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UBC Theses and Dissertations

Finite element analysis and cost/risk assessment of the flex cover system Fridriksson, Johann 2011

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FINITE ELEMENT ANALYSIS AND COST/RISK ASSESSMENT OF THE FLEX COVER SYSTEM  by Johann Fridriksson  B.A.Sc, Reykjavik University, 2010  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF  MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES (Civil Engineering)  THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver)  August 2011  © Johann Fridriksson, 2011  ii Abstract The flex cover system has two main components, the flex cover which is a flexible cover for coal transportation railcars, and a handling machine.  The flex cover system is currently being developed by Empire Dynamic Structures. The research presented in this thesis revolves around the flex cover itself and not the cover handling machine although the entire concept is introduced in some detail. The flex cover is described in detail and extensive analysis was performed to determine the behaviour of the cover under static and dynamic loading.  Additionally a cost/risk analysis was performed to break down and analyse the cover by components and determine the economical impacts involved with fabrication of the cover. The finite element analysis was performed by conventional methods using the software ANSYS.  The cost/risk analysis was done by creating a decision tree and using expected monetary value in the software DecisionPro. The finite element analyses showed that the original design of the cover holds up in the conditions imposed on it when it is undamaged but some types of damage can quickly impair on the performance. Due to a lack of solid data the cost/risk analysis did not yield useful results as most of the input values had to be made up.  The model was still constructed and is ready for future use when better data has been acquired.    iii Table of Contents Abstract .............................................................................................................................. ii Table of Contents ............................................................................................................. iii List of Tables .................................................................................................................. viii List of Figures ................................................................................................................... ix Acknowledgements ......................................................................................................... xii Chapter  1: Introduction ................................................................................................... 1 1.1 Motivations .......................................................................................................... 2 1.2 Objectives ............................................................................................................ 2 Chapter  2: The Flex Cover System ................................................................................. 3 2.1 Introduction .......................................................................................................... 3 2.2 Mechanical Design of the Cover Handling Machine ........................................... 5 2.3 The Flex Cover .................................................................................................... 8 2.4 Glass – Reinforced Plastics (GRP) .................................................................... 10 2.4.1 Fibreglass Production ................................................................................. 10 Chapter  3: Performance Assessment and Decision Making ....................................... 12 3.1 Introduction ........................................................................................................ 12 3.2 Decision Analysis and Making .......................................................................... 13 3.3 Framework of the Decision Problem ................................................................. 14 3.3.1 Decision Situation ...................................................................................... 14 3.3.2 Types of Decision Theories and Making ................................................... 15 3.3.3 Uncertainties in Engineering ...................................................................... 15 3.3.3.1 Introduction ......................................................................................... 15  iv 3.3.3.2 Decision Making under Uncertainty ................................................... 16 3.3.4 Utility and Risk .......................................................................................... 17 3.3.4.1 Introduction ......................................................................................... 17 3.3.4.2 Risk in Engineering ............................................................................. 17 3.3.4.3 Decision Making under Risk ............................................................... 19 3.4 Engineering Tools .............................................................................................. 21 3.4.1 Introduction ................................................................................................ 21 3.4.2 Decision Trees ............................................................................................ 21 3.4.3 Monte Carlo Simulations ........................................................................... 22 3.4.4 Expected Monetary Value .......................................................................... 23 3.5 Cost/Risk Analysis ............................................................................................. 24 3.5.1 Introduction ................................................................................................ 24 3.5.2 Major Components of the Flex – Cover ..................................................... 25 3.5.3 Important Considerations ........................................................................... 26 3.5.4 Model Construction .................................................................................... 27 3.5.5 Cost ............................................................................................................. 28 3.6 Inefficiency ........................................................................................................ 30 Chapter  4: Finite Element Analysis of the Flex Cover ................................................ 32 4.1 Flow Around Objects ......................................................................................... 32 4.1.1 Reynolds Number ....................................................................................... 33 4.1.2 Strouhal Number ........................................................................................ 34 4.1.3 Gust Factor ................................................................................................. 34 4.1.4 Drag Coefficient ......................................................................................... 36  v 4.1.5 Vortex Shedding and the Kármán Vortex Street ........................................ 36 4.2 Aerodynamics of Trains ..................................................................................... 37 4.3 Wind Speed in Canada ....................................................................................... 39 4.4 Operational Speeds of Coal Transportations Trains .......................................... 40 4.5 Computational Fluid Dynamics (CFD) .............................................................. 40 4.5.1 What is CFD ............................................................................................... 40 4.5.2 Validation of Existing CFD ........................................................................ 41 4.5.3 Implementation and Results of CFD .......................................................... 41 4.6 Tools .................................................................................................................. 46 4.6.1 ANSYS ....................................................................................................... 46 4.6.2 Operating ANSYS ...................................................................................... 46 4.7 Description of Model ......................................................................................... 47 4.7.1 Longitudinal Beams and Ribs .................................................................... 47 4.7.2 Skin ............................................................................................................. 48 4.7.3 End Hatches ................................................................................................ 48 4.7.4 Boundary Conditions .................................................................................. 48 4.8 Sources of Error ................................................................................................. 48 4.8.1 Skin ............................................................................................................. 49 4.8.2 Position of Beam Sections .......................................................................... 49 4.8.3 End Hatches ................................................................................................ 50 4.9 Finite Element Analysis ..................................................................................... 50 Chapter  5: Details and Results From Finite Element Analyses ................................. 51 5.1 Introduction ........................................................................................................ 51  vi 5.2 Shedding Frequencies ........................................................................................ 51 5.3 Changing the Boundary Conditions ................................................................... 52 5.3.1 Modal Analysis .......................................................................................... 52 5.3.2 Mass Participation ...................................................................................... 55 5.3.3 Static Analysis ............................................................................................ 57 5.3.4 Reactions at Supports ................................................................................. 60 5.4 Holes in the Cover ............................................................................................. 62 5.4.1 Modal Analysis .......................................................................................... 62 5.4.2 Static Analysis ............................................................................................ 63 5.5 Perpendicular Cuts ............................................................................................. 66 5.5.1 Modal Analysis .......................................................................................... 66 5.5.2 Static Analysis ............................................................................................ 68 5.6 Hatch Doors ....................................................................................................... 70 5.6.1 Dynamic Analysis ...................................................................................... 70 5.6.2 Static Analysis ............................................................................................ 70 5.7 Latch Failure ...................................................................................................... 73 Chapter  6: Conclusions .................................................................................................. 76 6.1 Cost/Risk Assessment ........................................................................................ 76 6.1.1 Uncertainties ............................................................................................... 76 6.1.2 Risk ............................................................................................................. 76 6.1.3 Inefficiencies .............................................................................................. 76 6.2 Finite Element Analyses .................................................................................... 77 6.2.1 Dynamic Analyses ...................................................................................... 77  vii 6.2.2 Static Analyses ........................................................................................... 77 Works Cited ...................................................................................................................... 78 Appendix A Drawings of Flex Cover ............................................................................ 80 Appendix B Drawings of Rail Car ................................................................................. 82 Appendix C Script files for ANSYS .............................................................................. 83 C.1 Cover with Simple Supports ...................................................................... 83 C.2 Modal analysis ............................................................................................ 89 C.3 Static Analysis ............................................................................................ 89 C.4 Cutting Holes in Your Model ..................................................................... 90 Appendix D Expected Monetary Value Example .......................................................... 91 D.1 Example from James Jones ........................................................................ 91 D.2 Actions ....................................................................................................... 91 D.3 States of Nature .......................................................................................... 91 D.4 Payoff Table ............................................................................................... 