A Cornputer Model of a Train Derailment
Elton Edward Toma
A thesis submitted to the Department of Mechanical Engineering in confonity with the requirements for the degree of Doctor of Philosophy
Queen's University Kingston, Ontario, Canada
October, 1998
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This thesis presents the development of a train derailment computer model. The planar model is based on coupled sets of 5 degree of freedom sub-system models for each rail car. and includes coupler body reaction forces, car-ground reaction forces. brake forces, and car-twcar collision forces. The mode1 allows constraint removal to model coupler failure and derailment conditions. The differentiai equations of motion for the train system are derived using Lagrange's equations with added multipliers.
The cornplete cornputer model was validated against the well known Mississauga derailment of 1979. The model resulted in 24 derailed cars, identical to the actual event, and a -11.7% error in accident site area. The major collisions produced by the model correlate well with the damage which occurred at the actual event.
The derailment model was also tested for sensitivity to parameter variation about a baseline train. The changes in the model outcome for variations of f20% and f-50% for CU mas, train length, train speed, braking force, ground reaction force. and derailment quotient (L/V ratio) were performed. The outcomes were compared by number of derailed cars, peak collision forces, and the length, width, and area of the accident scene. The results show the model to be numerically stable within the f50% boundaries, and to produce outcornes physicdy consistent with the variations in the parameters. Train speed, car mas, and train length were the parameters which had the greatest effect on the number of derailed cars and the peak collision force.
Findy, the Mississauga derailment model was tested wit h alt erations to the brak- ing model. The results show that instantaneous application of brake forces approach- ing wheel lockup significantly reduced the severity of the Mississauga derailment model results.
Recommended future research includes the study of the effects of braking, car placement, mass, and length, and the surrounding terrain near the rail lines on the severity of derailments. Acknowledgements
This work was supewised by Dr. R.J. Anderson, P.Eng. d the Deportment of Mechanical Engineering. 1 thank him very much for the freedom given to pursue this research, and for his guidance and advice as the work progressed. His expertise. experience, and ability to quickly accept or reject ideas and theories was invaluable throughout the work.
1 am very grateful to my family for t heir encouragement and words of advice throughout my years at Queen's, and to the Vernooy family for their continued wmth and support.
Thanks Andreas Schumann, Bruce Minaker, Geoff Rideout, and Rob Langlois for being thoroughly enjoyable Company in the Dynamics lab - 1 have never known a more diverse group of people. Thanks also to John Stewart and Kelly McKinley for their often needed words of advice and wisdom.
A thank you goes to Hilary Richardson for the fun part-tirne work over several yearç at the Mechanicd Engineering Library, and to the staff of the Mechanical Engineering machine shop for being good Company and help throughout the years.
1 would also like to thank Dr. Anderson, the Department of Mechanical Engineering, the Naturd Sciences and Engineering Research Council, and the School of Graduate Studies at Queen's University for their financial assistance. Dedicated to my wife Cathy, for her continuous support, helpfd advice, and endless patience. Contents
1 Introduction 1 1.1 Railway Accidents in Canada ...... 3. 1.2 RailEquipmentSurnrnary ...... 6 1.2.1 Locomotives ...... 6
7 1.2.2 Reight Cam ...... I 1.2.3 Couplers ...... 11 1.2.4 Trucks and Wheelsets ...... 14 1.2.5 Brakes ...... 15 1.3 Past Research Focusing on Derailments ...... 17 1.4 Previous Derailment Models ...... 19 1.4.1 The Yang and Manos Mode1 (1972) ...... 20 1.4.2 Anderson Mode1 (1990) ...... 22 1.4.3 Johnson Mode1 (1991); Guran Mode1 (1992) ...... 24 1.4.4 Gracie Mode1 (1991); Roorda and Gracie Model (1992) .... 26 1.5 Objectives of a State-of-the-Art Derailment Mode1 ...... 27
2 Theory 28 2.1 Solution Method ...... 28 2.1.1 Development of the Specid Purpose Mode1 ...... 30 2.1.2 Lagrange's Equations with Added Multipliers ...... 32 2.2 The Rigid Body Model ...... 36 2.2.1 Assumptions, Simplifications. and Characteristics ...... 37 2.2.2 Analysis of the Rigid Body Mode1 ...... 38 2.3 Rigid Body Equations of Motion ...... 44 2.3.1 Sub-System Equations of Motion ...... 44 2.3.2 Car Rai1 Constraints: Straight Track ...... 17 2.3.3 Car Rail Constraints: Constant Radius Cwed Track ..... 49 2.3.4 Coupler-tecoupler Pivot Constraints ...... 52 2.3.5 Constraint Stabilization ...... 53 2.3.6 Assembled Equations of Motion ...... 55 2.4 Generdized Forces ...... 55 2.4.1 Brake Force Mode1 ...... 55 2.4.2 Coupler Mode1 ...... 59 2.4.2.1 StrikerReactionModel ...... 60 2.4.2.2 Coupler-tecoupler React ion Mode1 ...... 62 2.4.3 Ground Reaction Model ...... 64 2.4.3.1 Basic Analysis of the Derailing Train System: Case 1. 66 2.4.3.2 Basic Andysis of the Derailing Train System: Case 2 . il 2.4.3.3 Basic Analysis of the Derailing Train System: Case 3. 74 2.4.3.4 Soi1 Mechanics Literature ...... 76 2.4.3.5 A Veloci ty Dependent Ground Reaction Force Funct ion . 78 2.4.3.6 Application of the Reaction Force ...... 81 2.4.4 Car Collision ...... 53 2.5 Computer Program Implementation ...... 90 2.5.1 Input Data ...... 94 2.5.2 Output Data ...... 95
3 Results 96 3.1 Mode1 Validation: Modelling of the Mississauga Derailment ...... 96 3.1.1 Input Data ...... 101 3.1.2 Results and Discussion ...... 103 3.1.3 Conclusions: Mode1 Validation ...... 110 3.2 Variation of Parameters ...... 111 3.2.1 Base System ...... 113 3.2.2 Variation of Initial Train Speed ...... 115 3.2.3 Variation of Car Mass ...... 118 3.2.4 Variation of Train Length ...... 122 3.2.5 Variation of Brake Force ...... 126 3.2.6 Variation of the Derailment Quotient (L/V) ...... 129 3.2.7 Variation of Ground Reaction Force ...... 132 3.2.8 Cornparison of the Parameter Variation Results ...... 135 3.3.9 Conciusion: Variation of Parameters ...... 140 Mississauga Derailment: Brake Application Tests ...... 142 3.3.1 Results and Discussion of Brake Application Tests ...... 143
4 Summary and ConcIusions 146 4.1 Surnmary ...... 136 4.1.1 Recommendat ions for Future Reseasch ...... 153
vii List of Tables
CI 1.1 Power output and mass of typical locomotives ...... 1 1.2 Freight car weights and dimensions...... Y 1.3 Dangerous goods tank car classification ...... 9
2.1 Coupler parameters used in the mode1...... 62
3.1 Derailed freight car list . Mississauga derailment .[l1...... 99 3.2 lnput data for Mississauga mode1...... 102 3.3 Peak coilision forces . Cornputer rnodel of the Mississauga derailment . 106 3.4 Resul ts of vrtriat ion of initial train speed ...... 115 3.5 Results of variation of car mass ...... 119 3.6 Results of variation of train length ...... 122 3.7 Results of variation of brake force...... 126 3.8 Results of variation of L/V ...... 129 3.9 Results of variation of ground reaction force ...... 132 3.10 Results of brake application tests to the Mississauga mode1...... 144 List of Figures
1.1 Train speed influence on number of cars derailing per accident ..... 4 1.2 Train length influence on number of cars derailing Fer accident .... .5 1.3 LPGfAmmonia tank car ...... 10 1.4 Vinyl chloride service tank cor ...... 10 1.5 Coupler components ...... 12 1.6 Freight car coupler heads: TypeE, F . SE and SF ...... 13 1.7 Tweaxle freight car truck ...... 15 1.8 Typical Result of the Yang and Manos Mode1 (1972)...... 21 1.9 Results of Birk, Anderson and Coppens Mode1 (1990)...... 24
2.1 Rigid body mode1 of the system...... 39 2.2 Coupled subsystems ...... 41 2.3 General definitions for a rail car sub-system ...... 13 2.4 General rigid body mode1 of a sub-system ...... 43 2.5 Car-rail constra.int geometry: Straight track ...... 48 2.6 Car-rail constraint geometry: Constant radius curved track ...... 50 2.7 Coupler-to-coupler constraint geometry...... 52 2.8 Brake force mode1...... 58 2.9 Design drawing of a typical E/F Type coupler ...... 59 2.10 Location of coupler striker force element ...... 61 2.11 Coupler striker force mode1...... 63 2.12 Coupler reaction moment ...... 65 2.13 Basic derailing train system: case 1...... 67 2.14 Results of basic rigid bar analysis: case 1...... 70 2.15 Basic derailing train system: case 2 ...... 72 2.16 Results of basic rigid bar analysis: case 2 ...... 73 2.17 Resdts of basic ngid bar andysis: case 3...... 76 2.18 Ground reaction force rnodei ...... 80 2.19 Nurnber of derailed cars for a given train speed and length ...... 82 2.20 Application of ground reaction forces ...... 83 2.2 1 Collision mode1 geometry- ...... Y6 2.22 Contact force variation with gap distance...... SS 2.23 Collision contact force variation wit h approach velocity and gap distance . 89 2.24 Flow chart of the derailment mode1 cornputer program ...... 91
