.3 Hub outer diameter calculations The AASHTO standards call for a hub radial thickness of 0.4 times the inner diameter of the hub. However, in the real practice, the hub radial thickness used is 0.1 to 0.2 times the inner diameter of the hub. It is interesting to study the whole range: lower limit (0.1), the upper limit (0.4), and the center point (0.25). So, we have. 40000.0 25000.0 10000.0 For example, the calculations of three inspected le vels of the hub outer diameter of the 17th Street Bascule bridge are shown below. Diameter Inner Hub )Diameter/2 Inner HubDiameter/2 Outer (Hub 63
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Solve for the outer diameter, we have 40000.025000.010000.0HID HID/22HOD 40000.025000.010000.012.944 12.944/22 ][23.29919.41615.533 in In the same manner, calculations are applied to other two bascule bridges and given in Table 5.3. Table 5.3 Specifications of levels of hub outer diameters 17 th Street Causeway Bascule bridge Levels D [in] Low 15.533 Middle 19.416 High 23.299 Actual hub outer diameter used [in] 17.760 Christa McAuliffe Bascule bridge Levels D [in] Low 21.600 Middle 27.000 High 32.400 Actual hub outer diameter used [in] 32.000 Hallandale Bascule bridge Levels D [in] Low 31.200 Middle 39.000 High 46.800 Actual hub outer diameter used [in] 35.000 64
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5.2 Fracture Toughness and Yield Strength The two responses that we are interested in this study are the critical crack length (CCL) and the stress ratio (SR). The critical crack length depends on the tangential stress developed and the fracture toughness IK of the material. The fracture toughness of the material is temperature dependent and it decreases with a decrease in temperature. The stress ratio depends on the total stress induced (VonMises Stress) and the yield strength sY of the material. The yield strength is also temperature dependent and it increases with a decrease in temperature. Greenberg (1969) introduced the dependence of fracture toughness and yield strength of material on temperature as sketched as in Figure 5.1. 02040608010025015050050020406080100Yield Strength (ksi) Yield StrengthFracture ToughnessFracture Toughness, KIc (ksiin1/2)Temperature (0F) Figure 5.1 Temperature dependence of fracture toughness and yield strength of ASTM E24 steel casting (Greenberg and Clark, Jr., 1969) 65
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5.2.1 The fracture toughness, K I Consider an edge radial crack in a hollow cylinder that is relatively small in comparison to the radial thickness of the cylinder as depicted in Figure 5.2. a Figure 5.2 Critical crack length The stress intensity factor or the fracture toughness at the crack tip is given by the Equation 5.1 afKeI (5.1) When is equal, the crack length reaches the maximum crack length allowable before a crack propagates catastrophically. This maximum crack length, a IK TKIC c is determined by Equation 5.2 (Kanninen and Popelar, 1985). 222)(eIccfTKa (5.2) 66
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5.2.2 The stress ratio The stress ratio is defined as the ratio of the yield strength of the material to the VonMises stress induced in the material. The VonMises stress is obtained from the finite element analysis (ANSYS). Mathematically, stress ratio can be expressed like as esTYSR (5.3) Discrete points for the yield strength and the fracture toughness at different temperatures were approximated from the curves and used to either interpolate or extrapolate to obtain the required values based on the temperature. The lower curve was used for fracture toughness. These points for yield strength and fracture toughness are listed in Tables 5.4 and 5.5, respectively. Table 5.4 Yield strength, Y s as a function of temperature (Greenberg and Clark, Jr., 1969) Temperature [F] Yield Strength [ksi] 340 102 320 95 300 89 280 83 260 78 240 73 220 68 200 64 180 60.5 160 58 140 56 120 54 100 52 80 50.5 67
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68 Table 5.4 (Continued) 60 49 40 48 20 47.5 0 47 20 47 40 47 60 47 80 47 Table 5.5 Fracture toughness, K I as a function of temperature Temperature [F] Fracture Toughness [ksi in] 250.0 28 200.0 29 150.0 30 100.0 34 50.0 39 0.0 51 50.0 68 70.0 77 5.3 Typical Results of the Critical Points Typical results are shown how the hoop stress and temperature vary as a function of time at the critical point (r= 9.0000 = 90.000 o z= 19.000 ) where the OMCCL occurs for the Christa McAuliffe Bascule bridge during the staged coo ling type 2 process (Figures 5.3 and 5.4). Similar plots are shown the same bridge for the critical point (r= 26.000 = 120.00 o z= 33.667 ) where OMSR occur (Figures 5.6 and 5.7). Since the fracture toughness and yield strength vary as a function of temperature (Figure 5.1), plots for the CCL and SR as a function of time are shown in Figures 5.5 and 5.8, respectively.