92 D.5 Expected Value Criterion ........................................................................... 94 D.6 Maximax Criterion ..................................................................................... 94 D.7 Maximin Criterion ...................................................................................... 94 D.8 Minimax Criterion ...................................................................................... 95 D.9 Putting it all Together ................................................................................. 95   viii List of Tables Table 1 Timeline for typical cover handling cycle ............................................................. 7 Table 2 Material properties for GRP components .............................................................. 9 Table 3 Typical properties for some common fibres ........................................................ 10 Table 4 Values of payoffs and probabilities ..................................................................... 19 Table 5 Wind speeds used for CFD .................................................................................. 42 Table 6 Average pressure over cover ................................................................................ 43 Table 7 Complete results for hatch doors. ........................................................................ 45 Table 8 Shedding frequencies of the covered rail car ....................................................... 51 Table 9 Comparison of frequencies for first 5 modes ....................................................... 52 Table 10 Mass participation along x – axis. ..................................................................... 55 Table 11 Mass participation along y – axis. ..................................................................... 55 Table 12 Mass participation along z – axis. ...................................................................... 56 Table 13 Mass participation per mode .............................................................................. 56 Table 14 Reactions for simple supports ............................................................................ 60 Table 15 Reactions for fixed supports .............................................................................. 61 Table 16 Frequencies of 5 modes under all modelled damage states (holes) ................... 63 Table 17 Frequencies of 5 modes under all modelled damage states (cuts) ..................... 66 Table 18 Frequencies of 5 modes under all modelled damage states (hatches) ............... 70 Table 19 Support reactions for simply supported cover ................................................... 73 Table 20 Support reactions with a single latch missing .................................................... 74 Table 21 Support reactions with two consecutive latches missing ................................... 75 Table 22 Support reactions with three consecutive latches missing ................................. 75  ix  List of Figures Figure 1 EDS´s idea of a cover handling machine ............................................................. 1 Figure 2 EDS´s concept for the cover handling machine ................................................... 3 Figure 3 Grabbing points .................................................................................................... 4 Figure 4 Range of motion ................................................................................................... 4 Figure 5 Size requirements of buildings ............................................................................. 5 Figure 6 Machine components ............................................................................................ 5 Figure 7 Drive carriage and machine track ......................................................................... 6 Figure 8 Lifting arms .......................................................................................................... 6 Figure 9 Grabber frame ....................................................................................................... 7 Figure 10 Lifting arms in upper and lower positions .......................................................... 8 Figure 11 Uncertainty and economic impact versus time ................................................. 12 Figure 12 Risk utility functions ........................................................................................ 18 Figure 13 Decision under risk ........................................................................................... 20 Figure 14 Decision tree layout .......................................................................................... 21 Figure 15 The flex cover attached to a railcar .................................................................. 25 Figure 16 Connection detail at bottom corners ................................................................. 26 Figure 17: Total project cost ............................................................................................. 27 Figure 18: Design decision branch ................................................................................... 29 Figure 19: Fabrication decision branch ............................................................................. 30 Figure 20: Material procurement decision branch ............................................................ 30 Figure 21 Vector plot of object moving through open air (α – yaw angle) ...................... 32  x Figure 22 Left: Streamlined body, flow follows contours.  Right: Bluff body, flow separates. ........................................................................................................................... 33 Figure 23 Calculated gust factors (Naessa, 2000). ........................................................... 35 Figure 24 Drag coefficients of various shapes plotted against Reynolds number ............ 36 Figure 25 The Kármán vortex street ................................................................................. 37 Figure 26 Schematic view of vortex development for moderate and small yaw angles ... 38 Figure 27 Vortex development with yaw angle exceeding 80° ........................................ 39 Figure 28 Map of design wind speed in Canada (50 year return period, NBCC) ............. 39 Figure 29 Pressure contour plot for yaw angle 90° and wind speed 33 m/s. (From CFD) 41 Figure 30 Load cases ........................................................................................................ 42 Figure 31 Pressure contour for load case 1 (Head wind) .................................................. 43 Figure 32 Pressure contour for load case 2 (Wind 90°) .................................................... 44 Figure 33 Pressure contour for load case 3 (Wind 29°) .................................................... 44 Figure 34 Pressure contour for hatch door (Case 3) ......................................................... 45 Figure 35 Extruded elements ............................................................................................ 49 Figure 36 Model of the flex cover .................................................................................... 50 Figure 37 First mode shape ............................................................................................... 52 Figure 38 Second mode shape .......................................................................................... 53 Figure 39 Third mode shape ............................................................................................. 53 Figure 40 Fourth mode shape ........................................................................................... 54 Figure 41 Fifth mode shape .............................................................................................. 54 Figure 42 von Mises stress contour for simple support condition .................................... 57 Figure 43 von Mises stress contour for fixed support conditions ..................................... 58  xi Figure 44 Deflections (Simply supported) ........................................................................ 59 Figure 45 Deflections (Fixed supports) ............................................................................ 59 Figure 46 Numbering of supports ..................................................................................... 60 Figure 47 Position of holes in cover ................................................................................. 62 Figure 48 Von Mises stress contour (3 holes) .................................................................. 64 Figure 49 Deflections (3 holes) ......................................................................................... 65 Figure 50 Position of cuts in cover ................................................................................... 66 Figure 51 Mode shape of second mode with cuts ............................................................. 67 Figure 52 von Mises stress contour with cuts in three bays ............................................. 68 Figure 53 Deflections with cuts in three bays ................................................................... 69 Figure 54 Stress contour (1 hatch missing) ....................................................................... 71 Figure 55 deflections (1 hatch missing) ............................................................................ 72 Figure 56 Support numbering ........................................................................................... 73   xii Acknowledgements I would like to thank my supervisor Dr. Siegfried F. Stiemer and Dr. Solomon Tesfamariam, co-signer for the funding of the project, for presenting me with a chance to work on an industry connected project, and their guidance.  I would also like to extend my gratitude to a few of my fellow colleagues: Dr. Wudi, assistant professor of civil engineering in the earthquake engineering test center at Guangzhou University in China, for his help with the finite element modeling. A special thanks goes to Kevin Spacey, masters student in the department of civil engineering at UBC, for his contribution to this thesis.  He is credited for the work presented in chapter 3 of this thesis.  Last but not least my loving wife Eygló and two beautiful sons Kolbeinn and Friðrik for their support and patience, and my parents Friðrik and Hildur for their support while I completed my education.   1 Chapter  1: Introduction Coal is most commonly transported to power plants and other mass consumers using railcars. These railcars typically do not have any covers during transport because the low value of coal cannot support the cost of a cover handling system (Figure 1).  In addition there are technical challenges associated with providing covers for coal cars. The coal loading is a continuous process where the train runs at a steady speed of 0.4 mph (Stobart, 2008) so there is very little time for any kind of cover handling operations. Therefore the covers and cover handling system must be extremely reliable since any slow down or stops could result in large economic impacts.  At the other end where the rail cars are emptied the train stops and gets positioned correctly on a roller dumper using several lever arms, still high reliability is required.  In this thesis the flex cover system will be presented.  The focus is mainly on evaluating the serviceability, safety and cost effectiveness of the cover.  The cover handling system will also be introduced.   Figure 1 EDS´s idea of a cover handling machine  2 1.1 Motivations There are several major benefits of putting covers on the railcars, both environmental and economical.  Blown coal dust results in wasted product and environmental contamination.  Coal dust is also known to cause major damage to rail lines, often resulting in train derailments: the coal dust fills into the track ballast, limiting water drainage, and subsequent freeze thaw cycles result in heaving of the rail track. Another issue is coal getting wet from rain and snow fall and then freezing to the side of the railcar so that a substantial amount of coal might be hauled back and forth.  Lastly, analysis and tests have shown that the fuel consumption is nearly equivalent when hauling both full and empty railcars due to the increased aerodynamic drag of the empty cars.  The savings in fuel consumption alone could support the cost of the cover handling system.  1.2 Objectives The main objectives of this project are to provide an initial assessment of strength and serviceability for the prototype cover and suggest improvements for industrialization. Secondly to provide a cost/risk analysis. This includes:  Development of wind loads that the cover might be subjected to during transport; examples include cover missing from adjacent car and wind gusts.  Analysis of vortex shedding from the covered railcar and natural frequencies of the cover.  Assessment of cover strength and capacity based on the developed load cases.  Cost/risk analysis where manufacturing costs are considered.   3 Chapter  2: The Flex Cover System 2.1 Introduction The flex cover system is comprised of a flexible cover made from fibreglass and a cover handling machine.  The flex cover is latched on to a rail car using steel latches. Empire Dynamic Structures (EDS) have taken on the design of the cover handling machine and have already come up with a conceptual design which is currently under further development.  The machine (figure 2) grabs the cover by both sides, lifts it off and hangs it flush on the side of the railcar.  The details of the machine design are reported on in the next few pages.   Figure 2 EDS´s concept for the cover handling machine  The details of the machine design are: 1. Reliability of cover handling (Figure 3)  Grabbing cover along both edges provides very good control of cover.  Little chance of cover becoming unhinged during opening/closing motion due to wind gusts or rapid train movements.  