3.1 Outcome of the Mississauga derailment ...... 98 3.2 Mississauga derailment model: ha1 outcome ...... 104 3.3 Mississauga derailment model: normalized peak collision forces for each car ...... 105 3.4 Mississauga derailment model. collision of car 8 into 7 at 5.70 seconds. 108 3.5 Mississauga derailment model. collision of car 8 into 6 at 5.90 seconds . 108 3.6 Mississauga derailment model. collision of car 11 into 9 at 8.00 seconds. 109 3.7 Mississauga derailment model. collision of car 11 into 10 and 12 into Il at 8.60 seconds...... 109 3.8 Variation of parameters: final car positions for the base system mode1. 1 14 3.9 Outcome with variation of -50% initial speed ...... 116 3.10 Outcome with variation of -20% initial speed ...... 117 3.1 1 Outcome with variation of +20% initial speed ...... 117 3.12 Outcome with variation of +50% initial speed ...... 118 3.13 Outcome with variation of -50% car mass ...... 120 3.14 Outcome with variation of -20% car mass ...... 120 3.15 Outcome with variation of +20% car mass...... 121 3.16 Outcome with variation of +50% car mass ...... 121 3.17 Outcome with variation of -50% train length ...... 124 3.18 Outcome with variation of -20% train length ...... 124 3.19 Outcome with variation of +20% train length ...... 125 3.20 Outcome with variation of +50% train length ...... 125 3.21 Outcome with variation of -50% braking force...... 127 3.22 Outcome with variation of -20% braking force...... 127 3.23 Outcome with variation of +20% braking force...... 128 3.24 Outcorne with variation of +50% braking force ...... 128 3.25 Outcome with variation of -50% derailment quotient ...... 130 3.26 Outcorne with variation of -20% derailment quotient ...... 130 3.27 Outcome with variation of +20% derailment quotient ...... 131 3.28 Outcome with variation of +50% derailment quotient ...... 131 3.29 Outcome with variation of -50% ground reaction force ...... 133 3.30 Outcome with variation of -20% ground reaction force ...... 134 3.31 Outcome with variation of +20% ground reaction force...... 134 3.32 Outcome with variation of +50% ground reaction force...... 135 3.33 Cornparison of changes in number of derailed cars ...... 136 3.34 Cornparison of changes in peak collision force...... 138 3.35 Cornparison of changes in the accident scene area...... 139 3.36 Cornparison of changes in the accident severity measure ...... 141 3.37 Outcome of brake application test 3 ...... 145 Notation
Baumgarte velocity stabilization parameter acceleration coefficient, ith constraint for the kth degree of freedom coefficient, it h constraint , t ime dependent
Baumgarte position stabilization parameter
static friction coefficient, ground reaction model dynamic friction coefficient, ground reac t ion model veloci ty friction coeficient , ground react ion model collision model quadratic coefficient collision model linear coefficient collision model constant coefficient
distance buffer distance, collision model control distance, collision model coupler body length distance, CG of front coupler to coupler pivot distance, CG of rail car to front coupler pivot distance, CG of rail car to front truck distance, CG of rail car to a coupler pivot distance, CG of rear coupler to coupler pivot distance, CG of rail car to rear coupler pivot distance, CG of rail car to rear truck distance, striker opening distance, coupler pivot to striker contact distance, CG of rail car to a truck
constraint function, or general function brake force contact force, rail car collision instantaneous displacernent dependent contact force ground reaction force, rail car contact force constant
xii striker plate reaction force unadjusted brake force ground reaction force, surface 1 ground reaction force, surface 2 force function, or general function gravity (context is clear) mass moment of inertia mass moment of inertia of coupler body mass moment of inertia of rail car stiffness coefficient first stage striker plate stiffhess second stage striker plate stiffness a constant brake force adjustment constant brake force timing parameter total ground reaction coefficient propagation parameter timing parameter length of bar length of a rail car number of constraints mass (context is clear) initial mass of bar mass of coupler body mas of rail car coupler reaction moment, Types EE coupled coupler reaction moment, Types EF coupled coupler reaction moment, Types F-F coupled number of generalized coordinates number of cars, general number of cars in a train nurnber of cars derailed normal force, surface 1 normal force, surface 2
kth generalized coordinate generalized applied force
curve radius
system kinetic energy t ime delay time
stopping distance function
velocity initial velocity of bu velocity of train local position generd displacement gap distance stopping distance of bar global position X coordinate of centre of curvature of track X coordinate of rail car X coordinate of front truck, relative to curve centre X coordinate of front coupler X coordinate of rear truck, relative to curve centre X coordinate of rear coupler X coordinate of truck local position global position Y coordinate of centre of curvature of track Y coordinate of rail car Y coordinate of front truck, relative to cuve centre Y coordinate of front coupler Y coordinate of rail YRC Y coordinate of rear truck, relative to curve centre YRCP Y coordinate of rear coupler YTR Y coordinate of truck
angular displacement of front coupler angular displacement of rear coupler a srnall positive number relative angle bet ween mated coup lers Lagrange multiplier angular displacement angular displacement of rail car friction coefficient friction coefficient, traction limited braking friction coefficient, surface 1 friction coefficient, surface 2
Vectors
{Fq ) external applied forces {Fc) Coriolis and centripital forces from constraint equations {FI generd force vector (9) general force vector {Cl} generalized coordinates vector {Qd generalized applied force vector
Mat rices [JI Jacobian of constraint equations [MI mass rnatrix [BI general matrix
Overscripts
first derivat ive wit h respect t O t ime second derivative with respect to time Chapter 1
Introduction
Freight trains carry a multitude of goods across most industrialized countries. with coal, grain. iron ore, lumber and automobiles being typical of the cargo on board these trains. However, a consist' may dso have tank cms carrying chernicals such as chlorine, toluene, liquefied petroleum gas ( LPG), or styrene. The derailment of a train with even one tonk car containhg these hazardous materids, officially known as dangerous goods, is a serious event which places many people at risk due to the potential for fire, explosion, chemical spill or gas cloud.
Improving the safety level of the railroads carrying these dangerous goods involves a study of ail aspects of the accident, from the causes to the outcorne. The research presented in this thesis focuses on advancing the study of the derailment event itself by modelling the motions and forces acting on the rail cars during the derailment. The development of a computer model capable of reproducing a derailment is the product of this research. The comprehensive model includes a detailed rai1 car and coupler model, grouod reaction force model, collision model, and allows uncoupling and derailment of cars, which are al1 improvernents upon previous models.
This thesis is divided into three main sections. In the first chapter, railway
'A consist is a raihoad term describing the group of cars making up a complete freight train. equiprnent and safety are reviewed, foîlowed by a summary of the previous work concerning the analysis and modelling of rail vehicles and derailments. In the second chapter, the objectives of the research are presented followed by a description of the relevant theory and a detailed description of the model. In the third chapter the results of the modelling of a major derailment are presented, wit h analysis and discussion of the results. Sensitivity analysis to variations in the important parameters are then presented, and the importance of these variables discussed.
Chapter One begins with an introduction to rail accidents in Canada. Following is a summary of the rail industry in North America and a description of the equipment used, with an emphasis on the information needed for a computer model. A brief discussion of the causes of a derailment and the resulting outcome is then presented. Past research concerning causes and prevention of derailments is then reviewed, followed by a comprehensive review of the previously published derailment models.
1.1 Railway Accidents in Canada
Although rail passenger safety is of the utmost importance, the largest risk to human life and property is from the derailment of a freight train containing a mixture of dangerous goods. An example of this risk is the well known derailment in November of 1979 of twenty-four freight cas near Mississauga, Ontario, which resulted in the evacuation of over a quarter of a million people from the densely populated surrounding area. Among the cars that derailed, nineteen contained dangerous commodities including chlorine, propane and toluene. The puncture of a chlorine car, the explosion of several propane cars and the intense fire al1 prompted the evacuation[l][2]. No fatalities resulted from this incident but the risk to public safety was very high, resulting in a govemment inquiry to study the event and make recommendat ions towards preventing future disasters[1] .