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35030025020015010050050100)(Fo Time Dry ice Liquid nitrogen Figure 5.3 Temperature versus time of the node (r= 9.0000, = 90.000 o z= 19.000) 69
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05000100001500020000)(psi 70 Figure 5.4 Hoop stress versus time of the node (r= 9.0000, = 90.000 o z= 19.000) Time Dry ice Liquid nitrogen
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0246810121416 )(in 71 Figure 5.5 CCL versus time of the node (r= 9.0000, = 90.000 o z= 19.000) Time Liquid nitrogen Dry ice
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35030025020015010050050100 72 Figure 5.6 Temperature versus time of the node (r= 26.000, = 120.00 o z= 33.667) Time F o Dry ice Liquid nitrogen
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0500010000150002000025000 )(psi Time Dry ice Figure 5.7 VonMises stress versus time of the node (r= 26.000, = 120.00 o z= 33.667) Liquid nitrogen 73
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0123456789 74 Figure 5.8 SR versus time of the node (r= 26.000, = 120.00 o z= 33.667) Time Liquid nitrogen Dry ice
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5.4 Sensitivity Analysis of 2 MixedLevel Factors We are interested to know the effects of the hub outer diameter factor (threelevel factor, D) and the radial interference (twolevel factor, C) on the critical crack length and the stress ratio. 5.4.1 Specification of geometries of the hub and the trunnion Based on the review in the Chapter 4, we are interested in three different values of hub outer diameter (low, middle and high) and two different values of radial interference (low and high). Table 5.6 shows the summary of the levels of these treatments (factors) for three different bridges. Table 5.6 Specifications of two mixed level factors 17 th street Causeway Bascule Bridge Level Code D [in] Low 1 15.533 Middle 0 19.416 Upper +1 23.300 Level Code C [in] Lower 1 0.0021250 Upper +1 0.0038600 Christa McAuliffe Bascule Bridge Level Code D [in] Low 1 21.600 Middle 0 27.000 Upper +1 32.400 Level Code C [in] Lower 1 0.0023718 Upper +1 0.0043085 Hallandale Bascule Bridge Level Code D [in] Low 1 31.200 Middle 0 39.000 Upper +1 46.800 Level Code C [in] Lower 1 0.0026811 Upper +1 0.0048704 5.4.2 Collected data and sensitivity analysis According to the general factorial design of experiment, we need to perform 23= 6 runs to perform the sensitivity analysis. The obtained data are listed in Table 5.7. 75
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76 Table 5.7 Collected data for sensitivity analysis 17 th St. Causeway Bascule Bridge Christa McAuliffe Bascule Bridge Hallandale Bascule Bridge OMCCL OMSR OMCCL OMSR OMCCL OMSR Run # C D [ ] [ ] [ ] [ ] [ ] [ ] 1 1 1 0.19084 1.7131 0.17097 1.6689 0.13175 1.4965 2 1 1 0.12449 1.3806 0.14429 1.3975 0.1024 1.3232 3 1 1 0.14814 1.2052 0.17210 1.6474 0.12245 1.0439 4 1 1 0.13008 1.2509 0.15150 1.5075 0.11139 0.99647 5 1 0 0.24065 1.7209 0.17152 1.6655 0.13434 1.1993 6 1 0 0.19838 1.5546 0.17123 1.5415 0.12636 1.1191 The theory of general factorial design from Chapter 4 is employed to analyze the effects the twolevel main factor (radial interference, C ) and the threelevel main factor (hub outer diameter, D ) and the interaction factor, CD. Summary of the re sults are given as follows in Table 5.8. Table 5.8 The contributions of factors on the OMCCL and the OMSR 17 th St. Causeway Bascule Bridge Christa McAuliffe Bascule Bridge Hallandale Bascule Bridge OMCCL OMSR OMCCL OMSR OMCCL OMSR [ % ] [ % ] [ % ] [ % ] [ % ] [ % ] C 25.848 13.402 49.185 80.460 51.252 8.6073 D 68.518 72.524 25.908 8.5127 31.222 88.961 CD 5.6341 14.074 24.907 11.027 17.525 2.4328 We can manipulate to have a graphical and comparative l ook of those treatments (hub outer diameter, radial interference and the interaction fact or) on the outputs ( OMCCL and OMSR ) for the three bascule bridges (Figure 5.9).
