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Sensitivity Analysis Of Design Parameters For Trunnion-Hub Assemblies Of Bascule Bridges Using Finite Element Methods by Jai P. Paul 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 Besterfield, Ph.D. Muhammad M. Rahman, Ph.D. Date of Approval: January 31, 2005 Keywords: Thermal Stresses, Design of Experiments, Non-Linear Material Proper ties, Interferences, ANSYS Copyright 2005, Jai P. Paul
DEDICATION To my mother, who has s upported me and has believed in me, and to my wife, Beena, who encouraged and motivated me at all times. They helped me, mentally and emotionally, to achieve my goals during my entire Masters program and especially during my thesis.
ACKNOWLEDGMENT It gives me immense pleasure and great pride to present my thesis report titled, Sensitivity Analysis of Design Paramete rs for Trunnion-Hub Assemblies of Bascule Bridges using Finite Element Methods. I express my sincere thanks and gratitude to Dr. Autar K. Kaw, whose guidance and direction helped me treme ndously to complete this work. It is my privilege that I had the chance to work with the Florida Prof essor of the Year, 2004. 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 for the financial support, and for the academic and computer resources he provided. I am also thankful for his patience and for his generosity. I wish to thank Dr. Glen Besterfield, for helping me perform the sensitivity analysis according to the design of experime nts standard and for his help with ANSYS. I also thank Dr. Muhammad M. Rahman, for his contributions in helping me understand the thermal processes and for being on my committee. I am deeply indebted to Dr. Niranjan Pa i, for his suggestions and his timely hints which helped me every time I faced problem s when working with ANSYS. I also thank the ANSYS users group who helped me w ith prompt responses to my questions. I want to thank my fellow graduate student, Nathan Collier, for helping me in analyzing the data obtained. My interactions with him have always served as a check of my work and helped me verify it. Above all, I want to thank God in Jesus Christ, whose presence in my life has always encouraged and comforted me. Because He lives, I can face tomorrow.
i TABLE OF CONTENTS LIST OF TABLES iv LIST OF FIGURES vi ABSTRACT viii CHAPTER ONE: INTRODUCTION 1.1 Bascule Bridges 1 1.2 Assembly Procedures 2 1.3 Overview 4 CHAPTER TWO: BACKGROUND 2.1 Where it All Started 6 2.2 Previous Work Done at USF 7 2.2.1 Literature Reviewed 7 2.2.2 Design Tools for THG Assemblies 9 2.2.3 Parametric Finite Element Analysis 11 2.2.4 Experimental Analysis 14 2.2.5 Conclusions 17 2.3 Objectives for Present Work 18 CHAPTER THREE: TECHNICAL DETAILS 3.1 Introduction 19 3.2 Geometrical Details 19 3.3 Analytical Details 21 3.3.1 Equilibrium Equations 22 3.3.2 Stress-Strain Equations 23 3.3.3 Strain-Displacement Equations 23 3.4 Boundary Conditions for AP1 24 3.4.1 Cooling of the Trunnion-Hub Assembly 27 3.5 Nonlinear Material Prop erties of the Metal 28 3.5.1 Youngs Modulus 29
ii 3.5.2 Coefficient of Thermal Expansion 29 3.5.3 Thermal Conductivity 30 3.5.4 Density 31 3.5.5 Specific Heat 32 3.6 Nonlinear Material Properties of Li quid Nitrogen 32 3.6.1 Coefficient of Convection of Liquid Nitrogen 33 3.6.2 Convection to Liquid Nitrogen at 0 F 33 CHAPTER FOUR: FINITE ELEMENT MODELING 4.1 Introduction 35 4.2 Coupled Field Analysis 35 4.2.1 Direct Coupled Field Analysis 36 4.2.2 Sequential Coupled Field Analysis (Indirect Coupled Field) 36 4.3 The Finite Element Model 36 4.4 Elements Used for Finite Element Modeling 37 4.4.1 ANSYS Element Library and Classification 37 4.4.2 Selection of Elements 38 4.4.3 Element Characteristics 40 4.5 Assumptions 4.5.1 Sequential Coupled Field Approach 43 4.5.2 Finite Element Method Assumptions 43 4.5.3 Material Properties 44 CHAPTER FIVE: DESIGN OF EXPERIMENTS 5.1 Introduction 45 5.2 Guidelines for Designing Experiments 45 5.2.1 Recognition of and Statement of the Problem 45 5.2.2 Choice of Factors, Levels and Range 46 5.2.3 Selection of Response Variables 46 5.2.4 Choice of Experimental Design 47 5.2.5 Performing the Experiment 47 5.2.6 Statistical Analysis of Data 47 5.2.7 Conclusions and Recommendations 48 5.3 Factorial Design 48 5.4 Calculations Involved in 2 3 Factorial Designs 49 CHAPTER SIX: SUMMARY OF RESULTS 6.1 Introduction 51 6.2 Fracture Toughness and Yield Strength 53
iii 6.3 Post-Processing to get Final Results 56 6.4 Results for 5% Variation Analyses 56 6.5 AASHTO Results 63 6.6 Explanation of Results 68 CHAPTER SEVEN: CONCLUSIONS AND RECOMMENDATIONS 7.1 Conclusions 69 7.2 Recommendations for Future Work 70 REFERENCES 71 APPENDIX A: PROGRAM FLOW AND USER INSTRUCTIONS A.1 Program Flow 74 A.2 Modeling the Experiment 77 APPENDIX B: VERIFICATIONS OF ANALYSES B.1 Introduction 79 B.2 Test 1 for Structural Analys is for Interference Stresses 79 B.2.1 Exact Solution 80 B.2.2 ANSYS Solution 82 B.2.3 Comparison of Actual Solu tion vs. ANSYS Solution 84 B.3 Test 2 for Structural Analys is for Interference Stresses 85 B.4 Test 1 for Thermal Analysis for Cooling in a Liquid Bath 85 B.4.1 Exact Solution 86 B.4.2 ANSYS Solution 87 B.4.3 Comparison of Actual Solu tion vs. ANSYS Solution 88 B.5 Test 2 for Thermal Analysis for Cooling in a Liquid Bath 89 B.6 Test 1 for Structural Analysis for Thermal Stresses 91 B.7 Test 2 for Structural Analysis for Thermal Stresses 93
iv LIST OF TABLES Table 2.1 Geometric Parameters 12 Table 2.2 Interference Values 12 Table 2.3 Critical Crack Length and Ma ximum Hoop Stress for Different Assembly Procedures and Different Bridges 13 Table 2.4 Nominal Dimensions of Full-Scale Trunnion and Hub 14 Table 2.5 Summary of Comparison of Results of AP1 and AP2 16 Table 2.6 Comparison of Results of the Two Assembly Procedures Obtained from Experimental and FEA Analyses 16 Table 5.1 Notations for Experiment Combinations 49 Table 6.1 Geometric Dimensions for 17 th Street Causeway Bridge 52 Table 6.2 Yield Strength as a Function of Temperature 55 Table 6.3 Fracture Toughness as a Function of Temperature 55 Table 6.4 Values of the Different Levels of the Factors for the 5% Variations 57 Table 6.5 Critical Crack Length Values for 5% Variations 58 Table 6.6 Stress Ratio Values for 5% Variations 59 Table 6.7 Notations and Values for 2 3 Factorial Design Data Analysis for CCL 61 Table 6.8 Percentage Contributions of the Factors for CCL 62 Table 6.9 Notations and Values for 2 3 Factorial Design Data Analysis for SR 62 Table 6.10 Percentage Contributions of the Factors for SR 63 Table 6.11 Values of the Different Levels of the Factors for the AASHTO Results 64 Table 6.12 Critical Crack Length Values for AASHTO Results 65 Table 6.13 Notations and Values for 2 3 Factorial Design for CCL (AASHTO) 65 Table 6.14 Percentage Contributions of the Factors for CCL (AASHTO) 66 Table 6.15 Stress Ratio Values for AASHTO Results 66
v Table 6.16 Notations and Values for 2 3 Factorial Design for SR (AASHTO) 67 Table 6.17 Percentage Contributions of the Factors for the Stress Ratio (AASHTO) 67 Table B.1 Coefficients for FN2 fit 80 Table B.2 Comparison of Temperatures from Maple and ANSYS for Specific Times 89
vi LIST OF FIGURES Figure 1.1 Bascule Bridge 1 Figure 1.2 Trunnion-Hub-Gird er (THG) Assembly 2 Figure 1.3 Two Different Assembly Procedures 3 Figure 2.1 Locations of Cracks on Hub 6 Figure 2.2 Introductory Screen for th e Interference Stresses Due to FN2 and FN3 Fits (Denninger 2000) 10 Figure 2.3 Positions of Gauges on Trunnion and Hub for AP1 15 Figure 2.4 Positions of the Gauges on Trunnion and Hub for AP2 15 Figure 3.1 Trunnion Dimensions 19 Figure 3.2 Hub Dimensions 20 Figure 3.3 Girder Dimensions 21 Figure 3.4a Trunnion Coordinates (side view) 24 Figure 3.4b Trunnion Coordinates (front view) 24 Figure 3.5a Hub Coordinates (side view) 25 Figure 3.5b Hub Coordinates (front view) 25 Figure 3.6a Girder Coordinates (side view) 26 Figure 3.6b Girder Coordinates (front view) 26 Figure 3.7 Youngs Modulus of Steel as a Function of Temperature 29 Figure 3.8 Coefficient of Thermal Expansion of Steel as a Function of Temperature 30 Figure 3.9 Thermal Conductivity of Steel as a Function of Temperature 31 Figure 3.10 Density of Steel as a Function of Temperature 31 Figure 3.11 Specific Heat of Steel as a Function of Temperature 32 Figure 3.12 Convective Heat Transfer Co efficient of Liquid Nitrogen as a Function of Temperature 33 Figure 3.13 Heat Flux versus Temperatur e Difference for Liquid Nitrogen 34
vii Figure 4.1 SOLID45--3D Structural Solid, and SOLID70--3D Thermal Solid 40 Figure 4.2 Trunnion-Hub Assembly with SOLID45 and Solid70 Elements 41 Figure 4.3 CONTA174 Overlaying the Trunnion Outer Diameter Surface 42 Figure 4.4 TARGE170 Overlaying the Hub Inner Diameter Surface 43 Figure 6.1 Trunnion dimensions 51 Figure 6.2 Hub dimensions 52 Figure 6.3 Fracture Toughness and Yield Strength of the Material 53 Figure 6.4 Critical Crack Length 54 Figure 6.5 Different Levels of Interference against CCL and SR Values 60 Figure 6.6 Different Levels of Hub-Outer Radius against CCL and SR Values 60 Figure 6.7 Different Levels of Trunnion Inner Radius against CCL and SR Values 60 Figure A.1 Flowchart for the ANSYS Program 75 Figure A.2 Flowchart for the Excel Program 76 Figure A.3 Flowchart for the 2 k Factorial Design Data Analysis 77 Figure B.1 Interference Stresses (Von-Mi ses) between 2 Cylinders (Isometric View) 82 Figure B.2 Interference Stresses (Von-M ises) between 2 Cylinders (Front View) 83 Figure B.3 Interference Stresses (Von-Mises) at the Interface 83 Figure B.4 Temperature Distribution of Copper Wire from Actual Solution 87 Figure B.5 Temperature Distribution of Copper Wire from ANSYS 88 Figure B.6 Comparison of Cooling Pro cesses with Constant and Varying Properties 90 Figure B.7 Temperature Distribution fo r Cooling of the Copper Cylinder 91 Figure B.8 Thermal Stresses during the Cooling of the Copper Cylinder 92 Figure B.9 Von-Mises Stresses in the Co mpound Cylinders after Interference 93 Figure B.10 Temperature Distri bution in Compound Cylinders 94 Figure B.11 Von-Mises Stresses in the Compound Cylinders after Cooling 95
viii SENSITIVITY ANALYSIS OF DESI GN PARAMETERS FOR TRUNNION-HUB ASSEMBLIES OF BASCULE BRIDGE S USING FINITE ELEMENT METHODS Jai P. Paul ABSTRACT Hundreds of thousands of dollars coul d be lost due to failures during the fabrication of Trunnion-Hub-Girder (THG) assemblies of bascule bridges. Two different procedures are currently utilized for the THG assembly. Crack formations in the hubs of various bridges during assembly led the Florida Department of Transportation (FDOT) to commission a project to investig ate why the assemblies failed. Consequently, a research contract was granted to the Mechanical Engineering department at USF in 1998 to conduct theoretical, numerical and experimental studies. It was found that the steady state stresses were well below the yield streng th of the material and could not have caused failure. A parametr ic finite element model was designed in ANSYS to analyze the transient stresses, temperatures and critical crack lengths in the THG assembly during the two assembly procedures. The critical poi nts and the critical stages in the assembly were identified based on the critical crack length. Furthermore, experiments with cryogenic strain gauge s and thermocouples were developed to determine the stresses and temperatures at critical points of the THG assembly during the two assembly procedures. One result revealed by the studies was that large tensile hoop stresses develop in the hub at the trunni on-hub interface in AP1 when the trunnion-hub assembly is cooled for insertion into the girder. These stresses occurred at low temperat ures, and resulted in low values of critical crack length. A suggestion to solve this was to study the effect of thickness of the hub and to understand its influence on critical stresses and crack lengths.
ix In addition, American Association of State Highway and Tran sportation Officials (AASHTO) standards call for a hub radial thickness of 0.4 times the inner diameter, while currently a thickness of 0.1 to 0.2 times the inner diameter is used. In this thesis, the geometrical dimensi ons are changed acco rding to design of experiments standards to find the sensitivity of these parameters on critical stresses and critical crack lengths during the assembly. Parameters ch anged are hub radial thickness to trunnion outer diameter ratio, trunnion outer di ameter to trunnion bore diameter ratio and variations in the inte rference. The radial th ickness of the hub was found to be the most influential parameter on critical st resses and critical crack lengths.