One of the conclusions of the inquiry was that a study be undertaken to analyze the effects of train length, velocity, and placement of dangerous goods cars on the outcome of a derailment. This resulted in the formation of The Toronto Area Rail Transportation of Dangerous Goods Task Force which studied nine areas of interest related to the transport of dangerous goods: a perception survey of residents living near rail lines, a route analysis, a dangerous goods risk assessment, dangerous goods risk management, effects of rail technology on risk, effects of speed on risk and derailment outcome, effectiveness of rail line bders, emergency response, and land use near rail lines [3][4].
Among the many findings of the Task Force, train speed and length were found to be significant factors related to the severity of a derailment. Figure 1.1 shows the effects of train speed on the number of derailed cars, for all train lengths. Figure 1.2 shows the effects of train length for three train speeds on the number of derailing cars. The effect of speed and train length on the number of derailed cas is clearly seen. Also reported, but not indicated by these figures, is an increase in accident severity with train speed. The recommendations of the Task Force were to reduce the speeds of dangerous goods trains travelling through populated areas. which resulted in a 50 km/hr speed restriction for trains travelling through the Toronto area.
On average between 1985 and 1994, 1029 railway accidents occurred per year in Canada[5][6]. The severity of t hese accidents ranges from derailments and railway crossing collisions to a single wheel lifting off the track in a yard. The more serious incidents are the main track derailments which typically involve a fully loaded train travelling at main line speeds approaching 100 km/hr.
Between 1985 and 1994, an average of 128.6 main track derailments occurred per year, accounting for twelve percent of al1 railway accidents. Most dangerous are the main track derailments that involve a dangerous goods train. An average of 37.5 dangerous goods trains derailed per year from 1985 to 1994, nearly thirty percent of al1 main track derailments, and almost four percent of all railway accidents[5][6].
- 40 MPH - - 30 MPH
**- 15 MPH
Note: This figure has been created from North Amerrcan: - technical research papers - university studies - research and development inves tiqa tions
-0 10 20 30 40 50 60 70 80 90 IO0 Train Length (Loaded Cors)
Figure 1.2: Train length influence on number of cars derailing per accident. Repro- duced in full from [3].
undetected intemal defects, inadequate track geometry maintenance, undetected roller bearing failures, worn truck components and wheel failures. A similar study(6l reports that the causes of main track derailments, excluding yard derailments, are spiït approximately 40% track, 40% equipment, and 20% operations2. As well, although the speed and length of a train have an effect on the number of derailed cars, neither has an effect on the point of derailment within a train[7].
The variety of these causes suggests that derailments cannot be eliminated, although they can be minimized by improved maintenance and the continuing development of detection and waniing systems. In addition to the efforts of preventing and detecting derailments, efforts towards reducing the severity of the outcome is also an important area of study. Research conceming the reduction of derailment outcome has to date focused on the analysis of past derailments and the
2Taken by this author to include buman error, among other operational causes.
5 resultant statisticai data[3][4][5][6][?].
To better understand the causes of a derailment, and the possible outcornes, a brief summary of rail equipment is presented in the next section.
1.2 Rail Equipment Summary
In North America, approximately 30 000 locomotives haul over 1.8 million freight cars over 500 000 kilometers of track, with an average train length of nearly 70 cars[8][9]. The variety of equiprnent in use is considerable, but because of industry standardization and the longevity of the equipment, most of the equipment can be described by a few representative exam ples.
1.2.1 Locomotives
The majority of freight locomotives are the diesel-electric types, in which the traction drive is supplied by electric motors mounted on the trucks above the axles. Power for the electric traction motors is provided by a large diesel engine-powered DC generator or AC dternator. The diesel engine is typically rated at 1000 to 3000 kW (1500 to 4000 HP).The electric transmission system allows the diesel engine to be used at fdl power at low train speeds, something that would be very difficult to achieve with ân al1 mechanical transmission of power. Until recently al1 traction motors were DC, but AC motors have been recently introduced [10][ll]. A typical locomotive has a mass of 110 000 to 170 000 kilograms, and has either eight or twelve traction wheels. AU routine operations for the train are controlled from the locomotive.
Two types of diesel-electric locomotive are in generd use in Canada: four-axle and six-axle units with DC traction motor units. The lighter four-axle units have less available tractive effort and are thus favoured for the faster, longer trains on the prairies. The heavier, and more cost ly, six-de units are reserved for use in areas where tractive effort demands are higher, such as on the steeper grades in the rnountains and for longer heavier trains[lO]. New locomotives wit h AC traction motors have recently ben introduced, with increased power and tractive effort over the older DC units. but these are not in general use yet. Table 1.1 sumrnarizes the minimum and maximum weights and available power for a few locomotives in use in Canada.
Table 1.1: Power output and mas of typical locomotives. From Wallace et al. [IO].
1 Maaufacturer Mode1 Type Power Min. Mas Max. Mass 1 (kW) (kg) (kg) Bombardier KR416 4-axle 2 400 114 600 127 O00 Bombardier KR616 6-axle 2 400 172 400 190 500 Electromotive GP-40-2 4-axle 2 245 116 100 125 900 Electromotive SD-40-2 &de 2 245 166 900 190 400 General Electric B-36-7 4-axle 2 700 118 000 127 O00 General Electric C-3û-7 6-axle 2 700 166 000 190 500
1.2.2 Freight Cars
A summary of freight car weights and measures for a few representative examples of the more common types of cars in service is shown in Table 1.2. The list is far from exhaustive, as most cars are built on a custom basis with design details specific to a particular region of operation. The dimensions of al1 freight cars used in North America must meet the standards set forth by the Association of American Railroads (AAR) in its Manual of Standards and Recommended Practices. The specifications do not dictate the car structure or design but they do set Limits on maximum stress levels, dimensional clearances, and the location of the centre of gravity[8][9].
The weight of an empty freight car is known as the light weight. The load limit is Table 1.2: Freight car weights and dimensions[8][9].
car type Light Gross length width truck Weight Weight cent ers (kW (kN) (4 (ml (4 box car 262 941 14.98 3.2 12.41 toluene 384 1170 14.10 3.2 11.50 (DOT 103) caustic soda 384 1170 12.44 3.2 9.86 (DOT 111) propane 384 1170 18.55 3.2 15.97 (DOT 105) propane 501 1170 19.89 3.2 16.82 (DOT 112) open top hopper 374 1170 15.40 3.0 12.34 covered hopper 246 926 11.96 3.2 8.00 (cernent ) covered hopper 293 872 17.0 3.2 13.75 (grain) pressurized t a.nk 280 1170 13.20 3.2 9.86 (chlorine) pressurized t ad 501 1170 19.30 3.2 15.97 (LW pressurized tank 384 1170 15.37 3.2 12.03 (vinyl chloride) non-pressure tank 361 1170 13.94 3.2 10.59 (caustic soda) high side gondola 274 897 15.30 3.2 12.34 mill gondola 317 851 16.00 2.9 13.30 the maximum allowed weight of the load, which is the difference between the maximum allowed gross weight on the rails and the light weight. The maximum aliowed gross weight is determined by the number of wheels, the spacing of the wheels, and the wheel md wheel bearing sizes. Freight cars of similar external Table 1.3: Dangerous goods tank car classification[9].
Service Safety Valve Setting (psi cryogetllc DOT 113 30 to 115 non-pressurized DOT 103 35 DOT 104 35 DOT 111 35 to 75 DOT 115 35 - -- pressurized DOT 105 75 to 450 DOT 206 375 to 600 DOT 107 determineci by pressure vesse1 design DOT 109 75 to 225 DOT 110 375 to 750 DOT 112 150 to 375 DOT 114 255 to 300 DOT 120 (proposed) 150 to 450 DOT 220 (proposed) 375
dimensions may have different load capacities due to differences in wheel and axle sizes and light weights. To ensure that overloading is prevented it is required that the load limit be stencilled onto the side of each freight cor in service.
Tank cars cause the most concern after a derailment has occurred as they typically t rassport dangerous goods. Al1 tank cars that transport hazardous commodit ies carry a United States Department of Transportation (DOT) classification, summarized in Table 1.3. As examples of typical tank cars, Figures 1.3 and 1.4 are drawings of liquefied petroleum gas (LPG)and vinyl chloride tank cars. Although most tank cars are similar looking in appearance, the tank wall thickness is significantly different for each type of tank car, resulting in different levels of punct ure resistance during a derailment. To improve puncture resistance head shields are stmdard on all DOT 112/114 tank cars. The head shields preveni; Figure 1.3: LPG/ Ammonia tank car.[8]
L-TOC. --- OvEii GRAM
Figure 1.4: Vinyl chloride service tank car.[8] punctures from occurring during shuntin8 of cars in switch yards or during a derailment. As well, refrigerated cars that carry Liquefied gases are required to be covered in a heat shield to slow the heating time of the tank contents should the car be engulfed in flarnes after a derailment.
The major components of d freight cars are the couplers, the trucks, and the brakes which are discussed in the next section. Following the rail equipment summary is a discussion of the causes and possible outcomes of a derailment.