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CDCD 17th St.Christa McAuliffeHallendale 0255075100%FactorsOMCCL CDCD 17th St.Christa McAuliffeHallendale 0255075100%FactorsOMSR Figure 5.9 The effects of the treatments on the OMCCL and OMSR 77
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78 As seen from Figure 5.9, some conclu sions can be stated as follows: For the OMCCL the hub outer diameter contributes up to 69% (17 th Street Bascule bridge), while in those other two bascule bridges, the radial interference play up to 51% of contribu tion to the outputs. The interaction seems also come into play in the Hallandale Bascule bridge and the Christa McAuliffe Bascule bridge (over 18%), but its role is just 6% comparatively to the other factors in the 17 th Street Bascule bridge. For the OMSR the radial interference contribute up to 80% in the Christa McAuliffe Bascule bridge, while the hub outer diameter play the most important role in the other tw o bascule bridge (up to 89%). The interaction factor does not seem to play a significant factor in the OMSR since it contributes less than 14% in all three bascule bridges. It is obvious that there are no general conclusion about the contributions of specific factor to the OMCCL and OMSR Since we have three geometrical different bascule bridges, we come up with different results for each individual bridge. This is in agreement to the previous USF works that each bascule bridge needs to be analyzed separately. Another way to have a thorough understanding of the effect of hub outer diameter and radial interference on the OMCCL and the OMSR is to look at Figures 5.10 and 5.11.
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OMCCL vs C0.100.150.200.2521012 OMCCL vs CDC LowC High0.100.150.200.2521012 17 th street Causeway Bascule Bridge OMCCL vs D0.100.150.200.2521012 OMCCL vs C0.10.150.20.2521012 OMCCL vs CDC LowC High0.100.150.200.2521012 Christa McAuliffe Bascule Bridge OMCCL vs D0.10.150.20.2521012 OMCCL vs C0.100.120.1421012 OMCCL vs CDC LowC High0.100.120.1421012 Hallandale Bascule Bridge OMCCL vs D0.100.120.1421012 Figure 5.10 The variations of the OMCCL versus the changing of selected factors 79
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From Figure 5.10 of OMCCL versus individual factor, we can conclude some issues. While the OMCCL decreases with an increase of radial interference, the OMCCL varies as the second order of hub outer diameter. It means that we prefer to have the hub outer diameter at its medium level. In other words, the AASHTO criteria should be a hub radial thickness to inner hub diameter ratio of 25.02/4.01.0 and that will give a higher OMCCL. Although we cannot control the radial interference, but it does contribute significantly to the outputs. This may explain why failures occur in some bascule bridge (radial interference factor at its high level), but do not occur in similar geometrical bascule bridges (radial interference at its low level). 80
