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Collier, Nathaniel Oren.
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Benefit of staged cooling in shrink fitted composite cylinders
h [electronic resource] /
by Nathaniel Oren Collier.
260
[Tampa, Fla.] :
University of South Florida,
2004.
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Thesis (M.S.M.E.)University of South Florida, 2004.
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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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ABSTRACT: To assemble the fulcrum of bascule bridges, a trunnion is immersed into liquid nitrogen so that it can be shrunk fit into the hub. This is followed by immersing the resulting trunnionhub assembly into liquid nitrogen so that it can be then shrunk fit into the girder. On one occasion in Florida, when the trunnionhub assembly was put into liquid nitrogen, development of cracks on the hub was observed. Experimental and numerical studies conducted since 1998 at University of South Florida show that the cracking took place due to combination of high interference stresses in the trunnionhub assembly, low fracture toughness of steel at cryogenic temperatures, and steep temperature gradients due to sudden cooling. In this study, we are studying the benefit of staged cooling to avoid cracking in the trunnionhub assembly when it is cooled down for shrink fitting. We looked at three cooling processes 1) Direct immersion into liquid nitrogen 2) Immersion into a refrigerated chamber, then liquid nitrogen 3) Immersion into a refrigerated chamber, then a dryice/alcohol bath, and finally liquid nitrogen. The geometry of the trunnionhub assembly was approximated by a composite made of two infinitely long hollows cylinders. The transient problem of temperature distribution and the resulting stresses was solved using finite difference method. Using critical crack lengths and VonMises stress as failure criteria, the three cooling processes were compared. The study showed that the minimum critical crack length and stress ratio is increased by as much as 200% when cooling first in refrigerated air followed by liquid nitrogen. However, there is little benefit from adding dryice/alcohol as an intermediate step in the cooling process.
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Adviser: Kaw, Autar K.
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thermal stress.
nonhomogeneous material properties.
transient.
bascule bridge.
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Dissertations, Academic
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x Mechanical Engineering
Masters.
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t USF Electronic Theses and Dissertations.
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u http://digital.lib.usf.edu/?e14.311
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Benefit of Staged Cooling In Shrink Fitted Composite Cylinders by Nathaniel Oren Collier 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. Glen H. Besterfield, Ph.D. Muhammad M. Rahman, Ph.D. Date of Approval: March 29, 2004 Keywords: bascule bridge, transient, thermal st ress, nonhomogeneous material properties Copyright 2004, Nathaniel Collier
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i TABLE OF CONTENTS LIST OF TABLES..............................................................................................................ii LIST OF FIGURES...........................................................................................................iii ABSTRACT....................................................................................................................... .v CHAPTER 1 Â– INTRODUCTION.....................................................................................1 Literature Survey............................................................................................................2 Current Study.................................................................................................................. 4 CHAPTER 2 Â– PROBLEM FORMULATION..................................................................6 Thermal Formulation......................................................................................................7 Thermal Boundary Conditions......................................................................................10 Elasticity Formulation...................................................................................................11 Elasticity Boundary Conditions....................................................................................20 Failure Criteria..............................................................................................................2 3 CHAPTER 3 Â– RESULTS................................................................................................26 Conclusions...................................................................................................................3 3 REFERENCES.................................................................................................................35 APPENDICES..................................................................................................................37 Appendix 1: Property Data for Cylinder Material........................................................38 Appendix 2: Convection Medium Property Data.........................................................43 Appendix 3: Program Flow...........................................................................................49 Appendix 4: Verification..............................................................................................50 Appendix 5: Convergence Test.....................................................................................52 Appendix 6: Tables of Results......................................................................................54
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ii LIST OF TABLES Table 1. Geometric Data for the Three Bridges 26 Table 2. Constants 2 FNC Used in Finding Cylinder Tolerances 27 Table A1. Elastic Properties of Stee l as a Function of Temperature 38 Table A2. Thermal Properties of Steel as a Function of Temperature 40 Table A3. Fracture Toughness of ASTM E24 Steel as a Function of Temperature 42 Table A4. Convection Coefficient for Liquid Nitrogen as a Function of Temperature 44 Table A5. Property Data for Air as a Function of Temperature 46 Table A6. Property Data for Isopropyl Alc ohol as a Function of Temperature 46 Table A7. Results for the Christa McAuliffe in Cooling Process 1 54 Table A8. Results for the Christa McAuliffe in Cooling Process 2 55 Table A9. Results for the Christa McAuliffe in Cooling Process 3 56 Table A10. Results for Hillsborough Ave. in Cooling Process 1 57 Table A11. Results for Hillsborough Ave. in Cooling Process 2 58 Table A12. Results for Hillsborough Ave. in Cooling Process 3 59 Table A13. Results for 17th St. Causeway in Cooling Process 1 60 Table A14. Results for 17th St. Causeway in Cooling Process 2 61 Table A15. Results for 17th St. Causeway in Cooling Process 3 62
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iii LIST OF FIGURES Figure 1. A Bascule Bridge 1 Figure 2. Completely Assembled Tr unnionHubGirder (THG) System 2 Figure 3. THG Assembly Procedures 3 Figure 4. Geometry of the Composite Cylinder 6 Figure 5. Thermal Conductivity and Specifi c Heat for Cylinder Material as a Function of Temperature 8 Figure 6. Nodal Locations along Cylinder Wall 9 Figure 7. Coefficient of Thermal Expa nsion as a Function of Temperature 18 Figure 8. Fracture Toughness and Yield Stre ngth as a Function of Temperature 25 Figure 9. Geometry of TrunnionHub Assembly 26 Figure 10. Overall Minimum Critical Crack Length as a Function of HubTrunnion Thickness Ratio for the Christa McAuliffe Bridge 29 Figure 11. Overall Minimum Stre ss Ratio as a Function of HubTrunnion Thickness Ratio for th e Christa McAuliffe Bridge 29 Figure 12. Overall Minimum Critical Crack Length as a Function of HubTrunnion Thickness Ratio for the Hillsborough Ave. Bridge 31 Figure 13. Overall Minimum Stress Ra tio as a Function of HubTrunnion Thickness Ratio for the Hillsborough Ave. Bridge 31 Figure 14. Overall Minimum Critical Crack Length as a Function of HubTrunnion Thickness Ratio for the 17th St. Causeway Bridge 32 Figure 15. Overall Minimum Stress Ra tio as a Function of HubTrunnion Thickness Ratio for the 17th St. Causeway Bridge 32
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iv Figure A1. YoungÂ’s Modulus of ASTM A203A Steel as a Function of Temperature 39 Figure A2. PoissonÂ’s Ratio of ASTM A203A Steel as a Function of Temperature 39 Figure A3. Tensile Strength of ASTM A203A Steel as a Function of Temperature 39 Figure A4. Yield Strength of ASTM A203A Steel as a Function of Temperature 39 Figure A5. Thermal Conductivity of AS TM A203A Steel as a Function of Temperature 40 Figure A6. Specific Heat of ASTM A203A Steel as a Function of Temperature 40 Figure A7. Density of ASTM A203A St eel as a Function of Temperature 41 Figure A8. Coefficient of Thermal Expa nsion of ASTM A203A Steel as a Function of Temperature 41 Figure A9. Fracture Toughness and Yield Stre ngth as a Function of Temperature 41 Figure A10. Heat Flux Versus Temperatur e Difference for Liquid Nitrogen 43 Figure A11. Convection Coefficient for Liquid Nitrogen as a Function of Wall Temperature 45 Figure A12. Convection Coefficient for Ai r as a Function of Wall Temperature for a Diameter = 1 in 48 Figure A13. Convection Coefficient for Dr yIce/Alcohol as a Function of Wall Temperature for a Diameter = 1 in 48 Figure A14. Flow Chart of Computer Program 49 Figure A15. Comparison of Exact and Progr ammed Solution to the Water Problem 50 Figure A16. Comparison of Exact and Progr ammed Solution to the Air Problem 51
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v BENEFIT OF STAGED COOLING IN SHRINK FITTED COMPOSITE CYLINDERS Nathaniel Collier ABSTRACT To assemble the fulcrum of bascule bri dges, a trunnion is immersed into liquid nitrogen so that it can be shrunk fit into the hub. This is followed by immersing the resulting trunnionhub assembly in to liquid nitrogen so that it can be then shrunk fit into the girder. On one occasion in Florida, wh en the trunnionhub assembly was put into liquid nitrogen, development of cracks on th e hub was observed. Experimental and numerical studies conducted si nce 1998 at University of S outh Florida show that the cracking took place due to combination of hi gh interference stresses in the trunnionhub assembly, low fracture toughness of steel at cryogenic temperatures, and steep temperature gradients due to sudden cooling. In this study, we are studying the benefit of staged cooling to avoid cracking in the trunnionhub assembly when it is cooled down for shrink fitting. We looked at three cooling processes 1) Direct immersion in to liquid nitrogen 2) Immersion into a refrigerated chamber, then liquid nitrogen 3) Immersion into a refrigerated chamber, then a dryice/alcohol bath, and finally liquid nitrogen. The geometry of the trunnionhub assembly was approximated by a composite made of two infinitely long hollows cylinde rs. The transient problem of temperature distribution and the resulting stresses was solved using fi nite difference method. Using critical crack lengths and VonMises stress as failure crite ria, the three cooling processes were compared. The study showed that the minimum critical crack le ngth and stress ratio is increased by as much as 200% when cooling first in refrigerated air followed by liquid nitrogen. However, there is little benefit fr om adding dryice/alcohol as an intermediate step in the cooling process.
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1 Figure 1. A Bascule Bridge CHAPTER 1 Â– INTRODUCTION Although the analysis of composite cylinde rs has many applications in various disciplines, this particular analysis was done as part of a specific study at the University of South Florida on the subject of bascule bridges. A bascul e bridge is what is more commonly known as a draw bridge. The draw bridges of the medieval times were designed to keep invaders out of a castle. The draw bridges of today are used to allow water traffic to pass underneath roadways. Although the use of these bridges is different, the basic concept is the same. They work like a lever that rotates around a central point known as the fulcrum. Depicted in Fig. 1, the lever is the roadway and the fulcrum is a trunnion. The critical point of this kind of bridge design is th e fulcrum. The fulcrum is inserted into the bridge girder. When the bri dge is raised, the motors apply torque to the fulcrum which rotates the bridge. This means that the fulcrum must be securely fastened to the bridge. For this reason, a trunnion is used and is supported by a hub, which is inserted to the bridge as well as bolted to the girder. This assembly is referred to as the trunnionhubgirder (THG) system and is seen constructed in Figure 2. Due to the need for strength, an interference f it such as FN2 and FN3 (Shigle y, 1986) is used to construct the THG assembly. Currently, two procedures are followed in the United States of America to make the THG assembly. AP#1: The trunnion is shrunk in liquid ni trogen (or some cold fluid, cold enough to provide enough shrinkage) and inserted into the hub. The same step is then repeated on the resulting trunnionhub asse mbly for insertion into the girder.