4  Figure 3 Grabbing points  2. Compact motion profile (Figure 4)  Shorter range of motion better enables the machine to operate at higher train speeds  Figure 4 Range of motion 3. Minimal space requirements (Figure 5)  Compact motion profile results in reduced building size requirements  5  Figure 5 Size requirements of buildings  2.2 Mechanical Design of the Cover Handling Machine  Figure 6 Machine components   6 In order for the cover handling machine to sync up with the train the machine is on separate tracks that run parallel to the train’s tracks.   Figure 7 Drive carriage and machine track The drive carriage is powered with electrical motors and will be equipped with sensors that give the position and speed of the incoming railcars.   Figure 8 Lifting arms  7 The lifting arms are equipped with hydraulic jacks that control the movement of the grabber frame.  The two visible jacks (Axis 1 and Axis 2) control the positioning of the frame and the hidden one (Axis 3) rotates the grabber frame.   Figure 9 Grabber frame The grabber frame is equipped with twelve grabber arms, six on each side, to match the number of latches.  Table 1 Timeline for typical cover handling cycle  8 As stated in the introduction the operational speed of the train during loading is 0.4 mph.  The engineers at EDS have chosen to design the machine for 1.6 mph, resulting in a 22 second cycle time (table 1) so they have implemented a safety factor of 4 for the cycle time.  Figure 10 Lifting arms in upper and lower positions  2.3 The Flex Cover The flex cover was designed specifically to cover coal transportation rail cars that utilize a rotary car dumper to empty the rail cars.  The flexibility of the cover is a major benefit when unloading the railcars as the cover hangs flush on the side of the rail car while it is emptied (figure 10 left).  The reason why the cover needs to be flush on the side is that there are only 12 inches of clearance to the side when the dumper rotates the rail car. The original design utilizes very little material where structural components are very slender and the skin is extremely thin.  The structural components, longitudinal beams and ribs, have the dimensions 3” x 3/8” (76.2mm x 9.5mm) and the skin is 3/32” (approx. 2.4mm).  The complete weight of a single cover is estimated to be around 500 kg.  Complete drawings of the flex cover are included under appendix A.  Both  9 components are made from glass-reinforced plastic (GRP), the beams only have fibres in the longitudinal direction and the skin in two perpendicular directions.  The cover is latched on to the rail car with 6 steel latches on each side that are connected to the ribs. At both ends there are curved hatches that lay flush with the cover when flat and are inclined at about 45° when the cover is latched on to a rail car. Table 2 shows the material properties for the composite fibreglass components.   Table 2 Material properties for GRP components  The longitudinal beams and ribs are made with a commercial product manufactured by Bedford reinforced plastics.  The skin is made at the fabricators production facility by laying out a woven fibre cloth, with fibres in two perpendicular directions, in polyethelyn laminate. The fabricator has designed a cover that has the flexibility needed to perform in terms of making a curved shape when placed on a railcar and hanging flat off the side Youngs modulus 1.1 Msi 4.5 Msi 7.3 GPa 31.0 GPa Ult. tens. stress 17.3 ksi 90 ksi 119.5 MPa 620.5 MPa Ult. comp. stress 24.1 ksi 50 ksi 166.1 MPa 344.7 MPa Ult. bend. stress - ksi 100 ksi - MPa 689.5 MPa skin beam Material properties  10 when it is taken off.  No attention has been paid to ultimate strength and dynamic behaviour.  2.4 Glass – Reinforced Plastics (GRP) Glass-reinforced plastic, more commonly referred to as fibreglass, is a composite material made of a plastic matrix reinforced by glass fibres.  Some of the things that make fibreglass an attractive option include its light weight, strength and durability.  When compared to carbon fibre the fibreglass has less strength and stiffness, but on the other hand is less brittle and the raw materials are far less expensive.   Table 3 Typical properties for some common fibres  2.4.1 Fibreglass Production Glass fibres are made by extruding liquid glass through bundles of tiny orifices, typically about 17 – 25 micrometers in diameter for E – Glass and 9 micrometers for S – Glass.  These filaments are then sized with a chemical solution and bundled together to make a roving.  The chemical solution used in the sizing is a sort of coating or primer that both helps protect the filaments during further processing as well as insure a proper Material Density (g/cm 3 ) Tensile strength (MPa) Young´s modulus (GPa) E-Glass 2.55 2000 80 S-Glass 2.49 4750 89 Alumina (Saffil) 3.28 1950 297 Carbon 2 2900 525 Kevlar 29 1.44 2860 64 Kevlar 49 1.44 3750 136  11 bonding between the filaments and plastic matrix.  The coating allows for transfer of shear forces to the plastic matrix and makes sure the fibres do not slip in the plastic which would cause a localized failure in the composite material. The rovings can be used directly for making a composite material or made into fabrics such as, woven fabrics, knit fabrics or uni – directional fabrics depending on what qualities the end product shall possess.  Other applications include gun roving where an automated gun chops up the filament into short lengths and mixes with resin.  The material is then sprayed into a mold (Fibreglass, 2011).   12 Chapter  3: Performance Assessment and Decision Making 3.1 Introduction As the knowledge of engineering increases and our tools for analysis grow stronger we find new challenges and improved ways to tackle old ones.  Performance based design has in the past decades become an increasingly popular tool in engineering design.  The concept has in recent years also been implemented as a tool in decision making.   Figure 11 Uncertainty and economic impact versus time  In today´s business driven world there is a certain monetary value attached to everything around us.  At the start of a new project the main target is usually to design and build a structure or product, without exceeding a certain amount of money, within a given timeline.  Figure 11 shows a plot of uncertainties and economic impact versus time. In the early stages both the uncertainties and the economic impact can be great but as the project progresses they both decrease (Ramadhas, 2005).  13 Uncertainties are usually determined through analyzing historical data and engineering judgement.  These uncertainties are defined based on cause of failure, design and manufacturing of products, maintenance and application so the reliability assessment therefore becomes less theoretical and more practical.  Hence it can be concluded that an engineer’s success in reliability analysis is based on implementation of the methods and tools available along with his own judgement (Preidt). In an environment where liability is a big concern, being able to support decisions with proven decision making theories and analysis is a great asset (Ramadhas, 2005).  3.2 Decision Analysis and Making Decision analysis is a process dealing with analytical techniques to help guide a choice, which varies between each decision maker based on their values.  Decision analysis is a combination of many things including, philosophy, theory, methodology and professional experience.  The input values are put forth as intervals, mean and standard deviation, meaning the outcome will also be an interval.  A common value that most decision makers agree upon is economic benefit, increasing profits or decreasing cost. Other personal values may also contribute to the decision process such as health, happiness or security.  By placing such values in a project, more informal decisions can be made.  By structuring a decision and breaking down the unknown events which may affect the results, more caution is provided.  Once a functional model is achieved the decision maker can modify the values to find how they affect the outcome.  By doing so the input values can be tested for sensitivity and the most critical parts can be identified.  14 Through intuition and logic, and by using this mathematical process, a better understanding of the relationship between actions and objectives can be reached.  3.3 Framework of the Decision Problem 3.3.1 Decision Situation The overall framework of decision making can be broken down into basic steps. Although these steps vary with between situations, a simple framework can help visualize the problem at hand.  The following framework is suggested in a paper by Vignesh Ramadhas (Ramadhas, 2005):  1.  Identification  of  an  array,  Ai  ( i  =  1,  2,  3…n  ),  which  consists  of  all mutually exclusive and  collectively exhaustive alternative strategies that the decision maker may adopt in the given situation. 2.  Identification of an array, Sj   ( j = 1, 2, 3…m ), which consists of all mutually exclusive  and  collectively  exhaustive  states  of  nature  within  which  each  of the alternatives will operate. 3.  Prediction  and  evaluation  of  outcomes,  Xij,  for  all  possible  combinations of alternatives and states of nature. Where Xi corresponds to the outcome of strategy Ai when the state of nature is Sj. 4.  Allocation of a probability Pj to each of the states of nature, there by forming a probability array. 5.  Evaluation of each alternative Ai, based on select criteria and arriving at the best possible alternative.  15 These major components can be used to describe the framework of a system, although they may not provide sufficient detail for all decision making processes.  3.3.2 Types of Decision Theories and Making Decision making and decision theories are highly dependent on the information available and the amount of alternatives given.  There are a variety of alternatives that arise depending on the amount of information.  Uncertainties, risk and a lack of information are a common part of modern engineering.  By using decision tools and engineering judgement it is the engineer´s job to determine the most proper course of action.  Several classic methods are available to help determine and make better decisions under uncertainties, risk, and incomplete knowledge (Ramadhas, 2005).  3.3.3 Uncertainties in Engineering 3.3.3.1 Introduction The conceptualization of real life structures, situations, or phenomenon aids an engineer in solving problems.  These conceptual models are replicated to the best of the engineer’s knowledge and experience.  Sometimes such models do not accurately replicate real life scenarios, bringing uncertainties into play (Ramadhas, 2005).  There are two types of uncertainties that result in a problem, aleatory and epistemic.  Aleatory uncertainty, or statistical uncertainty, arises from physical inconsistency that is inherent in the unpredictable nature of future events.  Epistemic uncertainty, or systematic uncertainty, arises from the differences in expert knowledge of modeling assumptions,  16 unknown or only partially known parameters, and errors in extrapolation (Finn, 2010). These two uncertainties help determine where clarification can be made, if available. The four decision theory criterions that will be examined will include optimist, pessimist, opportunist, and realist viewpoints (Jones, 2005).  3.3.3.2 Decision Making under Uncertainty There are several criteria available for engineers or decision makers to aid in decision making when a surplus of uncertainties is present.  The four criteria discussed here include the Maximax, Maximin, Minimax, and Expected Value criterions. The Maximax, an optimistic criterion, looks at the best that could happen under each action and then chooses the action with the largest value. The decision maker assumes the greatest outcome will occur and their decision will provide the best scenario. This type of decision maker sees the maximum of the maximums or the "best of the best" (Jones, 2005).  A key example of this type of decision maker is a lottery player; they seek large payoffs while ignoring the probabilities of success. The Maximin, a pessimistic criterion, looks at the worst that could happen under each action and then chooses the action with the largest payoff (Jones, 2005).  The decision maker acts under the assumption that the worst that can happen will.  They then choose the action with the best worst case scenario that arises.  This type of decision maker puts their money into a savings account because they could lose money at the stock market (Jones, 2005). Minimax is decision making based on opportunistic loss.  These are the kind of decision makers who look back after the state of nature has occurred and say "Now that I  17 know what happened, if I had only picked this other action instead of the one I actually did, I could have done better" (Jones, 2005).  This is similar to the maximin principle in theory only in this case the decision maker wants to minimize the loss of a certain action. The realistic criterion computes the expected value under each action and then picks the action with the largest expected value.  This is the only method of the four that incorporates the probabilities of the states of nature.  A detailed example showing the different decisions and payoffs by using the four previously stated criterions can be found in Appendix D.  3.3.4 Utility and Risk 3.3.4.1 Introduction Utility is a term used for the level or measure of relative satisfaction given a possible outcome based on monetary value.  A decision maker’s attitude towards risk directly affects their utility, this is also known as risk tolerance (Ramadhas, 2005).  An investor's risk tolerance varies according to age, income requirements, financial goals, undesirable results, among others.  For example, a 70-year-old retired widow has a lower risk tolerance than a single 30-year-old executive, who generally has a longer time frame to make up for any losses.  3.3.4.2 Risk in Engineering The utility functions in figure 12 represent the various levels of risk tolerance an investor may incur (Ramadhas, 2005).  