1.2.3 Couplers
At both ends of d freight cars is the coupler assembly. Figure 1.5 shows the major components of the assembly: the shank or body, the yoke, and the striker casting. The basic coupler consists of a pinned link, with the coupler mechanism at the head cast integral with the shank. The shank of the coupler passes through the opening in the striker casting to be joined to the front of the yoke by an 8.9 centimetre (3.5 inch) diameter pin. The striker casting mounts to the frame of the freight car at either end. The yoke in turn encloses the draft gear which is fixed to the frarne of the freight car. The draft gear acts as a shock absorbing device necessary to reduce loads transmitted to the freight during coupiing and normal service braking[8][9] .
The basic coupler design was originally formalized as the Type E coupler, which became the Standard4 coupler adopted by the AAR in 1932. The Type E was designed in response to the requirement for a coupler that would couple upon impact, could be uncoupled without requiring a worker to go between the cars, and could be used by dl railroads. Prior to the Type E coupler, over forty different coupler designs existed, with no assurance that one design was capable of coupling with another[8][9].
- ..- 3~hun& & the process of placing the fieight cars in order ~hilebuilding the consist. 4Referred to in capital Ietters as it is the standard industry design. AAR SIanaArd E Couplef. Ai~lSMnL Types E60C and
Coupler Tvoe W9C Grades Cor E Sleu
YOKE rrpe WSA uw wltn Convanrionai Oraft Gearr Grau- C or E Siel
Figure 1.5: Coupler components. From upper left to bottom right: Standard Type E; Standard Type E/F; Standard Type F; Y45A Yoke; Type F Striker Casting.[S] Standard 'E'
- Lower Shelf 'E'
Figure 1.6: Freight car coupler heads: Type-E, FI SF and SE.[9] Development of the Type E coupler continued after its adoption as the standard coupler design, with the aim of decreasing free slack and preventing verticd disengagement of couplen upon derailment or separation due to hard coupling during shunting of cars. The result was the Type F coupler, adopted as the Alternate Standard coupler by the AAR in 1954. The Type F differs from the Type E mainly in the addition of a bottom protrusion below the knuckle. changes to the shank pin design and in casting details to reduce the free slack. These differences are easily seen in Figure 1.6. As the Type F interlocking feature prevented uncoupling of cars during trairi derailment, contributed to the prevention of car overturning or telescoping, and reduced the rate of head punctures, the DOT/FRA mandated that hazardous material tank cars were to be equipped with F Type couplers beginning in lWO[8][9].
Continuing safety considerations brought the addition of a top shelf and hood to the Type F, resulting in the SF, and top and bottom shelves to the Type E, resulting in the SE. The different combinations are shown in Figure 1.6. The addition of the top shelf prevents a coupler that has detached at the shank pin from fding down onto the tracks and possibly causing a derailment, an improvement over the single bottom shelf units which do not prevent this. As of 1970 al1 dangerous goods were required to be manufactured with Type F couplers only, and as of 1979 dl dangerous goods cars were required to be manufactured with the Type SF couplen[8][9].
1.2.4 Tkucks and Wheelsets
The major components of freight trucks are the two wheelsets with bearing assemblies, two side frames, a bolster and the two suspension spring sets. The side frames rest on the beaxing assembly locating the wheelsets. The bolster rests perpendicular to the side fiames on the suspension springs completing the basic frame. The center plate on the bolster mates with the pin on the underside of the car body, with side bearings providing roll and yaw control. The complete truck assembly is not rigidly fixed to the freight car body as lifting the car body leaves the trucks on the track. As weil, lifting the tmck frarne leaves the wheelsets with bearings on the track. An assembled truck is shown in Figure 1.7.
The consequence of the piecemeal construction of the tmck bodies is that during a derailment the trucks typically separate hmthe freight cars as soon as the car has rolled over or bounced off the centre plate. The trucks then disperse into the various components, result ing in post-accident scenes wit h wheelsets and truck frarnes littering the accident area. and the freight cars rolled onto their side.
Figure 1.7: Two-axle freight car truck[lO].
1.2.5 Brakes
Al1 freight trains use air as the brake pressure medium. The locomotive carries a cornpressor which supplies pressurized air to a reserve tank on the locomotive. Each car has front and rear brake air lines which interconnect through the length of the train. In the original form of the air brake, the Westinghouse straight air brake, braking was activated with the direct application of pressurized air to the main air line and thus to the brake cylinder on each car. This system is now only used on locomotives as it does not provide braking in the event of a complete loss of locomotive air pressure. Al1 freight cars now use the automatic air brake[8][9].
In the automatic system, when the main air line pressure is full a the-way valve connects the main air fine to a reservoir tank mounted on each car, and the car brake cylinder is at atmospheric pressure. When the brakes are applied and the main air line pressure is reduced, or the main air line has broken, the reduction in line pressure below the reservoir tank pressure moves the bway valve to connect the reservoir tank to the brake cylinder. The pressure from the reservoir tank then acts on the brake cylinder piston, and the braking force is applied to the wheel treads by the brake pads. On older systems a single large brake cylinder actuates a system of rods and Links to transfer the brake force to the brake pads at each wheel. Newer systems have a pair of brake cylinders mounted on each truck, with each cylinder actuating a pair of brake pads directly to the wheels. The newer system has fewer moving parts to Wear out but requires two, slightly smaller, brake cylinders.
As the braking action depends on the reduction of air pressure in the system, brake actuation progresses from the cars nearest to the locomotives towards the rear of the train, or from a point of failure in the air line outwards. This creates a delay in the complete braking of the train, and cm create severe problems in emergency brake applications when the last car does not begin braking until several seconds after brake application[l 1].
When the brakes are retracted, main air line pressure is brought back to a level above the reservoir tank pressure, moving the valve to the initial position of brake release. To ensure that the brakes are applied to standing cars, which may [ose air reservoir tank pressure with time, a manual brake wheel is provided. 1.3 Past Research Focusing on Derailments
Understanding the cause of a train derailment is an important part of developing an accurate mode1 of the derailment. Many factors affect how the train will derail, at what point dong the train the derailment will occur, and how many cars will be involved. These factors include the failure of a wheel bearing, rail, roadbed or equipment [5][6] [7] [12] [13].
The causes that involve structural or mechanical component failure axe outside the field of computational dynamics. However, research has been done to study the effects of the dynamic behaviour of the rail car system and its components on the possibility of causing derailments. The literature contains many models to analyze wheel, rail, and car suspension design parameters in relation to their effects on causing a derailment. These models are useful in the design of new cars, wheels, trucks, and rail, as well as the planning of safe speed limits for trains. The models can be summarized as follows:
a models analyzing the hunting of tmcks and lateral stability that may cause a derailment: Blader and Kurtz (1973)[Ml, Fujii ( 1975)[15]
models analyzing the multibody dynamics of the car/truck/wheel/t rack interface to study curving or other behaviour that may lead to a derailment: Aknin et al. (1994)[16], Blader and Mealy (1985)[17], Duffek et al. (1986)[18], hori(1978) (191, Kawasaki (1981)[20], Matsu0 (1976)[21] and (1986)[22], and Zhang and Fuda(1988)[23]
models andyzing wheel climbing a.a cause of derailment: Karmel and Sweet (1982)[24],Sweet et al. (1979)[25], Sweet et al. (1983)[26],Sweet and Karmel (1984) [27]
models of derailments of wheelsets at grade crossings: Cherchas et al. (1979)(281 a expert system, or statisticd models: Hoag (1985)[29]
experimental determination of the loads needed to derail a wheelset: Kalaycioglu et al. (1987) [3O ]
a sensors to detect impending derailment: Nmce (1980)[31].
Although a significant amount of research has focused on the cause and prevention of derailments, contributing to a reduction in accident rates[5] [32][33], derailments continue to occur. Unfortunately, improved car design cannot prevent a derailment caused, for example, by a ballast washout following a flash flood, or the collision between a train and a vehicle on the tracks. In the cases when a derailment is not preventable, a reduction in the severity of the outcome is most desired. Prevention of the worst possible outcome of the derailment through the placement of dangerous goods cars in the locations of lowest risk in the consist, or the planning of evacuation procedures and emergency response guidelines would be greatly aided by the use of a detailed and accurate computer model of the derailment process.
However, the cause of a derailment must be taken into account in the development of a derailment model. This requires a brief look at the point of origin of a derailment and the cause, or initiation, of the event.
The point of origin of a derailment may be divided into two groups: the derailment of the first locomotive in the train, and the derailment of any other car dong the train. For a derailment originating with the first locomotive, the initiation of the event could be caused by the destruction of the rail ahead of a train, failure of the rail as the first locomotive passes over a weakened portion of rail, the lifting of a wheel over the rail, the failure of a wheel or axle on the locomotive, or through a collision with a foreign object on the rail. For a derailment of any other car dong the train the initiation of the derailment could be caused by a failure of the rail, a wheel or axle failure, or a wheel Lifting over the rai1[5] [6][7].