CHAPTER 1 INTRODUCTION 1.1 Bascule Bridges A bascule bridge is a type of movable bridge that can be opened or closed to facilitate the movement of water-borne traffic such as ships and yachts. Bascule is the French word for seesaw. It belongs to the first class of levers, where the fulcrum is located between the effort and the resistance. However, the bascule bridge belongs to the second or third class of levers depending on how the load is designated. The bascule bridge opens like a lever on a fulcrum (see Figure 1.1). The fulcrum that is fit into the girder of the bridge is made of a trunnion shaft attached to the leaf girder via a hub, as shown in Figure 1.2, and supported on bearings to permit rotation of the leaf. The trunnion, hub and girder when fitted together are referred to as a trunnion-hub-girder (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 bascule bridge, power is supplied to the THG assembly by means of a curved rack and pinion gear at the bottom of the girder. Leve r Fulcrum Figure 1.1 Bascule Bridge. 1
Trunnio n Hub Girde r Figure 1.2 Trunnion-Hub-Girder (THG) Assembly Assemblies of this type are generally constructed with interference fits between the trunnion and the hub, and between the hub and the girder. The interference fits allow the trunnion to form a rigid assembly with the leaf and permits the rotation of leaf through bearings. The two interference fits are supplemented by keys or dowel pins at the trunnion, and by structural bolts at the girder in some cases. Typical interference fits used in the THG assemblies for Florida bascule bridges are FN2 and FN3 fits (Shigley and Mishke, 1986). FN2 and FN3 fits are US Standard Fits. According to Shigley and Mishke (1986), FN2 designation is, Medium-drive 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 high-grade cast-iron external members. FN3 designation is, Heavy drive fits that are suitable for heavier steel parts or for shrink fits in medium sections. 1.2 Assembly Procedures The Trunnion-Hub-Girder assembly can be manufactured in two different ways; called Assembly Procedure 1 (AP1) and Assembly Procedure 2 (AP2), as shown in Figure 1.3. 2
Assembly Procedure 1 Assembly Procedure 2 Trunnion, Hub and Girder Trunnion, Hub and Girder Trunnion fitted into Hub Hub fitted into Girder Trunnion-Hub fitted into Girder Trunnion fitted into Hub-Girder Completed THG assembly Completed THG assembly Figure 1.3 Two Different Assembly Procedures 3
4 AP1 involves the following four steps: The trunnion is first shrunk by cooling in liquid nitrogen. This shrunk trunnion is then inserted into the hub and allowed to warm up to ambient temperature to develop an inte rference fit on the tr unnion-hub interface. The resulting trunnion-hub assembly is then shrunk by cooling in liquid nitrogen. The shrunk trunnion-hub assembly is then in serted into the gird er and allowed to warm up to ambient temperature to devel op an interference f it on the hub-girder interface. AP2 consists of the following four steps: The hub is first shrunk by c ooling in liquid nitrogen. This shrunk hub is then inserted into the girder and allowed to warm up to ambient temperature to develop an inte rference fit on the hub-girder interface. The trunnion is then shrunk by cooling in liquid nitrogen. The shrunk trunnion is then inserted into the hub-girder assembly and allowed to warm up to ambient temperature to deve lop an interference fit on the trunnionhub interface. During either of these assembly proce dures, the trunnion, hub and girder develop both structural stresses and th ermal stresses. The structural stresses arise due to interference fits betw een the trunnion-hub and the hub-gird er. The thermal stresses are a result of temperature gradients within the co mponent. These temperature gradients come into play when either the trunnion or the hub is immersed in liquid nitrogen or when a cold trunnion is inserted in to the hub, which is at r oom temperature. The term Transient Stress will be used to mean stresses dur ing the assembly procedure. The term Steady State Stress will be used to mean the stresses in the trunnion, hub and girder at the end of the assembly procedure. 1.3 Overview Chapter One, Introduction describes what a bascule bridge is, how it works, the construction and the assembly procedures invo lved in the manufacture of bascule bridges.
5 Chapter Two, Background, describes the history, prev ious work done at USF, which involves the literatures reviewed, FEA a nd experimental analyses and their results and conclusions along with recommendations s uggested. The objective of this thesis is also explained. Chapter Three, Technical Details describes the geometry of the THG assembly, the analytical backgr ound, and the non-linear material properties of the metal and liquid nitrogen. Chapter Four, Finite Element Modeling describes the modeling approach, the elements used, and assumptions used. Partic ular emphasis is given to the selection of elements and their proper orientations for the thermal and structural analyses. Chapter Five, Design of Experiments explains the guidelines for experiments, and simultaneously presents the systematic approach to this thesis in accordance with the guidelines established. It also describes the st atistical data analysis procedure used along with the calculations involved. Chapter Six, Results, presents the observations and the results obtained for the sensitivity of the trunnion inner diameter, hub outer diameter, and the interference at the trunnion-hub interface on the critical crack length and the critical stresses developed during the assembly. It also explains the results obtained. Chapter Seven, Conclusions and Recommendations presents the conclusions of this thesis based on the results obtained. Recommendations for future work are also suggested. Appendix A gives the flowchart of the programs used for this thesis. It also gives the procedures to be followed to start and run the different programs. Appendix B explains the various test analyses carried out to verify the different simulations. It uses various processes, whose results or behavior are already established, and simulates the same using ANS YS to obtain similar results.
CHAPTER 2 BACKGROUND 2.1 Where It All Started On May 3 rd 1995, during the Step 3 of AP1, described in chapter 1, a cracking sound was heard in the trunnion-hub assembly immediately after immersion in liquid nitrogen for the Christa McAuliffe Bridge in Brevard County, FL. When the trunnion-hub assembly was taken out of liquid nitrogen, the hub was found to be cracked near the inner radius of the hub (see Figure 2.1). In another case in February 1998, while inserting the trunnion into the hub during Step 2 of AP1, the trunnion began to stick to the hub in the Venetian Causeway Bascule Bridge. In this case, the trunnion had been cooled down in a dry ice/alcohol medium and the resulting contraction in the trunnion was probably not sufficient to slide into the hub. Figure 2.1 Locations of Cracks on Hub 1 6 1 Figure 2.1 is reprinted from an independent consultants report.
7 These failures during the assembly procedure became a concern for the Florida Department of Transportation and they wa nted to find the reas on. Why were these failures taking place and only on a few of the many THG assemblies carried out in Florida? Why were they not happening on the same THG assemblies again? How can we avoid losses of hundreds of thousands of dollars in material, labor and delay in replacing failed assemblies? Preliminary investigations done by independent consulting firms and the assembly manufacturers gave various r easons for the possible failure including high cooling rate, use of liquid nitrogen for cooling, residual stresses in the castings and the assembly procedure itself. FDOT officials wanted to carry out a complete numerical and experimental study to find out the reason for these failures, how they could be avoided in the future and develop clear specifica tions for the assembly procedure. 2.2 Previous Work Done at USF In 1998, the FDOT gave a two-year gran t (www.eng.usf.edu/~besterfi/bascule/) to the USF Mechanical Engineering Department to investigate the problem. 2.2.1 Literature Reviewed The study (www.eng.usf.edu/~besterfi/bascule/) primarily focused on analyzing transient stresses and failures caused due to them. This broad scope encompassed topics such as temperature-dependent material properties, thermoelastic contact, thermal shock and fracture toughness. Pourmohamadian and Sabbaghian (1985) modeled the transi ent stresses with temperature dependent material properties under an axisymmetric load in a solid cylinder. However, their model did not incorporate non-symmetric loading, complex geometries and thermoelastic contact, all of which are present in the THG assembly. The trunnion-hub interface and the hub-gird er interface are in thermoelastic contact. Attempts were made to model th ermoelastic contact between two cylinders by Noda (1985). However, the models were onl y applicable for cylinders and not for nonstandard geometries. In add ition, the issue of temperaturedependent material properties was not addressed in this study.
8 Transient stresses in a cylinder under nonaxisymmetric temper ature distribution were studied by Takeuti and Noda (1980). Du e to the complex geometry, temperaturedependent material properties of the THG assembly and non-axisymmetric temperature distribution, this study was relevant to the research efforts. However, the issues of thermoelastic contact and complex geometries were not addressed in this study. Noda also modeled a transient thermoelastic cont act problem with a position dependent heat transfer coefficient (1987) and transient ther moelastic stresses in a short length cylinder (1985). These efforts, although useful to und erstanding the thermoelastic modeling, did not address the issues of temperature de pendent material properties and complex geometries. The following studies aided in understanding the role of fracture toughness in the study. Thomas, et al. (1985) found the thermal stresses due to the sudden cooling of cylinder after heating due to convection. The results indicated the magnitude of stresses attained during the cooling phase increase s with increasing duration of heating. Consequently, the duration of application of the convective load can be a factor influencing the maximum stresse s attained in the assembly. Parts of the THG assembly are subjected to thermal shock when they are cooled down before shrink fitting. Oliveira and Wu (1987) determined the fracture toughness for hollow cylinders subjected to stress gradient s arising due to thermal shock. The results covered a wide range of cylinder geometries. It is clear that the drawb ack of all previous studies of transient thermal stresses is their inability to deal with non-standard geom etries. Further, previ ous research efforts address some of the issues (that is, te mperature dependent material properties, thermoelastic contact, non-axis ymmetric loading, thermal shock) but never all of them. At the outset, it became apparent that isolating and pinpointing the causes of failure intuitively is difficult for three reas ons. First, it was obser ved that cracks were formed in some bridge assemblies but not in others. Secondly, the cracks occurred in different parts of the hub for different bridges and at different loading times. Finally, the problem involves the interplay of several issues, that is,
9 Complex geometries, such as, gussets on the hub, make it a 3-D elasticity and heat transfer problem. Thermal-structural interaction, due to the cooling and warming of the THG components and the shrink fitting of these components, results in both thermal and mechanical stresses. In addition, conduction takes place along contact surfaces. Temperature-dependent material propertie s, such as, coefficient of thermal expansion, specific heat, thermal conduc tivity, yield strength and fracture toughness, can be highly nonlinear functions of temperature. Hence, an intuitive analysis is not merely di fficult but intractable. To research more into the problem, three studies were conducted. 2.2.2 Design Tools for THG Assemblies The first study for the grant was conduc ted by Denninger (2000) to find the steady state stresses in the THG assembly by developing several design tools. Due to the shrink fitting done to the system and based on the type of standard fits at the interfaces of the trunnion-hub and hub-gi rder, interferences ar e created at the two interfaces. These interf erences cause pressures at th e interfaces and, correspondingly, develop hoop (also called circumferential a nd tangential), radial, and Von-Mises stresses in the THG assembly. Tool 3, the relevant tool for this thesis, requires the user to specify the industry standard interference fit ( FN2 or FN3 ) at each of the tw o interfaces. The critical stresses as well as the radial displacements in each member (trunnion, hub, and girder) are calculated using the given inform ation (see Figure 2.2). This program allows the user to check what the approximate stead y state stresses would be after assembly. In this program, the user can see if the hoop stresses (both compressive and tensile, if applicable) are more than the yield stresses, which ma y cause hoop cracks in that respective cylinder. The VonMises stress is also given to show how these stresses directly compare with the yield strength of the material.
Figure 2.2 Introductory Screen for the Interference Stresses Due to FN2 and FN3 Fits (Denninger 2000) This study showed that the steady state stresses are well below the ultimate tensile strength and yield strength of the materials used in the assembly. Hence, these stresses could not have caused the failure. The first study concluded that the transient stresses needed to be investigated since they could be more than the allowable stresses. The stresses during the assembly process come from two sources thermal stresses due to temperature gradients, and mechanical stresses due to interference at the trunnion-hub and hub-girder interfaces. Are these transient stresses more than the allowable stresses? Since fracture toughness decreases with a decrease in temperature, do these transient stresses make the assembly prone to fracture? These are some of the questions to be answered. 10
11 2.2.3 Parametric Finite Element Analysis The second study for the grant was c onducted by Ratnam (2000). A parametric finite element model was designed in ANSYS called the Trunnion-Hub-Girder Testing Model (THGTM) to analyze the stresses, temper atures and critical crack lengths in the THG assembly during the two assembly procedures. The finite element approach was most suitable because it could handle the interplay of complex geometries, coupled thermal and structural fields, and temperature dependent properties. This study used critical crack length and hoop stress, for comparing the two assembly procedures. The relevant theory for crack formation in the Trunnion-HubGirder (THG) was formulated based on several observations. First, the steady state stresses after assembly were well below the yield point and could not have caused failure. Second, experimental observati ons indicate the presence of sm all cracks in the assembly. Third, the brittle nature of the material (ASTM A203-A) ruled out failure due to Von Mises stresses. Finally, cracks were formed during the immersion of the trunnion-hub assembly in liquid nitrogen. This observation is important as fracture toughness decreases with a decrease in temperature while yiel d strength increases with a decrease in temperature The results obtained are important from two perspectives, one of which is explicitly presented in the results and the ot her, is implicitly suggested. The explicit result is the comparison between the two assemb ly procedures. Implicitly presented in the results is a comparison of different bri dges explaining why some THG assemblies are more prone to cracking than others. The geometric parameters for the thr ee bridges, namely, Christa McAuliffe Bridge, Hillsborough Avenue Bridge and 17 th Street Causeway Bridge, are presented in Table 2.1. Interference values for FN2 fits were obtained from the Bascule Bridge Design Tools (Denninger, 2000). In this study the wo rst case, that is, maximum interference between the trunnion and hub, and minimum inte rference between the hub and the girder, is analyzed. These values of interference will cause the largest tensile hoop stress in the hub. The interference values, based on FN2 fits, used in this an alysis are presented in Table 2.2. These parameters are shown in Figures 3.1 through 3.3.