18  Figure 12 Risk utility functions  For a risk adverse person the utility value of each dollar is less than that of the previous dollar, meaning that they value the alternatives at less than their expected values (Stiemer, 2011).  An example of this is a person who puts his or her money into a bank account with a low but guaranteed interest rate instead of buying stock which could yield a huge payoff but also involves a risk of losing value.  Someone who is risk neutral is willing to play the long-run odds when making decisions, and will evaluate alternatives according to their expected values.  A key example for this type of risk tolerance involves a decision maker being indifferent to receiving an average payoff; say $1, or an alternative with equal chances of yielding $0 and $2.  If a person is risk seeking, then the utility value of each dollar is greater than that of the pervious dollar (Ramadhas, 2005). As opposed to the risk adverse person this person would buy the stock hoping for the big payoff. To make this definition of value precise, we define the certain equivalent of an alternative as the amount that the decision maker would be indifferent between (1) having that monetary amount for certain and (2) having the alternative with its uncertain outcome.  19 3.3.4.3 Decision Making under Risk Typically when an engineer makes decisions where risk is involved the probabilities of occurrence are specified.  These probabilities are derived from the engineering judgement, personal experience, historical data, combined discussions etc. (Ramadhas, 2005).  Once specified these probabilities can be applied to the expected payoffs giving a greater indication of the most suitable alternatives.  An example of this is applying probabilities to several parameters of a design to determine whether the project is feasible or non-feasible based on monetary value.   Table 4 Values of payoffs and probabilities  Using the values of payoffs and the subjective probability values the expected payoff values for each of the alternatives can be determined as shown. To quickly explain the calculations: for design option 3 you have 0.7 * 200000 + 0.3 * (-150000) = 95000. Payoff Probability Payoff Probability Design Option 1 100000 50% -50000 50% Design Option 2 150000 20% -100000 80% Design Option 3 200000 70% -150000 30% Available Alternatives States of Nature Feasible Not Feasible  20  Figure 13 Decision under risk  Using this method the alternative that represents the best option is expressed in terms of maximum profit obtained.  Design option 3 (figure 13) gives the largest payoff using this technique. However, this method does not exclude the fact that during some circumstances, the chosen decision may not be the best or most profitable alternative (Ramadhas, 2005).  The advantage of this technique includes the probabilities of state of nature and estimated payoffs.  Important considerations should be noted when using such techniques because selection of alternatives for unique or extremely important projects may be unfavourable.  The probabilities used are subjective to the fact of informed decision making.  If an engineer produces uninformed values some unfavourable results may occur.  For the decision maker to use intuition as the basis of determining probabilities he must first define the level of risk.    21 3.4 Engineering Tools 3.4.1 Introduction A variety of mathematical techniques and computer software are available to aid in making more accurate predictions.  By using decision tools, decision makers can not only improve the quality of their predictions but also perceive their predictions with more clarity.  The records can then be kept and referred to with ease for future work.  Decision trees, which are discussed in the next section, provide an excellent method of planning response to random events.  Monte Carlo simulations, on the other hand, are best suited for modeling uncertainty and volatility.  Expected Monetary Value provides a total of the weighted outcomes associated with a decision.  These three tools provide valuable aids in decision processes and are discussed in the next few sections.  3.4.2 Decision Trees  Figure 14 Decision tree layout   22 Decision trees provide some of the most useful tools in decision making.  Their graphical layout gives the decision maker a chance to visualize the decision process (figure 14).  The function of the decision tree is to aid in the selection of the best course of action in cases where you have uncertainty.  Once modeled the decision process and probabilities associated with alternatives are easily analysed, the model also provides a good way to communicate the risks involved (Stiemer, 2011).  Constructing a risk profile graph or table illustrating all possible outcomes and probabilities is a useful tool. Personal risk aversion can even be implemented when constructing a decision model (Stiemer, 2011). Making decision trees with DecisionPro is a fairly simple task, as the user interface is similar to most spreadsheet programs.  DecisionPro allows the user to perform Monte Carlo simulations, Expected Monetary Value analysis, and many other probabilistic decision functions. Although the actual construction of the trees maybe simple some complications are attached.  Decision trees act as tools to aid decision makers.  However the actual probabilities, outcomes, alternatives, and inputs are made based on common sense and intuition.  Without a strong background or intuition for reliability, the decision process may be misguided.  3.4.3 Monte Carlo Simulations Monte Carlo simulation provides a sophisticated way to model the inherent uncertainties involved with making estimations and engineering judgment.  Realistic estimates can be made through sampling of data using brute force methods (Wong, 2006). These types of simulations provide an aid in selecting the best course of action, as well as  23 giving a range of outcomes, probabilities of reaching targets and most probable outcomes.  Monte Carlo simulation methods vary, but tend to follow a particular pattern (Kalos & Whitlock, 2008): 1. Define a domain of possible inputs. 2. Generate inputs randomly from a probability distribution over the domain. 3. Perform a deterministic computation on the inputs. 4. Aggregate the results. Applying Monte Carlo simulation requires a set data where mean and standard deviation values are known or can otherwise be determined.  3.4.4 Expected Monetary Value In probability theory the expected value of a random variable is the weighted average of all possible values that random variable can take.  Expected Monetary Value (EMV) is therefore the average amount of money you stand to gain from choosing a certain path.  It is expressed mathematically as the product of an event's probability of occurrence and the gain or loss that will result (Stiemer, 2011).  EMV analysis is most efficiently used with decision tree event nodes.  EMV verifies that all individual probabilities lie in the range 0 to 1 and that all probabilities add up to be 1.  The Expected Monetary Value is calculated using the equation:   Equation 1 Where:  pi – probability of component i, vi – value of component i   24 Where    if risk aversion is not applied, or       if risk aversion is applied.  Using DecisionPro the various risk assessments can be set and calculated.  The constant k is set using the Set Risk Aversion tool, accessed by pointing to Decision Tree in the Tools menu and clicking Set Risk Aversion (Utility) (Stiemer, 2011).  3.5 Cost/Risk Analysis 3.5.1 Introduction Using the Expected Monetary Value and Decision Tree applications, a cost/risk analysis has been done for the flex cover.  Through this risk analysis the alternative that produces the most reliable outcome can be found.  The reliability will be dictated through monetary values.  The alternative resulting in the lowest monetary value will be sought after.    25 3.5.2 Major Components of the Flex – Cover For the purpose of this analysis the Flex-cover has been identified to have four major components: latches, skin, ribs/edge beams and end connections.  The cover can be manufactured in several different ways, as of now, The fabricator fabricates the skin in house using a woven glass fibre and a polyethelyn matrix.  The ribs and edge beams are a commercial product manufactured by Bedford Reinforced Plastics.  The skin of the cover is 2.4mm thick. The six ribs are attached to the outer diameter of the skin.  Two edge beams and one spinal beam run the length of the cover on the inner diameter of the skin, all with thicknesses of 9.5mm.  Figure 15 The flex cover attached to a railcar  The cover-to-car latches are attached at the ends of the ribs, as seen in figure 15. A total of twelve coated steel latches, six per side, are attached at the rib ends.   The end hatches are fibre glass trusses.  The connection at the top is a hinge that can also slide along the longitudinal axis (see appendix A) and the connections at the bottom corners are equipped with a pump action cylinder (figure 16) that allows it to extend out as the cover flattens out.  26  Figure 16 Connection detail at bottom corners  3.5.3 Important Considerations The model created for this project provides a simple cost/risk calculation based on the knowledge of several colleagues and personal intuition.  The model is a simplified version of reality and the monetary values obtained are fictitious.  The material and labour costs included in the model disregard all fluctuations and inflation rates. One important consideration involves quality of laminated fibre glass production. It is assumed that basic machined laminate products will be far superior to handmade fibre glass products.  Therefore, in reality, the rib components purchased from Bedford Reinforced Plastics are stronger than those assumed in this report. This simplification is made to give alternatives to third party purchases of major components.  By assuming the quality of a handmade component is equal to a third party machined component, alternatives are available.  This assumption allows for the rib and cover skin system to either be purchased or made in house by labourers.   27 3.5.4 Model Construction In an attempt to incorporate the previous sections involving decision analysis and tools, a cost tree was developed using DecisionPro.  The model was developed based on the four different components of the Flex-cover.  Each component was analysed separately for the purpose of simplifying the model. Figure 17 is a screen capture of the total project cost from DecisionPro.  The cost of the major rib and cover components are calculated as the minimum cost of either fabricating or purchasing, while the end connections and latch components are bought through a third party. For rib and cover components EMV is included with fabrication results; if fabrication of a cover does not perform well costs and time delays may cause financial loss. The fabrication costs will depend on, labour wage and work hours, design costs, material costs, etc.  If the cost of the project exceeds a certain value due to the probabilities of failure, then continuing will run the risk of economical loss.  Figure 17: Total project cost  28 The input parameters considered for this model are relatively basic events deduced from intuition and past fibre-glass construction problems.  The actual values that are to be produced require much more experience or additional help from third parties.  3.5.5 Cost The costs illustrated in this example do not necessarily represent the final outputs of the Flex-cover system.  As mentioned earlier, the four major components of the Flex- cover system were analysed.  The costs associated with each component varied with the amount of information released.  Given the dimension drawings of the Flex-cover, Appendix A, the rib and skin could be broken down into three components: design, fabrication, and material procurement.  Using personal judgement, the end connections were broken down into two components: design and material procurement.  The final component, the latch, consisted of a material procurement branch. Design: For the rib, end connection, and cover skin, three possible scenarios were considered.  It is assumed that during the design phase, and engineer’s first proposal could be successful.  This success would incur the least cost in the project, as the amount of time would also be minimal.  The probability of occurrence selected for the rib and cover skin components are high because of the simplicity of the products and experience of the design team (Ramadhas, 2005).  Another possibility is an unsuccessful initial design but sufficient modifications to meet requirements.  The cost to re-engineer and modify the initial design would be slightly higher than before.  To determine the probability of failure, the success and modification probabilities would be added and subtracted from unity.  The probability placed on failure in design is low since such an  29 outcome would be very rare; however, ruling out this probability of failure all together would not be common due to the highly unpredictable nature any project (Ramadhas, 2005). The incurred costs from the failure event would be greatest.  The design scenario is represented in the decision tree below.  Figure 18: Design decision branch  Fabrication: Different from the design step, four possible outcomes were predicted in the fabrication stage for rib and cover skin systems.  Despite the addition of one more scenario, the first outcome would be similar to before; success of fabrication would have the least cost. Since fabricating the rib and cover skin were not all that superior to machined components, a reasonable probability was placed on the success of this outcome. The event of modifying the initial fabrication was given a small probability in the range of 10-20%.  The third event incorporates a greater change to the fabrication process and was also given a small probability, within the range of 7-10%.  The finale event of failure is calculated the same as previously stated.  The event of failure should always be incorporated due to the high unpredictability.  