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2 AP#2: In this procedure, the hub is first sh rink fitted into the girder. This is then followed by the trunnion being shrink f itted into hubgirder assembly. It is this shrink fitting process that is th e subject of much study. As a material gets colder, it becomes more likely to crack. Ther e have been problems encountered during this THG assembly. According to the Florida Department of Transportation (FDOT), the first method of construction produced cracks in the THG assembling process for the Christa McAuliffe Bridge. The failure of this assembly is expensive to replace and delays the bridge construction. For this reason, a study is being conducted to determine why the THG assemblies failed during assembly and how the problem can be avoided in the future. Literature Survey Although the topic of compos ite cylinders has a much broader scope than that mentioned here, there are several individual studies conducted on this specific problemÂ— that of the cracking of the THG assembly dur ing assembly. The following describes their contribution. To aid in the design of these bridges, Denninger (2000) developed a series of design tools, which calculated the torque needed to lift th e bridge, found the stresses in the THG assembly, and developed a bolt pattern used to supplement the hubgirder fit. Although his work was mainly to develop tools for study of these bridges, Denninger concluded that the steady state stresses were well below the ultimate tensile strength of the material. Thus, it was determined that the cause of failure must come from the transient stresses of the assembly process. Figure 2. Completely Assembled TrunnionHubGirder (THG) System Trunnion Hub Girde r
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3 Ratnam (2000) conducted a finite element an alysis on the two THG assembly procedures shown in Fig. 3. He used this model to anal yze the stresses, temperatures and critical crack lengths for both procedures. His study concluded that assembly procedure AP#2 resolves the problems encountered with the AP#1. However, Ratnam also suggested that there could be a THG geometry where assemb ly procedure AP#1 is safer than AP#2. For this reason, each THG assembly must be an alyzed by a method detailed in his study (Ratnam, 2000) to determine which procedure is preferred. Nichani (2001) experimentally studie d the two currently used assembly procedures. The experiment was conducted with thermocouples and strain gauges at critical points in the design. The complete procedure was logged and the stresses calculated from the measured strains. These stresses were compared with the yield Figure 3. THG Assembly Procedures
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4 strength, and the critical cr ack length. The results of th is experiment agree with RatnamÂ’s conclusionsÂ—that there is a favorable method of assembly with respect to yield stress and critical crack length. Assembly pr ocedure AP#2 was safer based on the stated criteria. Current Study The studies conducted by Ratnam (2000) and Nichani (2001) focused on alternative methods of assemb ly due to the fact that assembly method one had produced failure in the past. They both concluded that assembly procedure AP#2 was preferred to AP#1. The problem with implementing AP#2, is that in many cases the trunnionhub assembly is manufactured elsewhere and then sent to the job site for insertion. Assembly procedure AP#2 would require that the comp lete trunnionhubgirder assembly is done on the job site. For this reason, the goal of th is study is to return to assembly procedure AP#1 and determine if staged cooling can alleviate the encountered problems. In the Christa McAuliffe Bridge, the trunnionhub (TH) cracked while being c ooled for insertion in to the girder. Therefore, only the step of assemb ly where the trunnionhub is cooled will be studied. The current method of cooling uses liquid nitrogen, which boils at 321F. This severe thermal gradient cause s the material to cool very quickly, inducing high thermal stresses. Cooling in stages by use of a refrigerated chamber at 30F and a dryice/alcohol bath at 108F w ould certainly be of benefit to the integrity of the THG assembly process. This method, however, is al so sure to cost more money and time. This study hypothesizes that staged cooling will significantly increase the overall minimum critical crack length, making the assembly procedure safer. Therefore, a numerical analysis wa s conducted, simulating the trunnionhub assembly in different stages of cooling. This is to represent the portion of the THG assembly process where the trunnionhub assembly is cooled for insertion into the girder. Three different cooling processes were studied. Process 1 Direct immers ion into liquid nitrogen
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5 Process 2 Immersion into a refrigera ted chamber, then liquid nitrogen Process 3 Immersion into a refrigerated chamber, then a dryice/alcohol bath, and finally liquid nitrogen The analysis was conducted by the method of finite differences to solve the governing differential equations. For simplicity, the trunnionhub assembly was modeled as a composite cylinder. This assumption is valid to make since the goa l is not to develop sp ecific numbers, but to quantify the benefit of one procedure over a nother. It is assu med that the trunnionhub assembly will behave similarly to a composite cylinder. This problem is not a complicated one to solve under the normal a ssumption of constant material properties. However, due to the wide range of temperat ures, the materials properties need to be assumed as functions of temperature.
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6 CHAPTER 2 Â– PROBLEM FORMULATION The geometry of a trunnionhub assembly was simplified as a composite cylinder. This simplification was made so that the fini te differencing would be a feasible method of solving for all unknowns. It should also be noted that the exact stresse s and critical crack ratios are not of interest he re. The purpose is to compar e cooling methods. This fact justifies for many such simplifications to be made. The more important question is relatively quantifying how much st aged cooling benefits the TH assembly process. Figure 4 shows these cylinders as well as explai ns some of the notation to be used. Figure 4. Geometry of the Composite Cylinder Cylinder 1 Cylinder 2 b c a
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7 Thermal Formulation To determine the effect of thermal st resses, it is first necessary to know the temperature distribution within the cylinde rs as a function of space and time. The composite cylinder is to be immersed in several media and will be cooled solely by convection. This problem is axisymmetric and so lved in a cylindrical coordinate system. Each cylinder is treated as a separate cylinder, linked by boundary conditions. The governing equation for each cylinder is as follows: r T T k r r r t T T C Tj j j j p j) ( 1 ) ( ) ( (1) where, ) ( t r f T and is the temperature distribution, subscript 2 and 1 j and designates Cylinders 1 and 2, respectively, is the material density, pC is the material specific heat, k is the material thermal conductivity, T is the radial temperature distribution, r is radial position, and t is time. Equation (1) (zi ik, 1993) is used to govern both cyli nders but is applie d separately so that the cylinders can differ in material proper ties. Since the cylinders will be subjected to cryogenic temperatures it is necessary to assu me that all material properties are functions of temperature. Note the variance of thermal conductiv ity and specific heat over temperature in Fig. 5. The ther mal conductivity doubles over the range of temperature of interest. The specific heat at room temperat ure is six times greater than that at the ambient temperature of liquid nitrogen. T hus the normal simplification of assuming constant material properties is not valid for Eq. (1).
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8 Thermal Conductivity 0.0 0.5 1.0 1.5 2.0 35015050 FBTU / (hr in F) Specific Heat 0.00 0.05 0.10 0.15 35015050 FBTU / (lbm F) Figure 5. Thermal Conductivity a nd Specific Heat for Cylinder Material as a Function of Temperature This analysis used finite differencing for the soluti on of differential equations. Finite differencing replaces derivatives with approximations using discrete points. A computer program was written to perform the necessary calculations rapidly. Because computers are never exact in their calculations, errors al ways exist. In transient problems, the radial temperature distribution is found at each time step and is usually based in part on the previous time step. This causes the small errors in calculations to propagate as further time steps are ca lculated resulting in a phenomenon called instability. Most methods of solving transi ent problems have a stability criterion which limits either the number of nodes or the time st ep in the solution. This is not desirable because a designer wishes to have complete control over the simulation and for the simulation to always be stab le. The CrankNicolson method is used in this simulation because it is unconditionally stable. (zi ik, 1993). n i j n i j i j n i j n i j i j j i n i j n i j i j n i j n i j i j j i n i j n i j i j p jT T r k T T r k r r T T r k T T r k r r t T T C 1 , 1 2 1 1 1 1 1 1 2 1 ,2 1 2 1 2 1 2 11 1 2 1 1 1 2 1 (2) where, subscript i designates a nodal location, and superscript n designates a time step.