Bridge Geometric Parameters Christa McAuliffe Hillsborough Avenue 17 th Street Causeway height of the girder flange(in) 1.5 1 0.75 height of the girder web(in) 82 90 60 Extension of the trunnion on the gusset side (in) 18.5 20 6 distance to hub flange(in) 4.25 8.5 4.25 total length of the girder(in) 82 90 60 total length of the hub(in) 16 22 11 total length of the trunnion, (in) 53.5 62 23 outer radius of the hub (minus flange) (in) 16 15.39 8.88 outer radius of the hub flange(in) 27 24.5 13.1825 inner radius of the trunnion(in) 1 1.125 1.1875 outer radius of the trunnion (inner radius of the hub) (in) 9 8.39 6.472 gusset thickness(in) 1.5 1.5 1.25 backing ring width(in) 1.75 1.75 0.78125 width of the girder flange(in) 17 14 1.25 width of the girder (web) (in) 1.5 1 0.75 width of hub flange(in) 1.75 1.75 1.25 Table 2.1 Geometric Parameters Bridge Diametrical Interference Christa McAuliffe Hillsborough Avenue 17 th Street Causeway Trunnion-Hub (in) 0.008616 0.008572 0.007720 Hub-Girder (in) 0.005746 0.005672 0.004272 Table 2.2 Interference Values 12
13 The THGTM is used to analyze the stresse s and critical crack length at possible locations of failure. The point with the greatest probability of failure was chosen and the critical crack lengths, hoop stress and temp erature against time for that point were plotted. A comparison of the hi ghest hoop stress and critical cr ack length is presented in Table 2.3 for all of the bridges. AP1 AP2 Bridge Critical Crack Length (in) Maximum Hoop Stress (ksi) Critical Crack Length (in) Maximum Hoop Stress (ksi) Christa McAuliffe 0.2101 28.750 0.2672 33.424 Hillsborough Avenue 0.2651 29.129 0.2528 32.576 17 th Street Causeway 0.6420 15.515 1.0550 17.124 Table 2.3 Critical Crack Length and Maximum Hoop Stress for Different Assembly Procedures and Different Bridges Examinations of the results revealed significant differences in the behavior of each bridge. In some bridges, a lower cri tical crack length was found to occur during AP1 (that is, Christa McAuliffe and 17 th Street Causeway) while in others (that is, Hillsborough Avenue) the opposite was true, however, only slightly. In addition, a slightly lower value of cr itical crack length during AP1 versus AP2 of Christa McAuliffe Bridge was observed. A phenomenon called crack arrest that prevents these cracks from growing catastrophically was studied to explai n how in some cases crack formation could be arrested in spite of low values of critical crack length during the assembly process. It was concluded that crack arrest is more likely to occur during AP2 than during AP1 because the possibility of crack arrest is greater when thermal stresses alone are present, as they are transient and change rapidly, as compared to procedures when a combination of both thermal and interfer ence stresses are present. The maximum hoop stress was found to be le ss than the yield strength in all the bridge assemblies indicating that they will not fail due to large stresses.
2.2.4 Experimental Analysis The earlier two studies (Denninger (2000) and Ratnam (2000)) had provided theoretical estimates of steady state and transient stresses. Also, the latter study had presented a comparison of the stresses in the two assembly procedures. The theoretical values of stresses, from these two studies needed to be validated against experimental values of stresses obtained from full-scale models. This formed the basis of the third study, Nichani (2001), which was to experimentally determine transient and steady state stresses and temperatures during both assembly procedures. Trunnion-hub-girders were instrumented with cryogenic strain gauges and thermocouples to determine stresses and temperatures at critical points. These sensors monitor strains and temperatures during all steps (cool down in liquid nitrogen and warm up in ambient air) as explained in Chapter 1. Two identical sets of trunnion, hub and girder were assembled, one using assembly procedure 1 (AP1) and the other using assembly procedure 2 (AP2). The stresses developed during these two procedures were compared against each other. The aim of these studies was to determine which of the assembly procedures was safer in terms of lower stresses and/or larger critical crack lengths. Nominal dimensions of the trunnion, hub and girder are shown in Table 2.4. Component Inner Diameter (in) Outer Diameter (in) Length or Thickness 2 (in) Interference (in) Trunnion 2.375 12.944 23 Hub 12.944 17.760 11 Girder 17.760 60.00 3 0.75 0.0077 0.0047 Table 2.4 Nominal Dimensions of Full-Scale Trunnion and Hub 2 The trunnion and hub are expressed in terms of length and the girder is expressed in terms of thickness. 14 3 The girder was approximated by a flat plate (60 60 0.75) with a hole of diameter 17.76.
The details of the positions of the gauges on the trunnion and the hub are shown in Figure 2.3 and in Figure 2.4 for AP1 and for AP2, respectively. Figure 2.3 Positions of Gauges on Trunnion and Hub for AP1 Figure 2.4 Positions of the Gauges on Trunnion and Hub for AP2 The thermocouples were mounted about half an inch from each strain gauge. Therefore, each mark in Figure 2.3 represents a set of one strain gauge and one thermocouple. One strain gauge and one thermocouple were placed on the diameter of the hole in the girder. This gauge would find the stress in the girder at the hub-girder interface. Table 2.5 summarizes the comparisons of the results of AP1 and AP2 obtained from the experimental analysis based on all three criterions, hoop stress, CCL and Von Mises stress. The critical crack length (CCL) and factor of safety (FOS) in Table 2.6 are based upon the fracture toughness and yield strength. The maximum hoop and Von15
16 Mises stresses are calculated by finding these stresses at the strain gauge locations throughout the assembly procedure. The factor of safety is calcu lated from finding the minimum of the ratio between the yield stre ngth and hoop stress at the strain gauge locations. Procedure Hoop Stress (ksi) Critical Crack Length ( CCL ) (in) Von Mises Stress (ksi) Factor of safety (FOS) AP1 25.7 0.3737 49.2 2.95 AP2 19.5 0.7610 30.9 3.29 Table 2.5 Summary of Comparison of Results of AP1 and AP2 Assembly Procedure CCL (in) Yield Strength (ksi) Hoop Stress (ksi) Temp ( 0 F) Time AP1 0.3737 96 25.7 -278 8 th minute into trunnion-hub cool down (step 3 of AP1) Exptl Analysis AP2 0.7610 65 19.5 -171 3 rd minute into hub cool down (step 1 of AP2) AP1 0.2037 53 37.0 -92 3 rd minute into trunnion-hub cool down (step 3 of AP1) FEA Analysis AP2 0.6196 53 21.5 -92 1 st minute of trunnion warm up into hub (step 3 of AP2) Table 2.6 Comparison of Results of the Tw o Assembly Procedu res Obtained from Experimental and FEA Analyses Table 2.6 gives the comparison of results obt ained from the experimental analysis and the FEA analysis. It can be seen that the results are similar and largely agree.
17 2.2.5 Conclusions The hypothesis at the begi nning of the study that AP2 resolves the problems associated with AP1 was validated largely by the results obtained from the experimental and the FEA analyses. It was found that large tensile hoop stresses develop in the hub at the trunnion-hub interface in AP1 when the trunnion-hub assembly is cooled for insertion into the girder. These stresses occur at low temperatures, and result in low values of critical crack length. Peak stresses during AP2 occur when the hub is cooled for insertion into the girder. Note that the critical crack le ngth allowed under AP1 is less than half that could be allowed under AP2. In other words, the critical crack length for AP2 could be more than double that could be allowed under AP1. The conclusions of this report were that for the given full-scale geometry and interference values, AP2 was safer than AP1 in terms of lower hoop stresses, lower VonMises stresses and larger critical crack lengt hs. However, since each bridge is different, there can be possible situations where AP1 may turn out to be a better process. One common problem associated with both assembly processes studied was thermal shock. In AP2, the sharp thermal gradient sometimes led to very low values of critical crack length ( CCL ). In AP1, a combination of high thermal and interference stresses results in possibility of crack formation. A lower th ermal gradient can impr ove both the assembly processes. Based on the results, the followi ng were some recommendations: Consider heating the outer component as an alternative to cooling the inner component. Consider staged cooling wherein the trunn ion or hub is first cooled from room temperature to 0 0 F, then dry-ice/alcohol is used to cool it down further to -109 0 F, before being cooled to -321 0 F (liquid nitrogen). Study the effect of warming one component while cooling the other component in a medium other than liquid nitr ogen, such as dry ice/alcohol. Studying the effect of thickness of hub on the hoop stress developed. Conduct sensitivity analysis of geometry of th e bridge to understand their influence on stress distribution.
18 2.3 Objectives for Present Work In an effort to continue with the studies on the THG assemblies, this thesis focuses on one of the results revealed by the studies that large tensile hoop stresses develop in the hub at the trunnion-hub interface in AP1 when the trunnion-hub assembly is cooled for insertion into the girder. As a suggestion to solve this, it was proposed to develop a sensitivity study of the geometrical parameters of the THG assembly. These mainly include developing specifications for the hub radial thickness cu rrently 0.1 to 0.2 times the inner diameter is used in Florida, while American Association of State Highway and Transportation Officials (AASHTO) sta ndards call for a hub thickne ss of 0.4 times the inner diameter, the inner diameter of the trunnion the trunnions are presently made hollow and the inner diameter is made to be about 1/5 th the outer diameter. the variations in the interference fits. It would be imperative to fi nd the effect of these three parameters as they have a significant effect on the transient stresses and transient fracture resistance of the assembly.
CHAPTER 3 TECHNICAL DETAILS 3.1 Introduction This chapter deals with the geometrical details of the trunnion, the hub and the girder. The thermal and structural boundary conditions on the trunnion-hub-girder assembly during the assembly procedure AP1 are discussed. The material properties of the metal and liquid nitrogen are also discussed. 3.2 Geometric Details The geometric specifications of the trunnion, hub and girder are shown in Figures 3.1, 3.2 and 3.3, respectively. lt lh l contact region with hub on outside of trunnion rto rti Figure 3.1 Trunnion Dimensions The dimensional parameters of the trunnion are 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), 19
lh = total length of the hub, rti = inner radius of the trunnion, and rto = outer radius of the trunnion (inner radius of the hub) rhg rho lf whf wgw wbrr ex Hub-girder contact Hub-backing ring contact a rea lh Figure 3.2 Hub Dimensions. The dimensional parameters of the hub are rhg = outer radius of the hub (minus flange), rho = outer radius of the hub flange, wbr = backing ring width, wgw = width of the girder (web), whf = width of hub flange, lf = distance to hub flange, lh = total length of the hub, tg = gusset thickness, and ex = distance from the end of the backing ring to the end of the hub. 20
wgw hgw rhg wgf lg hgf Figure 3.3 Girder Dimensions. The dimensional parameters of the girder are wgf = width of the girder flange, hgw = height of the girder web, lg = width of the girder, hgf = height of the girder flange, and wgw = width of the girder web. 3.3 Analytical Details The equations of equilibrium, the strain-displacement equations and the stress-strain equations for the trunnion-hub-girder assembly are discussed in this section. To develop these equations, the following symbols are used, where i = 1, 2 and 3 represents the trunnion, hub, and girder, respectively. ir = radial stress i = hoop stress iz = axial stress ir = shear stress in r plane 21
ir = radial strain i = hoop strain iz = axial strain iru = radial displacement iu = hoop displacement izu = axial displacement ir = shear strain in the r plane iz = shear strain in the z plane izr = shear strain in the z r plane Ti = temperature dependent Poissons ratio TKi = temperature dependent thermal conductivity TGi = temperature dependent shear modulus Thc = temperature dependent heat transfer coefficient of cooling medium Thw = temperature dependent heat transfer coefficient of warming medium 3.3.1 Equilibrium Equations The equations of equilibrium are given by: 01zrrrirziiririr (3.1) 021 rzrririziir (3.2) 01 rrrzizrizizriz (3.3) 22
3.3.2 Stress-Strain Equations The stress-strain equations are given by: TTiziiiriirodTTTTE)()()(1 (3.4) TTiziriiiidTTTTE0)()()(1 (3.5) TTiriiiziizdTTTTE0)()()(1 (3.6) TGiirir (3.7) TGiizrizr (3.8) TGiiziz (3.9) 3.3.3 Strain-Displacement Equations The strain-displacement equations are given by: ruirir (3.10) rururirii1 (3.11) zuiziz (3.12) ruruuriiirr1 (3.13) ruzuizirrz (3.14) zuuriizz1 (3.15) 23
3.4 Boundary Conditions for the Trunnion-Hub Assembly The boundary conditions for the THG assembly during each step of AP1 are discussed in this section. To study the boundary conditions, the end co-ordinates are established as shown in Figures 3.4, 3.5 and 3.6. Because of the complexity of the geometry, the boundary conditions given are only at the surfaces of contact (before and after contact) and those imposed for limiting rigid body motions in the finite element analysis. Areas for which the boundary conditions are not specified are stress free. 1el 2el 2sl 1sl z trunnion-hub contac t surface Figure 3.4a Trunnion Coordinates (side view) 1ir 1or Figure 3.4b Trunnion Coordinates (front view). 24
2el 2sl Hub-backing ring contact Hubg irder contact z from end of trunn ion Figure 3.5a Hub Coordinates (side view) 2or 2ir Figure 3.5b Hub Coordinates (front view) 25
y 3el 3 s l 3oby 3iby 3ity z from end of trunnion 3 rz Figure 3.6a Girder Coordinates (side view) 3rx 3lx 2or x Figure 3.6b Girder Coordinates (front view) 26
3.4.1 Cooling the Trunnion-Hub Assembly 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 the whole assembly. The most critical part of the assembly, from previous studies, was identified as the trunnion-hub 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 trunnion-hub contact, the thermal conditions during cooling of the trunnion-hub and the structural conditions immediately after the cooling, and before sliding it into the girder. The trunnion-hub assembly is immersed in a cooling medium at temperature until it approaches steady state at time. cT ct The thermal boundary conditions at the inside radius of the trunnion, are: 1irr ))(()(1ccTTTh r TTK ,, 1irr 11eslzl ctt 0 (3.16) At the outer radius of the trunnion,, there are non-contact and contact surfaces. 1orr At the non-contact surface, ))(()(1ccTTTh r TTK ,,,(3.17) 1irr 21sslzl 12eelzl ctt0 At the contact surface, r TTK r TTK )()(21 ,, 1orr 22eslzl ctt 0 (3.18) At the outer radius of the hub, 2orr ccTTTh r TTK2 ,, 2orr 22eslzl ctt 0 (3.19) The structural boundary conditions on the inside radius of the trunnion, are 1irr ;0),,,(11tzrui 20 ,, (3.20) 11eslzl ctt0 0,,,11tzrir 20 ,, (3.21) 11eslzl ctt0 27
At the outer radius of the trunnion, where there is no contact 1orr ;0),,,(11tzror 20 ,,, (3.22) 21sslzl 12eelzl ctt0 0,,,11tzror 20 ,, (3.23) 12eelzl 12eelzl ctt0 At the surface in contact at the trunnion outer radius, 1orr tzrtzriror,,,,,,2211 20 ,, (3.24) 22eslzl ctt0 tzrtzriror,,,,,,2211 20 ,, (3.25) 22eslzl ctt0 tzrutzruiror,,,,,,2211 20 ,, (3.26) 22eslzl ctt0 tzrutzruio,,,,,,2211 20 ,, (3.27) 22eslzl ctt0 At the outer radius of the hub, 2orr 0,,,22tzror 20 ,, (3.28) 22eslzl ctt0 0,,,22tzror 20 ,, (3.29) 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 (3.30) ctt0 0,,,22tlrsrz 22oirrr 20 (3.31) ctt0 0,,,22tlrsz 22oirrr 20 (3.32) ctt0 3.5 Nonlinear Material Properties of Metal The nonlinear material properties for a typical steel, Fe-2.25 Ni (ASTM A203-A) are plotted in the next several pages. Though nonlinear material properties in general are explored, particular emphasis is given to properties at low temperatures. 28
3.5.1 Youngs Modulus The elastic modulus of all metals increases monotonically with increase in temperature. The elastic modulus can be fitted into a semi-empirical relationship: TE 10TTTeeSEE (3.33) where = elastic constant at absolute zero, 0E = constant, and S = Einstein characteristic temperature. eT The Youngs modulus remains stable with change in temperature, i.e., the variation is not very large (see Figure 3.7), and hence is assumed to remain constant throughout this analysis. 05101520253035-400-300-200-1000100Temperature (oF)Young's Modulus(Mpsi) Figure 3.7 Youngs Modulus of Steel as a Function of Temperature 3.5.2 Coefficient of Thermal Expansion The coefficient of thermal expansion at different temperatures is determined principally by thermodynamic relationships with refinements accounting for lattice 29
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 0 F to 80 0 F as shown in the Figure 3.8. 01234567-400-300-200-1000100Temperature (oF)Coefficient of Thermal Expansion(10-6 in/in/oF) Figure 3.8 Coefficient of Thermal Expansion of Steel as a Function of Temperature 3.5.3 Thermal Conductivity The coefficient of thermal conductivity (see Figure 3.9) increases with an increase in temperature by a factor of two from 0 F to 80 0 F. 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). 30