Figure 19 shows the fabrication scenario of the cover component.  30  Figure 19: Fabrication decision branch  Material Procurement: The Flex-cover system had several components where the material and costs were known and purchased.  The cover-to-car latch system had few issues in variability in unit cost and other variations, but for simplicity a deterministic approach was adopted (Ramadhas, 2005).  The simplistic scenario is shown below in figure 20.  Figure 20: Material procurement decision branch  3.6 Inefficiency A simple way to determine inefficiency is to analyse the most likely costs with the most favourable costs.  The method is taken from Ramadhas paper and implemented here help conclude overall results. The inefficiency definition as shown in Ramadhas paper is defined as:  31 100100 ____ ____       scenariofavorablemostofCost scenariolikelymostofCost  Equation 2  The cost of the most likely scenario is given as the EMV from the DecisionPro model, and the cost of most favourable scenario is typically the value from the success of a scenario.  When the two values are close the inefficiency is low resulting in little or no required changes.  If inefficiency is on the other hand high improvement is needed. Referring back to figure 19 the most likely scenario resulted in $316 cost, while the most favourable scenario resulted in $275 cost.  Calculating the inefficiency using the above equation gives 14.9%. This value shows that improvements can be made to the fabrication process.  Less uncertainties during fabrication and higher success rates amount to less inefficiency.   32 Chapter  4: Finite Element Analysis of the Flex Cover 4.1 Flow Around Objects Fluid mechanics deal with the study of all fluids under static and dynamic situations.  In fluid mechanics air is considered a fluid with very low density and objects moving through the air are said to be immersed or submerged.  When looking at objects moving through air individual components are expressed with vectors where you have the object, the incoming flow and a resulting vector.  The resulting vector represents the force (wind speed or pressure) acting on the object and the angle this force makes with the object is called yaw angle.  In most modern aerodynamic problems the objects speed is far greater than the wind speed, i.e. passenger jets and high speed trains, which results in small yaw angles.  In the case shown in figure 21 the incoming flow has a greater velocity than the object so a yaw angle of 90° is encountered (Gawthorpe, 1994).  Figure 21 Vector plot of object moving through open air (α – yaw angle)  Generally objects can be defined as streamlined or bluff.  Streamlined bodies are designed to offer the least resistance to fluid flow.  Bluff bodies are on the other hand  33 more resistant to fluid flow and cause separation of flow.  The amount at resistance an object has is quantified in the drag coefficient (CD).  The geometrical characteristics and velocity of the object/wind govern the Strouhal (St) and Reynolds (Re) numbers (Finnemore, 2002).  Figure 22 Left: Streamlined body, flow follows contours.  Right: Bluff body, flow separates.  4.1.1 Reynolds Number The Reynolds number (Re) is a dimensionless number that gives measure to the ratio of inertial forces to viscous forces and quantifies the importance of these forces for a given flow condition.  Reynolds numbers are often encountered when doing dimensional analysis in fluid dynamics problems and characterize whether flow is laminar or turbulent.  Laminar flow occurs at low Re, where viscous forces are governing, and is characterized by a smooth, constant fluid motion.  Turbulent flow occurs at high Re, where inertial forces are governing, and is characterized by flow instability and vortices (Finnemore, 2002).  The Reynolds number is typically given by:      Equation 3 Where:  Re = Reynolds Number;   = density of fluid; V = mean velocity of the object relative to the fluid; L = characteristic linear dimension;   = dynamic viscosity  34 4.1.2 Strouhal Number The Strouhal number (St) is a dimensionless number useful for analysing oscillations in unsteady fluid dynamics problems.  Aero-dynamically bluff cross sections shed vortices at a frequency governed by the Strouhal number.  The shedding of vortices generates a periodic variation in the pressure over the surface of the structure. When the frequency of this variation approaches one of the natural frequencies of a structure, vortex-induced vibration can occur.  The Strouhal number is typically given by the equation:      Equation 4 Where:  St = Strouhal Number;    = frequency of vortex shedding;   = velocity of the fluid; L = characteristic linear dimension  The Strouhal number is dependent on the shape of an object, the Reynolds number and the angle of attack although the dependency has been shown to decrease with higher Reynolds numbers (Takagawa, 1987).  Tests have shown this number to be in the range of 0.12 – 0.15 for rectangular and a combined circular/rectangular cylinders (Fleck, 2001) which is a good representative of the covered rail cars.  4.1.3 Gust Factor The wind velocity at any location can vary significantly with time.  In addition to steady wind the effects of wind gusts, which can last for up to a few seconds, need to be  35 considered.  The gust factor accounts for an increased dynamic amplification of loading in the direction of the wind.  In general the gust factor is the ratio of the mean wind velocity to the highest gust speed.  Often these values are the 3 second gust and the 30 minute mean wind velocity.  The effects are more extreme in the case of a wind gust with shorter rise time.  The gust factor does not include across wind effects, vortex shedding or instability due to galloping or flutter (Naessa, 2000). The data plotted in figure 23 comes from Norway but one can readily assume that air behaves the same way around steep mountain ridges anywhere in the world.  Figure 23 Calculated gust factors (Naessa, 2000).      Equation 5  Note:  The subscript 1 denotes that it was measurements taken from station number 1.  36 4.1.4 Drag Coefficient In fluid dynamics the drag coefficient Cd is a dimensionless quantity that is used to quantify the aerodynamic drag or resistance of a submerged object.  Drag is in most cases highly dependent on Reynolds number but for square cylinders the value evens out at 2 for Reynolds numbers exceeding about 10 4  (Finnemore, 2002).  A plot showing the drag coefficient for various shapes is presented in figure 24.  Figure 24 Drag coefficients of various shapes plotted against Reynolds number  4.1.5 Vortex Shedding and the Kármán Vortex Street As stated earlier aerodynamically bluff bodies shed vortices at a certain frequency governed by the Strouhal number.  As the Reynolds number increases the flow starts separating from the object and a phenomenon called eddies start developing.  At some limiting Reynolds number, depends on the shape of the cross section and the turbulence in the stream, the eddies break off and flow away with the stream.  This is the beginning  37 of the Kármán vortex street (figure 25) where vortices are shed first from one side of the cylinder and then the other (Finnemore, 2002).  The frequency of the separation of the eddies is what can cause vibrations on structures.   Figure 25 The Kármán vortex street  4.2 Aerodynamics of Trains Aero dynamics of trains have been widely reported on through the years.  Most of the work has focused on high speed trains.  In general the ideas and concepts remain the same as they would for lower train speeds.  With lower train speeds we encounter bigger yaw angles so the effective wind can be perpendicular to the train. From the fluid mechanics standpoint a train is best described as a moving bluff body with a high slenderness ratio.  It differs from the flow around aeroplanes and other air and space vehicles because the interaction of the underside flow and ground cannot be ignored.  Geometrical characteristics, operational speeds and wind speeds result in a three dimensional turbulent flow field which can in most cases be treated as incompressible. Cases where compressibility needs to be considered are; at extremely high speeds, during  38 tunnel entry and when a train passes at high speed.  Side wind results mainly in the disturbance of the flow symmetry around the train and causes a rise in aero dynamic forces and moments.  This results in an impaired rollover threshold of the train and causes a major threat to operational safety (Khier, 2000).   Figure 26 Schematic view of vortex development for moderate and small yaw angles  Khier et al. reported that for small yaw angles (Figure 26) the first vortex (A) develops from the lower leeward edge at the front of the railcar.  This vortex has been observed to quickly drift away from the train with the flow.  Further downstream the next vortex (B) is generated and grows steadily in the axial direction.  The last vortex (C) originates from the top leeward corner and grows steadily along the length of the train. Increasing the yaw angle results in an advance in the build up and break down of vortex A.  A different flow pattern is observed when the yaw angle exceeds 80° where flow separation is induced from both upper and lower leeward edges (Figure 27).  39  Figure 27 Vortex development with yaw angle exceeding 80°  4.3 Wind Speed in Canada According to the NBCC the highest design wind speed used in Canada is 140 km/h (approx. 90mph).  A map of design wind speed in Canada is presented in figure 28.   Figure 28 Map of design wind speed in Canada (50 year return period, NBCC)  40 4.4 Operational Speeds of Coal Transportations Trains In the case of coal transportation trains the wind speed plays the bigger role as the average operational speed of the train is only 20.6 mph (33.2 km/h) when the train is fully loaded (Canadian Pacific - Driving the digital railway).  The operational speed goes up to 50 mph (80.5 km/h) when the coal cars are empty (Canada, 2004) which is still considerably lower than the speed of conventional passenger trains.  4.5 Computational Fluid Dynamics (CFD) 4.5.1 What is CFD Computational fluid dynamics (CFD) is a branch of fluid mechanics where numerical methods are used to generate flow simulations with the help of computers. The equations for fluids are non-linear and quite complex and therefore hard to solve.  By using computers these equations of motion can be solved for nearly any arbitrary situation.  The Navier – Stokes equations are the fundamental basis for most CFD problems.  The equations are derived from applying Newton´s second law to fluid motion making the assumption that fluid stress is the sum of a pressure term and a diffusing viscous term. Many studies have been conducted in recent years where CFD is compared with physical tests, most of which show good agreement between the two, establishing CFD as an attractive alternate to physical testing (Bruno, 2009) (Khier, 2000) (Okajima, 1990).  It shall be noted that as with all other computer simulation the input has to make sense in order to get reasonable output.   41 4.5.2 Validation of Existing CFD A CFD analysis was done by Voytek Klaptocz and Dan Vyselaar of STX Europe in 2009.  To use the results of this analysis with confidence the work needed to be verified.  In the preceding chapter all the relevant parameters have been reviewed. The CFD was verified by calculating the maximum pressure on the side of the train with an apparent angle of 90°.  At this angle the incoming wind speed is 33 m/s, the result compares very well to the highest value on the pressure contour shown in figure 29.   Figure 29 Pressure contour plot for yaw angle 90° and wind speed 33 m/s. (From CFD)  4.5.3 Implementation and Results of CFD Figure 30 shows a schematic of the three load cases that were considered.  The arrows indicate the direction of the incoming wind.  Table 4 shows the results from the calculations for wind speed and yaw angle (Apparent angle in table 4).  The train´s  42 velocity is 50 mph and the incoming wind is 90 mph.  The calculations are then made using the vector calculations from chapter 4.1.   Figure 30 Load cases  The model was setup as two and a half railcars to get a good idea of what the load would be on the hatch doors under head wind conditions if the cover was missing from the adjacent rail car.  Table 5 Wind speeds used for CFD  The average pressure over the surface was calculated for each load case.  The highest load on the cover was found to be when the incoming wind was 90° to the side of the railcar (Table 5).  This suction was applied to the cover in the static analysis.  43 The CFD showed that the covers are exposed to the highest suction pressure when the air flow is perpendicular to the side of the rail car.  This is because the flow accelerates over the cover creating a low pressure region.  Table 6 Average pressure over cover  The pressure contour for load case 1 is shown in figure 31.  This case did not yield a maximum load on any part of the cover.   Figure 31 Pressure contour for load case 1 (Head wind)  Figure 32 shows the pressure contour for case 2, where the incoming wind is perpendicular to the rail cars.  This load case produced the highest suction on the cover and the average value of 2140 Pascal will be used in the static analysis of the cover.  44  Figure 32 Pressure contour for load case 2 (Wind 90°)  Figure 33 shows the pressure contours on the covers for load case 3.  Load case 3 yielded the highest load on the hatch doors.  Figure 33 Pressure contour for load case 3 (Wind 29°)  45 The highest load on the hatch doors was found to be from load case 3.  The incoming wind speed is higher for load case 1 (head wind) but it still results in less pressure on the hatch doors.  Because of a lack of in depth background information the reasons for this result cannot be explained.  Figure 34 Pressure contour for hatch door (Case 3)   Table 7 Complete results for hatch doors.  The highest pressure on the hatch doors is 3447 Pascal and comes from load case 3.  