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9 The properties at each node are evaluated at the temperature of the previous time step. This is incorrect but insigni ficant if the time step is small enough. This method is unconditionally stable and second order accura te. The cylinder was discretized as shown below in Fig. 6 and this equation was used for all interior nodes (nodes 1 to N 1 of Cylinder 1 and nodes 1 to M 1 of Cylinder 2). N represents the number of nodes in Cylinder 1, and M represents the number of nodes in Cylinder 2. The boundary nodes are discussed below. Figure 6. Nodal Locations along Cylinder Wall This generates a system of linear equations which are solved us ing matrix algebra to determine the nodal temperatures. Befo re this can happen, the boundary conditions must be applied to the problem. Cylinder 1 Cylinder 2 0 1 2 N 2 N 1 N 012M2 M1M r =a r =br =c
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10 Thermal Boundary Conditions In each case the cylinder is rapidly cooled by various media. Cylinder 1 is experiencing convection on its inner surf ace. Therefore the boundary condition on the inner edge at a r is: amb a rT t r T T h r T T k ) ( ) ( ) (1 1 1 (3) where, h is the convection coefficien t of the cooling medium, and ambT is the ambient temperature of the cooling medium. A second order accurate forward divided differe nce approximation (Boresi, 1991) is used for the derivative term. The thermal conductivity is evaluated at the wall temperature and the convection coefficient is evaluated at the average of the wall and ambient temperature. amb n amb n n n n nT T T T h T T T r T k 1 1 1 1 1 1 1 1 1 1 1 10 0 2 1 0 0) 2 ( 4 3 2 1 ) ( (4) Similarly, Cylinder 2 is also experien cing convection on its outer surface at c r A second order accurate backward divided difference approximation is used for this derivative term. amb c rT t r T T h r T T k ) ( ) ( ) (2 2 2 (5) amb n amb n n n n nT T T T h T T T r T kM M M M M M 1 2 2 1 2 1 2 1 2 2 2 2) 2 ( 4 3 2 1 ) (2 1 (6) The two interface conditions are due to the fact that the temperature and the heat flux must be equal at the interface, b r This means that, b r b rt r T t r T ) ( ) (2 1 (7) 1 2 1 10 n nT TN (8) and, r T T k r T T k 2 2 1 1) ( ) ( (9)
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11 In this case a backward difference approxima tion must be used for the derivative of 1T and a forward difference approximation for 2T. 1 2 1 2 1 2 2 2 2 1 1 1 1 1 1 1 1 12 1 0 2 1 04 3 2 1 ) ( 4 3 2 1 ) ( n n n n n n n nT T T r T k T T T r T kM N N N (10) These interface conditions, Eqs. (8,10), link th e differential equation of each cylinder. Thus the governing equation is used for interior nodes and the boundary condition equations are used for the boundary nodes. What results is a system of linear equations where the unknowns are the nodal temperatures (at time step,1 n). This system of linear equations is solved progressively for each ti me step. The nodal temperatures obtained for one time step are used to calculate the nodal temperatur es for the next time step. Elasticity Formulation Once the temperature distribution at a part icular time step is known, the thermal stresses can be calculated. The composite cylinder experiences stress from two sources: thermal gradient and the preimposed interface fit. The inner cylinder is actually too large to be placed in the outer cyli nder. It is shrunk in a cold medium and inserted into the outer cylinder. In this study, this process is already assumed to have taken place. What results is a pressure at the interface that becomes a source of stress in the composite cylinder. For this portion of the simulation, equilibrium must be satisfied. When formulating a problem, it is sometimes difficult to choose the form in which one wants the equations. Many times the boundary conditions can help make that decision. In this case, it is desirable to have the equations only in terms of displacements. All other desired quantities are calculated directly fr om displacements. The following are the three equations of equilibrium (Timoshenko, 1951): 0 1 z r r rj rz j j r j r j r (11)
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12 0 2 1 r z r rj r j z j r j (12) 0 1 1 j rz j z j zr j zr r r z (13) where, r is the radial stress, is the hoop stress, z is the longitudinal stress, and r, rz and z are the shear stresses in the r rz and z planes, respectively. Equations (1113) are all in te rms of stress. To get them into displacements as desired they must be first related to strains. The following are the stressstrain equations. They have been modified to account for thermal stresses (Timoshenko, 1951). ) ( ) ( 1 ) (j z j j j r j j j rT T E T (14) ) ( ) ( 1 ) (j z j r j j j j jT T E T (15) ) ( ) ( 1 ) (j r j j j z j j j zT T E T (16) ) ( T Gj j r j r (17) ) ( T Gj j z j z (18) ) ( T Gj j rz j rz (19) where, ) () ( ) (r T T j jj initialT d T T (20) r is radial strain, is strain in the hoop strain,
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13 z is longitudinal strain, r, z and rz are shear strains in the r z and rz planes, respectively, is the coefficient of thermal expansion, E is the modulus of elasticity, G is the modulus of rigidity, and is PoisonÂ’s ratio. Then the strains are, by definition, expressed as follows (Timoshenko, 1951): r uj r j r (21) r u r u rj r j j 1 (22) z uj z j z (23) r u r u u rj j j r j r 1 (24) r u z uj z j r j rz (25) j z j j zu r z u1 (26) where, ru is the radial displacement, u is the angular displacement, and zu is the longitudinal displacement. Solving these fifteen equations (Eqs. 1126) can be intract able, but there are several simplifications that can be made in this st udy. The problem is axisymmetric. This means that at any constant radial location from the center, the displacements, stresses, and strains will be the same. There will be no displacement in the direction either. Therefore, all equations which contain u or any derivative of are eliminated. The equations then look more manageable. Notice th at Eqs. (27), (28), and (34) have become simplified as well.
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14 0 z r rj rz j j r j r (27) 0 1 1 j rz j z j zr j zr r r z (28) ) ( ) ( 1 ) (j z j j j r j j j rT T E T (29) ) ( ) ( 1 ) (j z j r j j j j jT T E T (30) ) ( ) ( 1 ) (j r j j j z j j j zT T E T (31) ) ( T Gj j rz j rz (32) r uj r j r (33) r uj r j (34) z uj z j z (35) r u z uj z j r j rz (36) The problem is also assumed to be a case of generalized plane strain. This means that shear stresses will be zero (rz = z = 0). This affects Eqs. (2736) in the following way. 0 r rj j r j r (37) 0 zj z (38) ) ( ) ( 1 ) (j z j j j r j j j rT T E T (39) ) ( ) ( 1 ) (j z j r j j j j jT T E T (40)
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15 ) ( ) ( 1 ) (j r j j j z j j j zT T E T (41) r uj r j r (42) r uj r j (43) z uj z j z (44) 0 r u z uj z j r j rz (45) Furthermore, z is assumed to be a constant valueÂ—that is to say that the composite cylinder will longitudinally elongate (or actually shrink in this case), but that it will do so by a constant amount. This assumption eliminates the system dependence on Eqs. (38, 45). While this simplifies the problem, it still introduces the need for another boundary condition as discussed belo w due to the fact that z equals an unknown constant. Thus the equations change as follows: 0 r rj j r j r (46) ) ( ) ( 1 ) (j z j j j r j j j rT T E T (47) ) ( ) ( 1 ) (j z j r j j j j jT T E T (48) ) ( ) ( 1 ) (j r j j j z j j j zT T E T (49) r uj r j r (50) r uj r j (51) Cj z (52) where C is a constant.
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16 Equations (5052) are substituted into Eqs. (4749). This removes all the strain terms from the system. 0 r rj j r j r (53) ) ( ) ( 1 ) (j z j j j r j j j rT T E T r u (54) ) ( ) ( 1 ) (j z j r j j j j j rT T E T r u (55) ) ( ) ( 1 ) (j r j j j z j jT T E T C (56) Equations (5456) are solved simulta neously for the three stress terms (r , and z ). The equations were solved us ing symbolic manipulation feat ures of Maple 8 (Maplesoft, 2003). ) ( 1 ) ( ) ( ) ( ) ( 1 1 ) ( ) ( 2 ) ( ) (2r r r C r r r u r r dr du r r r r E rj j j j j r j j r j j j j r (57) ) ( 1 ) ( ) ( ) ( 1 ) ( 1 ) ( ) ( 2 ) ( ) (2r r r C r r r u r r dr du r r r r E rj j j j j r j j r j j j j (58) ) ( 1 ) ( ) ( 1 ) ( ) ( 1 ) ( ) ( 2 ) ( ) (2r r r r r C r u r r dr du r r r r E rj j j j j r j j r j j j j z (59) By substituting the stresses in terms of displacements (Eqs. (5759)) into Eq. (53), the equilibrium equation is rewritten in terms of displacements only. This final substitution is lengthy and wa s also done by using Maple. The result was organized by grouping terms which contained a second derivative of radi al displacement, a first derivative of radial displacement, radial displacement, and the constant C. Thus the long
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17 result of writing the equilibrium equation in terms of only radial displacements can be expressed in this form: 0 ) ( ) ( ) ( ) ( ) (5 4 3 2 2 2 1 r D C r D u r D dr du r D dr u d r Dj j j r j j r j j r j (60) where 1D, 2D, 3D 4D, and 5D are all coefficients made up of material properties and are functions of radial locati on. They have also been si mplified by using Maple. As a further precaution, the code generator option of Maple was used to generate the FORTRAN code from the simp lified Maple equations. The fo llowing are the coefficient functions: 1 ) ( ) ( 2 ) 1 ) ( )( ( ) (2 1 r r r r E r Dj j j j j (61) 1 ) ( ) ( 2 1 ) ( 2 ) ( ) ( 2 ) ( 1 ) ( ) ( 2 ) ( 4 ) ( 2 ) ( ) ( 1 ) ( ) ( 2 1 ) ( 2 ) ( ) ( 2 ) ( ) (2 2 3 2 2 2 2 3 2 r r r r r r r E r r r r dr r d r E r r r r r dr r dE r Dj j j j j j j j j j j j j j j j j j j (62) 2 2 2 2 3 2 2 2 2 2 2 3 31 ) ( ) ( 2 1 ) ( 2 ) ( ) ( 2 ) ( 1 ) ( ) ( 2 1 ) ( 2 ) ( ) ( 1 ) ( ) ( 2 ) ( ) ( ) ( 2 ) ( ) ( r r r r r r r E r r r r dr r d r E rr r r r r dr r dE r Dj j j j j j j j j j j j j j j j j j (63) 2 2 2 2 2 2 3 41 ) ( ) ( 2 1 ) ( 2 ) ( ) ( 1 ) ( ) ( 2 ) ( ) ( ) ( 2 ) ( ) ( r r r dr r d r E r r r r r dr r dE r Dj j j j j j j j j j j j (64) 2 51 ) ( 2 ) ( ) ( ) ( 2 1 ) ( 2 ) ( ) ( 1 ) ( 2 ) ( ) ( ) ( r dr r d r r E r dr r d r E r dr r dE r r Dj j j j j j j j j j j (65)
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18 Note that the material properties are shown to be functions of radius when in reality they are functions of temperature. In this study, th e temperature at each time step is a function of radius. For simplicity, the material propertie s are shown as functions of radius. When a material property was calculated, the radius input was used to calcu late the corresponding temperature at that point and the proper ty was evaluated at said temperature. As with the thermal formulation, th e material properties are functions of temperature and must be kept as functions a nd not constants. Note in Figure 7 how the coefficient of thermal expansion varies over the specified temperature range. 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 3502501505050 FE06 in / (inF) Figure 7. Coeffi cient of Thermal Expansion as a Function of Temperature Equation (60) was used for all interior nodes in both cylinders. Care must be taken to keep the material properties of each cylinder completely separate. The derivatives of ru were approximated using the followi ng second order accurate central approximations: 1 0 1 2 2 22 1j r j r j r j j ru u u r dr u d (66) 1 1 ,2 1j r j r j j ru u r dr du (67)
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19 In Eqs. (6265) there are derivatives of material properties. The material properties used are discrete data points gi ven in Appendix 1. Th ese data points were interpolated using cubic splines. The deri vatives were obtained using second order accurate central approximations.