The front-of-t rain origin, regardless of cause, will produce similar derailment outcornes; the locomotive wiU run off the rails and initiate the derailment process. The emergency braking signal would initiate and propagate from the fiont of the train due to the actions of the locomotive operator or due to the failure of the brake pipe.
The derailment resulting from a rniddle-of-train origin may take two forms. In the fmt case, the cars would uncouple at the derailment point, splitting the train into two systems and leaving what is essentidy a fiont of train derailment to proceed. The emergency braking signal would propagate from the point of derailment at the rupture in the brake line pipe. In the second case the cars would remain coupled and the ernergency brake application signal will either be delayed until the cars have travelled enough to rupture the brake pipe or the locomotive operator initiates emergency braking. The resulting derailment may be quite different from a front of train derailment. If the cars do not uncouple, the derailment may proceed in a far less destmctive manner with cars derailing but not uncoupling. This result is more likely to occur for derailments at low speed in switch yxds rather than derailments occurring at operating speeds on mainline track.
Given the above, it can be concluded that the outcome of a derailment is generally not significantly dependent on the type of initiation or the point of origin. The exception is for a low speed derailment, with a middle-of-train origin, of a train which remains coupled. Thus the initiation of a derailment can be easily modelled through the artificial derailment of the first car, or the removal of the rail at any other point dong the train.
1.4 Previous Derailment Models
The prediction of the motion of rail cars during a derailment of a train, either by numerical or other means, has received relatively little attention. Yang and Manos ( 1972)[34j5 and Anderson (1990)[3616 have produced the only models that are complete enough to provide useful results and t hat advanced the state-of-the-art significantly at the time of their publication. Johnson (1991)[39], Guran et al. (1992)[40],Gracie (1991)[41],and Roorda and Gracie (1992)[42]all produced models that analyze the basic kinematics of the derailment process, but do not improve upon the existing state-of-the-art models.
1.4.1 The Yang and Manos Mode1 (1972)
Yang and Manos(341 published the first attempt at a computer model of the derailment of a complete train. A summary of the work was later published by Yang, Manos and Johnstone[35]. The model assumes that the first car always derails, or that cars ahead of the derailment point have uncoupled and are not involved in the derailment. The group of cars that remain on the track behind the derailing cars is modelled as a single body representing the total mass of those cars. Cars derail when they reach a specified track location, thus the model of N cars starts with one derailed car and a lumped mass of N-1 cars still on the track. Once the front wheel location of the second car passes the point of derailment, there are two derailed cars, and N-2 cars represented by the lumped mass. As the train is assumed to be braking, the bodies still on the track produce a braking force which is applied through the coupler to the derailed cars.
Ground reaction forces frorn the wheels of the derailed cars are modelled by a simple Coulomb friction model. The forces are applied to the derailed car at the locations of the truck-to-car comection, the applied direction being opposi te to the motion of the truck. A coupler reaction moment between cars, a function of the angle between adjacent cars, is modelled by a polynomial equation.
The solution begins by manipulating the equations such that the unknown coupler Spublished also under Yang, Manoe and Johtone[35] 6results were originally published in 1990 as part ofI371[38] forces and the angular accelerations of the derailed cars are solved for. The acceleration of the cars remaining on the rails is then found. The two accelerations axe integrated and used in the constraint equations to solve for the accelerations, velocities, and positions of the derailed cars. This procedure continues, with more cars leaving the track as time progresses, until all bodies corne to a stop.
The cars are not permitted to uncouple in this model, and the couplers are not limited in their range of angulor displacement. A reaction moment between cars is the only restraint on relative rotation between bodies. The results of this model typicdy have a zig-zag appearance, with some cars overlapping each other, as seen in Figure 1.8. The cars do not typically deviate fax from the rail and the overlapping of the cars produces a compact appearing final result. However, for the velocity range studied by Yang and Manos, and with the soi1 coefficient of friction adjusted to between 1.0 and 2.0 times the normal force of the derailed car, the number of cars predicted to derail by the model is in good agreement with derailment statistics.
Figure 1.8: Typical Result of the Yang and Manos Mode1 (1972)[34j. The main weaknesses of the Yang and Manos[34] mode1 are as follows:
a does not allow coupler failure
a relative angle between cars is not restricted to a Limiting dueas occurs with real couplen, allowing extreme relative rotation angles between cars
a does not model the collision of cars
a does not permit a car to derail due to high lateral forces generated during the derailment
a is restricted to straight track
a uses a constant coefficient Coulomb friction rnodel for the ground reaction model .
1.4.2 Anderson Model (1990)
An improved derailment model has been published by Anderson[36]. The model was origindy part of an overall derailment disaster model which included the representation of trackside geography, such as road and building locations, and a burning tank car model to estimate the extent of damage to the surrounding area due to a derailment involving dangerous goods [37][Xi].
The rnodel differs from the Yang and Manos model in that each car has three degrees of freedom with the coupling between cars modelled by very stiff springs. The model is essentially a chah of spring-mass systems in which the springs are removed from the model when the force applied to the spring has reached a limiting value, allowing the cas to separate. The trucks and wheel sets are not modelled, and the ground reaction forces are applied to the cars at the locations of the trucks using a simple constant coefficient Coulomb friction model. An added feature of the model is the estimation of tank-car roll after derailing. Once a car has uncoupled from the cars leading and trailing it, the kinetic energy of the car is used to calculate the amount of roil the car will undergo. Because of the construction of tank cars, roll is dowed only until the top cap of the tank car has reached the ground, after which only sliding is modelled. The height of the centre of mass of the car, the car track width, the velocity and ground friction force are the parameters required for the roll model.
The model also determines whether two cars have collided and the angle at which the collision took place, so that tank car punctures can be predicted and used as input to the burning tank car model. The type of tank rupture is estimated based on the relative approach velocity and the incident angle, allowing for the type of content spill or tank explosion to be modelled by the buming tank car rnodel. The collision is rnodelled assuming the freight cars are point masses and, following classical impact theory, uses an estimate of the coefficient of restitution to determine the post-impact velocities of the bodies. The shape of the freight cars is not considered in the calculation.
Any car on the track may derail, unlike the assumption made by Yang and Manos that the first car of the train will derail. To follow the prescribed path of the track, the cars ahead of and behind the derailment location are given velocities and accelerations that will keep them on the track. This dows for curved track to be modelled, as the specified veloci t ies and accelerat ions cont rol the direction of the train still on the rail. As cars derail they are no longer guided by the given velocities and accelerations (which represent rail forces), but by the gound reaction.
The main advancements of this model over the Yang and Manos model is that coupler failure, uncoupling and the collision of cars are alI modelled. An estimation of car roll and the modelling of curved track are also added. The result of the mode1 when given initial conditions similar to the Mississauga derailment is shown in Figure 1.9. The main drawback of this model is that it sders from frequent numerical instabilities. These occur when the very stiff coupler springs are removed from the model when the couplen are deemed to have broken[36]. When less stiff springs, which do not create these numerid problems, are used the couplers stretch far beyond what is realistically expected. Similar instabilities occur when a car reaches the point of derailment on the track and the constraint forces guiding the cars on the rail are removed [36]. Despite these problems, the rnodel does produce a realistic result, and was the best mode1 to date.
Figure 1.9: Results of Birk, Anderson and Coppens Model (1990)[38].
1.4.3 Johnson Model (1991); Guran Model (1992)
The simplest model, by Johnson[39], is based on the study of the kinematics of linked chahs published by Samuel Vince in 1780. Published in the International Journal of Mechanical Engineering Education, Johnson's analysis is presented as a teaching tool as well as a research aid.
The train is modelled as a series of rigid bars of equd mass attached together by frictionless pins, resting on a frictionless surface. The system is excited by a transverse impulse applied to the front of the first rod. The velocities at each joint just after impact axe interpreted as an indication of the severity of jack-knifing of the cars. As the system is assumed to be resting on a frictionless surface, there is not a final resting position of the cars, thus the initid velocity of each joint is compared, relative to the velocity of the first car, and the results interpreted from t hese veloci t ies.
Johnson compares the kinematic results of this simple model to the outcome of a January 1968 collision between a highway freight truck transporting a 135 tonne transformer and a 450 tome passenger train travelling at 120 km/hr. The model does produce results that mimic the appearance of the actual accident. Johnson suggests the model serves as a usefûl demonstration of the power of basic analysis to study complex events.
Guran et al. [40] extended Johnson7s[39]analysis by dowing wiations in the masses of the cars in the train. The effect of the position of cars with high mass on the initial velocities was studied. As with the Johnson model, the train is resting on a frictionless surface, with no forward velocity, such that only the jack-knifing behaviour is studied.