This pressure will be used to calculate how much force the connections between the cover and hatch doors need to be able to withstand.   46 4.6 Tools A few decades ago most engineering work was done by simplifying problems, usually to two dimensional problems, and using hand calculations to solve them.  Today we use more complex models and computers to solve the problems.  In the past few years great advances have been made in computer technology.  As a result computation times have gone down but at the same time the capacity to solve complex problems has gone up.  The finite element analysis for this project was done using ANSYS™.  4.6.1 ANSYS ANSYS™ is a program that allows the user to create a virtual prototype of a design operating under real – world multiphysics conditions.  Multiphysics meaning that any physical situation can be imposed on the object whether it be heat fluctuations, uniform pressure or dynamic loading of any sort.  These multi physics functions are what separate ANSYS from other commercial FE software.  Along with conventional beam and area elements the program also contains various types of solid elements.  4.6.2 Operating ANSYS There are two main methods of operating ANSYS; using the command line or writing a script file that carries out the commands, or using the graphical user interface (GUI).  There is one big disadvantage to using the GUI.  This disadvantage is that there is no undo button so the user must save the progress under a new name every time he wants the option of undoing a step.  The best way to use the program is to use the script option. This involves writing the commands consecutively in a .txt file and running the file  47 through ANSYS.  The advantages of doing this are that there is no need to save your work and making changes to the model is easy.  A list of commands is accessible in the help menu where every command is explained in detail.  A number of scripts used in this project can be found in Appendix C along with some instructions on how they are used.  4.7 Description of Model The model geometry represents the flex cover in its curved shape when attached to a railcar.  First a file creating the cover with fixed supports was written.  There are many details that need to be correctly executed in order for the model to work.  The model is comprised of beam and shell elements, beam elements at the longitudinal edges need to be rotated to fit the angle of the skin at the supports.  The area elements that make up the skin of the cover need to be split along the axis of the beams for them to mesh together and function as a whole.  When a functional model had been achieved many variations where made to test for sensitivity to various circumstance for example, holes or cuts in cover, latches missing and one or both of the hatch doors missing.  4.7.1 Longitudinal Beams and Ribs The beams and ribs where modeled as lines and given dimensions and properties. The BEAM 189 element was chosen to model all beam elements.  The element is based on the Timoshenko beam theory and therefore takes shear deformations into account.  It is suitable for modelling slender to moderately thick beams.  The BEAM 189 is a quadratic, 3 node, beam element with 6 degrees of freedom at each node, translations and rotations in the x, y and z directions.  48 4.7.2 Skin The skin of the cover was modeled by creating the curved shape in the form of a line and dragging that line along a path, creating an area.  The SHELL 63 element was chosen to model the skin.  The element has both bending and membrane capabilities, both in – plane and axial loads are allowed.  The element has 4 nodes and 6 degrees of freedom at each node, translations and rotations in the x, y and z directions.  4.7.3 End Hatches The end hatches where created using areas as well.  The SHELL 63 element was also chosen for the hatches the.  The hatches where given a greater thickness than the skin.  4.7.4 Boundary Conditions The boundary conditions at the latch positions will be simple supports for all analyses.  The reason for this is that even though the design of the latches aims towards a fixed end no guarantees can be made to that extent so any additional stiffness gained from the latching system is considered a bonus.  4.8 Sources of Error In most cases when modelling an object some simplifications are needed.  When making simplifications some error could be associated with it and that is something engineers need to be aware of acknowledge and accept.  There are a few things about the model that need to be identified as potential sources of error.  49 4.8.1   Skin From the cover drawings in appendix A the thickness of the skin is given as 1/100inch or 2.4mm.  Because the skin is hand crafted there is no guarantee that the skin has a uniform thickness of 2.4mm, in fact it is more likely that there is some variability to it where 2.4mm is a possible mean.  4.8.2 Position of Beam Sections Because of the complexity of modelling an arch with variable radius the beams and ribs are not positioned as shown in the drawing.  In the model the skin and beams all share a centerline and the various cross sections are extruded from those center lines (figure 35).   Figure 35 Extruded elements  50 4.8.3 End Hatches The end hatches connect to the top of the cover where the first ribs are positioned (see figure 36) when counted from the end.  This is about 250mm from where it connects in reality.  The connections between the cover and hatch are all done through nodes and meshed together so the connections are essentially rigid in the model.  In reality they do allow some rotations and translations but implementing these sorts of functions in the model is complicated.  4.9 Finite Element Analysis A series of analyses, both static and dynamic, where performed to determine if there would be any problems associated with the covers.  First an analysis was run for the undamaged cover.  Then by cutting pieces out and removing latches the cover was deconstructed in an organized way to see how much damage the cover can sustain before it poses a threat to the train and the surrounding environment.  The details and results of these analyses are presented in chapter 5.  Figure 36 Model of the flex cover  51 Chapter  5: Details and Results From Finite Element Analyses 5.1 Introduction A series of finite element analyses has been performed, both static and dynamic, each with a purpose of checking for sensitivity to a certain change in conditions.  The results of these analyses are presented here in an organized matter where each change in condition and its effects are reported on separately.  5.2 Shedding Frequencies The shedding frequencies where calculated for a range of Strouhal numbers and wind speeds.  An average height of 4 meters was used as the characteristic dimension of the covered railcar.  The results of these calculations are presented in table 8.   Table 8 Shedding frequencies of the covered rail car  Natural frequencies in the range of 0.300 – 1.238 Hz need to be avoided in order to prevent dynamic issues related to vortex shedding.  V 33 m/s V 10 m/s L 4.00 m L 4.00 m St f T St f T Hz s Hz s 0.12 0.990 1.01 0.12 0.300 3.33 0.13 1.073 0.93 0.13 0.325 3.08 0.14 1.155 0.87 0.14 0.350 2.86 0.15 1.238 0.81 0.15 0.375 2.67  52 5.3 Changing the Boundary Conditions 5.3.1 Modal Analysis Changing the boundary conditions from simply supported to fixed has a significant effect on the first, fourth and fifth modes.  Table 9 shows a comparison of the first 5 modes of the cover for simply supported and fixed boundary conditions.  Table 9 Comparison of frequencies for first 5 modes  Figure 37 shows the first modeshape which is a lateral sway of the cover.  The cover swings from side to side along the x – axis.  Figure 37 First mode shape Mode Pinned Fixed 1 2.121 2.969 2 5.566 5.586 3 5.734 5.587 4 5.952 6.729 5 6.999 7.790 Frequency (Hz)  53 The second mode (Figure 38) is a vertical mode where the end hatches and the top of the cover move in sync with each other along the y – axis.  Figure 38 Second mode shape The third mode (Figure 39) is a vertical mode of the end hatches where the hatches move up and down along the y – axis.  As one hatch is going up the other one is coming down.  Some minor deflections are also occurring in the cover itself.  Figure 39 Third mode shape  54  Figure 40 Fourth mode shape Figure 40 shows the fourth mode shape which is a vertical mode of the cover where the end hatches remain motionless.  Figure 41 Fifth mode shape Figure 41 shows the fifth mode shape which is a lateral mode where the cover swings back and forth in a rotating motion about the y – axis.  55 The order of the modes is not affected by changing the boundary conditions only their corresponding frequencies.  The results shown in tables 8 and 9 indicate there will be no dynamic problems associated with vortex shedding when the cover is undamaged.  5.3.2 Mass Participation The effective mass in the x direction is about 66% of the total mass.  The effective mass in the y direction is about 25% of the total mass and the remaining 9% goes to the z direction.  Tables 10 to 12 show the participation for each mode by direction.  Table 10 Mass participation along x – axis.  Table 10 shows the participation of mass in each mode along the x – axis.  All the effective mass goes to mode one as expected as mode one is a lateral movement along the x – axis.  Table 11 Mass participation along y – axis.  Mode Ratio Effective mass Cumulative mass fraction 1 1.000000 250.938 1.000000 2 0.000008 1.75E-08 1.000000 3 0.000072 1.31E-06 1.000000 4 0.000004 5.03E-09 1.000000 5 0.000242 1.46E-05 1.000000 Mode Ratio Effective mass Cumulative mass fraction 1 0.000000 1.13E-09 0.000000 2 0.839586 78.868 0.839586 3 0.000327 3.07E-02 0.839913 4 0.160086 15.038 1.000000 5 0.000000 2.19E-05 1.000000  56 Table 11 shows the participation of mass in each mode along the y – axis.  84% of the effective mass is participating in the second mode and 16% is participating in the fourth mode both of which are vertical modes.   Table 12 Mass participation along z – axis.  Table 12 shows the participation of mass along the z – axis.  The third mode is the mode where the end hatches show out of plane movement which accounts for most of the mass acting in the z direction.  The mass participation will not be discussed for any other cases, this was simply done to underline that the largest portion of mass contributes to the first mode and as you get to higher modes the mass participation decreases. Table 13 summarizes the information from tables 10 – 12.  The table shows the ratio of total mass to effective mass for each of the five modes.   Table 13 Mass participation per mode Mode Ratio Effective mass Cumulative mass fraction 1 0.000000 1.23E-09 0.000000 2 0.000390 1.39E-02 0.000390 3 0.999610 35.591 1.000000 4 0.000000 0.000 1.000000 5 0.000000 2.90E-07 1.000000 Mode Ratio Effective mass Cumulative mass fraction 1 0.659608 250.938 0.659608 2 0.207310 78.868 0.866918 3 0.093553 35.591 0.960472 4 0.039528 15.038 1.000000 5 0.000000 2.19E-05 1.000000  57 5.3.3 Static Analysis For the static analysis a uniform suction of 2140 Pascal was applied to the cover (load case 2).  This had previously been identified as the highest load the cover will be subjected to. Figure 42 shows the total stress (von Mises stress) for the cover.  The plot shows that the stress concentrations are mainly around the latch areas and where the end hatches connect to the cover.  Figure 42 von Mises stress contour for simple support condition  From figure 42 it can be seen that the critical regions are around the latches and the end hatches.  All critical regions are supported with the beam material and therefore have stress capacities well above the applied stresses.  In the top left corner of figure 42 a maximum stress of .894E+08 Pascal (89.4 MPa) is indicated.  FE models are known to show some spikes of high stresses that can be ignored.  Although not obvious on this  58 figure the stresses do reach the level indicated by the orange color where the top of the end hatch connects to the cover.  The highest stresses observed in the skin are around .199E+08 (19.9 MPa) which is well below the capacity of the skin.  Referring back to table 2 it is obvious that the stress capacities of the cover are not tested under the highest static load.  Figure 43 von Mises stress contour for fixed support conditions  Changing the boundary conditions from simply supported to fixed seems to not have a big effect on the development of stresses in the cover.  Comparing figures 42 and 43 there is not much variation between them. Contour plots of total displacements are presented in figures 44 and 45.  In both cases the displacement vector sum gives the same max values.  The displacements are spread over a larger area with simple supports.  The deflections shown for the hatch doors  59 should not be taken too seriously as the modelling of the hatch doors is not completely accurate.  The scale is in meters so the maximum displacement is 9.45 cm.  Figure 44 Deflections (Simply supported)  Figure 45 Deflections (Fixed supports)  60 5.3.4 Reactions at Supports Figure 46 shows a plot of the numbering of the supports.  For reference the reactions will be listed as FX 1, FY 1 and FZ 1 for support 1 and so on.  The co – ordinate system is also shown in the figure.  Figure 46 Numbering of supports  Table 14 shows the support reactions for the simple supports.  All values are Newton (N).   Table 14 Reactions for simple supports 1 2 3 4 5 6 FX 1897 -1897 6765 -6766 6421 -6421 FY -3481 -3480 -9358 -9359 -8894 -8894 FZ 5240 5239 -7966 -7965 -1058 -1056 7 8 9 10 11 12 FX 6421 -6421 6766 -6765 1897 -1897 FY -8894 -8894 -9359 -9358 -3480 -3480 FZ 1056 1058 7965 7966 -5239 -5240  61 Table 15 shows all support reactions for the fixed condition.  All forces are as before Newton (N) and all moments are Newton meters (N*m).   Table 15 Reactions for fixed supports  Changing the support conditions has a significant effect on how the forces distribute to the supports.  