PAGE 26
20 Elasticity Boundary Conditions The loads in this problem are the stresses due to the interface and the thermal stresses. Due to the absence of other extern al loads acting on the composite cylinder, the radial stress on the inner and outer edges must be zero. The inner radius (a r ) of Cylinder 1 was substituted into the expression found for the ra dial stress in terms of displacements only, Eq. ( 57). The outer radius (c r ) of Cylinder 2 was also substituted into Eq. (57). The radial stress term has a first derivative of ru in it which requires approximating. As before in the case of the thermal boundary conditions, a forward approximation was used for the inner edge and a backward approximation for the outer edge. 0 ) (1 ar (68) 0 ) ( 1 ) ( ) ( ) ( ) ( 1 1 ) ( ) ( 2 ) (1 1 1 1 1 1 1 1 2 1 10 a a a C a a a u a a dr du a a a a Er r (69) and 0 ) (2cr (70) 0 ) ( 1 ) ( ) ( ) ( ) ( 1 1 ) ( ) ( 2 ) (2 2 2 2 2 2 2 2 2 2 2 c c c C c c c u c c dr du c c a c EMr r (71) The two interface conditions as with the thermal boundary conditions, come from the conditions of the interface at r = b. In the thermal case, the temperatures and flux at node N of cylinder 1 and node 0 of cylinder 2 were equal. This does not correspond to the displacements. The inner cylinder was orig inally too large for th e outer cylinder, and so it was shrunk into the outer cylinder. This causes an interference which is based on just how tight a fit is needed. The displacements at the interface nodes (node N of cylinder 1 and node 0 of cylinder 2) will differ by a specified factor of this interference, denoted by
PAGE 27
21 02 1r ru uN (72) The remaining boundary condition is that the radial stresses must be equal at the interface as well. Again, Eq. (57) is used b ecause it represents the radial stress in terms of displacements only. ) ( ) (2 1b br r (73) ) ( 1 ) ( ) ( ) ( ) ( 1 1 ) ( ) ( 2 ) ( ) ( 1 ) ( ) ( ) () ( 1 1 ) ( ) ( 2 ) ( 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 2 1 10b b b C b b b u b b dr du b b b b E b b b C b b b u b b dr du b b b b EMr r r r (74) As stated before with the thermal portion, two separate systems of differential equations are being solved, linked by boundary conditions. There ar e second derivatives in each cylinder so a total of four boundary conditions are needed, two for each cylinder. The derivatives in all of these equations are discretized using the above mentioned methods. What results is a nother system of linear equa tions where the unknowns are displacements at various nodes. It would appear that the problem is ready to be solved, but another equation is needed. In Eq. (52) it was stated that z is equal to a constant C. The purpose of leaving z a constant was to be more accurate, allowing for the cylinder to contract longitudinally a constant amount. This intr oduces another unknown in to the problem, for which another equation is needed. Although the composite cylinder will contract, there will be no force acting on the face. By summi ng up the force on the face of the composite cylinder and setting it equal to zero, we deve lop another equation for finding the value of C. 0 2 22 1 dr r dr rc b z b a z (75)
PAGE 28
22 The area multiplied by the stress, z over the entire area of the end of the cylinder is the force. The expression for z in terms of only radial displa cements is used. The constant is not immediately seen in the formul a, but is part of the expression for z For the assumption of Cz to be possible, this condition must be satisfied. Equation (59) was used to express the longitudinal stress becau se it is in terms of displacements only. Satisfying this boundary condition proved to be the most difficult aspect of the program. The problem consists in this: z is a large expression which contains radial displacement terms. The very unknowns being solved for are inside the integral. To solve this problem, the integrals were approximated using multiplesegment SimpsonÂ’s 1/3 Rule (Chapra, 1998). They were expanded and like terms were collected. This completes another system of linear equations which is solved simultaneously to calculate the displacement at each node. These displacements were substituted into Eqs. (5759) to get the value of the stresses at each node. The next section details how the stress values are used to predict failure.
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23 Failure Criteria The purpose of the study was to compar e cooling procedures, not to find exact stresses. The stresses calculated determine if and when the cylinder fails. This quantifies how well different cooling methods function. This study based failure on two quantities: the overall minimum stress ratio and the overall minimum critical crack length. As the cylinders get colder, there is an increased chance that they will crack. It is also possible that the transient stress values exceed the al lowable. That is why both criteria are being examined. The stress ratio is a comparison of the V on Mises stress with the yield stress. It can also be viewed as the factor of safety. Failure occurs when the stress ratio falls below one (i.e. the Von Mises stress es become larger that the yield stress). The following formula was used (Timoshenko, 1951). r r YST SR 2 ) (2 2 (76) where, SR is the stress ratio, and YS is the yield strengt h of the material. Yield strength is not a constant value, but a function of temperature. Therefore, the Von Mises stress is calculated from the radial a nd hoop stresses at each node and compared to the yield strength at the partic ular temperature of said node For a given time step, the minimum stress ratio was found in the cylinder wall. That minimum stress ratio was compared to the minimum stress ratio of the previous time step. Only the lowest stress ratio was stored. The stress ratio that is ultimately generated is then a worst case stress ratio because it is the lowest in space and time. For this study, it was desired to know what the lowest stress ratio was and where on the two cylinders did it occur. The critical crack length is defined as the crack length after which failure occurs. The formula involves another physical prope rty called fracture toughness. In most instances, the critical crack is a specified le ngth based on what kind of equipment is being used for inspection. Based on that crack leng th and the fracture toughness of the material,
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24 a stress level can be found after which failure is assumed to occur. The following formula is used to determine that stress level (Kanninen, 1985). crack IC crackL T K 25 1 ) ( (77) where, crack is the hoop stress after which failure occurs, ICK is the fracture toughness of the material, and crackL is the length of the smallest de tectable crack in the assembly. In this study, the actual stress as well as the fracture t oughness is known. The stress used is the hoop stress at a specific point and the fracture toughness is evaluated at the temperature at that point. Thus the formula is rearranged to show the length of the critical crack. 225 1 crack IC crackK L (78) What this is interpreted to mean is that any crack larger than this length will produce failure. As with the overall minimum stress ratio, the overall minimum critical crack length is the absolute minimum crack length in time and space for the cylinder. This overall minimum crack length was recorded as well as the location it occurred. Both failure criteria are implemented because it is not known exactly what the cause of failure is. As the cylinder cools dow n, the yield strength increases which makes the stress ratio higher yet the fracture toughness drops making the cylinder more susceptible to cracking. Figure 8 shows a graph of the fracture toughness and yield strength as a function of temperature.
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25 Figure 8. Fracture Toughness and Yield Strengt h as a Function of Temperature (Source: Greenberg, 1969)
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26 CHAPTER 3 Â– RESULTS The concepts presented in the previous chapter were implemented to write a program in FORTRAN. Although the program allo ws for infinite number of cases to be run, three cases were taken into considerat ion for this thesis. A lthough each bridge has many important parameters, for this study we are merely interested in the geometric parameters listed in Table 1. For clarifica tion, Fig. 9 shows the correlation of these dimensions to the actual body. Table 1. Geometric Data for the Three Bridges Bridge Geometric Parameters Christa McAuliffe Hillsborough Avenue 17th Street Causeway a in 1.0 1.125 1.1875 b in 9.0 8.39 6.472 c in 16.0 15.39 8.88 Figure 9. Geometry of TrunnionHub Assembly Trunnion Hub b c a
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27 It can be seen from Table 1 that the di mensions of the girder and hub assembly can vary a large amount. According to the original research statement produced by FDOT, hub thicknesses (b c) can range from 1.5 inches to 8 inches. The trunnion dimensions are less flexible because they must meet certain criteria for the bridge. The hub thickness, however, is more arbitrary. While AASHTO Specifications (AASHTO, 1998) call for a hub thickness of 0.4 times the diameter (in this case b2 4 0 ), the standard practice in industry is to use a hub thickness of b2 2 0 to 1 0 (Request for Proposal, 1998). The hub thickness has a great effect on the thermal behavior of the assembly. To see the effect of the hub thickness, the trunnion dimensions ( a and b ) of each bridge were used and the outer dimens ion (which controls the hub thickness) was varied from b b 2 4 0 to 1 0. This was done for each of the three bridges as well as for each of the three cooling methods. For each br idge configuration and cooling method the overall minimum critical crack length and stress ratio were recorded. The results of that study are given below. Some comment should be made on the or ganization of the data. The cylinders have an interface fit, specifical ly a standard interface fit called FN2. If the trunnion is fit into the hub, there is an upper and lower limit by which th e outer diameter of the trunnion and the inner diameter of the hub vary. Thes e limits are calculated using the following expression (Shigley, 1986). 3 12 2 2 FN FN FND C L (79) where, 2 FNL is the limit in thousandths of an inch, 2 FNC is a constant specified in the table below, and 2 FND is the cylinder diameter in inches. Table 2. Constants 2 FNC Used in Finding Cylinder Tolerances Class of fit Cylinder A (hub) Cylinder B (trunnion) Lower Upper Lower Upper FN2 0 0.907 2.717 3.288
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28 The limits for the TrunnionHub used on the Christa McAuliffe Bridge are calculated using its radius in 9 b (See Table 1), Eq. (79), and the constants listed in Table 2. in 0 ) 9 2 ( 03 12 FNL (80) in 002377 0 ) 9 2 ( 907 03 12 FNL (81) in 007121 0 ) 9 2 ( 717 23 12 FNL (82) in 008617 0 ) 9 2 ( 288 33 12 FNL (83) The outer diameter of the trunnion, acco rding to the standard for FN2 fits, is 008617 0 007121 018 inches and the inner diameter of the hub is 002377 0 000000 018 inches. The difference between the inner diameter of the hub and the outer diam eter of the trunnion is called the in terference and is denoted as The interference will have a high and low limit. These limits are found by subtracting the diam eters at their extremes. 008617 0 0 18 008617 18 in (84) 006240 0 002377 18 008617 18 in (85) 007121 0 0 18 007121 18 in (86) 004744 0 002377 18 007121 18 in (87) Equations (8487) show that th e diametric interfer ence is as much as 0.008617 inches and as little as 0.004744 inches. These limits are small, but significant. In this study, the interface stress is the main sour ce of stress within the body. For this reason, each bridge configuration and cooling met hod was run twice, once at the high end of the diametrical interface limit and once at the low end. Thus the outer radius of Cylinder 2, th e diametrical interface, and the cooling process were varied, the program run, and the overall minimum critical crack and overall minimum stress ratio recorded. As mentioned in the introduction and repeated here for convenience, the three cooling proce sses considered are the following: Process 1 Direct immers ion into liquid nitrogen Process 2 Immersion into a refrigera ted chamber, then liquid nitrogen Process 3 Immersion into a refrigerated chamber, then a dryice/alcohol bath, and finally liquid nitrogen The following are the results from the Christa McAuliffe Bridge.