They concluded that the closer the concentration of cars with high mass is to the point of impact, the less diastic is jack-knifing. The model does not provide any information as to whether the fiont or rear cars of a freight train are more or less likely to derail, or the severity of the derailment. Both the Guran and Johnson models are only appropriate for cases when the lead car is struck from the side by a massive vehicle moving at high speed, and the wheel-ground reaction forces are ignored. 1.4.4 Gracie Model (1991);Roorda and Gracie Model (1992)
Gracie[4l] presented a model in which the positions of the cars are calculated from the amount of energy dissipated in a given tirne step. The model is described as being pseudestatic, in that the inertial effects are included as forces applied to the centre of mass of each car. Friction forces are applied to the truck locations opposite to the direction of the velocity. The motion of the cars continues until the initial kinetic energy is dissipated. The cars are not able to uncouple.
Gracie concluded that the model was incomplete, as numerical instabilities in the computer program couid not be avoided. Reasons for the instability were not found. but Gracie suspected that the calculation of the accelerations using previous increment position data may have caused a feedback type effect to cause the solution to drift with time. This suspicion was, of course, correct as Gracie numeridy differentiates the vehicle positions twice to arrive at the accelerations, which are then used to provide inertia forces. The resulting inertia forces are then used to End new positions. The process of numerically differentiating position data to determine velocities and accelerat ions, which are then used to determine new position data, is well known to be numerically unstable.
To solve the instability problem, Roorda and Gracie[42] ignore the inertial effects. The motion of the train is based on the initial kinetic energy, and the energy lost through frictional work. Positions are fouud at each position increment in the solution based on the geometry of the train in the previous two increment steps. Time is not a variable in the solution, although an estimate of the time per increment is found based on the energy lost to frictional work and the initial kinetic energy of the train.
By carefd tuning of the ground friction coefficients, Roorda and Gracie were able to reproduce the expected number of derailed cars for a given train speed and length, in cornparison with statisticai data[3]. This tuning process essentially matched the average energy lost to dissipative forces throughout the derailment with the initial energy of the system.
1.5 Objectives of a State-of-the-Art Derailment Mode1
Based on the information regarding the origin and initiation of a derailment, previous derailment rnodels, the statistical data of past derailments and the construction and operation of a typical freight train, a state-of-t he-art derailment mode1 should be capable of:
O modeiling a typical freight train with realistic Limits on train length, speed or track geometry
modelling the kinematics of the couplers and the forces generated by the couplers against the cars
r allowing for the uncoupling of cars to be governed by known load and displacement limits for couplers
O modelling the reaction forces of the car bodies with the ground
O modelling the collision of cars against each other, including estimates for the peak collision forces generated
permitting the derailment to initiate at any point on the moving train, and to be controlled either by track failure or exceeding lateral loads between the wheels and the rail
0 reproducing the outcome of a known derailment scene.
The next chapter presents the computer mode1 developed to achieve these objectives. Chapter 2
Theory
The objective of this research is to produce a computer rnodel of a train derailment. This chapter describes the theory underlying the solution process, the rigid body model of the system, and the computer implementation.
First, the solution methods available are reviewed and the method of choice described. Following this, the rigid body model is described and the equations of motion for the system are introduced. Then the equations of motion and the constraint equations for a single car sub-system are derived. Models for train braking, couplers, the car-ground interface, and car- tecar collision are t hen presented.
2.1 Solution Method
Two courses of action may be taken to study the derailment of a train by cornputer-generated model. A general purpose multibody software package could be used, or a specid purpose model could be developed based on a dynamic analysis of the system. Vehicle suspension onalysis, aerospace systems, robot rnechanisms. and manufacturing machines axe examples of the systems in the field of multibody dynamics typicaily studied using a general purpose (GP) software package[43][44] [45]. These systems d have one aspect in common: t hey remain linked as a complete system during operation. Mathematicdy this means the system degrees of freedom remain constant, and the equations of motion that describe the system remain unchanged in number and basic form. This characteristic makes these systems well suited for analysis with a GP multibody dynamics software package, and for the most part the complexity of these system warrants the use of one of these packages. However, the GP multibody dynamics programs generally have the following drawbacks[46]:
O the inherent generality of the software may result in less than ideal computational efficiency for a given problem.
the generality of the code may result in a more complicated user interface, requiring the user to have some knowiedge of multibody system theory.
r the GP code may not contain specific components required for the model. and programming the necessary components into the program may not be possible or even desirable.
These drawbacks are not insurmountable and general purpose multibody dynarnics software packages, such as AGEM, ADAMS, NEWEUL, MEDYNA or numerous others[47] have the potential to be used to model a train derailment. However, no single GP package can easily model the progression of uncoupling of cars and collisions between cars. As the intent of these packages is to rnodel linkages, mechanisms and other complete systems, they become difficdt to work with in modelling a system that requires complex force functions or systems in which the degrees of freedom rnay change throughout the solution.
Special purpose models, on the other hand, are not subject to most of the limitations cited for GP software models. The main drawback of a special purpose model is the inflexibility of the model to changes in the system. For linkage design, or the analysis of several different mechanisms, it becomes tirne consuming to develop models for each new system studied. However, for a comprehensive study of a single system, a special purpose model is adequate. For the study of a train derailment, past models have all been special purpose in nature. For this research, a specid purpose mode1 wa. developed.
2.1.1 Development of the Special Purpose Mode1
The type of analysis method used to develop a special purpose model rnust be carefully selected. For a system with a constant number of degrees of freedom throughout the life of the model, such as in vehicle suspension or robot rnanipulator analysis, either Newton's or Lagrange's methods may be used. A train derailment is unusual in compôrison with traditional problems in multibody dynamics in that a derailing train will break apart, which dters the nurnber of degrees of freedom of the system. To use a traditionai minimum coordinate solution method would require a reformulation of the equations of motion after each change in the degrees of freedom of the system. Although this is computationally possible, a more efficient method presents itself for the analysis of the system.
A rigid body system may also be andyzed assuming more than a minimum number of degrees of freedom for each rigid body, up to the maximum available, i.e. all bodies are assumed to be fiee of alI other bodies, and the "free" bodies are coupled with equations that describe the constraints. The equations for the constraints are added to the system through the use of additional variables, called added multipliers, which represent the intemal constraint forces. This arrangement of the system equations of motion is known as Lagrange's equations with added, or Lagrange, multipliers. By assuming a number of degrees of freedom greater thân the system minimum for a system of coupled bodies, and adding constraint equations to keep bodies joined together, the equations of motion of a system are greater in number but simpler in fom[43][48].
The use of Lagrange's equations with added multipliers is well described in advanced text books in multibody dynamics, robotics and mathematics such as Shabana[43][44], Greenwood[49], Amirouche[45], and ~Zmarkand Fufaev[50]. The method is often used to generate equations of motion for nonholonomic systemsl, into which the Lagrange multipliers are introduced to provide the extra equations needed to produce a complete set of differential equations of motion. When applied to holonomic systems, the extra constraint equations provide the values of the constraint forces as part of the solution process[44][45][49]. This solution technique also removes the requirement t hot the analyst ident ify the independent coordinat es, making it usefd as a solution algorithm in general purpose multibody dynarnic prograrns.
Typically the number of constraints in a system are assumed to be constant, and the increased number of degrees of freedom are used to simplify the analysis of the holonomic system. However there is no requirement that the nurnber of constraints remains constant for a numerical solution using this method. Given this fact, the method lends itself quite well to the development of a train derailment model. The freight cars and couplen can be modelled as free bodies coupled by constraint equations. As cars uncouple and derail, constraints are removed, changing the number of constraints while leaving the number of unconstrained degrees of freedom unchanged.
The method does have drawbacks. By assurning a maximum number of degrees of freedom and adding equations for constraints, the system of equations increases in size considerably. For example, a planar two body slider-crank mechanism, which is a one degree of freedom system, requires six equations describing the generalized coordinates and five describing the const raints: eleven equations in total. Secondly
lNonhoIonomic systems are those that have constraints which are non-integrable functions of velocity, and thus require more generalized coordinates than there are degrees of freedorn availabIe[49][50]. An example of a nonholonomic system is a rolling coin on a flat surface, describeci in ~e'unarkand Fufaev[SO] dong with numerous other examples. the constraints, usuaily positional, are not easily incorporated into the second order ordinary differential equations of motion that describe the generalized coordinat es. For holonomic systems the constraint equat ions, functions of the posi tional information of the system, rnay be differentiated and directly incorporated into the equations of motion, but oniy with a loss of the positionai information. Mathematically the loss of information does not pose a problem. but manifests itself numericdy as an increase in the global position error of the constraints with time as the numerical solution progresses[45][5 1] [52].
These drawbacks can be partidy overcome. The number of equations may be reduced by arranging the degrees of freedom and constraints to provide an intermediate number of equations of motion[43][44]. To accomplish t his. not al1 bodies are assumed free, and not al1 of the constraints are handled by added equations. General purpose multibody progams using the added multiplier method typically make use of this equation reduction step. The positional error, shouid the error be a problem, can be reduced through anaiyticai and numerical met hods [44][45] [52].