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OMSR vs C 1.2 1.4 1.6 1.8 21012 OMSR vs CD C Low C High 1.2 1.4 1.6 1.8 21012 17thstreet Causeway Bascule Bridge OMSR vs D 1.2 1.4 1.6 1.8 21012 OMSR vs C 1.2 1.4 1.6 1.8 21012 OMSR vs CD C Low C High 1.2 1.4 1.6 1.8 21012 Christa McAuliffe Bascule Bridge OMSR vs D 1.2 1.4 16 1.8 21012 OMSR vs C 1 1.25 1.5 21012 OMSR vs CD C Low C High 1.0 1.3 1.5 21012 Hallandale Bascule Bridge OMSR vs D 1 1.25 1.5 21012 Figure 5.11 The variations of the OMSR versus the changing of selected factors 81
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82 From Figure 5.11 of the OMSR versus individual factor we can also draw some conclusions. The radial interference ha s the same effect on the OMSR as it does to the OMCCL that the OMSR decreases with an increase of radial interference. Again, it is confirms that while failures occurred in some bascule bridge (radial interference factor at its high level), it did not occur in the similar geometrical bascule bridges (radial interference at its low level). The hub outer diameter also affects the OMSR as the second order of hub outer diameter. Again, for diff erent geometrical dimension bascule bridges give different results, so we have to do analysis for each bascule bridge. 5.5 Sensitivity Analysis of 3 MixedLevel Factors Since, we cannot draw a general rule for the TH assembly procedure based on investigation on the hub outer diameter and radi al interference, we like look at the effect of staged cooling (fourlevel factor, X ). We want to inspect whether the treatment X significantly contributes to th e critical parameters during the assembly procedure. 5.5.1 Specification of levels of each factor The details of levels of fact ors are given in the Table 5.9. Table 5.9 Specifications of thr ee mixed level factors 17 th Street Causeway Bascule Bridge Level Code X 1 x 1 Type 1 2 x 2 Type 2 3 x 3 Type 3 4 x 4 Type 4 Level Code D [in] Low 1 15.533 Middle 0 19.416 Upper +1 23.300 Level Code C [in] Lower 1 0.0021250 Upper +1 0.0038600
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83 Table 5.9 (Continued) Christa McAuliffe Bascule Bridge Level Code X 1 x 1 Type 1 2 x 2 Type 2 3 x 3 Type 3 4 x 4 Type 4 Level Code D [in] Low 1 21.600 Middle 0 27.000 Upper +1 32.400 Level Code C [in] Lower 1 0.0023718 Upper +1 0.0043085 Hallandal e Bascule Bridge Level Code X 1 x 1 Type 1 2 x 2 Type 2 3 x 3 Type 3 4 x 4 Type 4 Level Code D [in] Low 1 31.200 Middle 0 39.000 Upper +1 46.800 Level Code C [in] Low er 1 0.0026811 Uppe r +1 0.0048704 5.5.2 Collected data and sensitivity analysis We perform 24 runs for each bascule bridge, which means we need to make 72 runs to get the data that will be used for the se nsitivity analysis. The overall minimum critical crack length ( OMCCL ) and overall minimum stress ratios ( OMSR ) for those three bridges are given in Table 5.10. Table 5.10 Collected data for sensitivity analysis 17 th St. Causeway Bascule Bridge Christa McAuliffe Bascule Bridge Hallandale Bascule Bridge OMCCL OMSR OMCCL OMSR OMCCL OMSR Run # X C D [ ] [ ] [ ] [ ] [ ] [ ] 1 x 1 1 1 0.19084 1.7131 0.17097 1.6689 0.13175 1.4965 2 x 2 1 1 0.71244 2.2914 0.76874 2.2465 0.61141 1.9085 3 x 3 1 1 0.40833 2.7843 0.41079 2.8043 0.33743 2.6555 4 x 4 1 1 0.76677 2.8055 0.86087 3.0703 0.79923 2.9939 5 x 1 1 1 0.12449 1.3806 0.14429 1.3975 0.1024 1.3232 6 x 2 1 1 0.34329 1.6595 0.43280 1.7181 0.40052 1.5969 7 x 3 1 1 0.22398 1.8947 0.25121 2.1398 0.22804 2.1546 8 x 4 1 1 0.34329 1.8947 0.43280 2.1398 0.44564 