0.00.40.81.21.62.0-400-300-200-1000100Temperature (oF)Thermal Conductivity(BTU/hr/in/oF) Figure 3.9 Thermal Conductivity of Steel as a Function of Temperature 3.5.4 Density For the range of temperatures of interest to our study the density remains nearly constant, as shown in Figure 3.10. 0.000.050.100.188.8.131.52-400-300-200-1000100Temperature (oF)Density(lb/in3) Figure 3.10 Density of Steel as a Function of Temperature 31
3.5.5 Specific Heat Lattice vibrations and electronic effects affect the specific heat of a material. The contribution of two effects can be shown by TTC3 (3.34) where, = volume coefficient of thermal expansion, 3T = lattice contribution, = normal electronic specific heat, and T = electronic contribution. Note that specific heat decreases by a factor of five over the temperature range in question, as shown in Figure 3.11. 0.000.020.040.060.080.100.12-400-300-200-1000100Temperature (oF)Specific Heat(BTU/lb/oF) Figure 3.11 Specific Heat of Steel as a Function of Temperature 3.6 Nonlinear Material Properties of Liquid Nitrogen The temperature dependent convective heat transfer coefficients for liquid nitrogen is plotted next. 32
33 3.6.1 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 3.12 (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. 0.480.500.520.540.560.580.60-400-300-200-1000100Wall Temperature (oF)Convective Heat Transfer Coefficient(BTU/hr/oF/in2) 17.37 0.069 Figure 3.12 Convective Heat Transfer Coefficient of Liquid Nitrogen as a Function of Temperature. 3.6.2 Convection to Liquid Nitrogen at 0 F The phenomenon of convection to liquid nitrogen is quite complex and involves multi-phase heat transfer. Whenever an object at ambient temperature (that is, 80 0 F) comes into contact with liquid nitrogen, film boiling occurs until the temperature of the object reaches approximately 0 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 0 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 nucleation sites on the cooling surface. When nucleate boiling starts the object cools very rapidly. 1000100001000001101001000Change in Temperature, Twall Tsaturation (oF)Heat Flux, q/A (BTU/hr-ft2) Transition Boiling Film Boiling N ucleate Boiling Maximum heat Minimum heat flux Figure 3.13 Heat Flux versus Temperature Difference for Liquid Nitrogen (Barron 1999) 34
35 CHAPTER 4 FINITE ELEMENT MODELING 4.1 Introduction This chapter explains the modeling a pproach used in ANSYS to model the trunnion-hub assembly including the loads and the different processes. An understanding of this is necessary to comprehend and apprec iate the results obtaine d. The analysis used is called the Sequential Coupled Field analysis which is one of the types of Coupled Field analysis (ANSYS Coupled Field Analysis Guide, Release 7.0). The inherent assumptions and some problems associated with the Fini te Element Modeling of thermo-structural analysis and their resolutions using some non-conventional approaches are described. Assumptions made in the model are justif ied based on the physic s of the problem, computational time versus accuracy trade-off, limitations of finite element method, and the need for simplicity. 4.2 Coupled Field Analysis A coupled-field analysis is one that consists of the interactions between two or more disciplines or fields of engineering. For example, a piezoelectric analysis, handles the interaction between the structural and electric fields: it solves for the applied displacements due to voltage distribution, or vice versa. Thermal-stress analysis, thermalelectric analysis, fluid-structure analysis, magnetic-thermal analys is, magneto-structural analysis and micro-electromechanical system s (MEMS) are other examples of coupledfield analysis. This study involves the coupling of the th ermal and structural fields. ANSYS features two types of Coupled Fiel d analysis: Direct and Sequential.
36 4.2.1 Direct Coupled Field Analysis The direct method often consists of just one analysis that uses a coupled-field element type (for example, SOLID5, PLANE 13, or SOLID98) containing all necessary degrees of freedom. Coupling is handled by cal culating element matrices or element load vectors that contain all nece ssary terms, simultaneously. This method is used when the responses of the two phenomena are dependent upon each other, and is computationally more intensive. 4.2.2 Sequential Coupled Field Analysis (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 following analysis each belonging to a different field. This method is used where th ere is one-way interaction between the two fields. There are two types of sequential analysis: sequentially coupled physics and sequential weak coupling. In a sequentially coupled physics analysis, the 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 sequen tial thermal-stress analysis wher e nodal temperatures from the thermal analysis are applied as body force loads in the subsequent stress analysis. In a sequential weak coupling analysis the solution for the fluid and solid analysis occurs sequentially, and the load transfer between the fluid and th e solid region occurs internally across a similar or dissimilar mesh interface. An example of this type of analysis is a fluid-structure in teraction analysis requiring transf er of fluid forces and heat flux from the fluid to the structure and disp lacements and temperature from the structure to the fluid. 4.3 The Finite Element Model This thesis concentrates on step 3 of AP1 (see section 1.2) in which the trunnionhub assembly is cooled in liquid nitrogen. Prior to this step, the asse mbly has interference stresses from the 2 nd step, in which the shrunk trunnion is inserted into the hub to form an
37 interference fit. Hence, it becomes imperative to have th e interference stresses present in the assembly before subjecting it to cooling in liquid nitrogen. To incorporate this, the interference values are calculated based on the trunnion outer diameter or the hub inner diameter, using FN2 fit specifications (see section B.2). These values are then added to the diameter s and the geometry is constructed in ANSYS. A structural analysis to determine the in terference stresses is done by allowing the interference fit to take place. The problem is solv ed with no additional displacement constraints or external forces. The trunnion is constrained within the hub due to its geometry. Stresses are generated due to the general misfit between the target (hub) and the contact (trunnion) surfaces. The trunnion-hub assembly is hence obtaine d with the interference stresses and this assembly is now subjected to cooling in a liquid nitrogen ba th. This is done in ANSYS by subjecting the exposed areas of the assembly to convection to a cooling medium whose properties are the same as th at of liquid nitrogen. The result of this thermal analysis is the temperature distribution in the trunnion-hub assembly. The temperature distribution thus obtained is applied as the load to the subsequent structural analysis, to obtain the thermal-stresses in the trunnion-hub assembly. It is important to understand that the stresses obt ained after this analysis is the combination of the stresses due to the interference between the trunnion an d the hub (interference stresses), and the stresses due to the temperature gradient (thermal stresses). 4.4 Elements Used for Finite Element Modeling The elements for the finite element m odel are chosen from the ANSYS element library (ANSYS Element Reference Manual, Release 7.0), which co nsists of various elements to represent the different p hysical materials used in real life. 4.4.1 ANSYS Element Library and Classification They are grouped based on the following characteristics to make element type selection easier.
38 Two-Dimensional versus Th ree-Dimensional Models: ANSYS models may be either two-dimensional or three-dimensi onal depending upon the element types used. Axisymmetric models are considered to be two-dimensional. Element Characteristic Shape: In general, four shapes are possible: point, line, area, or volume. Degrees of Freedom and Discipline: The degrees of freedom of the element determine the discipline for which the elemen t is applicable: structural, thermal, fluid, electric, magnetic, or coupled-f ield. The element type should be chosen such that the degrees of freedom are sufficient to characterize the model's response. 4.4.2 Selection of Elements The elements used in this model are chosen based on all of the characteristics described in the previous se ction, including the different physical analyses the model undergoes. The method of selection of the elemen ts is briefly described in this section. The geometry of the trunnion-hub assembly is 3-dimensional 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 first analysis that the trunnion-hub assembly undergoes is a structural analysis which is done to include the interferen ce stresses that develop at the end of Step 2 (section 1.2), caused when the trunnion is shrink fit into th e hub. Since it is a structural analysis, a structural solid element (SOLID45) is chosen. The interference between the trunnion and the hub is simu lated with the help of special elements called Contact Elements. ANSYS supports both rigid-to-flexible and flexible-to-flexible surface-to-surface cont act elements. 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 re al constant set. These surface-to-surface elements are well-suited for applications such as interference fit assembly contact or entry contact, forging, and deep-drawing pr oblems. Since, the trunnion and the hub are expected to undergo deformation; the contact is identified as flexible-to-flexible contact.
39 In problems involving contact between tw o boundaries, one of the boundaries is conventionally established as the target surface, and the other as the contact surface. Contact elements are constrained against pene trating the target surface. However, target elements can penetrate through the contact surf ace. For flexible-to-flexible 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, Release 7.0, which can be followed when designating the surfaces. The most relevant guideline for this model reads, If one surface is markedly larger than the other surface, such as in the instance where one surface surrounds the other surf ace, the larger surface should be the target surface. Using the above guideline, the hub is designated as the target surfa ce 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 3-dimensional. They behave as structural contact having structural degrees of freedom in the first analysis. The interference fit trunnion-hub assembly, then, undergoes a thermal analysis when it is cooled in liquid nitrogen. A thermal solid element is required for this analysis. However, it is not required to select another element from the ANSYS element library as ANSYS automatically change s the structural element 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 other elements associated with them. Hence, their degrees of freedom ar e changed to make them behave as thermal contact. The final analysis the trunnion-hub assembly undergoes is a st ructural analysis where the total stress, that is, the combinati on of interference stresses and the stresses due to the temperature gradient (thermal stresses) is obtained. Since th is 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 model; SOLID45, SOLID70, TARGE170 and CONTAC174. The following section gives a brief description of each of these elements. 4.4.3 Element Characteristics The structural solid element used for the structural analyses is SOLID45. It is generally used for the three-dimensional 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. The thermal solid element used for the thermal analysis is SOLID70. It has a three-dimensional thermal conduction capability. The element has eight nodes with a single degree of freedom, temperature, at each node. The element is applicable to a three-dimensional, steady-state or transient thermal analysis. Figure 4.1 shows an 8-node (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. Figure 4.1 SOLID45--3D Structural Solid, and SOLID70--3D Thermal Solid 40
Figure 4.2 shows the trunnion-hub assembly with the hexahedral solid elements. The element type is SOLID45 for the structural analyses and SOLID70 for the thermal analysis. Figure 4.2 Trunnion-Hub Assembly with SOLID45 and Solid70 Elements The contact surface for the trunnion at the trunnion-hub contact is modeled using CONTA174. CONTA174 is used to represent contact and sliding between 3-D target surfaces and a deformable surface (trunnion), defined by this element. This element is located on the surfaces of 3-D 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. Figure 4.3 shows the 41
element CONTA174 overlaying the outside diameter surface of the trunnion at the trunnion-hub interface. Figure 4.3 CONTA174 Overlaying the Trunnion Outer Diameter Surface The target surface for the hub at the trunnion-hub contact is modeled using TARGE170. TARGE170 is used to represent various 3-D 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 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 4.4 shows the target element 42
TARGE170 overlaying the inside diameter surface of the hub at the trunnion-hub interface. Figure 4.4 TARGE170 Overlaying the Hub Inner Diameter Surface 4.5 Assumptions 4.5.1 Sequential coupled field approach The assumption in this approach is that the structural results are dependent upon the thermal results but not vice-versa. This is a fair assumption as the effect of strains on the thermal analysis is negligible. 4.5.2 Finite element method assumptions The standard inaccuracies associated with any finite element model due to mesh density, time increments, number of sub-steps, etc. are present in this model (Logan, 1996). 43
44 4.5.3 Material properties The material properties of the trunnion hub assembly and the cooling medium are temperature dependent and are evaluated at specified temperature increments. The properties in between or outsi de the extremes of these values are interpolated and extrapolated, respectively.
45 CHAPTER 5 DESIGN OF EXPERIMENTS 5.1 Introduction To efficiently perform experiments involving more than one factor that affect the desired result, a scientific approach to planning the experiment must be employed. Design of Experiments (Montgomery, 2001) re fers to the proces s of planning the experiment so that appropriate data that ca n be analyzed by sta tistical methods will be collected, resulting in valid and objective conclusions. The statistical approach to experimental design is necessary if we wish to draw meaningful conclusions from the data. Thus, there are two aspects to any experimental problem: the design of the experiment and the statistical analysis of th e data. The two subjects are closely related because the method of analysis depends directly on the design employed. 5.2 Guidelines for Designing Experiments When performing any scientific procedures it is important and recommended that proper guidelines be laid out that will help in achieving the desired goals without wavering, and in less time. The guidelines for efficiently desi gning experiments are explained in the following sections. 5.2.1 Recognition of and Statement of the Problem A clear statement of the problem ofte n contributes substantially to better understanding of the phenomenon being studied and the final solution of the problem. The trunnion-hub assembly presents a clear and well defined problem. In making the trunnion-hub-girder assembly for the fulcrum of bascule bridges, Assembly Procedure 1 (section 1.4) is used in many parts of the country, including Florida. During cooling of the trunnion-hub assembly to shri nk fit it into the girder, cracks devel oped that ultimately
46 caused failure in the assembly process and hence, loss of thousands of dollars. While AASHTO (American Association of State Highw ay and Transportation Officials) has its own specifications for the geometry, different standards are made use of in the actual assemblies. Therefore, it is important to study and analyze the effects of important geometric factors on the stresses produced and failure criteri a in the trunnion-hub assembly during cooling. 5.2.2 Choice of Factors, Levels and Range This is often done simultaneously with se lection of response variable (see section 5.2.3), or in the reverse order. Many times after performing the first trial, the results give a good idea to determine which factors affect it more and which ones have little effect on the results. After obtaining the results for the first FDOT grant, the principal investigators at the University of South Fl orida suggested that the outer radius of the hub, the inner bore of the trunnion, and the interference fit were key fact ors that affected the cracks developed during the cooling of the trunnionhub assembly. The levels and ranges for these factors are obtained from AASHTO st andards (see section 6.5) and the presently used values. 5.2.3 Selection of Response Variables In selecting the response variables, it should be made certain that the variables really provide useful information about the process under study. As cracks were developed during the assembly processes, the ma in variable to be studied is the failure criteria. Failure in metals can occur either when the stresses developed exceed the allowable yield stress of the material or wh en the length of the cracks developed exceed the allowable crack length. To verify the fo rmer, the minimum stress ratio is calculated. Stress ratio can be defined as the ratio of the yield strength of the material to the stress induced. To verify the latter, the minimum critical crack length is determined.