The next section describes the method of Lagrange's equations with added muitipliers, and presents solutions to the potential trouble spots.
2.1.2 Lagrange's Equat ions wit h Added Mult ipliers
The well known Lagrange equation of motion for holonomic systems takes the forrn
where T is the system kinetic energy as a function of the derivatives of the generalized coordinates (ql,q2, ,qn), and Q, are the generalized applied forces for each generalized coordinate. Application of these equations resuits in the well known equations of motion in vector form, where [Ml is the inertia matrix for the system, {ij} is the acceleration vector of the generalized coordinates, {Q,) is the vectot of the generalized extemally applied forces, and {g(q,q)) is the vector of forces that arises from differentiating the kinetic energy with respect to time and with respect to the generalized coordinates of each body (often called Coriolis forces for rigid body analysis)[44].
The derivation of Lagrange's equations with added multipliers is described fully in advanced dynamics text s, such as Shabana(441, Amirouche[45], Goldstein[53], Greenwood[M], Griffit hs [55] and Lanczos[56]. The equations, often known as the standard nonholonomic form of Lagrange's equations are
and combined with velocity constraint equations,
form a complete system of (n+ m) equations for the (n + rn) unknown variables qk and A;. The qk's are the generalized coordinates for the system, and the Ai's are the added Lagrange multipliers. Note that the aik terms are common to both Equations 2.3 and 2.4. For nonholonomic systems, the expressions Cg, Xiaik represent the generalized reaction forces of the nonholonomic constraints, which is also tme for most holonomic systems[44]. As a characteristic of a nonholonomic system is one in which the constraint equations are non-integrabie functions of tirne. the constraint equat ions are different idequat ions of the generalized velocities.
The equations of constraint for a holonomic system are integrable and may be aigebraic functions of displacement. The general form is
for m constraint equations[53]. The form required for use in the Lagrange equations with added multipliers, Equation 2.4, is the differential form of the constraint equations and is axrived at by defming the coefficients aii and ait as.
for each constraint equation, (i = 1,2, ,m).
Note that the coefficients aik and ail may be functions of the q's and time. For holonomic systems with scleronomous2 constraints the ait terms are zero and the complete set of equations of the constraint coefficients (aik)then forms an (nxm) rnatrix often known as the Jacobian of the eonstra.int equations;
for rn constraints and n generaiized coordinates.
The application of Equations 2.3 and 2.4 on the system energy function and the holonomic constraint equations resuft in the equations of motion given in the generd vector form,
Equations 2.8 and 2.9 are a system of differential equations. To solve for the generalized coordinates Equation 2.9 may be differentiated with respect to time to produce M{6 = -[&3 (2.10) and combined with Equation 2.8 to produce the system of equations
Using this combination of the equations of motion and the constraint equations to solve for the system accelerations and constraints is known as the augmented *A time invariant constraint is known as scleronornous, whereas a time variant constraint is known as rheonomous[53] method[43][45]. The solution for the unknown (6) and {A) can be found directly or through standard numerical equations solvers, and then numericdy integrated to produce the velocity and position results.
The augmented method may be reduced such that the constraints can be factored out to arrive at the system of equations for the generalized coordinates. This is done as follows. Rearranging Equation 2.8 leads to,
where (41 = {Qd + {d9, dl* Substituting tj back into Equation 2.10 results in
where
WC) = {-M{~H- Solving for X results in,
where [BI = [J][M]-'[JT]. (2.17) Substituting Equation 2.16 back into equation 2.8 leads to the underlying ordinary differential equation for the system[43][45],
The use of these equations of motion for consenmtive holonornic systems is straightforward. The generd degrees of freedorn are selected, which may assume that some or dl of the bodies in the system are unconstrained. The system energy is defined, and the constraints are put into a form equivalent to that of Equation 2.4. The values of the coefficients that make up the Jacobian of the constraint equations for a holonomic system are noted. The augmented technique is applied, resdting in Equation 2.11 which may be solved using standard numencal methods to produce a time history for the solution.
For certain systems being solved numerically the use of Equations 2.16 and 2.18 may bring about some time savings cornpared to the use of Equation 2.1 1 in the solution of the constraints, {A}, and the generalized coordinates, (6). Specifically, for systems with a diagonal mass matrix the numerical calculation of [Ml-' is a simple operation, allowing Equation 2.16 to be used to solve for {A), and Equation 2.18 to solve for {e).
2.2 The Rigid Body Mode1
A computer mode1 of any mechanicd system must be based upon a model that is an approximation of the red system, retaining the important features but ignoring ot hers in the interest of ob taining simplicity wit h completeness. Deciding which parameters are important is iduenced mainly by the value of any potential information obtained from including a particular parameter, and by what is Iost by excluding it. The available computing capacity also influences the decision.
Al1 previous derailment models have been planar, and the mode1 presented in this work does not stray from this tradition. A train derailment is essentially a planar process, as the variation in the ground elevation near the rails is typically much less t han the lengt h of a single freight car body. Of course, for situations when the landscape is significantly variable, such as in mountainous regions, the planar model does not apply.
A three dimensional model would add information concerning the vertical, roll, and pitch motions for each car. However, as the mode1 presented in this thesis was planned from the outset to include a model for the collision of the cars, the added complexity of t hree extra motions would add more complexity than required at this stage in the model developrnent. It was felt that including coupler kinematics and reaction forces, and modelling collisions between cars was far more important than the extra three degrees of freedom a fdy three dimensional model would provide. Therefore it was decided to proceed with a planar model that is as detailed as possible. The assumptions and simplifications made during the development of the model, and the important characteristics of the complete model, are surnmarized in the next section.
2.2.1 Assumptions, Simplifications, and Characteristics
The assumptions and simplifications made in the development of the derailment model are as follows:
motion is planar, involving only lateral motions and yaw rotations
truck bodies, including wheelsets, are ignored
rail car body is rectangular
a symmetrical rail car bodies, with the centre of mass at the geometric centre
identical couplers at the front and rear of a car
mass of the coupler bodies has little effect on the overall motions of the rail cws
a non-derailing cars can be modelied as a lumped mass remaining on the rails
a brake signai propagation from first car towards the last3.
3This restriction is not a fundamental requirement .
37 The model also incorporates the following characteristics:
a length, width, and inertid properties are defined for each car in the train
coupler length and kinematics are defined for each car type
0 kinematics and dynarnics between cars and couplers are modelled
a reaction forces at coupler-tecar interface are modeIled
coupler force and displacement limitations are irnposed, with uncoupling if limits are exceeded
straight or curved track constraints are possible
a lateral rail force limitations are imposed, with derailment if force is exceeded
velocity dependent car-to-ground reaction forces for derailed cars
car-tecar collision is modelled
the derailment may originate at my car on the rail4
a overall brake signal application delay
brake type function and application defined for each câr
braking forces me applied at each car
2.2.2 Analysis of the Rigid Body Mode1
An example of the system in the form used in the analysis is shown in Figure 2.1. The system as shown is in a partidy derailed state, with cars 1 to 3 fully derailed, in order to show the general state of the motions of the bodies and the configuration of the model. The break in the rail for this example is placed at the origin of the
4A srnall lateral disturbing force is required to initiate the derailment process. car N N-1 N-2 ... 4 3 2 11 . ,I 1 rail , 1 ,X 1 v I I
Figure 2.1 : Rigid body model of the system. coordinate system, although for the derailment model the position can be set to any value dong the X-&S. The complete train is grouped into two sections: the cars that are expected to derail and those t hat do not. In Figure 2.1, car N represents the group of cars that remain on the tracks and do not derail, while cars 1,2,* -.N - 1 represent the cars that are free to derail.
Grouping the train in this manner is justified by the fact that for any derailment the complete train seldom derails. Generdy only a portion of the train derails while the cars that remain on the track are stopped by the emergency application of the brakes or by the derailed cars on the rails. However, as the number of cars that derail is not known beforehand, the number expected to derail is estimated from the statisticd data amilable for real derailments[3], and a number of free cars greater than is expected to derail is selected as the number of bodies in the model. Grouping together the cars that do not derail simplifies the solution by reducing the number of equations necessary to describe the model.
The system has the fom of an open loop chain of bodies, which allows a method to be developed to generalize the numericd model of the system to handle a variable number of cars without refomulating the system energy or the constraint Jacobian for trains of diffenng lengths. For a system of Linked bodies in which body (k) is only coupled to bodies (k - l), (k+ l), or both the equations of motion with Lagrange multipliers can be manged into the form:
The inertial matrix is a banded diagonal matrix, partitioned with independent sections for each linked system. Similady, the constraint equations will take the form, for the augmented method:
As the constraints are limited to act between two linked systems, the constraint equations also take on a partitioned forrn. The partitioned nature of Equations 2.19 and 2.20 dows the complete system to be analyzed as individual and generd sub-systems which are connected to other sub-systems t hrough the constraint equations. The form of the constraints is identical for each similar type of constra.int .