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29 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.00.20.40.60.81.01.21.4 (cb)/(ba) nondimCrack Length in Process 1, Low Delta Process 1, High Delta Process 2, Low Delta Process 2, High Delta Process 3, Low Delta Process 3, High Delta Actual Bridge Dimensions AASHTO Specification Figure 10. Overall Minimum Critical Crack Length as a Function of HubTrunnion Thickness Ratio for the Christa McAuliffe Bridge 1 2 3 4 5 6 7 8 9 0.00.20.40.60.81.01.21.4 (cb)/(ba) nondimStress Ratio nondim Process 1, Low Delta Process 1, High Delta Process 2, Low Delta Process 2, High Delta Process 3, Low Delta Process 3, High Delta Actual Bridge Dimensions AASHTO Specification Figure 11. Overall Minimum Stress Ratio as a Function of HubTrunnion Thickness Ratio for the Christa McAuliffe Bridge
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30 Figures 10 and 11 show the overall minimu m critical crack length and the overall minimum stress ratio, respectively. In Fi gure 10, the overall minimum critical crack length is plotted versus the hubtrunnion thickness ratio. Note the curves representing Process 1. The overall minimum critical crack length is low for the full range of hubtrunnion thickness ratio. The crack length varies slightly fr om high interface values to low interface values. The curves representi ng Process 2 indicate an overall minimum critical crack length that is on average 150% longer. Not onl y that, but the crack length varies three times as much from high to in terface values as in process one. Adding a refrigerated air stage has definite advantages. Note the curves representing Process 3. The usefulness of adding a cooling stage of dryice/alcohol is limited. Process 3 only adds on average an additional 1% of benefit to process two. In Figure 11, the effect of adding a stage of dryice/alcohol can be more clearly seen. Process 2 increases the overall minimum stress ratio by an average of 20% and Process 3 increases it by an additional 7%. Although the overall minimum stress ratio does not fall below one in these experiment s, for larger hubtrunnion thickness ratios it does become as low as 1.5. The actual bridge configuration and the AASHTO specifications can be seen in Figures 10 and 11 by vertical lines. This s hows that the Christa McAuliffe was designed following the AASHTO standards and where the overall minimum critical crack length was about 0.15 inches. Figures 12 and 13 show the same results for the Hillsborough St. Bridge. The trunnionhub assembly of this bridge is si milar in size to the Christa McAuliffe (see Table 1) and therefore the results are simila r. Figures 14 and 15 show results for the 17th Ave. Causeway Bridge, whose trunnionhub asse mbly is roughly half the size of the other two bridges considered. Figure 14 shows almost no benefit to adding the dryice/alcohol as a cooling stage. This is most likely due to the fact that the rate at which heat is convected into the fluid is pa rtially based on the diameter of the cylinder. Since the 17th Ave. Causeway is half the size of the othe rs, its convection coefficient is smaller and therefore does not make a signi ficant contribution. This can also be seen in Figures 10 and 12 when the hubtrunnion th ickness ratio is smaller.
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31 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.00.20.40.60.81.01.21.4 (cb)/(ba) nondimCrack Length in Process 1, Low Delta Process 1, High Delta Process 2, Low Delta Process 2, High Delta Process 3, Low Delta Process 3, High Delta Actual Bridge Dimensions AASHTO Specification Figure 12. Overall Minimum Critical Crack Length as a Function of HubTrunnion Thickness Ratio for the Hillsborough Ave. Bridge 1 2 3 4 5 6 7 8 9 0.00.20.40.60.81.01.21.4 (cb)/(ba) nondimStress Ratio nondim Process 1, Low Delta Process 1, High Delta Process 2, Low Delta Process 2, High Delta Process 3, Low Delta Process 3, High Delta Actual Bridge Dimensions AASHTO Specification Figure 13. Overall Minimum Stress Ratio as a Function of HubTrunnion Thickness Ratio for the Hillsborough Ave. Bridge
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32 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.00.20.40.60.81.01.21.41.6 (cb)/(ba) nondimCrack Length in Process 1, Low Delta Process 1, High Delta Process 2, Low Delta Process 2, High Delta Process 3, Low Delta Process 3, High Delta Actual Bridge Dimensions AASHTO Specification Figure 14. Overall Minimum Critical Crack Length as a Function of HubTrunnion Thickness Ratio for the 17th St. Causeway Bridge 1 2 3 4 5 6 7 8 9 0.00.20.40.60.81.01.21.41.6 (cb)/(ba) nondimStress Ratio nondim Process 1, Low Delta Process 1, High Delta Process 2, Low Delta Process 2, High Delta Process 3, Low Delta Process 3, High Delta Actual Bridge Dimensions AASHTO Specification Figure 15. Overall Minimum Stress Ratio as a Function of HubTrunnion Thickness Ratio for the 17th St. Causeway Bridge
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33 Conclusions As expected, Processes 2 and 3 have a definite advantage over a range of thickness ratios and bridge sizes. A curious result is that Process 3 did not yield significantly improved results from Process 2. However, at higher diameters (the 17th Street Causeway Bridge is smaller than the other two) the effect of the dryice/alcohol from Process 3 is seen, but it does not seem to grant a high level of benefit. The benefit it does give is to the upper end of the critical crack length. Th is does not improve the worst case scenario (the lower end). Of course this information is based on the quality of the property data used, but it does suggest that prerefrigerati ng is recommended as it is relatively cheap and an easy procedure to perform. The graphs also depict th e effect of the hubtrunnion thickness ratio in assembly process. Making the hubtrunnion thickness ra tio larger is good for the increasing the overall minimum critical crack length, but ha s the opposite effect on the overall minimum stress ratio. Although the stre ss ratio never falls below un acceptable values in these numerical experiments, it does approach it s low limit of one at high values of hubtrunnion thickness ratio. Thus, a trade off si milar to the one between fracture toughness and yield strength occurs. Staged cooling not only raises the overall minimum critical crack length and stress ratio, it also incr eases the gap between the mi nimum and maximum possible values. This is not necessarily an improve ment. However, as the difference between minimum and maximum values increases, the grea ter the statistical chan ce that the actual value will be further from an extreme. In addition to the information graphed, the location of the overall minimum stress ratio and critical crack length was recorded. The values can be viewed in Appendix 6. A couple of interesting observations can be ma de. The minimum stress ratio always falls on the inside surface of Cylinder 1. This suggest s that the use of a material with a higher yield strength for the trunnion is of benefit. The overall minimum critical crack length most often falls on the outside surface of Cylinder 2, but geometries it fell on the inner surface of Cylinder 2 (at the interface). This poses another problem because cracks on the interface cannot be easily detected. A
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34 cylinder can visually pass a crack test, but fa il due to unseen cracks at the interface. This seems to happen in a few data points at hi gh interface values on the smaller cylinders when the thickness ratio is high. Therefore, smaller cylinders with high interface stress and thickness ratio are more likely to have the crack at the interface instead of the outer surface. This phenomenon does not occur for lo w interference values or in the larger THG geometries.
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35 REFERENCES AASHTO Mobile Bridge Insp ection, Evaluation, and Maintenance Manual, 1998. AspenTech, Â“Aspen PlusÂ”, [online] http://www.aspentech.com/ (Accessed 03/31/2004). Barron, R. F., 1999, Cryogenic Heat Transfer, Taylor & Francis, Ann Arbor. Boresi, A. P., and Chong, K.P., 1991, Approximate Solution Methods in Engineering Mechanics, Elsevier Applied Science Pub lishers LTD. New York, pp. 5153. Chapra, S. C., and Canale R. P., 1998, Numerical Methods for Engineers, WCB McGrawHill, New York. Denninger, M.T., 2000, Â“Design Tools fo r TrunnionHub Assemblies for Bascule Bridges.Â” MS Thesis, Mechanical Engin eering Department, University of South Florida, FL. Greenberg, H.D., and Clark, Jr. H.G., 1969, Â“A Fracture Mechanics Approach to the Development of Realistic Acceptanc e Standards for Heavy Walled Steel CastingsÂ”, Metals Engineer ing Quarterly, 9(3), 3033. Incropera, F. P., and DeWitt, D.P., 1996, Introduction to Heat Transfer, John Wiley & Sons, Inc., NewYork. Kaka, S., and Yener Y., 1995, Convective Heat Transfer, CRC Press, Inc., Boca Raton. Kanninen, M. F. and Popelar, C. H., 1985, Advanced Fracture Mechanics, Oxford Engineering Science Series, Oxford University Press, New York. Kreith, F., and Bohn, M., 1986, Principles of Heat Transfer, Harper & Row, New York, pp. 98. Logan, D. L., 1992, A First Course in Finite Element Method, PWSKENT Series in Engineering, PWSKENT Publishing Company, Boston. Maplesoft, 2003, Â“Maple 9.0: Command the BrillanceÂ”, [online] http://www.maplesoft.com/ (Accessed on 3/30/2004). Nichani, S., 2001, Â“Full Scal e Testing of TrunnionHubGird er Assembly of a Bascule Bridge.Â” MS Thesis, Mechanical Engin eering Department, University of South Florida, FL.
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36 zi ik, M. N., 1993, Heat Conduction, John Wiley & Sons, Inc., NewYork, pp. 8, 491. Ratnam, B., 2000, Â“Parametric Finite El ement modeling of trunnion Hub Girder Assemblies for Bascule Bridges.Â” MS Thesis, Mechanical Engineering Department, University of South Florida, FL. Request for Proposals for Parametric Fi nite Element Analysis of TrunnionHub Assemblies for Bascule Bridges, Februa ry 11, 1998, Florida Department of Transportation. Shigley, J.E., and Mischke, 1986, Standard Handbook of Machine Design, McGrawHill, New York. Timoshenko, S. P., and Goodier, J. N., 1951, Theory of Elasticty, McGrawHill Book Company, New York. pp. 406408.