2.2656 9 x 1 1 1 0.14814 1.2052 0.17210 1.6474 0.12245 1.0439 10 x 2 1 1 0.87990 1.9023 0.84360 4.6821 0.80329 1.2132 11 x 3 1 1 0.41756 2.5837 0.37457 2.9911 0.31554 2.0274 12 x 4 1 1 0.87990 4.6174 0.84360 4.8177 0.80329 3.0838 13 x 1 1 1 0.13008 1.2509 0.15150 1.5075 0.11139 0.99647 14 x 2 1 1 0.46121 2.1564 0.49661 2.9568 0.5244 1.1378 15 x 3 1 1 0.29336 2.2949 0.26392 2.3977 0.24017 1.8675 16 x 4 1 1 0.46121 2.8443 0.49661 3.1858 0.5244 2.6320 17 x 1 1 0 0.24065 1.7209 0.17152 1.6655 0.13434 1.1993 18 x 2 1 0 0.90157 3.6157 0.91257 3.5578 0.83675 1.3466 19 x 3 1 0 0.50934 3.1192 0.44160 3.0647 0.34769 2.2946 20 x 4 1 0 0.90156 3.9596 0.91257 4.4284 0.83675 2.8609 21 x 1 1 0 0.19838 1.5546 0.17123 1.5415 0.12636 1.1191 22 x 2 1 0 0.43991 2.3199 0.49802 2.4320 0.51426 1.2120 23 x 3 1 0 0.29517 2.2481 0.28945 2.3884 0.25155 2.0311 24 x 4 1 0 0.43991 2.4526 0.49802 2.7628 0.51426 2.2926
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84 Applying the general factorial de sign of experiment to analyze collected data for three bascule bridges, we have the contributions of the factors and their interaction factors on the critical crack length and stress ratios (neglect the small threefactor interaction factor) as shown in Tables 5.11. Table 5.11 The contributions of factors on OMCCL and OMSR 17 th St. Causeway Bascule Bridge Christa McAuliffe Bascule Bridge Hallandale Bascule Bridge OMCCL OMSR OMCCL OMSR OMCCL OMSR [ % ] [ % ] [ % ] [ % ] [ % ] [ % ] X 57.613 48.531 69.766 48.242 78.908 84.182 D 2.7903 7.8303 0.69916 14.132 1.1532 4.0830 C 27.588 17.499 19.599 18.836 12.062 4.9853 XD 1.0521 13.161 0.37089 10.501 1.4643 4.0810 XC 10.609 7.2668 9.2818 5.8899 6.0828 1.9909 DC 0.16866 1.6198 0.10583 0.84524 0.045525 0.61313 By depicting those contribution data from th e Table 5.11, we obtain a clear picture of the effects of each factor to the TH assembly procedure in Figure 5.12.
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XDCXDXCDC 17th St.Christa McAuliffeHallendale 0.025.050.075.0100.0%FactorsOMCCL XDCXDXCDC 17th St.Christa McAuliffeHallendale 0.025.050.075.0100.0%FactorsOMSR Figure 5.12 The effects of the interested factors on the OMCCL and OMSR 85
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86 In all three bridges, we see that. The staged cooli ng type (factor X ) plays the most important role on both OMCCL (up to 79% in the Halla ndale bascule bridge) and OMSR (up to 84% in the Hallanda le bascule bridge). The second in contribution is th e radial interference (factor C ) that contributes to the OMCCL up to 28% (in the 17 th St. Causeway bascule bridge) and to the OMSR up to 19% (in the Chri sta McAuliffe bascule bridge). The third in contribution to the OMCCL is the XC interaction factor (up to 11% in the 17 th St. Causeway bascule bridge), but the third in contribution to the OMSR are the outer hub diameter (factor D ) and the interaction factor XD (up to 14%). In general, with the contri bution of over 48% on either OMCCL or OMSR the staged cooling method plays th e most important role in all the three geometrical dimension bascule bridges. We are also interested to show the variations of the OMCCL and the OMSR with the changing of individual fa ctors (Figures 5.13 and 5.14).