47 5.2.4 Choice of Experimental Design In selecting the design, it is important to keep the experimental objectives in mind. In many engineering experiments, some of the factor levels wi ll result in different values of the response. Consequentl y, we are interested in identifying which factors cause this difference and in estimating the magnitude of the response change. Choice of design involves the consideration of various experimental conditi ons including the time for cooling, mesh, and the number of runs. The time for cooling was determined by knowing that the most stresses are produced during the initial cooling stages when the temperature gradient is the largest. Therefore, the tim e for cooling was set to 300 seconds (5 min). The cooling time was justified as the values for the critical crack length and stress ratio increased after sudden decreases. The numb er of trials was calculated using the 2 k factorial method. 5.2.5 Performing the Experiment When running the experiment it is vital to monito r the process care fully to ensure that everything is being done according to pla n. Prior to conducting the experiment, a few trial runs or pilot runs are often helpful. These runs provide information about the consistency of the experiment, a check on measurement system, a rough idea of experimental error, and a chance to practice th e overall experimental technique. This also provides an opportunity to revisi t the decisions made in the previous steps. A lot of trial runs were run to check the experiment and to verify the different analyses. Appendix B contains a detailed explanation of each of the different runs and their results. 5.2.6 Statistical Analysis of the Data Statistical methods should be used to analyze the data so that results and conclusions are objective rather than judgm ental in nature. The primary advantage of statistical methods is that they add objectivity to the decisi on making process. Statistical techniques coupled with good engineering or process know ledge and common sense will usually lead to sound conclusions. The statisti cal method used to obtain the results is the ANOVA 2 k factorial method.
5.2.7 Conclusions and Recommendations Once the data have been analyzed, practical conclusions must be drawn about the results and a course of action recommended. The following chapter explains the results and conclusions with recommendations, in detail. 5.3 Factorial Design Many experiments involve the study of the effects of 2 or more factors. Factorial Designs are most efficient for such type of experiments. Factorial Designs (Montgomery 2001) could be defined as experiments in which each trial or run contains all possible combinations of the levels of the factors that are investigated when it is necessary to study the joint effect of the factors on the response(s). There are several special cases of the general factorial design that are important because they are widely used in research work. The most important of these special cases is that of k factors, each at only two levels. It is called the 2 k factorial design. The levels of the factors are arbitrarily called low and high. The effect of a factor is defined to be the change in response produced by a change in the level of the factor. When the difference in response between the levels of one factor is not the same at all levels of the other factors, then it can be said that there is an interaction between the factors. Factorial Designs are advantageous and could be considered as the best method when there are 2 or more factors involved. The number of experiments required to determine the effect of each factor is reduced (2 k where k is the number of factors), misleading conclusions can be avoided when interaction is present, and the effects of a factor at several levels of the other factors can be estimated yielding conclusions that are valid over a range of experimental conditions. Three factors are considered in this thesis to be of considerable importance; the variations in the interference fits, the hub radial thickness, and the trunnion bore diameter. Since it involves 3 factors, it is called 2 3 Factorial Design and the total number of experiments required is experiments. There are three different notations that are widely used for the runs in the 2 823 k design. The first notation is the + and notation. The 48
second is the use of lowercase letter labels and the final notation uses 1 and 0 to denote high and low factor levels, respectively, instead of + and . 5.4 Calculations Involved in 2 3 Factorial Designs If the factors involved are A, B, and C, then the 8 experiments are named as shown in table 5.1. 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 Table 5.1 Notations for Experiment Combinations The main effects of the factors A, B, and C, for n replicates, are found using the following formulae (Montgomery 2001), respectively. ])1([41bcabccacbabanA (5.1) ])1([41accaabcbcabbnB (5.2) ])1([41abbaabcbcaccnC (5.3) The two factor interaction effects AB, AC, and BC, for n replicates, are found using the following formulae, respectively. ])1([41cacbcabcbaabnAB (5.4) 49
])1[(41abcbcaccabbanAC (5.5) ])1[(41abcbcaccabbanBC (5.6) The overall interaction effect ABC, for n replicates, is found using the following formula. )]1([41ababcacbcabcnABC (5.7) In equations 5.1 through 5.7, the quantities in brackets are called Contrasts of the treatment combinations. The sum of squares for the effects are calculated using nContrastSS82 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, aibjcknlijklTabcnyySS1112....12 ,...2 ,1 ; ,2,...c 1 ,...2 ,1 ; ,...2 ,1nlkbjai 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. aibjcknlijklyy1111.... The property that the treatment sum of squares plus the error sum of squares equals the total sum of squares is utilized to compute the error sum of squares. Hence, it is usually calculated by subtraction. )(ABCBCACABCBATESSSSSSSSSSSSSSSSSS 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 percentage contribution of effect A is calculated as 100TASSSS 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. 50
CHAPTER 6 RESULTS 6.1 Introduction The results obtained from the finite element analysis and the statistical analysis of the data are presented in this chapter. As discussed in earlier chapters, the purpose of this thesis is to study the effect of important parameters on the critical stresses and the critical crack length in the trunnion-hub assembly when it is cooled in liquid nitrogen, and also to optimize the geometry of the assembly in accordance with the AASHTO standards. Previous studies done at USF performed analyses on three different bridges; Christa McAuliffe Bridge, Hillsborough Avenue Bridge and 17 th Street Causeway Bridge. Since we are studying the trunnion-hub assembly for bascule bridges in general, it will suffice to study the effects on one bridge. For the thesis, the 17 th Street Causeway Bridge was chosen as the specimen bridge. The geometric parameters used are as shown in figures 6.1 and 6.2, and Table 6.1. The material properties of the metal used and the thermal properties of liquid nitrogen are presented in Chapter 3. 51 Figure 6.1 Trunnion dimensions rti rto lt l lh Contact region with hub on outside of trunnion
Figure 6.2 Hub dimensions 52 Table 6.1 Geometric Dimensions for 17th Street Causeway Bridge Bridge Geometric Parameters 17th Street Causeway rto (in) = outer radius of the trunnion (inner radius of the hub) 6.472 rti (in) = inner radius of the trunnion 1.1875 lt (in) = total length of the trunnion, 23 l (in) = Extension of the trunnion on the gusset side 6 lh (in) = total length of the hub 11 lf (in) = distance to hub flange 4.25 whf (in) = width of hub flange 1.25 rhg (in) = outer radius of the hub 8.88 rho (in) = outer radius of the hub flange 13.1825 tg (in) = gusset thickness 1.25 (in) = Interference between trunnion-hub (max) 0.00386 rho rhg rto whf lh lf
53 6.2 Fracture Toughness and Yield Strength e Critica Length a length depends on the hoop stress The two responses that we are interested in studying are th l Crack nd the Stress Ratio. The critical crack dev eloped ae material. The fracture toughness o material irease in temperature Figure 6.3. The stress ratio depends on the total stress induced (Von-Miseress) and tterial. The yield strength is also temperature dependent and it increases with a decrease in temperature, also shown in Figure 6.3. nd the fracture toughness ICK of th f the s temperature dependent and it decreases with a dec as shown in s St he yield strength sY of the ma Figure 6.3 Fracture Toughness and Yield Strength of the Material (Greenberg 1969) For an edge radial crack in a hollow cylinder that is small in comparison to the radial thickness of the cylinder (see Figure 6.4), the stress intensity factor or the fracture toughness at the crack tip is given by afKeI (6.1) where = crack length, a
ef 54 = edge effect factor4, = stress intensity factor, and IK = hoop stress; it is obtained straight from the finite element model. Figure 6.4 Critical Crack Length If where a TKKICI TKIC is the temperature dependent critical stress intensity factor or fracture toughness of the material, then the critical crack length (that is, determed by equation 6.2 (Kanninen and Popelar, 1985). the maximum crack length allowable before a crack propagates catastrophically) is in 222)(eIccfTKa (6.2) where ca= critical crack length. Tned as the ratio of the yield strength of the material to the he stress ratio is defiu he yield strength is temperature dependent and increases ithdecr it is calculated using the graph in figure 6.3. The Von-Mises stress is used as the induced stress and it is obtained straight from the finite elem can be expressed as, stress indced in thematerial. Since t w a ease in temperature, ent model. Mathematically, stress ratio sTY, where e ratioStress e is the Von-Mises stress. (6.3) N o exact data is found to calculate the values of yield strength and fracture toughness at different temperatures. The only information available is the graph from figure 6.3. Discrete points were approximated from the curves and used to either the required values based on the temperature. The interpolate or extrapolate to obtain 4 f e equals 1.25 for an edge crack which would be the worst case scenario.
55 lower ce was used fo tough F ksi F ksi urv r fractureness. Th ese points for yield strength and fracture toughness are listed in Table 6.2 and 6.3, respectively. Temperature Yield Strength Temperature Yield Strength -0 1 34 02 -120 54 -320 95 -100 52 -300 89 -80 50.5 -280 83 -60 49 -260 78 -40 48 -240 73 -20 47.5 -220 68 0 47 -200 64 20 47 -180 60 4 .5 40 7 160 58 60 47 -140 56 80 47 Table 6.2 Yield Strength as a Function of Temperature (Greenberg 1969) 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 Table 6.3 Fracture Toughness as a Function of Temperature (Greenberg 1969)
56 6.3 -Processing to Final Results ANSYS progr lts each second of the coolin in the working directory. The f iles contain the x, y and z values of the nodes in the polar co-ordinate system their temperature, total hoop stress and total Von-Mises stress. The temperature of the node is us ed to find its fracture toughness and yield rength as described in s ection 6.2. Using the fracture toughness and the hoop stress of ation 6.2. Similarly, using the yield strength and the Von-Mises stress of the node, the stress ratio (SR) at the location is f seconds (see section 5.2.4), 300 excel files ar e created one for each second. Hence, we now have the data for all locations at all tim es. Using basic functions in excel to find maximum and minimuCCL and SR are the least, is found for every second of cooling, a nd a new excel sheet is cr eated with these values. Subsequently, from this data the minimum values oL and SR are found. These values will be the least valuesL and SR for the whole run. Once the least values of CCL and SR for all the cases are found, st atistical data a nd the percentage contributions of each fa .4 Results for 5% Variation Analyses outer ete r itical crack length and the stress ratio, are of impo Post get The am creates excel files with the resu for g time st the node, the critical crack length (CCL) at the location is found using equ ound using equation 6.3. Since the cooling is allowed for 300 m, the location at which the f CC for CC nal ysis is done to fi ctor. 6 As discussed earlier, the effects of the trunnion i nner diameter, the hub diamr and the interference values, on the c rtance in this thesis. The trunnion i nner diameter, and the hub outer diameter are varied by +5% and -5%, and the interference va lue, is varied from maximum interference to minimum interference. This set of experi ments is primarily done for two reasons; first of all, to estimate the percen tage contributions of each factor and to study their effects on the critical crack length and th e stress ratio. Secondly, they are done to verify the results obtained from the analyses. This is done from the fact that the data obtained from the different levels of one factor for constant valu es of the other factors, forms a straight line when plotted against the response variable.