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37 APPENDICES
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38 Appendix 1: Property Data for Cylinder Material The following tables detail the properties used in this experiment. Although this is not always the exact steel us ed in the THG assembly, the numbers are representative of the steels one would encounter. Table A1. Elastic Properties of Stee l as a Function of Temperature Fe 2.25 Ni (ASTM A203A), Normalized Temperature Young's Modulus Poisson's Ratio Tensile Strength Tensile Yield Strength F Msi nondim ksi ksi 340.00 31.490 0.2756 115.0 102.0 320.00 31.440 0.2758 111.0 95.0 300.00 31.380 0.2760 108.0 89.0 280.00 31.320 0.2763 105.0 83.0 260.00 31.260 0.2765 102.0 78.0 240.00 31.200 0.2768 99.0 73.0 220.00 31.140 0.2770 96.0 68.0 200.00 31.070 0.2773 93.5 64.0 180.00 30.990 0.2776 91.0 60.5 160.00 30.910 0.2779 89.0 58.0 140.00 30.830 0.2781 87.0 56.0 120.00 30.750 0.2784 85.0 54.0 100.00 30.670 0.2787 83.0 52.0 80.00 30.590 0.2790 81.0 50.5 60.00 30.500 0.2793 79.0 49.0 40.00 30.410 0.2796 77.0 48.0 20.00 30.320 0.2799 75.5 47.5 0.00 30.230 0.2802 74.0 47.0 20.00 30.140 0.2805 73.0 47.0 40.00 30.050 0.2808 72.0 47.0 60.00 29.960 0.2811 71.0 47.0 80.00 29.870 0.2815 70.0 47.0
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39 Appendix 1 (Continued) 0.0 10.0 20.0 30.0 40.0 35015050 FE+03 ksi 0.00 0.05 0.10 0.15 0.20 0.25 0.30 35015050 F[nondim] 0 50 100 150 35015050 FE+03 psi 0 20 40 60 80 100 120 35015050 FE+03 psi Figure A1. YoungÂ’s Modulus of ASTM A203A Steel as a Function of Temperature Figure A2. PoissonÂ’s Ratio of ASTM A203A Steel as a Function of Temperature Figure A3. Tensile Strength of ASTM A203A Steel as a Function of Temperature Figure A4. Yield Strength of ASTM A203A Steel as a Function of Temperature
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40 Appendix 1 (Continued) Table A2. Thermal Properties of St eel as a Function of Temperature Fe 2.25 Ni (ASTM A203A), Normalized Temperature Thermal Conductivity Specific Heat Density Thermal Expansion F BTU / (secinF)BTU / (lbmF)lb / in3 x106 in / (inF) 340.00 0.0002825 0.0250 0.284 2.450 320.00 0.0002939 0.0360 0.284 2.760 300.00 0.0003103 0.0460 0.284 3.070 280.00 0.0003306 0.0535 0.284 3.330 260.00 0.0003508 0.0605 0.284 3.580 240.00 0.0003714 0.0670 0.284 3.830 220.00 0.0003917 0.0720 0.284 4.080 200.00 0.0004097 0.0770 0.284 4.300 180.00 0.0004244 0.0810 0.284 4.520 160.00 0.0004369 0.0850 0.284 4.720 140.00 0.0004494 0.0890 0.284 4.910 120.00 0.0004619 0.0920 0.284 5.090 100.00 0.0004744 0.0950 0.284 5.280 80.00 0.0004814 0.0980 0.284 5.430 60.00 0.0004883 0.1000 0.284 5.580 40.00 0.0004953 0.1020 0.284 5.720 20.00 0.0005022 0.1040 0.284 5.860 0.00 0.0005092 0.1055 0.284 6.000 20.00 0.0005125 0.1070 0.284 6.120 40.00 0.0005158 0.1080 0.284 6.240 60.00 0.0005194 0.1090 0.284 6.360 80.00 0.0005231 0.1100 0.284 6.470 0.0 0.5 1.0 1.5 2.0 35015050 FBTU / (hrinF) 0.00 0.02 0.04 0.06 0.08 0.10 0.12 35015050 FBTU / (lbmF) Figure A5. Thermal Conductivity of ASTM A203A Steel as a Function of Temperature Figure A6. Specific Heat of ASTM A203A Steel as a Function of Temperature
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41 Appendix 1 (Continued) 0.0 0.1 0.1 0.2 0.2 0.3 0.3 35015050 Flb / in3 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 35015050 FE06 in / (inF) The fracture toughness is also an important parameter in this study. No exact data was found for this quantity. The data wa s extracted from the following graph: Figure A9. Fracture Toughness an d Yield Strength as a Function of Temperature (Source: Greenberg, 1969) Figure A7. Density of ASTM A203A Steel as a Function of Temperature Figure A8. Coefficient of Thermal Expansion of AS TM A203A Steel as a Function of Temperature
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42 Appendix 1 (Continued) Discrete points were approximated by usi ng the lower fracture toughness curve. These points are listed in Table A3. Although this is the fract ure Toughness of ASTM E24 Steel, it is a representative number of all steels. Table A3. Fracture Toughness of ASTM E24 Steel as a Function of Temperature Temperature Fracture Toughness F 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
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43 Appendix 2: Convection Medium Property Data The heat transfer coefficients for convecti on into liquid nitrogen are not constants. The following graph was used to get these coefficients. Figure A10. Heat Flux Versus Temperature Difference for Liquid Nitrogen (Source: Barron, 1999) The curve is a plot of the amount of heat flux versus the temperature difference between the wall and the fluid. The convection coe fficient is found by taking points from this graph and dividing the heat flux at a point by the temperature difference at the wall. This yields a table of values which are presented below.
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44 Appendix 2 (Continued) Table A4. Convection Coefficient for Liqui d Nitrogen as a Function of Temperature Wall Temperature Coefficient Wall Temperature Coefficient F BTU / (in2sec) F BTU / (in2sec) 320 0.000579 240 5.88E05 318 0.000579 230 5.67E05 316 0.000849 220 5.59E05 314 0.001213 210 5.47E05 312 0.001586 200 5.31E05 310 0.002067 190 5.15E05 308 0.00256 180 5.00E05 306 0.003022 170 4.88E05 304 0.003426 160 4.78E05 302 0.003696 150 4.71E05 300 0.003776 140 4.65E05 298 0.003697 130 4.59E05 296 0.003549 120 4.53E05 294 0.003384 110 4.46E05 292 0.003195 100 4.38E05 290 0.002967 90 4.30E05 288 0.002704 80 4.23E05 286 0.002425 70 4.16E05 284 0.002136 60 4.10E05 282 0.001818 50 4.04E05 280 0.001448 40 3.99E05 278 0.001047 30 3.95E05 276 0.00067 20 3.92E05 274 0.000362 10 3.90E05 272 0.000161 0 3.88E05 270 9.46E05 10 3.87E05 268 8.20E05 20 3.86E05 266 7.72E05 30 3.86E05 264 7.38E05 40 3.86E05 262 7.11E05 50 3.86E05 260 6.93E05 60 3.86E05 250 6.28E05 70 3.86E05 80 3.86E05
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45 Appendix 2 (Continued) Convection Coefficient BTU / (in2sec) 0.0E+00 5.0E04 1.0E03 1.5E03 2.0E03 2.5E03 3.0E03 3.5E03 4.0E03 315265215165115 Wall Temperature F Figure A11. Convection Coefficient for Liquid Nitrogen as a Function of Wall Temperature The convection coefficients for the othe r media were calculated within the program at runtime. This is based on the fo llowing method. First property data must be obtained for air and alcohol.
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46 Appendix 2 (Continued) Table A5. Property Data for Air as a Function of Temperature Refrigerated air was assumed to be at 30 F. Temperature Volumetric Expansion Kinematic Viscosity Thermal Diffusivity Thermal Conductivity F 1/R in2/sec in2/sec BTU / (secftF) 238 4.56E03 4.82E11 6.49E11 1.61E07 148 3.23E03 9.24E11 1.28E10 2.21E07 58 2.51E03 1.48E10 2.08E10 2.76E07 32 2.04E03 2.06E10 2.90E10 3.24E07 68 1.91E03 2.34E10 3.32E10 3.44E07 104 1.78E03 2.63E10 3.70E10 3.62E07 140 1.67E03 2.93E10 4.14E10 3.81E07 176 1.57E03 3.25E10 4.59E10 4.00E07 212 1.49E03 3.57E10 5.08E10 4.20E07 248 1.42E03 3.91E10 5.60E10 4.39E07 284 1.35E03 4.27E10 6.15E10 4.59E07 320 1.29E03 4.63E10 6.67E10 4.79E07 356 1.23E03 5.00E10 7.24E10 4.98E07 392 1.17E03 5.37E10 7.83E10 5.16E07 482 1.06E03 6.38E10 9.35E10 5.63E07 572 9.72E04 7.42E10 1.09E09 5.22E07 662 8.94E04 8.53E10 1.26E09 6.49E07 752 8.28E04 9.69E10 1.42E09 6.90E07 The alcohol used in the dryice/alcohol mixture is isopropyl These properties were found using AspenPlus (AspenTech, 2004). Table A6. Property Data for Isopropyl Alcohol as a Function of Temperature The alcohol was assumed to be at 108 F Temperature Volumetric Expansion Kinematic Viscosity Thermal Diffusivity Thermal Conductivity F 1/R in2/sec in2/sec BTU / (secftF) 100 4.50E04 2.27E01 1.23E04 2.11E06 90 4.50E04 1.63E01 1.21E04 2.09E06 80 4.50E04 1.19E01 1.19E04 2.07E06 70 4.50E04 8.87E02 1.17E04 2.05E06 60 4.50E04 6.69E02 1.16E04 2.04E06 50 4.50E04 5.12E02 1.14E04 2.02E06 40 4.50E04 3.96E02 1.12E04 2.00E06 30 4.50E04 3.10E02 1.10E04 1.99E06 20 4.50E04 2.46E02 1.08E04 1.97E06 10 4.50E04 1.97E02 1.06E04 1.95E06 0 4.50E04 1.59E02 1.04E04 1.94E06