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OMCCL vs X 0.1 0.3 0.5 0.7 012345 OMCCL vs C 0.1 0.3 05 0.7 21012 17thstreet Causeway Bascule Bridge OMCCL vs D 0.1 0.3 05 0.7 21012 OMCCL vs XD Type I Type II Type III Type IV 0.1 0.3 0.5 0.7 21012 OMCCL vs X 0.1 0.3 0.5 0.7 012345 OMCCL vs C 0.1 0.3 05 0.7 21012 Christa McAuliffe Bascule Bridge OMCCL vs D 0.1 0.3 05 0.7 21012 OMCCL vs XD Type I Type II Type III Type IV 0.1 0.3 0.5 07 21012 OMCCL vs X 0.1 0.3 0.5 0.7 012345 OMCCL vs C 0.1 0.3 0.5 0.7 21012 Hallandal e Bascule Bridge OMCCL vs D 0.1 0.3 0.5 0.7 21012 OMCCL vs XD Type I Type II Type III Type IV 0.1 03 0.5 07 21012 Figure 5.13 The variations of the OMCCL versus the changing of selected factors 87
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88 From Figure 5.13, it is obvious that. The OMCCL vary to the third order of the cooling type factor (major contributor) in the way that the cooling type 4 (putting TH cooling inside the fridge air chamber (32 o F), then immerse it into the dry ice/ alcohol bath (108 o F), and followed by dropping into the liquid nitrogen (321 o F)) and type 2 (immersing the TH into the dry ice/ alcohol bath (108 o F) then drop it into the liquid nitrogen (321 o F)) approximately gives the same value of OMCCL So, cooling type 2 is recommended for the TH assembly procedure to obtain a large OMCCL since it tremendously reduces the cost and time comp ared to the cooling type 4. The OMCCL decreases with an increase of radial interferences (factor C ) which is identical to the re sults we got from Section 5.2. The OMCL vary in the manner of second order of the hub outer diameter where the OMCCL has its maximum value at the middle point. So, we prefer to have the hub radial thickness at its middle level of 0.25 times the hub inner diameter. And for the interaction factor XD as we can see that when we have the hub outer diameter at its middle ra nge, the cooling type 2 or type 4 also give the same maximum value of OMCCL compare to type 1 and 3. It also confirms that the choice of cool ing type 2 and the hub outer diameter at its middle value is the good treatment for the TH assembly procedure to optimize the value of OMCCL
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OMSR vs X 1.0 2.0 3.0 4.0 012345 OMSR vs C 1.0 2.0 3.0 4.0 21012 17th street Causeway Bascule Bridge OMSR vs D 1.0 2.0 3.0 4.0 21012 OMSR vs XD Type I Type II Type III Type IV 1.0 2.0 30 4.0 21012 OMSR vs X 1 2 3 4 012345 OMSR vs C 1.0 2.0 3.0 4.0 21012 Christa McAuliffe Bascule Bridge OMSR vs D 1.0 2.0 3.0 4.0 21012 OMSR vs XD Type I Type II Type III Type IV 1.0 2.0 3.0 4.0 21012 OMSR vs X 1 2 3 012345 OMSR vs C 1.0 20 3.0 21012 Hallandal e Bascule Bridge OMSR vs D 1.0 2.0 3.0 21012 OMSR vs XD Type I Type II Type III Type IV 1.0 2.0 3.0 21012 Figure 5.14 The variations of the OMSR versus the changing of selected factors 89
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90 From Figure 5.14, we draw the following conclusions. In the 17 th Street Causeway and Halla ndale Bascule bridges, the OMSR increases with the cooling type 1, 2, 3, 4, respectively, but for the Christa McAuliffe Bascule bridge doe s not act in the same manner. For example, the cooling type 2 gives the higher OMCCL than the cooling type 2. So, we see that the OMCCL is not always increase with the cooling type because of the difference in dimensions of each bascule bridge. In all three bascule bridges, the lower the radial interference, the higher the value of the OMSR The variation of OMSR versus the radial interference is the same as the OMCCL Since the contribution of th e hub outer diameter to the OMSR is relatively low (lower than 14%), it is also good to choose the middle value for the hub outer diameter for the