57 as shown in Table 6.4. Since there are 3 levels for the 3 actors, a total of 33 = 27 experiments are done. e above values and the results obtained were process l umbers 1, 2 and 3 are used. Si milarly, in figures 6.6(a) an d 6.6(b), the hub outer radius is made to vary while keeping the other two constant; trial numbers 3, 6 and 9 are used. And in figures 6.7(a) and 6.7(b), the trunnion inne r radius is ma hig the other 1, 10 an9 are The original values for the trunnion i nner radius, hub outer radius and the interference value is obtained from Table 6.1. The differe nt levels for th e trunnion inner radius and the hub outer radius are obtained by taking -5% and +5% of the original values. The different levels for the inte rference value are obtained by finding the minimum, the maximum, and the average (mid) value (see Appendix B, section B2). The levels of the factors used are f Table 6.4 Values of the Different Levels of the Factors for the 5% Variations The experiments were done using th 1 2 3 ed as explained in sect ion 6.3. Table 6.5 gives the cr itical crack length values, its locations and the time it occu rs for the 27 cases. Similarly, Table 6.6 gives the stress ratio values, its locations and the time it occurs for the 27 cases. This data is first verified by plotting it against diffe rent levels of one factor while keeping the other two constant. The results would be accurate if each of the plots obtained is a straight line. This is shown in figures 6.5 through 6.7. The interference value is made to vary in figures 6.5(a) and 6.5(b) while keeping the trunnion inner radi us and hub outer radius constant; tria n de to vary w le keepin two constant; trial numbers d 1 used. Variable Parameters (-5%) (nominal) (+5%) Inner radius of the trunnion T i 1.1281 1.1875 1.2468 Outer radius of the hub (minus flange)H i 8.436 8.88 9.324 Radial Interference (in) IN i 0.0021249 (min) 0.0029925 (mid) 0.00386 (max)
58 Trial No Y (De((secC(in Treatment Combinations X (in) g) Z T in) ime ) CL ) 1 T1-H1-IN1 1.1281 -150 1140.22998 23 2 T1-H1-IN2 6.472 -145 17 161 0.19648 3 T1-H1-IN3 6.472 -150 17 163 0.16011 4 T1-H2-IN1 1.1281 -90 23 114 0.23111 5 T1-H2-IN2 6.472 -150 17 180 0.21960 6 T1-H2-IN3 6.472 -150 17 183 0.17851 7 T1-H3-IN1 1.1281 -90 23 113 0.23266 8 T1-H3-IN2 9.324 -125 17 95 0.22157 9 T1-H3-IN3 6.472 -150 17 194 0.19326 10 T2-H1-IN1 1.1875 -90 23 111 0.23501 11 T2-H1-IN2 6.472 -145 17 161 0.19866 12 T2-H1-IN3 6.472 -150 17 163 0.16176 13 T2-H2-IN1 1.1875 -90 23 111 0.23619 14 T2-H2-IN2 6.472 -150 17 179 0.22225 15 T2-H2-IN3 6.472 -150 17 183 0.18051 16 T2-H3-IN1 1.1875 -90 23 111 0.23779 17 T2-H3-IN2 9.324 -125 0.22267 17 95 18 T2-H3-IN3 6.472 -150 17 193 0.19549 19 T3-H1-IN1 1.2468 -150 23 109 0.24031 20 T3-H1-IN2 6.472 -145 17 161 0.20093 21 T3-H1-IN3 6.472 -150 17 162 0.16345 22 T3-H2-IN1 1.2468 -150 23 109 0.24154 23 T3-H2-IN2 8.88 -125 17 93 0.22451 24 T3-H2-IN3 6.472 -150 17 183 0.18259 25 T3-H3-IN1 1.2468 -90 23 108 0.24321 26 T3-H3-IN2 9.324 -125 17 95 0.22381 27 T3-H3-IN3 9.324 -125 17 95 0.19719 Table 6.5 Critical Crack Length Values for 5% Variations
59 Trial Combinatio ) No Treatment ns X (in) Y (Deg Z (in) Time (sec) SR (in) 1 T1-H1-IN1 8.436 -120 16 1.80818 2.41 68 2 T1-H1-IN2 6.472 -150 17 81 1.65213 3 T1-H1-IN3 6.472 -150 17 76 1.47924 4 T1-H2-IN1 8.88 -120 16 1.76913 2.41 71 5 T1-H2-IN2 8.88 -120 16 1.65414 2.41 70 6 T1-H2-IN3 6.472 -150 17 78 1.52274 7 T1-H3-IN1 9.324 -120 16 1.75506 2.41 74 8 T1-H3-IN2 16 1.65464 9.324 -120 2.41 73 9 T1-H3-IN3 6.472 -150 17 80 1.55340 10 T2-H1-IN1 8.436 -120 16 1.81288 2.41 68 11 T2-H1-IN2 6.472 1.65819 -150 17 79 12 T2-H1-IN3 6.472 -150 17 74 1.48405 13 T2-H2-IN1 8.88 -120 16 1.77317 2.41 71 14 T2-H2-IN2 8.88 -120 16 1.65791 2.41 70 15 T2-H2-IN3 6.472 -150 17 78 1.52812 16 T2-H3-IN1 9.324 -120 12.416 74 1.75862 17 T2-H3-IN2 9.324 -120 126 1.65802 .41 73 18 T2-H3-IN3 6.472 -150 17 80 1.55915 19 T3-H1-IN1 8.436 -120 126 .41 67 1.81769 20 T3-H1-IN2 6.472 -90 17 78 1.66455 21 T3-H1-IN3 6.472 -150 17 73 1.48916 22 T3-H2-IN1 8.88 -120 12.416 71 1.77734 23 T3-H2-IN2 8.88 -120 126 .41 70 1.66179 24 T3-H2-IN3 6.472 -150 17 77 1.53357 25 T3-H3-IN1 9.324 -120 12.416 74 1.76230 26 T3-H3-IN2 9.324 -120 126 .41 73 1.66152 27 T3-H3-IN3 6.472 -150 17 80 1.56511 Table 6.6 Stress Ratio Values for 5% Variations
00 0.05 0.1 0.15 0.2 0.25 0..003.005ceCCL 0010.0020 0.0040 Interferen 0 0.S 515200.000.00305Innce 1.atio 0.001 2 0.0040.0 terfere tress R ( b) Figure 6.5 Different Levels of Interference against and Value a) ( CCL SR s 0.2295 0.230.2305 0.231 0.23150.232 0.2325 0.233 8.2.899.4r RadiusCCL 8.48.68 9.2 Hub Oute 1.471.48 1.491.5St 1.511.52ress R 1.53atio 1.541.55 1.56 8.28.69Hub Oadius 8.4 8.8uter R 9.29.4 (b) Figure 6.6 Different Levels of Hub-Outer Radius against CCL and SR Values (a) 0.2280.230.2320.2340.2360.2380.240.24184.108.40.206.251. 3 60 Trunnion Inner Radius CCL 1.8061.8081.811.8121.8141.8161.8181.8220.127.116.11.251.3Stress Ratio Trunnion Inner Radius Figure 6.7 Different Levels of Trunnion Inner Radius against CCL and SR Values (a) (b)
61 From the plots, we can clearly see that the data obtained are accurate since they form straight lines when plotted against different levels of each factor. After the data obtained is verified to be accurate, statistical analysis can be done to study the effects of the factors on the stress ratio and the critical crack length. As explained in section 5.3, we use the 2k factorial design for our statistical data analysis, where k is the number of factors. Since it involves 3 factors, it is called 23 Factorial Design and the total number of experiments required is experiments. The eliminated. This leaves us with 8 trials; trial numbers 1, 3, 7, 9, 19, 21, 25, and 27. The values for critical crack length are as shown in Table 6.7. For the purpose of the analysis, the three factors are labeled as shown in the table. trunnion inner radius has a very small effect (1%) when compared to the interference. Hi (Hub Outer Radius) B 823 experiments required are the trials in which each of the factors is either at the low level or at the high level. Therefore, all the trials in which the factors have nominal values are -5% (8.436) +5% (9.324) Overall Minimum Critical Crack Length INi (Interference) A INi (Interference) A Variations Min Max Min Max 00386) Table 6.7 Notations and Values for 2 3 Factorial Design Data Analysis for CCL Using the values from Table 6.7, the calculations for the 2 3 Factorial Design are done as explained in section 5.4 to obtain the effects of the factors on the critical crack length. The effects are shown in Table 6.8. We can see that the interference between the trunnion and the hub has the most effect on the critical crack length (84%), followed by the hub radius (8%), and the interaction of the interference and the hub radius (5%). The (0.0021249) (0.00386) (0.0021249) (0. T i -5% (1.1281) (1) 0.22998 a 0.16011 b 0.23266 ab 0.19326 (Trunnion Inner c 321 abc 0.19719 Radius) C +5% (1.2468) c 0.24031 ac 0.16345 b0.24
62 able 6.8 Percentage Contributions of the Factors for CCL Similar analysis is done for the Stress Ratio. The notations and values used are shown T in Table 6.9 and the effects of the factors are shown in Table 6.10. The interference between the trunnion and the hub has the most effect on the stress ratio (94%), followed by the interaction of the interference and the hub outer radius (5.64%). H i (Hub Outer Radius) B -5% (8.436) +5% (9.324) Overall Minimum Stress Ratio INi (Interference) A INi (Interference) A Variations Min (0.0021249) Max (0.00386) Min (0.0021249) Max (0.00386) -5% (1.1281) (1) 1.80818 a 1.47924 b 1.75506 ab 1.55340 Ti(Trunnion Inner Radius) C +5% (1.2468) c 1.81769 ac 1.48916 bc 1.76230 abc 1.56511 Table 6.9 Notations and Values for 23 Factorial Design Data Analysis for SR es Contribution Factor Effect Estimate Sum of SquarPercent p-values Interference, A -0.058 6.736 x 10-384.358 <0.00001 Hub Radius, B 0.018 6.<0.00001 566 x 10 -4 8.223 T041 x 10-310-51.242 <0.00001 runnion Radius, C 7. 9.914 x AB 0.0101 5 4.7 x 10 -4 5.886 <0.000 AC -3.402 x 2.31501 10 -3 x 10 -5 0.29 <0.000 BC -1.238.2 6 x 10 -4 92 x 10 -8 1.038 x 10 -3 <0.00001 AB C 9.655 x 1-52.061 x-62.335 x 10-4<0.00001 0 10 Pure E rror----33 ---0 .565 x 10 -1 -------Total -------7.985 -3--------------x 10
63 able 6.10 Percentage Contributions of the Factors for SR r, hile currently a thickness of 0.1 to 0.2 times the inner diameter is used. Therefore, it is imperative to study how the critical crack leange when the AASHTO standards are employed, anr than the curree low level of the hub diamer is choser diampractice. The high level is chosen aAASHTO recommended standard, which is 1.4 times the inner dce T 6.5 AASHTO Results The American Association of State Highway and Transportation Officials (AASHTO) standards call for a hub radial thickness of 0.4 times the inner diamete w ngth and the stress ratio ch d if it is bette nt practice. Th et n as 1.1 times the inne eter, which is the current s the iameter. Hen 7.1192 2 )2472.1.0()2 Inner iameeve 6 472.6( 2 x .1ter Diameter) (0 DInner l L Low Factor t ate Contribution p-Values Effec Estim Sum of Squares Percent Interference A 64 7 -0.2 0.139 94.08 <0.00001 Hub Diameter, B -4 0.01 2.161 x 10 0.146 <0.00001 Trunnion Diameter, C 9.599 10-30-4 x 1.843 x 1 0.124 <0.00001 A B 0.065 8.36 x 10 -3 5.64 <0.00001 A C -3-6 1.216 x 10 2.956 x 10 1.994 x 10 -3 <0.00001 B C -4-8 -1.23610 3.057 x 10 2.062 x 10 -5 <0.00001 ABC 10-32.061 x 10-6<0.00001 1.015 x 1.39 x 10 -3 Pure Error --1590-13 -----1.425 x 10 .612 x 1 -------Total -------0.148 --------------
9.0 608(6.472 2Diameter)Inner (0.4 Diameter Inner LevelHigh 26.472(0.42) 2) vaximum, and the trunnion inner diameter is varied by 10%. Sineed onments fo3 factorial design, only the high and low values are used. The ve as le Table 6 riable Paigh) The interferences ary from m um to minim ce we n ly 8 experi r the 2 alues used ar shown in Tab 6.11. Va rameters 1 (Low ) 2 (H Inner radiu s of the tr1.0680625 unnion T i 75 1.3 Outer radius of the hub (minus flange) Hi7.1192 9.0608 Radial Interference (in) IN0.002124 i9 (min) 0.00386 (max) .11 Values of the Different Levels of the Factors for the AASHTO Results The experiments were conducted and the data obtained were post-processed as explained in section 6.3. The calculations for the statistical data analysis are done using the equations given in section 5.4. The results obtained for the Critical Crack Length are shown in Table 6.12 through Table 6.14. 64
65 Trial No Treatment Combinations X (in) Y (Deg) Z (in) Time (sec) CCL (in) 1 T1-H1-IN1 7.1192 -125 17 90 0.09735 2 T1-H1-IN2 7.1192 -125 17 96 0.06799 3 T1-H2-IN1 1.06875 -90 23 116 0.22689 4 T1-H2-IN2 6.472 -150 17 191 0.18302 5 T2-H1-IN1 7.1192 -125 17 85 0.09983 6 T2-H1-IN2 7.1192 -125 17 91 0.0695 4 7 T2-H2-IN1 1.30625 -90 23 106 0.24783 8 T-150 18819167 2-H2-IN2 6.472 17 0. Table 6.12 Critical Crack Length Hi (Hub Outer Radius) B Values for AASHTO Results Low (7.1192) High (9.0608) Overall Minimum INi (Interference) C INi (Interference) C Critical Crack Length Variations Min (0.0021249) Max (0.00386) Min (0.0021249) Max (0.00386) Low (1.1281) (1) 0.09735 c 0.06799 b 0.22689 bc 0.18302 T i (Trunnion Inner Radius) A High (1.30625) a 0.09983 ac 0.06954 ab 0.24783 abc 0.19167 Table 6.13 Notations and Values for 23 Factorial Design for CCL (AASHTO)
66 able 6.14 Percentage Contributions of the Factors for CCL (AASHTO) From the above results, the hub outer radius is observed to have the most effect on the critical crack length (90%), followed by th The tress Rat Table 6.15 thr Tria Tr ns X Y eg) T T e interference (8.67%). results obtained for the S io ar e shown in ough Table 6.17 l No Combinatio eatment (in) (D Z (in) ime (sec) SR 1 T1-H16.472 -115 17 60 1.36273 -IN1 2 T1-6-115 1.10838 H1-IN2 .472 17 45 3 T1-H9.0608 -120 12.416 73 1.75767 2-IN1 4 T1-6-150 1.53136 H2-IN2 .472 17 80 5 T2-H1-IN1 6.472 -115 17 59 1.37644 6 T2-H1-IN2 6.472 -115 17 45 1.1162 1 7 T2-H2-IN1 9.0608 -120 12.416 72 1.77314 8 T2-H2-IN2 6.472 -150 17 78 1.55365 Table 6.15 Stress Ratio Values for AASHTO Results F Effect Estimate ares Percent Contribution p-Values a ctor Sum of Squ Trnion D 8.4 3 x 100.384 <0.00001 un iameter, A 04 x 10 -3 1.41 -4 Hu b Dia .033 90.067 meter, B 0.129 0 <0.00001 In terfer 7 x 10 ence, C -0.04 3.18 -3 8.669 <0.00001 A 6.38-33 x 10 B 9 x 10 8.16 -5 0.222 <0.00001 A -3. 2 x 100.059 <0.00001 C 303x 10 -3 2.18 -5 BC -0.01 9 x 100.555 <0.00001 2.03 -4 A -2 3 x 10 BC .84 x 10 -3 1.61 -5 0.044 <0.00001 Pure-0 --------------Error ------Total -------0.037 --------------
67 Huter R i (Hub O adius) B Lo w (7.608) 1192) High (9.0 Overall Minimum nterINrfere Stress Ratio IN i (I ference) C i (Inte nce) C Varia tions n 1249) ) (0.0021249) ( Mi (0.002 Max (0.00386 Min Max 0.00386) Low (1.06875) 3 1 (1) 1.3627 c 1.10838 b .7567 bc 1.53136 T i (Trunnion Inner Radius) A High 44 1.77314 (1.30625) a 1.376 ac 1.11621 ab abc 1.55365 Table 6.16 Notations and Values for 23 Factorial Design for SR (AASHTO) Tabl.17 Pentrib of tctors he Stress Ratio (AASHTO) the above results, the hub outer radius is oved toe the mt on e stress ratio (74%), followed by the interference (25%). Factor Effect Sum of Percent p-Values Estimate Squares Contribution Trunnion Diameter, A 0.015 4.397 x 10 -4 0.096 <0.00001 Hub Diameter, B 0.413 0.341 74.3565 <0.0000 1 Interference, C -0.24 0.115 25.198 <0.00001 AB 4.054 x 10-33.287 x 10-57.185 x 10-3<0.00001 AC 2 .38-4 x 102.40-5<0.00001 2 x 10 1.135 -7 81 x 1 BC 0.017 7 x 100.129 <0.00001 5.91 -4 A3.17-35 x 104.405 x 10-31 BC 4 x 10 2.01 -5 <0.0000 Pure x 15.91-13 Error -------2.705 0 -15 2 x 10 -------To-.458 --------------tal ------0 e 6 rcentage Co utions he Fa for t From bser hav ost effec th
68 6.6 Explanation of the Results tained for n analyses, ithat the interference between the trunnion or that the critical crack length and the stress ratio. The values for critical crack length and stress ratio chge theth vs, hry sm trunnner diameter or the hub outer diameter is changed (Tables 6.7 and 6is isctione of the conclusions obtained fromus studies done at USF which consequently led to the recommendation that AP2 is better than APn the trunnion and the hub are absent. owever, the results obtained from the AASHTO analyses show that the hub outer or of the esults can be attributed to two important reasons. The first reason is that the percentage variations of the hub diameter in the AASHTO are differenhe perceriations anae the variations are only 5% in the latter case, tout er. Since the varASalyseage conn is athe moortant readue to th of t. The touced in nnion-hubly are a the e stresses andhermal strterfes ar byf fits used and the diameters of the trunnion aor the se minimum, the hub outer diameds to nity (Und Fe). In other words, the interference stresses decrb o inSince the variation of tuter eater in the AASHTO analyses, the differenthe values o for CCL betweeo levso reater. Hence, the percentage contribution of the hub outer diameter is greater in the From the results ob the 5% variatio is clearly seen t and the hub is the fact has the most effect on an most when e interference alue change whereas the c ange is ve all when the.9). Th nion i in conjun wn ith o pre vio 1, as the interference stresses betwee H diameter is the most significant factor followed by the interference. This behavi r analyses t from t ntage vahey are ab in the 5%12% in the form lyses. Whil iations are more in the A HTO an s, the percent tributio lso more. The second and re imp son is e physics he problem tal stresses prod the tru b assem sum of interferenc the t esses. The in erence stress e influenced the kind o nd the hub. F e interferenc tresses to b ter nee approach infi gural a nster 1995 ease as the hu uter diameter creases. he hub o diameter is gr ce in btained and SR n the tw els is al g AASHTO analyses.