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47 Appendix 2 (Continued) From this data, and the specific wall temp erature and cylinder diameter, the Grashof, Prandtl, and Raleigh number were calculated using the following formulas. (Incropera, 1996) 2 3) ( Grk fluid wallD T T g (88) kPr (89) Pr Gr Ra (90) where, g is the gravitational constant, is the volumetric thermal expansion coefficient, wallT is the temperature of the wall, fluidT is the temperature of the fluid, D is the diameter of the surface from convection is taking place, k is the kinematic viscosity of the fluid, Gr is the Grashof number, P r is the Prandtl number, and Ra is the Raleigh number. From these quantities, the Nusselt number wa s calculated. A correlation for a vertical cylinder was not found. The correlation for a ve rtical plate functions similarly and thus the following Nusselt number was calculated from which the convection coefficient is estimated. 227 8 16 9 6 1Pr 492 0 1 387 0 825 0 Ra Nu (91) D k Nu h (92)
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48 Appendix 2 (Continued) where, Nu is the Nusselt number, h is the convection coefficient, and k is the thermal conductivity of the fluid. This process was done to approximate the conv ection coefficient for refrigerated air and dryice/alcohol. Convection Coefficient BTU / (in2sec) 0.0E+00 5.0E07 1.0E06 1.5E06 2.0E06 2.5E06 301010305070 Wall Temperature F Figure A12. Convection Coefficient for Air as a Function of Wall Temperature for a Diameter = 1 in Convection Coefficient BTU / (in2sec) 0.0E+00 5.0E06 1.0E05 1.5E05 2.0E05 2.5E05 3.0E05 3.5E05 4.0E05 108886848 Wall Temperature F Figure A13. Convection Coefficient for DryIce/Alcohol as a Function of Wall Temperature for a Diameter = 1 in
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49 Appendix 3: Program Flow Figure A14. Flow Chart of Computer Program Start Read cylinder dimensions, analysis parameters, property data Calculate the cubic spline coefficients for inter p olation o f the p ro p ert y data Assemble the temperature matrix, set boundary conditions and solve linear s y ste m Assemble the displacement matrix, set b oundar y conditions and solve linear s y ste m From displacements, calculate stresses, stress ratio and critical crack len g th Write the minimum critical crack and the minimum stress ratio found in cylinderwall Switch cooling media? Reached steady state? En d no yes Go to next time ste p
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50 Appendix 4: Verification Several tests were run to ensure the pr ogram written was correct for simpler cases. Although this does not prove that the more co mplex cases will be correct, it does suggest that the approximations were handled correctly. First the temperature portion was verified A problem was taken out of a text book for heat transfer where the diameter was very small and thus the te mperature distribution virtually constant. The temperature as a functi on of time is desired in this case. There are two cases involving placing the cy linder in water and also in air. Both cases were run. The problem comes from (Kreith, 1986). Be ing an example problem, the solution is worked out and plotted. Due to the fact that the problem i nvolves a solid cylinder, it was necessary to change the program. This is actually only involved changing one number, the one indicating the flux at the center. Belo w is the plot of the exact solution and the program. Temperature F 20 40 60 80 100 120 140 160 020406080100120 Exact Solution Programmed Solution Time s Figure A15. Comparison of Exact and Pr ogrammed Solution to the Water Problem
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51 Appendix 4 (Continued) Temperature F 20 40 60 80 100 120 140 160 020406080100120 Exact Solution Programmed Solution Time s Figure A16. Comparison of Exact and Pr ogrammed Solution to the Air Problem The maximum true error from Figs. A13 and A14 is 4%. This suggests that the thermal portion is approximating correctly. The stresses were also verified. Initially the elasticity part was written separated form the thermal part. A function was used to input a temperature matrix as if it were the steady state temperature distribution. This al lowed for checking in multiple ways. If the material properties are made constant, a nd a temperature distribution specified, the stresses can be written explicitly.
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52 Appendix 5: Convergence Test Two convergence tests were run to determ ine how many radial divisions and time divisions were necessary to obtai n good results. This test also confirms that the solution is converging. A node near the inner surface was chosen for study. The temperature at that node was recorded using three increasingly fine radial meshes. This is a good choice of node because the temperature gradient is hi gh there. This will be the greatest chance to study how the displacement changes as the number of radial divi sions increase. The following details the method for such a test and gives the results for this analysis. The temperature, NR, at a point (Logan, 1992) is given by: ) (N B A RN (93) where B and = constants, A = extrapolated result for infinite mesh density, and N = number of elements. Note that if is greater that one, as N becomes la rger (infinite) the temperature becomes equal to a value A. For the soluti on to converge, it is necessary that > 1. The solution will converge on the value of A. Three temperat ures were calculated using three different radial divisions. This generates a syst em of equations for which A, B, and are solved. 16 1895 6916B A R (94) 32 2526 7332B A R (95) 64 6831 7464B A R (96) Simultaneously solving these three equations for A, B and yields A=75.4605 in, B=408.0736, and =1.5060. Since >1, the results will converge. If 64 radial divisions are used, the answer obtained is 1% different from the value of A. This indicates that 64 radial divisions are sufficient for good results. This is valid for the cylinder dimensions used in this experiment. As stated in the result s section, in this analysis the outer diameter
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53 Appendix 5 (Continued) of the hub is to be varied. In other cases, 64 radial divisions may not be ample to ensure good results. Therefore a proportional amount of radial divisions were used so that the distance between radial divisi ons would remain the same. The same test was run for the number of time divisions. As in the previous experiment, the same node was chosen and st udied using 64 radial divisions, this time changing the time step between iterations. The following are those results. 2 6831 742B A R (97) 4 1777 744B A R (98) 8 9309 738B A R (99) Simultaneously solving these three equations for A, B and yields A=73.6955 in, B=2.02286, and =1.0345. Again, the solution is c onverging. Using 8 time divisions, the answer obtained is <1% different from the value of A. This indicates that the answer has indeed converged. Therefore 8 time divisi ons were used which corresponds to a time step of 0.5 seconds.
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54 Appendix 6: Tables of Results Table A7. Results for the Christa McAuliffe in Cooling Process 1 Outer Radius of Cylinder 2 Interference Critical Crack Length Location of Critical Crack Minimum Stress Ratio Location of Minimum Stress Ratio in in in in nondim in 10.800 4.74E03 0.173 10.800 4.779 1.000 11.736 4.74E03 0.182 11.736 3.965 1.000 12.672 4.74E03 0.186 12.672 3.464 1.000 13.608 4.74E03 0.187 13.608 3.123 1.000 14.544 4.74E03 0.186 14.544 2.872 1.000 15.480 4.74E03 0.182 15.480 2.678 1.000 16.416 4.74E03 0.177 16.416 2.521 1.000 17.352 4.74E03 0.172 17.352 2.390 1.000 18.288 4.74E03 0.165 18.288 2.277 1.000 19.224 4.74E03 0.159 19.224 2.178 1.000 10.800 8.62E03 0.101 10.800 3.511 1.000 11.736 8.62E03 0.113 11.736 2.759 1.000 12.672 8.62E03 0.122 12.672 2.333 1.000 13.608 8.62E03 0.130 13.608 2.072 1.000 14.544 8.62E03 0.134 14.544 1.896 1.000 15.480 8.62E03 0.137 15.480 1.769 1.000 16.416 8.62E03 0.138 16.416 1.674 1.000 17.352 8.62E03 0.137 17.352 1.599 1.000 18.288 8.62E03 0.135 18.288 1.539 1.000 19.224 8.62E03 0.133 19.224 1.487 1.000
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55 Appendix 6 (Continued) Table A8. Results for the Christa McAuliffe in Cooling Process 2 Outer Radius of Cylinder 2 Interference Critical Crack Length Location of Critical Crack Minimum Stress Ratio Location of Minimum Stress Ratio in in in in nondim in 10.8000 4.74E03 0.3879 10.8000 6.8616 1.0000 11.7360 4.74E03 0.4436 11.7360 5.3135 1.0000 12.6720 4.74E03 0.4905 12.6720 4.4956 1.0000 13.6080 4.74E03 0.5276 13.6080 3.9903 1.0000 14.5440 4.74E03 0.5523 14.5440 3.6467 1.0000 15.4800 4.74E03 0.5589 15.4800 3.3975 1.0000 16.4160 4.74E03 0.5597 16.4160 3.2085 1.0000 17.3520 4.74E03 0.5562 17.3520 3.0601 1.0000 18.2880 4.74E03 0.5494 18.2880 2.9407 1.0000 19.2240 4.74E03 0.5402 19.2240 2.8426 1.0000 10.8000 8.62E03 0.1800 10.8000 4.3065 1.0000 11.7360 8.62E03 0.2190 11.7360 3.2831 1.0000 12.6720 8.62E03 0.2571 12.6720 2.7607 1.0000 13.6080 8.62E03 0.2926 13.6080 2.4449 1.0000 14.5440 8.62E03 0.3245 14.5440 2.2337 1.0000 15.4800 8.62E03 0.3520 15.4800 2.0830 1.0000 16.4160 8.62E03 0.3746 16.4160 1.9702 1.0000 17.3520 8.62E03 0.3917 17.3520 1.8829 1.0000 18.2880 8.62E03 0.4000 18.2880 1.8133 1.0000 19.2240 8.62E03 0.4052 19.2240 1.7568 1.0000