Having fixed supports also introduces moments into the system that, although are very small and even negligible in most cases, the latches would need to be able to transmit to the railcar.  The script MX means moment about x – axis.  In all analysis that follows the support conditions are always simple supports.   1 2 3 4 5 6 FX 2268 -2268 6484 -6486 6164 -6164 FY -3936 -3935 -9093 -9094 -8703 -8704 FZ 3889 3889 -5601 -5600 -800 -796 MX -589 -589 -15 -15 -14 -14 MY -437 438 -12 12 -12 10 MZ 25 -25 33 -33 40 -40 7 8 9 10 11 12 FX 6164 -6164 6486 -6484 2268 -2268 FY -8704 -8703 -8704 -9093 -3935 -3936 FZ 796 800 796 5601 -3889 -3889 MX 14 14 14 15 589 589 MY 10 -10 10 -12 438 -437 MZ 40 -40 -40 -33 25 -25  62 5.4 Holes in the Cover 5.4.1 Modal Analysis Figure 47 shows the position of the three holes that were made in the cover.  The damage is something you might expect if the cover would get hit with rock fall from a mountain ridge.  The first hole made is the one to the far left in figure 47, the second is the one in middle and the third is the one on the far right.  All holes are the same size, 0.64 m 2 .   Figure 47 Position of holes in cover  The first thing to mention here is that by cutting those rectangular holes in the cover the modes got switched around a bit.  The second mode of the model with 3 holes  63 is the vertical mode (fourth mode) of the undamaged model.  Table 16 shows a comparison of the frequencies of the first five modes for all levels of damage and the undamaged cover.   Table 16 Frequencies of 5 modes under all modelled damage states (holes)  Because there is a reduction in both stiffness and mass from cutting the pieces out the modes are each affected in a different way.  The first mode is relatively unaffected by the holes which means that the ratio of lateral stiffness to mass essentially remains constant.  The modes that switch places have been color coded in table 16.  The holes have a greater impact on all other modes especially the mode marked with red and the fifth mode but as the mass participation is minimal for these modes there is no point in discussing why this occurs.  5.4.2 Static Analysis Performing static analysis on the cover with this kind of damage does not bring any unexpected results.  There are of course stress concentrations around the openings but the stresses don not come close to the capacity of the cover.  The result from this analysis is not complete and needs to be taken with caution.  For the first part making the Mode Pinned 1 hole 2 holes 3 holes 1 2.121 2.116 2.119 2.101 2 5.566 5.450 5.423 4.991 3 5.734 5.563 5.560 5.494 4 5.952 5.583 5.580 5.561 5 6.999 6.663 6.037 5.941 Frequency (Hz)  64 holes in the cover could well change the pressure on the cover and the outcome of any kind of gust wind on a damaged cover is subject to whether the holes are on the windward or leeward side.  Figure 48 Von Mises stress contour (3 holes)  Figure 48 shows the von Mises stress contour for the cover with three holes.  The stress around the supports and hatch connections do not seem to be affected in a big way. There are some stress concentrations around the holes which can be expected, especially around the corners.  The stresses appear to come close to the capacity of the skin at this stage but do not quite exceed the capacity.  65 By comparing the deflections with the undamaged cover (Figure 44) the maximum deflection at center span has increased.  Deflections tend to increase more around the holes because of the loss of stiffness in those areas.  Figure 49 Deflections (3 holes)  The scale is in meters so the largest displacement is 9.96 cm.   66 5.5 Perpendicular Cuts 5.5.1 Modal Analysis Figure 50 shows the positions of all cuts made in the cover.  Two perpendicular cuts were made in the skin at a time starting from left to right.  Figure 50 Position of cuts in cover Table 17 shows the frequencies of the first five modes for simple supports and all damage states.  The script 1 cut means 1 set of perpendicular cuts.  Table 17 Frequencies of 5 modes under all modelled damage states (cuts) Mode Pinned 1 cut 2 cuts 3 cuts 1 2.121 2.081 2.067 2.052 2 5.566 4.674 4.035 3.698 3 5.734 5.557 4.767 4.417 4 5.952 5.572 5.557 4.752 5 6.999 5.854 5.572 5.511 Frequency (Hz)  67 The first mode is in every case the same lateral mode as before.  All higher modes are vertical modes and involve movement of the pieces inside the cuts (see figure 51). Comparing the frequencies of the higher modes is tricky now as they are not all the same as for the undamaged cover.   Figure 51 Mode shape of second mode with cuts  The second mode is still a vertical mode but now most of the movement is on the inside of the cuts.     68 5.5.2 Static Analysis Like before a uniform suction of 2140 Pascal is applied to the model.  Figure 52 shows the von Mises stress for a heavily damaged cover.  Some stress concentrations form around the ends of the cuts as expected.  The stresses in those areas far exceed the capacity of the skin so if this sort of damage occurs its likely to keep growing as long as there is wind loading.  The stresses are in the magnitude where they could potentially damage the beam sections as well.  When the damage was modelled the assumption was that the cuts could not extend across the ribs so the cuts are only applied to the skin.   Figure 52 von Mises stress contour with cuts in three bays   69 The high stresses where the top of hatch connects to the cover are still present the contour plot cannot be adjusted to show all stress concentrations at once. Figure 53 shows the deflections of the cover with cuts in three bays.  The cover still deflects in a similar way as before the cuts were present.  The plot cannot be adjusted to show them as the deflections of the cut pieces are so large.  The scale on the plot is meters so the largest deflection is 1.6 meters.  The cuts are all the same lengths and the load is distributed uniformly over the cover but still there is not uniformity in the deflections.  The first piece deflects about 1.1 meters while the last one deflects 1.6 meters.   Figure 53 Deflections with cuts in three bays   70 5.6 Hatch Doors 5.6.1 Dynamic Analysis The absence of the hatch doors does not have a big impact on the first mode of the cover.  It does however have a considerable effect on higher modes.  Table 18 shows a comparison of the first five modes for three different cases, both hatches in place, one hatch missing and both hatches missing, all on simple supports.   Table 18 Frequencies of 5 modes under all modelled damage states (hatches)  The mode shapes are the same for all these cases.  The absence of one or both hatches decreases the stiffness in modes 1 to 3 but increases the stiffness in modes 4 and 5.  5.6.2 Static Analysis Static analysis does not yield any surprising results.  The cover has greater deflections where the end hatches are missing.  Bigger deflections also mean higher stresses in and near the regions of these displacements.  Figures 54 and 55 show the stresses and deflections for the case where one hatch is missing.  When both are missing Mode Pinned 1 hatch No hatch 1 2.121 2.086 2.007 2 5.566 4.190 4.167 3 5.734 5.573 4.243 4 5.952 7.282 8.781 5 6.999 8.027 8.948 Frequency (Hz)  71 there is symmetry in the stress and deflections and the values seen at the right end of figures 54 and 55 run all along the center of the cover from end to end.  Figure 54 Stress contour (1 hatch missing)  Figure 54 shows von Mises stress for the cover with one hatch missing.  The highest stress indicated in the plot is 86.4 MPa and has the red color assigned to it.  The red color is not visible on the plot.  The highest stresses on the plot are around 20 to 30 MPa.  When both hatches are missing there is symmetry in the stress distribution and the values are approximately the same as they are at the end where the hatch is missing in figure 44.  72 Figure 55 shows the deflections of the cover when one hatch is missing.  The largest deflection is on the hatch itself.  Where the hatch is missing the deflection is around 5 – 6 cm.  When both hatches are missing there is a uniform deflection along the center of the cover with a value of about 5 – 6 cm.  Figure 55 deflections (1 hatch missing)    73 5.7 Latch Failure This part will focus on latch failure.  Latch failure is defined as either latches breaking or unhooking.  Unhooking could occur in transit, because of excessive deflections, or if the cover handling machine fails to hook all latches.  A figure showing the numbering of supports is presented and will be referenced in the coming pages.   Figure 56 Support numbering   Table 19 Support reactions for simply supported cover 1 2 3 4 5 6 FX 1897 -1897 6765 -6766 6421 -6421 FY -3481 -3480 -9358 -9359 -8894 -8894 FZ 5240 5239 -7966 -7965 -1058 -1056 7 8 9 10 11 12 FX 6421 -6421 6766 -6765 1897 -1897 FY -8894 -8894 -9359 -9358 -3480 -3480 FZ 1056 1058 7965 7966 -5239 -5240  74 The highest load appears on the latches second from the ends on both sides. Given that one of these latches is the first to fail the static analysis was run again with one of those latches removed from the model.  The latch with the highest load in that run was the next one removed and so on. The question of dynamic and static analyses is perhaps not relevant in the case of latches failing because if one latch fails there is a tremendous increase in loading on the one next to it.  This creates a domino effect leading to all the latches on that side failing. The only thing that can be learned from this analysis is how much the load increases on the next latch in case of latch failure.  Given that latch at support number 10 is the first to fail the order would be as follows: 8, 6, 4 and finally the end latches would fail.   Table 20 Support reactions with a single latch missing  Table 20 shows the support reactions after the latch at position 10 has failed, all values are in Newton (N).  The values affected most have been highlighted.  Notice the increase in forces in the z direction which are not easily explained.  The sum of forces acting in the z directions is zero as it should be. 1 2 3 4 5 6 FX 1920 -1845 6732 -6601 6540 -5199 FY -3458 -3472 -9337 -9191 -9038 -7419 FZ 5083 5074 -8439 -8091 -860 -11016 7 8 9 10 11 12 FX 6483 -11934 6696 0 1803 -4594 FY -8953 -16003 -9277 0 -3402 -7379 FZ 1632 34742 8294 0 -5116 -21304  75  Table 21 Support reactions with two consecutive latches missing  Table 21 shows the support reactions with two consecutive latch failures.  The forces on the latch at the end and the latch at position 6 have increased a great deal.  From the information in the table it is obvious that the latch at position 6 is the next to go.   Table 22 Support reactions with three consecutive latches missing  Table 22 shows the force distribution when three consecutive latches have failed. From the data presented here it is obvious that latch failure is an unwanted failure mode to be avoided at all cost.  1 2 3 4 5 6 FX 1808 -748 6867 -3483 6793 -18888 FY -3158 -2462 -9535 -5449 -9332 -24498 FZ 4494 2070 -10476 -34051 -528 71215 7 8 9 10 11 12 FX 6767 0 6730 0 1384 -7230 FY -9253 0 -9288 0 -2899 -11055 FZ 3250 0 10581 0 -4975 -41581 1 2 3 4 5 6 FX 1548 7559 7172 -28095 7045 0 FY -2524 6933 -9926 -35431 -9657 0 FZ 3839 -42538 -14749 106650 -1065 0 7 8 9 10 11 12 FX 7059 0 7155 0 653 -10094 FY -9587 0 -9796 0 -1975 -14966 FZ 4558 0 14627 0 -3681 -67642  76 Chapter  6: Conclusions 6.1 Cost/Risk Assessment Giving the high uncertainty and incomplete knowledge of the components in this project, most of the data was determined through intuition and peer discussions.  The use of some decision analysis tools helped create more realistic results through uniform distributions, utility, and expected value.  6.1.1 Uncertainties To account for uncertainties, uniform distributions were applied to give minimum and maximum cost values, resulting in a range of possible outcomes rather than a fixed value.  Distribution curves can be implemented according to a client’s requirements thereby helping the engineer produce better results.  6.1.2 Risk Modeling risk using DecisionPro allows for a complete visualization of the engineer’s perspective of the project.  Decision trees are easy to follow which makes it possible to pass work on to other engineers. Some engineers may seek more risk than others so the option of adjusting the risk tolerance gives engineers a chance to set the program up for their preference.  6.1.3 Inefficiencies Determining inefficiency through most likely cost and most favourable cost helps determine which components need attention and where unnecessary work can be avoided.  77 6.2 Finite Element Analyses The finite element analyses showed that the original design of the cover holds up in the conditions imposed on it when it is undamaged but some types of damage can quickly impair on the performance.  6.2.1 Dynamic Analyses Dynamic analyses conducted under various damage states showed that the cover can take a substantial amount of damage without impairing the dynamic response.  The change in frequencies was mainly in higher modes where mass participation is very low.  6.2.2 Static Analyses The static analyses show that stress concentrations occur around the damaged areas.  The models of the cuts show that the stresses observed at the ends of the cuts are high enough for the damage to perpetuate further given the initial cut is long enough. Through static analyses the potential danger of latch failure was also identified.  If a single latch fails it has a multiplying effect on the forces acting on the adjacent latches. This can potentially cause a domino effect resulting in all latches on one side failing.   78 Works Cited Bruno, L. F. (2009). 