69 critical crack length and the critical stresses ly, followe ). The values fo r stress ratio were also higher ( 30% for low interference and 40% for high interference) for the AASHTO cases (Table 6.15). herefore, it can be concluded that the AASHTO standards are safer than the current practices used in the manufacturing of trunnion-hub-girder assemblies for bascule bridges. If employed, the AASHTO standards will yield higher critic al crack length and CHAPTER 7 CONCLUSIONS AND RECOMMENDATIONS 7.1 Conclusions The main objective of this thesis is to study the effects of geometric parameters such as the hub outer diamet er, the trunnion inner diamet er, and the interference, on critical stresses and critical crack length during the assembly. The results obtained by varying the parameters by a value of % re vealed that the inte rference between the trunnion and the hub had the most effect on the developed during the assembly. For cases with low or minimum interference, the critical crack length and stress ratio were higher than in cases with maximum interference. Although this set of analyses revealed importa nt and valid results, these values are not used in real life nor are they recommended by AASHTO. The AASHTO standards call for a hub radial thickness of 0.4 times the diameter. However, in real life practi ce, the hub radial thickness us ed is 0.1 to 0.2 times the diameter. Hence, using the extremes as the hub thickness, % vari ation in the trunnion inner diameter, and interfer ences ranging from minimum possible value to maximum possible value, it is found that the hub outer diameter is the most significant factor that affects the critical crack length and the crit ical stresses developed during the assemb d by the interference between the tr unnion and the hub (Tables 6.14 and 6.17). The critical crack length values obtained fo r the AASHTO specifications were more than two times the critical crack length values obtai ned for the specifications used in real life practice (Table 6.12 T
70 stress ratio. It can also pr ove out to be more economi cal during the manufacturing process by reducing the failures and hence, saving hundreds of thousands of dollars. Comparing the two sets of analyses, that is, the 5% variations and the AASHTO cases, we see that while the interfere final results the most in the former, the hub outer diam terferences are same for bot h the analyses and they vary by approximately %. The and the hub outer di ameter vary by % in the first set of analyse ions made after comple nce affect s the eter has the most significant e ffect in the latter. Th e variations in the in trunnion inner diameter s; the trunnion inner diameter varies by % and the hub outer diameter varies by approximately % in the AASHTO cases. Therefore, the results obtained cannot be attributed entirely to the percentage variati ons in the factors and it cannot be concluded that the factor varying the most will have the most effect on the final results. As explained in section 6.5, the physics of the problem is a very impor tant factor to be considered. 7.2 Recommendations for Future Work Although this thesis evolved as a result of the recommendat ting the studies to understand the fail ures that occurred during the manufacturing of the trunnion-hub-girder assembly for bascul e bridges, there is still scope for future work to completely optimize the geometry and the manufacturing process. This thesis focused at the sensitivity of important parameters including the hub outer diameter, the trunnion inner diameter, and the interference, on the critical stresses and the critical crack length. It would be in teresting to study and an alyze the sensitivity of other parameters such as the hub length and the gusset thickness on the critical stresses and the critical crack length. Based on this, the entire geometry of the trunnion-hub assembly could be optimized and made availa ble to the manufacturers to ensure safe manufacturing and effective use in the bridges that would be required to support more lanes of traffic in the years to come.
71 2000, Parametric Finite Element modeling of trunnion Hub Girder ssemblies for Bascule Bridges, MS Thesis, Mechanical Engineering Dept, USF, Ugura zi ik, M. N., 1993, Heat Conduction, John Wiley & Sons, Inc., New York REFERENCES Glen Besterfield, Autar Kaw & Roger Crane, 2001, Parametric Finite Element Modeling and Full-Scale Testing Of Trunnion-Hub-Girder Assemblies for Bascule Bridges, Mechanical Engineering Dept, USF, FL Denninger, M.T., 2000, Design Tools for Trunnion-Hub Assemblies for Bascule Bridges, MS Thesis, Mechanical Engineering Dept, USF, FL Ratnam, B., A FL Nichani, S., 2001, Full Scale Testing of Trunnion-H ub-Girder Assembly of a Bascule Bridge, MS Thesis, Mechanical Engineering Dept, USF, FL l, A.C., and Fenster, S.K., 1995, Advanced Strength and Applied Elasticity, Prentice Hall PTR, New Jersey ANSYS Inc., 2002, ANSYS Release 7.0 Documentation Montgomery, D.C., 2001, Design and Analysis of Experiments, John Wiley & Sons, Inc., New York
72 higley, J.E., and Mischke, 1986, Standard Handbook of Machine Design, McGrawHill, New York imoshenko, S. P., and Goodier, Jy of Elasticity, McGraw-Hill Book ompany, New York Kreith, ogan, D. L., 1992, A First Course in Finite Element Method, PWS-KENT Publishing arron, R. F., (1999), Cryogenic Heat Transfer, Taylor and Franci s Company, PA, pp. Greenberg, H.D., and Clark, Jr. H.G., 1969, A Fracture Mechanics Approach to the Development of Realistic Acceptance Standards for Heavy Walled Steel Castings, merican Association of State Hig hway and Transportation Officials, 1998, Mobile echanical Engineering Dept, USF, FL, 2003, Scope of Work Proposal for b-Girder Assembly in Bascule Bridges S T N., 1951, Theor C F., and Bohn, M., 1986, Principles of Heat Transfer, Harper & Row, New York L Company, Boston B 161-172 Metals Engineering Qu arterly, 9(3), 30-33 A Bridge Inspection, Evaluation, and Maintenance Manual M Optimization of Geometrica l Parameters of Trunnion-Hu
74 Appendix A: Program Flow and User Instructions A.1 Program Flow The flowchart of the programs is given in figures A.1 through A.3. It gives the flowchart of the ANSYS program, followed by the Excel program, and finally, the 2k factorial design calculations, which is done using MathCAD. Read Trunnion and Hub parameters, and Interference Choose Element type, read material and liquid nitrogen properties Create 1/6th Solid and Mesh it Create Contact Pair with Contact Elements Start ANSYS program
75 Appendix A (Continued) Figure A.1 Flowchart for the ANSYS Program Appand ly symmetric displacement boundary conditions p erform structural anal y sis for Interference Stresses Change elements from thermal to Clear all loads; apply convection loads, symm etric thermal boundar y conditions and solve for tem p erature Write x, y, z locations and tem p erature of nodes for ever y time Change elements from structuralto Clear all loads; apply interference stresses, thermal loads, and s y mmet r ic dis p lacem ent boundar y conditions and solve f or total Write hoop stress and Von-Mises stress of nodes for ever y time in Create Excel files and write in valu es f rom Stop ANSYS
Appendix A (Continued) 76 Figure A.2 Flowchart for the Excel Program Stop No Start Excel Program Calculate Fracture Toughness and Yield Strength based on temperature Calculate Critical Crack Length and Stress Ratio for all locations Find minimum CCL an d SR Write these values into new excel sheet Is it end of cooling time? Find minimum from the new Exct el shee Go to next time step Yes
Appendix A (Continued) Perform calculations for the 2 k factoria l design as explained in section 5.4 Ocontributr Stop utput percentage ions of each facto Read critical crack length or stress ratio values Start 77 igu.2 Modeling the Experiment The commands for the ANSYS program are stored in a file called 7th_Causeway_Compland file read in the input arameters including the dimensions of the unnion and the hub, interference, divisions for mesh, and the cooling time. They can be changed according to the need of the user portant to make sure that the command le for ANSYS, and the excel program, Data Analyzer, for calculating the CCL and SR sing fracture toughness and yield strength are stored in the same directory. After starting F re A.3 Flowchart for the 2k Factorial Design Data Analysis A 1 ete. The first few lines in the comm p tr and saved before starting the experiment. It is im fi u
Appendix A (Continued) 78 NSYS, pull down the File menu and select Change Directory. When the window opens ith all the in which the command file is stored and click K. Now, this directory is set as the working directory and all the files that result from e program will be stored in this directory, including the excel sheets with the results. he experiment can be started in two ways; pull down the File menu, click on Read Input rom, select the command the Open ANSYS file tab, from e drop down menu for file typat opens, choose ANSYS Commands, lect the command file and click OK. ANSYS starts reading the commands in the file nd performs the appropriate functions. The total run time depends on the time for ooling, and the compime of 300 seconds in the test P4 processor with 5ately 1.5 hrs. The number f the excel files created also depend on the cooling time, and are equal to the cooling me. Once the ANSYS run i excel files are stored in the working irectory, the excel program SR can be started. The file, Data nalyzer, is an excel macro that calculates the Critical Crack Length and Stress Ratio. he first two rows of the file are for the input. The input is the number of files that the NSYS program creates. Hence, the input will be that same number as the cooling time the ANSYS program. After inputting the number of files, click the Analyze tab. The rogram starts calculating the fracture toughness, yield strength, CCL and SR for every g. It then finds the inimum values for CCL and SR from each file and stores it in the Data Analyzer file me it occurs. From the values stored, it finds the minimu A w directories, select the directo ry O th T F file and click OK, or click ones in the window th th se a c uter configurations used. For a cooling t la 12MB RAM, the run time is approxim o ti s completed and the for calculating the CCL and d A T A in p location in each file, which represents each time-step for coolin m along with its location and the ti m and outputs at the beginning of the worksheet. The program for statistical data analysis, called per_con, is a MathCAD program that performs calculations as explained in section 5.4. The inputs are manually entered and the percentage contributions are shown in the same MathCAD sheet.