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56 Appendix 6 (Continued) Table A9. Results for the Christa McAuliffe in Cooling Process 3 Outer Radius of Cylinder 2 Interference Critical Crack Length Location of Critical Crack Minimum Stress Ratio Location of Minimum Stress Ratio in in in in nondim in 10.8000 4.74E03 0.3880 10.8000 8.4047 1.0000 11.7360 4.74E03 0.4440 11.7360 6.4427 1.0000 12.6720 4.74E03 0.4912 12.6720 5.3498 1.0000 13.6080 4.74E03 0.5288 13.6080 4.7070 1.0000 14.5440 4.74E03 0.5568 14.5440 4.2855 1.0000 15.4800 4.74E03 0.5758 15.4800 3.9889 1.0000 16.4160 4.74E03 0.5870 16.4160 3.7696 1.0000 17.3520 4.74E03 0.5916 17.3520 3.6012 1.0000 18.2880 4.74E03 0.5908 18.2880 3.4682 1.0000 19.2240 4.74E03 0.5859 19.2240 3.3606 1.0000 10.8000 8.62E03 0.1801 10.8000 4.7867 1.0000 11.7360 8.62E03 0.2191 11.7360 3.5551 1.0000 12.6720 8.62E03 0.2573 12.6720 2.9577 1.0000 13.6080 8.62E03 0.2931 13.6080 2.6075 1.0000 14.5440 8.62E03 0.3254 14.5440 2.3786 1.0000 15.4800 8.62E03 0.3535 15.4800 2.2183 1.0000 16.4160 8.62E03 0.3769 16.4160 2.1003 1.0000 17.3520 8.62E03 0.3957 17.3520 2.0102 1.0000 18.2880 8.62E03 0.4101 18.2880 1.9395 1.0000 19.2240 8.62E03 0.4204 19.2240 1.8820 1.0000
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57 Appendix 6 (Continued) Table A10. Results for Hillsborough Ave. in Cooling Process 1 Outer Radius of Cylinder 2 Interference Critical Crack Length Location of Critical Crack Minimum Stress Ratio Location of Minimum Stress Ratio in in in in nondim in 10.068 4.63E03 0.178 10.068 5.100 1.125 10.941 4.63E03 0.190 10.941 4.193 1.125 11.813 4.63E03 0.196 11.813 3.642 1.125 12.686 4.63E03 0.199 12.686 3.271 1.125 13.558 4.63E03 0.199 13.558 3.004 1.125 14.431 4.63E03 0.197 14.431 2.801 1.125 15.303 4.63E03 0.192 15.303 2.639 1.125 16.176 4.63E03 0.187 16.176 2.505 1.125 17.048 4.63E03 0.181 17.048 2.393 1.125 17.921 4.63E03 0.174 17.921 2.295 1.125 10.068 8.42E03 0.101 10.068 3.653 1.125 10.941 8.42E03 0.114 10.941 2.856 1.125 11.813 8.42E03 0.126 11.813 2.393 1.125 12.686 8.42E03 0.134 12.686 2.114 1.125 13.558 8.42E03 0.140 13.558 1.928 1.125 14.431 8.42E03 0.144 14.431 1.795 1.125 15.303 8.42E03 0.146 15.303 1.696 1.125 16.176 8.42E03 0.146 16.176 1.619 1.125 17.048 8.42E03 0.146 17.048 1.558 1.125 17.921 8.42E03 0.144 17.921 1.507 1.125
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58 Appendix 6 (Continued) Table A11. Results for Hillsborough Ave. in Cooling Process 2 Outer Radius of Cylinder 2 Interference Critical Crack Length Location of Critical Crack Minimum Stress Ratio Location of Minimum Stress Ratio in in in in nondim in 10.0680 4.63E03 0.3846 10.0680 7.3378 1.1250 10.9406 4.63E03 0.4454 10.9406 5.5702 1.1250 11.8131 4.63E03 0.4983 11.8131 4.6679 1.1250 12.6857 4.63E03 0.5419 12.6857 4.1214 1.1250 13.5582 4.63E03 0.5757 13.5582 3.7549 1.1250 14.4308 4.63E03 0.5951 14.4308 3.4919 1.1250 15.3034 4.63E03 0.5984 15.3034 3.2937 1.1250 16.1759 4.63E03 0.5967 16.1759 3.1386 1.1250 17.0485 4.63E03 0.5911 17.0485 3.0140 1.1250 17.9210 4.63E03 0.5825 17.9210 2.9118 1.1250 10.0680 8.42E03 0.1739 10.0680 4.4071 1.1250 10.9406 8.42E03 0.2138 10.9406 3.3173 1.1250 11.8131 8.42E03 0.2537 11.8131 2.7731 1.1250 12.6857 8.42E03 0.2918 12.6857 2.4483 1.1250 13.5582 8.42E03 0.3269 13.5582 2.2331 1.1250 14.4308 8.42E03 0.3576 8.3900 2.0807 1.1250 15.3034 8.42E03 0.3800 8.3900 1.9673 1.1250 16.1759 8.42E03 0.4004 8.3900 1.8799 1.1250 17.0485 8.42E03 0.4190 8.3900 1.8105 1.1250 17.9210 8.42E03 0.4279 17.9210 1.7540 1.1250
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59 Appendix 6 (Continued) Table A12. Results for Hillsborough Ave. in Cooling Process 3 Outer Radius of Cylinder 2 Interference Critical Crack Length Location of Critical Crack Minimum Stress Ratio Location of Minimum Stress Ratio in in in in nondim in 10.0680 4.63E03 0.3847 10.0680 8.2885 1.1250 10.9406 4.63E03 0.4456 10.9406 6.1590 1.1250 11.8131 4.63E03 0.4987 11.8131 5.1261 1.1250 12.6857 4.63E03 0.5427 12.6857 4.5204 1.1250 13.5582 4.63E03 0.5770 13.5582 4.1246 1.1250 14.4308 4.63E03 0.6020 14.4308 3.8474 1.1250 15.3034 4.63E03 0.6185 15.3034 3.6433 1.1250 16.1759 4.63E03 0.6276 16.1759 3.4862 1.1250 17.0485 4.63E03 0.6304 17.0485 3.3579 1.1250 17.9210 4.63E03 0.6283 17.9210 3.2543 1.1250 10.0680 8.42E03 0.1739 10.0680 4.5627 1.1250 10.9406 8.42E03 0.2139 10.9406 3.3909 1.1250 11.8131 8.42E03 0.2538 11.8131 2.8224 1.1250 12.6857 8.42E03 0.2921 12.6857 2.4891 1.1250 13.5582 8.42E03 0.3275 13.5582 2.2713 1.1250 14.4308 8.42E03 0.3576 8.3900 2.1188 1.1250 15.3034 8.42E03 0.3800 8.3900 2.0066 1.1250 16.1759 8.42E03 0.4004 8.3900 1.9209 1.1250 17.0485 8.42E03 0.4190 8.3900 1.8536 1.1250 17.9210 8.42E03 0.4360 8.3900 1.7996 1.1250
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60 Appendix 6 (Continued) Table A13. Results for 17th St. Causeway in Cooling Process 1 Outer Radius of Cylinder 2 Interference Critical Crack Length Location of Critical Crack Minimum Stress Ratio Location of Minimum Stress Ratio in in in in nondim in 7.766 4.25E03 0.184 7.766 5.298 1.188 8.439 4.25E03 0.204 8.439 4.303 1.188 9.113 4.25E03 0.219 9.113 3.717 1.188 9.786 4.25E03 0.230 9.786 3.326 1.188 10.459 4.25E03 0.236 10.459 3.019 1.188 11.132 4.25E03 0.240 11.132 2.790 1.188 11.805 4.25E03 0.240 11.805 2.619 1.188 12.478 4.25E03 0.238 12.478 2.486 1.188 13.151 4.25E03 0.235 13.151 2.381 1.188 13.824 4.25E03 0.230 13.824 2.295 1.188 7.766 7.72E03 0.094 7.766 3.602 1.188 8.439 7.72E03 0.111 8.439 2.686 1.188 9.113 7.72E03 0.127 9.113 2.224 1.188 9.786 7.72E03 0.141 9.786 1.951 1.188 10.459 7.72E03 0.153 10.459 1.771 1.188 11.132 7.72E03 0.162 11.132 1.644 1.188 11.805 7.72E03 0.169 11.805 1.549 1.188 12.478 7.72E03 0.174 12.478 1.477 1.188 13.151 7.72E03 0.177 13.151 1.419 1.188 13.824 7.72E03 0.178 13.824 1.372 1.188
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61 Appendix 6 (Continued) Table A14. Results for 17th St. Causeway in Cooling Process 2 Outer Radius of Cylinder 2 Interference Critical Crack Length Location of Critical Crack Minimum Stress Ratio Location of Minimum Stress Ratio in in in in nondim in 7.7664 4.25E03 0.3491 7.7664 6.8990 1.1875 8.4395 4.25E03 0.4206 8.4395 5.1363 1.1875 9.1126 4.25E03 0.4890 9.1126 4.2811 1.1875 9.7857 4.25E03 0.5517 9.7857 3.7795 1.1875 10.4588 4.25E03 0.6070 10.4588 3.4516 1.1875 11.1318 4.25E03 0.6537 11.1318 3.2178 1.1875 11.8049 4.25E03 0.6916 11.8049 3.0329 1.1875 12.4780 4.25E03 0.7206 12.4780 2.8902 1.1875 13.1511 4.25E03 0.7414 13.1511 2.7769 1.1875 13.8242 4.25E03 0.7398 13.8242 2.6846 1.1875 7.7664 7.72E03 0.1403 6.4720 3.7989 1.1875 8.4395 7.72E03 0.1694 6.4720 2.8284 1.1875 9.1126 7.72E03 0.1978 6.4720 2.3575 1.1875 9.7857 7.72E03 0.2213 6.4720 2.0814 1.1875 10.4588 7.72E03 0.2408 6.4720 1.9009 1.1875 11.1318 7.72E03 0.2587 6.4720 1.7746 1.1875 11.8049 7.72E03 0.2751 6.4720 1.6816 1.1875 12.4780 7.72E03 0.2900 6.4720 1.6107 1.1875 13.1511 7.72E03 0.3037 6.4720 1.5550 1.1875 13.8242 7.72E03 0.3161 6.4720 1.5102 1.1875
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62 Appendix 6 (Continued) Table A15. Results for 17th St. Causeway in Cooling Process 3 Outer Radius of Cylinder 2 Interference Critical Crack Length Location of Critical Crack Minimum Stress Ratio Location of Minimum Stress Ratio in in in in nondim in 7.7664 4.25E03 0.3491 7.7664 6.9086 1.1875 8.4395 4.25E03 0.4206 8.4395 5.1422 1.1875 9.1126 4.25E03 0.4890 9.1126 4.2856 1.1875 9.7857 4.25E03 0.5518 9.7857 3.7834 1.1875 10.4588 4.25E03 0.6072 10.4588 3.4552 1.1875 11.1318 4.25E03 0.6541 11.1318 3.2254 1.1875 11.8049 4.25E03 0.6922 11.8049 3.0563 1.1875 12.4780 4.25E03 0.7216 12.4780 2.9273 1.1875 13.1511 4.25E03 0.7428 13.1511 2.8259 1.1875 13.8242 4.25E03 0.7567 13.8242 2.7446 1.1875 7.7664 7.72E03 0.1403 6.4720 3.8018 1.1875 8.4395 7.72E03 0.1694 6.4720 2.8302 1.1875 9.1126 7.72E03 0.1978 6.4720 2.3589 1.1875 9.7857 7.72E03 0.2213 6.4720 2.0826 1.1875 10.4588 7.72E03 0.2408 6.4720 1.9020 1.1875 11.1318 7.72E03 0.2587 6.4720 1.7756 1.1875 11.8049 7.72E03 0.2751 6.4720 1.6826 1.1875 12.4780 7.72E03 0.2900 6.4720 1.6116 1.1875 13.1511 7.72E03 0.3037 6.4720 1.5559 1.1875 13.8242 7.72E03 0.3161 6.4720 1.5112 1.1875