3D flow around a rectangular cylinder: A computational study. Journal of Wind Engineering and Industrial Aerodynamics (98). Canada, T. s. (2004). Rail Reports R04Q0006. Canadian Pacific - Driving the digital railway. (n.d.). (Canadian Pacific Railway) Retrieved June 25, 2011, from http://www.cpr.ca/en/invest-in-cp/key- metrics/Documents/cp-train-speed-2011.pdf) Cruikshank, J., & Loewen, N. (2011). Proposal for FlexCover System Trial. Fibreglass. (2011, August 15). Retrieved August 15, 2011, from Wikipedia: http://en.wikipedia.org/wiki/Fiberglass Finn, L. (2010). A brief guide to GSC Open File 4459. UBC. Finnemore, J. F. (2002). Fluid Mechanics with Engineering Applications Third Edition. New York: McGraw - Hill Higher Education. Fleck, B. (2001). Strouhal numbers for flow past a combined circular–rectangular prism. 89. Gawthorpe, R. (1994). Wind effects on ground transportation. Journal of Wind Engineering and Industrial Aerodynamics , 73-92. Jones, J. (2005). Decision Theory. Richland Community College., Mathematics. Kalos, M. H., & Whitlock, P. A. (2008). Monte Carlo Methods. WILEY-VCH. Khier, W. B. (2000). Flow structure around trains under side wind conditions: a numerical study. 29. Lewin, J., Ballard, G., & David S, B. (2003). Spillway gate reliability in the context of overall dam failure risk. USSD Annual Lecture. Charleston, South Carolina. Naessa, A. C. (2000). Gust factors for locations downstream of steap mountain ridges. Journal of Wind Engineering and Industrial Aerodynamics (87), 131-146. Okajima, A. (1990). Numerical Simulation of Flow Around Rectangular Cylinders. (33). Preidt, C. A Tool For Design and Decision Making: Reliability.  79 Ramadhas, V. (2005). A Contribution To Innovative Methods For Enhancing The Economy of Steel Structures Engineering. MASc Thesis, UBC, Department of Civil Engineering. Stiemer, S. F. (2011). Risk Evaluation and Feasibility Study. UBC. Stobart, D. (Director). (2008). Extreme Trains (The Coal Train) [Motion Picture]. Takagawa, N. (1987). Vortex shedding behind a square cylinder in transonic flows. 178. Wong, V. (2006). Risk of Collapse and Damage of Steel Bridges in the GVRD. University of British Columbia, Department of Civil Engineering. Zou, T., Madadevan, S., Mourelatos, Z., & P, M. (2002). Reliability analysis of automotive body-door subsystem. Reliability Engineering and System Safety (78), 315- 324.   80 Appendix A  Drawings of Flex Cover   81     82 Appendix B  Drawings of Rail Car  83 Appendix C  Script files for ANSYS C.1 Cover with Simple Supports This script draws up the cover with simple supports and meshes the model.  Text that comes after an exclamation point is comments for the user and is not read by the program.  After this file has been run the analysis files can be run.  ! Open pre - processor /PREP7  ! Define parameters TSHELL = 0.0024                   !Thickness of shell TBEAM = 0.0095                    !Thickness of beam WBEAM = 0.0762                    !Width of beam 6" YBEAM = TSHELL/4 + TBEAM/4        !Offset distance of beams up/down XBEAM = WBEAM/2                   !Offset distance of beams side MESH = 0.1  ! Draw arc ! Define keypoints K,1,-1.524,0 K,2,-0.8599,0.5767 K,3,0,0.800 K,4,0.8599,0.5767 K,5,1.524,0  ! Keypoints to define arcs K,6,-1.19195,0.3542 K,7,1.19195,0.3542  ! Define lines LARC,2,4,3,-1.7673 LARC,1,2,6,-1.9326 LARC,4,5,7,-1.9326  ! Combine lines LCOMB,ALL,0    84 ! Make path lines K,8,0,0.800,14.751 L,3,8 K,9,-1.524,0,14.751 L,1,9 K,10,1.524,0,14.751 L,5,10  ! Create surface from arc ADRAG,1,,,,,,2  ! Change view to isometric /VIEW,,1,1,1  ! Divide area and lines at rib intersections WPOFFS,0,0,1.0255 ASBW,ALL LSBW,ALL WPOFFS,0,0,2.540 ASBW,ALL LSBW,ALL WPOFFS,0,0,2.540 ASBW,ALL LSBW,ALL WPOFFS,0,0,2.540 ASBW,ALL LSBW,ALL WPOFFS,0,0,2.540 ASBW,ALL LSBW,ALL WPOFFS,0,0,2.540 ASBW,ALL LSBW,ALL  ! Create ribs LGEN,2,1,,,0,0,1.0255,0,0 LGEN,6,10,,,0,0,2.540,0,0  ! Create keypoints for end pieces K,53,0,0.800,1.0255 K,54,0,0.800,13.7255    85 ! Split the shell in longitudinal direction WPROTA,,,90 ASBW,ALL WPROTA,,,-90  ! Move working plane to make end hatch KWPLAN,,9,10,54  ! Draw rectangle and subtract to make end hatch RECTNG,0,3.048,0,2 ASBA,1,ALL,,KEEP,KEEP ADELE,1 ADELE,3  ! Move working plane to make end hatch KWPLAN,,5,1,53  ! Draw rectangle and subtract to make end hatch RECTNG,0,3.048,0,2 ASBA,1,ALL,,KEEP,KEEP ADELE,1 ADELE,4  ! Move working plane back to position KWPLAN,,39,37,38 WPOFFS,1.524,0,0  ! Define element type and material properties for beams ET,1,BEAM189 MP,EX,1,7.29E9        !N/m^2 MP,PRXY,1,0.4 MP,DENS,1,1937.6      !kg/m^3  ! Section properties of center beam SECTYPE,1,BEAM,RECT SECDATA,TBEAM,WBEAM + WBEAM SECOFFSET,CENT  ! Select center beam and assign attributes LSEL,S,LINE,,2 LSEL,A,LINE,,44 LSEL,A,LINE,,38 LSEL,A,LINE,,32  86 LSEL,A,LINE,,26 LSEL,A,LINE,,21 LSEL,A,LINE,,7 LATT,1,,,,,,1 LESIZE,ALL,MESH LMESH,ALL  ! Section properties of other beams SECTYPE,2,BEAM,RECT SECDATA,TBEAM,WBEAM SECOFFSET,CENT  ! Select left beam and assign attributes LSEL,S,LINE,,23 LSEL,A,LINE,,45 LSEL,A,LINE,,39 LSEL,A,LINE,,33 LSEL,A,LINE,,27 LSEL,A,LINE,,20 LSEL,A,LINE,,8 LATT,2,,1,,6,,2 LESIZE,ALL,MESH LMESH,ALL  ! Select left beam and assign attributes LSEL,S,LINE,,43 LSEL,A,LINE,,46 LSEL,A,LINE,,40 LSEL,A,LINE,,34 LSEL,A,LINE,,28 LSEL,A,LINE,,22 LSEL,A,LINE,,13 LATT,2,,1,,7,,2 LESIZE,ALL,MESH LMESH,ALL  ! Select ribs and assign attributes LSEL,S,LINE,,49 LSEL,A,LINE,,48 LSEL,A,LINE,,47 LSEL,A,LINE,,37 LSEL,A,LINE,,12 LSEL,A,LINE,,11  87 LATT,2,,1,,,,2 LESIZE,ALL,MESH LMESH,ALL  ! Select skin ASEL,S,,,9,21 ASEL,A,,,5  ! Define element type, material properties and real constants for skin ET,2,SHELL63 MP,EX,2,31E9            !N/m^2 MP,PRXY,2,0.4 MP,DENS,2,1450          !kg/m^3 R,2,TSHELL,TSHELL,TSHELL,TSHELL  ! Assign element type to shell AATT,2,2,2 AESIZE,ALL,MESH AMESH,ALL  ! Select hatch doors ASEL,s,,,2,3  ! Define element type, material properties and real constants for end hatch ET,3,SHELL63 MP,EX,3,31E9            !N/m^2 MP,PRXY,3,0.4 MP,DENS,3,1450          !kg/m^3 R,3,0.015,0.015,0.015,0.015  ! Assign element type to shell AATT,3,3,3 AESIZE,ALL,MESH AMESH,ALL  ! Make circular bar at bottom of hatches LSEL,S,,,1 LSEL,A,,,31  ! Define element type ET,4,BEAM189 MP,EX,4,200E9            !N/m^2  88 MP,PRXY,4,0.3 MP,DENS,4,7800          !kg/m^3  ! Section properties SECTYPE,4,BEAM,CSOLID SECDATA,0.0125 SECOFFSET,CENT  ! Assign element to circular bar LATT,4,,,,,,4 LESIZE,ALL,MESH LMESH,ALL  ! Merge intersecting nodes and keypoints NUMMRG,ALL  ! Select keypoints at supports KSEL,S,KP,,36 KSEL,A,KP,,37 KSEL,A,KP,,31 KSEL,A,KP,,32 KSEL,A,KP,,26 KSEL,A,KP,,27 KSEL,A,KP,,21 KSEL,A,KP,,22 KSEL,A,KP,,16 KSEL,A,KP,,17 KSEL,A,KP,,11 KSEL,A,KP,,12  ! Set boundary condition to pinned DK,ALL,UX,0,,,UY,UZ   89 C.2 Modal analysis This is a simple script that sets the analysis type to modal and solves the system of equations.  To view results the use general post processor in the GUI.  ! Open pre - processor /PREP7  ! Set up modal analysis ANTYPE,2,NEW MODOPT,SUBSP,5  ! Close pre-processor FINISH  ! Start solution phase and solve system /SOLU SOLVE  C.3 Static Analysis This script applies a 2140 Pascal uniform pressure on the surface of the cover and does a static analysis based on the applied load.  To view results the use general post processor in the GUI.  ! Open pre - processor /PREP7  ! Set analysis type ANTYPE,0,NEW  ! Select skin and apply uniform pressure on cover ASEL,ALL SFA,ALL,1,PRES,-2140           !Load is 2140 Pa  ! Close pre-processor FINISH  ! Start solution phase and solve system /SOLU SOLVE     90 C.4 Cutting Holes in Your Model In order to cut holes in the model the user must create the shape of the hole by block or cylinder and use the boundary of the new object to subtract from the cover.  By inserting the following commands, right after where the end hatches are created, two perpendicular cuts are generated in the shell.  After ! Move working plane back to position KWPLAN,,39,37,38 WPOFFS,1.524,0,0  Insert ! Offset working plane to location of tear WPOFFS,1.2,0,-1.2 WPROTA,0,-90,0  ! Create block to subtract from area BLC5,0,0,0.02,2.2,1  ! Punch hole in shell ASBV,21,1  ! Offset working plane to location of tear WPROTA,0,90,0 WPOFFS,-0.4,0,-1 WPROTA,0,-90,0  ! Create block to subtract from area BLC5,0,0,1,0.02,1  ! Punch hole in shell ASBV,23,1    91 Appendix D  Expected Monetary Value Example D.1 Example from James Jones Zed and Adrian and run a small bicycle shop called "Z to A Bicycles". They must order bicycles for the coming season. Orders for the bicycles must be placed in quantities of twenty (20). The cost per bicycle is $70 if they order 20, $67 if they order 40, $65 if they order 60, and $64 if they order 80. The bicycles will be sold for $100 each. Any bicycles left over at the end of the season can be sold (for certain) at $45 each. If Zed and Adrian run out of bicycles during the season, then they will suffer a loss of "goodwill" among their customers. They estimate this goodwill loss to be $5 per customer who was unable to buy a bicycle. Zed and Adrian estimate that the demand for bicycles this season will be 10, 30, 50, or 70 bicycles with probabilities of 0.2, 0.4, 0.3, and 0.1 respectively. D.2 Actions There are four actions available to Zed and Adrian. They have to decide which of the actions the best one under each criterion is. 1. Buy 20 bicycles 2. Buy 40 bicycles 3. Buy 60 bicycles 4. Buy 80 bicycles  Zed and Adrian have control over which action they choose. That is the whole point of decision theory - deciding which action to take. D.3 States of Nature There are four possible states of nature or outcomes. 1. The demand is 10 bicycles 2. The demand is 30 bicycles 3. The demand is 50 bicycles 4. The demand is 70 bicycles  92 Zed and Adrian have no control over which state of nature will occur. They can only plan and make the best decision based on the appropriate decision criteria. D.4 Payoff Table After deciding on each action and state of nature, create a payoff table. The numbers in parentheses for each state of nature represent the probability of that state occurring.  Action State of Nature B uy 20 B uy 40 B uy 60 B uy 80 Deman d 10 (0.2) 5 0 - 330 - 650 - 970 Deman d 30 (0.4) 5 50 7 70 4 50 1 30 Deman d 50 (0.3) 4 50 1 270 1 550 1 230 Deman d 70 (0.1) 3 50 1 170 2 050 2 330  Ok, the question on your mind is probably "How the [expletive deleted] did you come up with those numbers?". Let's take a look at a couple of examples. Demand is 50, buy 60: They bought 60 at $65 each for $3900. That is -$3900 since that is money they spent. Now, they sell 50 bicycles at $100 each for $5000. They had 10 bicycles left over at the end of the season, and they sold those at $45 each of $450. That makes $5000 + 450 - 3900 = $1550. Demand is 70, buy 40:  93 They bought 40 at $67 each for $2680. That is a negative $2680 since that is money they spent. Now, they sell 40 bicycles (that's all they had) at $100 each for $4000. The other 30 customers that wanted a bicycle, but couldn't get one, left mad and Zed and Adrian lost $5 in goodwill for each of them. That's 30 customers at -$5 each or -$150. That makes $4000 - 2680 - 150 = $1170. Opportunistic Loss Table The opportunistic loss (regret) table is calculated from the payoff table. It is only needed for the minimax criteria, but let's go ahead and calculate it now while we're thinking about it. The maximum payoffs under each state of nature are shown in bold in the payoff table above. For example, the best that Zed and Adrian could do if the demand was 30 bicycles is to make $770. Each element in the opportunistic loss table is found taking each state of nature, one at a time, and subtracting each payoff from the largest payoff for that state of nature. In the way we have the table written above, we would subtract each number in the row from the largest number in the row.  Remember that the numbers in this table are losses and so the smaller the number, the better.  Buy 20 Buy 40 Buy 60 Buy 80 Demand 10 0 380 700 1020 Demand 30 220 0 320 640 Demand 50 1100 280 0 320 Demand 70 1980 1160 280 0 Action State of Nature  94 D.5 Expected Value Criterion Compute the expected value for each action. For each action, do the following: Multiply the payoff by the probability of that payoff occurring. Then add those values together. Matrix multiplication works really well for this as it multiplied pairs of numbers together and adds them. If you place the probabilities into a 1x4 matrix and use the 4x4 matrix shown above, then you can multiply the matrices to get a 1x4 matrix with the expected value for each action. Here is an example of the "Buy 60" action if you wish to do it by hand. 0.2(-650) + 0.4(450) + 0.3(1550) + 0.1(2050) = 720 The expected values for buying 20, 40, 60, and 80 bicycles are $400, 740, 720, and 460 respectively. Since the best that you could expect to do is $740, you would buy 40 bicycles. D.6 Maximax Criterion The maximax criterion is much easier to do than the expected value. You simply look at the best you could do under each action (the largest number in each column). You then take the best (largest) of these. The largest payoff if you buy 20, 40, 60 or 80 bicycles is $550, 1270, 2050, and 2330 respectively. Since the largest of those is $2330, you would buy 80 bicycles. D.7 Maximin Criterion The maximin criterion is as easy to do as the maximax. Except instead of taking the largest number under each action, you take the smallest payoff under each action (smallest number in each column). You then take the best (largest of these).  95 The smallest payoff if you buy 20, 40, 60 or 80 bicycles is $50, -330, -650, and - 970 respectively. Since the largest of those is $50, you would buy 20 bicycles. D.8 Minimax Criterion Be sure to use the opportunistic loss (regret) table for the minimax criterion. You take the largest loss under each action (largest number in each column). You then take the smallest of these (it is loss, after all). The largest losses if you buy 20, 40, 60, and 80 bicycles are $1980, 1160, 700, and 1020 respectively. Since the smallest of those is $700, you would buy 60 bicycles. D.9 Putting it all Together Here is a table that summarizes each criteria and the best decision.  Criterion Buy 20 Buy 40 Buy 60 Buy 80 Best Action Expected Value 400 740 720 460 Buy 40 Maximax 550 1270 2050 2330 Buy 80 Maximin 50 -330 -650 -970 Buy 20 Minimax 1980 1160 700 1020 Buy 60 Action

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