sake of the OMCCL as in the previous discussion. The larger the radial interference, the smaller the OMSR is, but we can not control this factor. Furthe r more, in all three bascule bridges the cooling type 2 gives the OMSR value larger than 1.4, meaning that the yield strength is always 1.4 times the maximum stress exists in the TH assembly in during the cooling procedur e. So it good to choose the cooling type 2 in practice for saving time and cost. In summary, from the sensitivity analysis of two mixed level factors, we have radial interference factor ( C ), hub outer diameter factor ( D ), then D plays a significant role in some bridges but C contributes the most to the critical factors in the other bridges. There are no general conclusion can be dr aw in the term of contribution. We also have from the sensitivity analysis of three mixed level f actors that the staged cooling types factor ( X ): type 1 (liquid nitrogen only), type 2 (dry ice followed by liquid nitrogen), type 3 (refrigerator air followed by liquid nitrogen), and type 4 (fridge air, then immersing into dry ice, and th en followed by liquid nitrogen). X factor increase the CCL and SR and is the most contri butes to the critical elements: up to 79% on the OMCCL
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91 and up to 84% on OMSR. Staged cooling type 2 and 4 are the most beneficial one, but cooling type 2 can be employed in the TH assembly procedure since it is the best efficient in the term of time and cost consumption.
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92 REFERENCES ANSYS Inc., 2005 ANSYS Release 10.0 Documentation. AspenTech, Aspen Plus, [online] http://www.aspentech.com Accessed 03/31/2004. Barron, R. F., (1999), Cryogenic Heat Transfer Taylor and Francis Company, PA, 161172. Besterfield, G., Kaw, A. and Crane, R., 2001, Parametric Finite Element Modeling and FullScale Testing Of TrunnionHubGirder Assemblies for Bascule Bridges Mechanical Engineering Dept, USF, FL. Brentari E.G., and Smith R.V., 1964, Nucleate and Film Pool Boiling Design Consideration for O 2 N 2 H 2 and He International Advanced in Cryogenic Engineering, 10b, pp.325341. Chiang, C.L., 2003, Statistical Methods of Analysis World Scientific Publishing Co. Pte. Ltd., Singapore, 467468. Collier, N.O., 2004, Benefit of Staged Cooling in Shr inkFitted Composite Cylinders MS Thesis, Mechanical Engineering Dept, USF, FL. Denninger, M.T., 2000, Design Tools for TrunnionHub A ssemblies for Bascule Bridges MS Thesis, Mechanical Engineering Dept, USF, FL. Greenberg, H.D., and Clark, Jr. H.G., 1969, A Fracture Mechanics Approach to the Development of Realistic Acceptance Standar ds for Heavy Walled Steel Casting Metals Engineering Quarterly, 9(3), 3033. Incropera, F.P., and DeWitt, D.P., 1996, Introduction to Heat Transfer, John Wiley & Sons, Inc., New York. Montgomery, D.C., 2001 Design and Analysis of Experiments 5 th edition, John Wiley & Sons, Inc., New York. Nichani, S., 2001, Full Scale Testing of TrunnionH ubGirder Assembly of a Bascule Bridge MS Thesis, Mechanical Engineering Dept, USF, FL.
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93 zi ik, M. N., 1993 Heat Conduction, John Wiley & Sons, Inc., New York. Paul, J., 2005, Sensitivity Analysis of Design Para meter for TrunnionHub Assemblies of Bascule Bridges Using Finite Element Methods MS Thesis, Mechanical Engineering Dept, USF, FL. Ratnam, B., 2000, Parametric Finite Element m odeling of trunnion Hub Girder Assemblies for Bascule Bridges MS Thesis, Mechanical Engineering Dept, USF, FL. Reed, R.P., 1983, Material at Low Temperature. ASM International, Materials Park, Ohio. Ugural, A.C., and Fenster, S.K., 1995 Advanced Strength and Applied Elasticity, Prentice Hall PTR, New Jersey.