Appendix B: Verifications of Analyses B.1 Introduction To verify the various analyses and to check if the results are true, sample analyses were done for problems whose solutions were known. The solutions were first calculated with the help of analytical equations, and then simulated using ANSYS to obtain similar solutions. The final verification includes the application of the knowledge of thermal stresses that a member will not experience any stresses due to temperature, if the temperature gradient in the member is equal to zero. Following are the various test analyses that were done. 79 B.2 Test 1 for Structural Analysis for Interference Stresses At the end of step 2 of AP1 (see section 1.2), the trunnion is fit into the hub to form an interference fit. This interference fit produces interference stresses in the trunnion-hub assembly. To verify the structural analysis to find the interference stresses, an interference fit between two cylinders was simulated and the stresses thus produced were found. The specifications of the 2 cylinders are: Cylinder 1 Inner Radius = 2" Outer Radius = 8.7" Cylinder 2 Inner Radius = 8.7" Outer Radius = 16.7" It is assumed that both the cylinders are made of steel (E = 29 Msi, = 0.3), FN2 fit is used at the interface, the radial displacement is of the form for axisymmetric problems, and that plane stress conditions apply. The interference values on the cylinders for the FN2 fit are calculated using the formula: 3 1 CDL, (B.1) where L is the limit in thousandths of an inch
80 Appendix B (Continued) C is the coefficient of the fit whose values are given in table B.1, and D is the diameter. Hole Limits Shaft Limits Class of Fit wer Upper Lower Upper Lo F N2 0 +0.907 +2.717 +3.288 Table B.1 Coefficients for FN2 fit B.2.1 Exact Solution The exact solution of the problem was obtained using Maple. The lower limit and the upper limit of the cylinders are first found, using equation B.1 and the coefficients from Table B.1. Hence, 00704054.01lim l Lower limit of cylinder 1: Upper limit of cylinder 1: 00852017.01lim u Lower limit of cylinder 2: 02lim l Upper limit of cylinder 2: 00235030.02lim u The maximum diametrical interference in inches is: 00852017.0 2lim1lim1 ludel 008 520170 .0 The Radial Interferenc e in inches is: 00426008.0211del The radial deflection in cylinder 1 is giv en by: r CrCUr2.1 (B.2) 1The radial deflection in cylinder 2 is given by: rCCUr432 (B.3)
Appendix B (Continued) 81 Radial stress on Cylinder 1: 11)1(CCE 22121rr (B.4) Radial stress on Cylinder 2: 21Cr 24321)1(rCE (B.5) Hoop stress on cylinder 1: 22211rCCE (B.6) 1)1( 1 Hoop stress on cylinder 2: 21E 24321)1(rCC (B.7) for this problem are: @ r = 8.7"; 8.7"; Ur2-Ur1 = 1The boundary conditions are substituted in equations B.2 through B.7 and solved for the constan, 280001266869.0431 The boundary conditions @ r = 2"; 01r @ r = 16.7"; 02r 021rr @ r= ts C 1 C 2 C 3 and C 4 Hence 023149675.090000446957.0 00094110.02 CCC The constants C1, C2, C3 and C4 are used to calculate the stresses at the interface where terface, = C r = 8.7, from the equations B.4 through B.7. Therefore, at in 1r psi09.4971 = 2r psi09.4971 1 =psi82.5525 2 =The Von-Mises stress is found using the formula: psi45.8674 2,12,122,122,1)()()2,1(rre (B.8) The Von Mises Stress for Cylinder 1 and 2 are: Psie39.5270)1( Psi58.11961)2 ( e
Appendix B (Continued) 82 tresses at the interface due to the interference fit are calculated to be 11.96 ksi for cylinder 2 and 5.27 ksi for cylinder 1. The same problem was solved in ANSYS. A structural analysis to determine the one by allowing the interference fit to take elements as described in Chapter 4 are used at the interface. The problem is solved with constraints or external forces. Cylinder 1 is constrainednerated due to the general misfit between the target (cylinder 2) and the contact (cylinder 1) surfaces. The Von-Mises stresses obtained due to the interference fit are shown. It can be seen that the maximum stresses occur at the interface. The interface can be seen as a thin white line. Hence, the actual Von-Mises s B.2.2 ANSYS Solution interference stresses was dplace. Contact no additional displacement within cylinder 2 due to its geometry. Stresses are ge Figure B.1 Interference Stresses (Von-Mises) between 2 Cylinders (Isometric View)
Appendix B (Continued) Figure B.2 Interference Stresses (Von-Mises) between 2 Cylinders (Front View) 83 Figure B.3 Interference Stresses (Von-Mises) at the Interface
Appendix B (Continued) 84 Hence, the Von-Mises stresses obtained from ANSYS at the interface due to the interference fit are found to be between 11.34 ksi and 12.36 ksi for cylinder 2, and between 4.22 ksi and 5.24 ksi for cylinder 1. B.2.3 Comparison of Actual Solution vs. ANSYS Solution To verify the accuracy of the result obtained from ANSYS, the percentage difference between the exact solution, obtained using Maple, and the solution obtained from ANSYS is calculated by making use of the formula, Percentage Difference = 100SolutionExact Solution ANSYS -Solution Exact (B.9) The maximum value of the stress from ANSYS is used to find the percentage difference. For cylinder 2, Percentage Difference 100 x 1196112363-11961 %36.3 For cylinder 1, Percentage Difference = 100 x 52705244-5270 %49.0 The percentage differences between the actual solution and the ANSYS solution for the stresses for the 2 cylinders are very small (3.36% and 0.49%) and hence, it can be concluded that the structural analysis to find the interference stresses is accurate.
85 ub assembly as the model, with parameters and properties as explained in Chapter 3. The interference stress was compared with the that Ra Appendix B (Continued) B.3 Test 2 for Structural Analysis for Interference Stresses This test was done using the trunnion-h interference stress that Ratnam (2000) obtained in his work. The value for the hoop stress tnam (2000) obtained after interference was approximately 14121 psi while the Hoop stress obtained in this test is 13678 psi. The formula from equation B.9 was used and the percentage difference was found to be %14.3100 14121 13678-14121 Difference Percentage B.4 Test 1 for Thermal Analysis for Cooling in a Liquid Bath Step 3 of AP1 (see section 1.2) involves cooling the trunnion-hub assembly to shrink it so that it can be fit into the girder with an interference fit. This is done by immersing the assemb. The initial tempe of the trunnion-hub assembly is 800F and the temperature of liquid nitrogen is -3210F, and its convection coefficient changes as a function ure. This is simulated in ANSYS to obtain 0F. The mperature distribution against time was obtained and plotted, and the temperature of the ire at specific times was found using ANSYS. These results were compared with the ctual solutions, calculated using the analytical equations. The specifications of the roblem are given below: Diameter of the copper wire ly in a liquid nitrogen batheratur of temperat the temperature distribution, which will then be used to find the stresses due to temperature gradients. This thermal analysis was verified using the following example. A copper wire at 300 0 F is cooled in a water bath maintained at 100 te w a p ind )32/1( Initial temperature of wireThermal conductivity of copper FT00300 FfthrBtuk / 216
86 Appendix B (Continued) Specific heat of copper C FlbBtu / 091.0 Density of copper ftculb / 558 Temperature of water bath FT0100 Convection coefficient of water FfthrBtuh / 152 The formula to find the temperature distribution is given by, VCthAeTTTT)(0 (B.10) where A and V are the surface area per unit inch and volume per unit inch, respectively. The Biot number was calculated using the formul a khdBi4 Hence, 510521.4216412132115Bi It was found to be much less than 0.1, implying that the interna l resistance may be neglected. B.4.1 Exact Solution The exact solution of the problem is obtained using Maple, by substituting the variables in equation B.10. The units are kept consistent. The temperature distribution plot for time t = 0 to 0.012 hr is as shown in figure B.4.
Appendix B (Continued) Figure B.4 Temperature Distribution of Copper Wire from Actual Solution YS Solution The same problem was solved in ANSYS. A thermal analysis was done to find n, and the results were plotted against time. A long copper wire as firnst time plot from ANSYS for time t = 0 to 0.012 hr is shown below. B.4.2 ANS the temperature distributio wst built with the given diameter. The properties, and the loads, that is, initial temperature and convection on areas, in this case, were specified. Temperatures at specific times were also obtained. The temperature agai 87
Appendix B (Continued) Figuree times from the solution obtained from ANSYS is calculated by making use of the formula given in equation B.9. The time, temperatures from Maple and ANSYS, and the percentage difference, are shown in Table B.2. Figures B.4 and B.5, are very identical showing a similar temperature distribution solution in both Maple and ANSYS. In additi Table B.2, we can see that the maximum percentage difference is only 1.722% (4.10x10-4 hr). Hence, it can be concluded that the thermal process to cool the trunnion-hub assembly is accurate. B.5 Temperature Distribution of Copper Wire from ANSYS B.4.3 Comparison of Actual Solution vs. ANSYS Solution To verify the accuracy of the result obtained from ANSYS, the percentage difference between temperatures at specific times from the actual solution, obtained using Maple, and the temperatures at the sam on from 88
89 Temperature ( F ) Appendix B (Continued) Time ( hr ) ANSYS Solution Maple Solution Difference Percentage Difference 1.00x10-5298.929 299.095 0.165 0.055 2.00x10-5297.846 298.193 0.348 0.117 5.00x10-5294.634 295.514 0.879 0.298 1.40x10-4285.452 287.690 2.238 0.778 4.10x10-4261.468 266.049 4.582 1.722 1.22x10-3211.694 214.980 3.286 1.528 2.13x10-3174.463 176.085 1.622 0.921 3.04x10-3149.642 150.347 0.706 0.469 3.95x10-3133.095 133.316 0.222 0.166 4.86x10-3122.063 122.046 0.017 0.014 5.76x10-3114.709 114.655 0.054 0.047 6.67x10-3109.806 109.698 0.108 0.099 7.58x10-3 106.537 106.417 0.120 0.113 8.49x10-3104.358 104.246 0.112 0.107 9.40x10-3102.905 102.810 0.095 0.093 1.03x10 -2 101.937 101.868 0.069 0.068 1.12x10 -2 101.291 101.242 0.050 0.049 1.20x10 -2 100.903 100.864 0.039 0.039 Table B.2 Comparison of Temperatures fr om Maple and ANSYS for Specific Times erature changes. These properties include thermal conductivity, specific heat, and thermal expansion of the material, and heat transfer co efficient of liquid nitr ogen. This test was B.5 Test 2 for Thermal Analysis for Cooling in a Liquid Bath The properties of the mate rial used (ASTM A203-A St eel), and that of liquid nitrogen vary as a function of temperatur e and are not constant as the temp
Appendix B (Continued) 90 done to compare the cooling of a cylinder with constant properties against the cooling propce, the properties of the different teme values is known, talue wrty. The hypothesis for that botila the mean of the varying ps choseonstant vever, conroperties cannot be used for the trunnion-hub assembly process because the thermnsion doefect the cignificaffects thestresses. resultsn in figund it can be clearly seen that both the ccedures similar, ting the acof the simn of the ccess wiature dependent properties. with varying erties. Sinhe mean v material for peratur as chosen as the constant prope his test is t h the cooling procedures will be sim r since roperties i n as the c alue. How stant p al expa s not af ooling as s antly as it thermal The are show re B.6, a ooling pro are very hus, prov curacy ulatio ooling pro th temper Figure B.6 Comparison of Cooling Processes with Constant and V arying Properties
Appendix B (Continued) 91 was found. According to our hypoth B.6 Test 1 for Structural Analysis for Thermal Stresses This test is done to verify the structural analysis for thermal stresses. The hypothesis for this test is that the body will not experience any stresses due to temperature when the temperature gradient in the body is close to or equal to zero. A copper cylinder was chosen for this test. A thermal analysis was done in ANSYS, where it was cooled in a liquid bath until it reached steady state. The stresses developed in the cylinder due to the temperature gradient while cooling esis, as the cylinder reaches steady state, the stresses in the cylinder should also approach zero. The cooling of the cylinder and the thermal stresses developed are shown in figures B.7 and B.8, respectively. F igure B.7 Temperature Distribution for Cooling of the Copper Cylinder
Appendix B (Continued) Figure B.8 Thermal Stresses during the Cooling of the Copper Cylinder The cylinder reaches steady state of 300K after approximately 11,000 seconds. When comparing this with the thermal stress plot, the stresses rise sharply during the first few seconds of cooling but it decreases and it approaches zero, and is equal to zero after approximately 11,000 seconds. Thus, we can be sure that the thermal stress is accurate. 92
Appendix B (Continued) 93 B.7 Test 2 for Structural Analysis for Thermal Stresses For this test, the compound cylinder problem from section B.1 is made use of. The same hypothesis is used that the stresses due to temperature gradient approach zero as the body approaches steady state. In this case, the compound cylinders are first shrink fitted with an interference fit, thus, producing interference stresses. This compound cylinder is then cooled in a liquid bath of -321 0F, till it reaches steady state. The stresses in the cylinders are then found. According to our hypothesis, the stresses should approach values close to the interference stresses as the cylinders approach steady state. This happens because the total stress increases during cooling because of the added stresses due to the temperature gradient. However, the temperature gradient approaches zero as the cylinder reaches steady, thus, nullifying the thermal stresses. Figure B.9 shows the Von-Mises stress in the cylinders, which is purely interference stresses; the maximum being 12363 psi and the minimum being 3210 psi. Figure B.9 Von-Mises Stresses in the Compound Cylinders after Interference
Appendix B (Continued) 94 till it reaches steady state. Figure The cylinder is now cooled in a liquid bath of -321 0 F B.10 shows the temperature distribution of the cylinders. The maximum temperature is -49.614 0 F and the minimum temperature is -326.924 0 F 5 It can be seen that only parts of the cylinder attain the temperature of the liquid at this time-step and not the entire cylinder. Hence, there is some temperature gradient in certain parts of the cylinder and consequently, those parts of the cylinder will have stresses greater than the interference stresses. Figure B.10 Temperature Distribution in Compound Cylinders 50 The temperature is less than -321 F because ANSYS extrapolates to get values for nodes that do not have specific values.
Appendix B (Continued) 95 Figure B.11 Von-Mises Stresses in the Compound Cylinders after Cooling The Von-Mises stress is again plotted for the compound cylinders after the cooling, as shown in figure B.11. According to our hypothesis, the stresses are expected to be close to the interference stresses. The minimum stress here is 3270 psi, which is very close to the minimum interference stress value, 3210 psi. The maximum stress here is 18438 psi, where as the maximum interference stress value was 12363 psi. The difference in the maximum stresses is due to the temperature gradient in parts of the cylinder that have not yet reached steady state. However, if the cylinders would be cooledck to the interference stress values. This test analysis confirms the thermal stress analysis and also the entire simulation for the trunnion-hub assembly as the trunnion-hub assembly undergoes identical conditions. till it totally reaches steady state, we could see the stresses going ba
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Paul, Jai P.
Sensitivity analysis of design parameters for trunnion-hub assemblies of bascule bridges using finite element methods
h [electronic resource] /
by Jai P Paul.
[Tampa, Fla.] :
b University of South Florida,
Thesis (M.S.M.E.)--University of South Florida, 2005.
Includes bibliographical references.
Text (Electronic thesis) in PDF format.
System requirements: World Wide Web browser and PDF reader.
Mode of access: World Wide Web.
Title from PDF of title page.
Document formatted into pages; contains 107 pages.
ABSTRACT: Hundreds of thousands of dollars could be lost due to failures during the fabrication of Trunnion-Hub-Girder (THG) assemblies of bascule bridges. Two different procedures are currently utilized for the THG assembly. Crack formations in the hubs of various bridges during assembly led the Florida Department of Transportation (FDOT) to commission a project to investigate why the assemblies failed. Consequently, a research contract was granted to the Mechanical Engineering department at USF in 1998 to conduct theoretical, numerical and experimental studies. It was found that the steady state stresses were well below the yield strength of the material and could not have caused failure. A parametric finite element model was designed in ANSYS to analyze the transient stresses, temperatures and critical crack lengths in the THG assembly during the two assembly procedures.The critical points and the critical stages in the assembly were identified based on the critical crack length. Furthermore, experiments with cryogenic strain gauges and thermocouples were developed to determine the stresses and temperatures at critical points of the THG assembly during the two assembly procedures.One result revealed by the studies was that large tensile hoop stresses develop in the hub at the trunnion-hub interface in AP1 when the trunnion-hub assembly is cooled for insertion into the girder. These stresses occurred at low temperatures, and resulted in low values of critical crack length. A suggestion to solve this was to study the effect of thickness of the hub and to understand its influence on critical stresses and crack lengths.
Adviser: Dr. Autar K. Kaw.
Design of experiments.
Non-linear material properties.
x Mechanical Engineering
t USF Electronic Theses and Dissertations.