The ease of constraint removal may now be clearly seen. The rows from Equation 2.20 that represent a constraint, the columns from the Jacobian of Equation 2.19, and the associated constra.int multiplier, A, are all removed from the set of equations. Al1 other equations remain unchanged. No new initial conditions are required, as the system is not dected by the removal of the constraint until after the constraint is gone. No impulse force is present as no instantaneous change in velocity takes place. This ease of the removal of the constraints is the key feature that allows the derailment model to split at the couplen as required without disturbing the numerical solution. This is a direct result of the fact that constraint removal does not change the fundamental form of the equations of motion, as the number of unconstrained degrees of fieedom is not altered at any time during the solut ion.
However, the addition of a constraint does requires the correct initial conditions for q and q for all degrees of freedom, as the application of the constraint implies the application of the constraint force. The addit ion of a const raint essentially requires an impulse andysis of the system to arrive at the proper initiai conditions for the post-constrained state. For this reason constraints are only removed from the system as required, but not added. This prevents the use of adding constraints to the system to represent the discontinuous reactions that occur in the limiting of the rotation of a coupler body as it stops against the striker plate on a car body, or in the collisions of the bodies against each other.
The cornplete system may now be analyzed as a more manageable single car sub-system to arrive at the equations of motion, and as a constrained sub-system set to arrive at the constraint equations. Figure 2.2 shows a coupled set of sub-systems, where the rail is defined by a single line representing the centre of the track, with the contact for the cars at the truck pivot locations. The global X-Y axis is fixed. The rear car (k) is fully constrained to the rails at the front and rear truck pivot locations, the front car (i) is constrained at the rear truck pivot only, and the two sub-systems are coupled together at the common point on the front and rear couplers. The two types of const raint s, car-t *rail and coupler-to-coupler, are the only types used in the model. Either set of constraints cm be removed at any time from any car in the consist to model both derailment and coupler failure. The CAR - RAI L CONSTRAINTS
L RAI L T
COUPLER-COUPLER CONSTRAINT
Figure 2.2: Coupled sub-systems. car-to-rail constraints are assumed to act at the truck pivot locations as the wheelsets and the truck frarnes are ignored5.
Figure 2.3 shows notation used to define the pertinent characteristics used in the mode1 of each sub-system. Each car is modelied as an identical sub-system with characteristics such as length and mass unique for each car. Note that the car-to-coupler pivot within each sub-mode1 is not modelled as a removable constraint. The coupler bodies act as single degree of freedom links attached to the cor body, with the coupler constraint removed at a single location. Free coupler bodies are not used, as modelling the couplers as free bodies would increase the degrees of freedom per sub-system from five to nine and the number of constraints linking sub-systems from two to six, significôntly increasing the number of equations to solve numerically.
Figure 2.4 shows a schematic of the mode1 for a rai! car sub-system, showing the
'In a derailment, the trucks and wheelsets are typically shed fiom the car aRer leaving the rails and rolling over. Including the trucks and wheelsets as added bodies in the subsystems would provide Little added information to the results, but with a high cost of added cornplexity. COUPLER-COUPLER STRIKER PUTE i
PNOT FORCE ELEIlENT 8 COUPLER -CAR CENTRE OF !us
mCK PNOT LOCAtlON I i FROGHT CAR l I BODY i
Figure 2.3: General definitions for a rail car sub-system.
------dry? Y
Figure 2.4: General rigid body mode1 of a sub-system. important dimensions and variable definitions used in the general development of the sub-system equations of motion. The sub-system for each freight car, consisting of the car body and the couplers, is defined with three degrees of freedom for the car body, (XCR,YCR, and a single rotational degree of freedom for each of the two coupler bodies (a,B). All coordinates are relative to the fixed reference frame. A single constraint equation is defined for each active car-terail constraint and two constraint equations for each active coupler-to-coupler constraint. The coupler constraints essentidy connect two sub-systems at a common point, while the rail path constraint comects the earth system to the sub-system.
The next section describes the equations of motion for a general sub-system and the necessaxy constraint equations to cornplete the derailment model.
2.3 Rigid Body Equations of Motion
The cornplete system, sbown in Figure 2.1, is described in several sections. The equations of motion are developed for a single car sub-system, followed by the constraint equations for a single sub-system on straight track and curved track, and for a coupled set of sub-systems. The models describing the generdized forces involved in the braking, coupler reactions, ground reactions, and collisions of the sub-systems are then developed. The complete system is then finally assembled.
2.3.1 Sub-System Equations of Motion
As shown in Figure 2.4, the single car sub-system consists of the car body with three degrees of freedom and the two coupler bodies each with a single degree of freedom, which yields a total of five degrees of freedom for a sub-system. The car body is assigned mass rnc~and moment of inertia ICR. The couplers are assumed to be identical for front and rem locations (as they are on real freight cars), with mass mcp and moment of inertia ICP- The positions of the mass centres for the freight car body are shown as (XC~,YCR), the front coupler body as (XFCP7YFCP) and the rear coupler body as (XRCp,YRcP). The rotations of the bodies relative to the fuced global reference frame is (O) for the main body and (a)and (B) for the front and rear coupler bodies respectively. The distance from the car body CG to the front and rear coupler body pivot points is dFp and dRp. The coupler body centre of mass (CG) to pivot distance is given by ~F~Mand d~c~-
Following the procedure for the derivation of equations of motion using Lagrange's equations presented in Section 2.1.2, the system energy is found to be,
w here xFCP= xCR- dFPésin 0 - dFcM&sin a (2.22)
Gp= iCR+ dFPB COS B + dFChfciCOS a (2.23) xRCp= xcR+ dwé sin 0 + dRCMBsin fl (2.24) yRCP= SR- dRPé COS @ - dRCMICOS P. (2.25)
Considering the system without external applied forces and in the absence of constraints, the application of Lagrange's equations produces the system of equations of motion for a single sub-system: where
Simplification of the mass matrix provides significant reductions in the numerical solution complexity. As freight cars are essentiaily symmetrical, setting dp = dFP= dRP reduces the ml3, mal, rnz3 and mn terms to zero. If it is then assumed that the coupler mass and inertia have little effect on the overail motion of the freight car and therefore act at the car-to-coupler pivot point, we may set dFCM= dRCM = O. This reduces al1 the remaining terms in the mass matrix to zero except for the diagonal elernents, as weU as reducing the right hand side of Equation 2-26 to zero.
The resulting non-zero coefficients for the mass rnatrix are:
Note that for a real system, the values defined for the mil, m22, and m~ûterms representing the mass and inertia of the rail cm will dominate over m44and mss, which represent the coupler body mass moment of inert ia. The resulting equations of motion, less external applied forces and constraints. for a single car sub-systern thus takes the form,
[MIta) = 0. With the addition of applied external forces, the result is the familiar fonn
[MHQ = {Qd- (2.28) To develop the complete systern, the sub-systems are joined together by the constraints at the couplers and are kept to the path of the track by the car-terail constraints. The equations t hat describe these constraints are presented next .
Car Rail Constraints: Straight
Figure 2.5 shows the front portion of one car and the rear portion of a second car to identify the geometry for front and rear truck pivot constraints on straight track. Not show is the situation for a car that has not derailed, where both trucks are on the track.
For straight track the position constraint for the front truck pivot point of the car and the rail is given by,
It is assumed that the freight car is symmetrical, thus dRT = dm, represented by dT (unique for each car), is the distance frorn the centre of mass to either the front or rear truck pivot point.
The first derivative of the c0nstra.int equation is,
yCR+ eCRdTCOS eCR= 0. The second derivative, after grouping acceleration terms onto the left hand side and velocity terms onto the right, is
PcR + &RdT COS OCR = @RdT sin BCR. (2.31 ) The resulting Jacobian (see the order of the generalized coordinates in Equation 2.26) is then
Similarly for the rear truck pivot point, the resulting position, velocity and acceleration const raint equations are
YCR- dT sin OcR - YiArL= O
FRONT TRUCK PIVOT YCR w
Figure 2.5: Car-rail constraint geometry: Straight track. Top: front truck pivot geometry. Bottom: Rear truck pivot geometry. and the Jacobian is given by
2.3.3 Car Rail Constraints: Constant Radius Curved Track
A curve in a rail line is typically of constant radius, with a transition portion connecting the straight track to the portion with constant curvature. The t ransistion portion is not accounted for by the constant radius constraints.
Figure 2.6 shows the geometry for the two possible car-terail situations for a single car partidy derailed on curved track; front or rear truck on the track. As before, a car with both trucks on i;he track uses both of these two constraints.
Curved track is assumed to be circular, with a radius of curvature R centered at the position (Xc, Yc).The equation for the track is then,
The position of a tmck that must be on the track is given as (XTR,YTR) For the front truck, XTR - XC = XCR + dT COS BCR - XC (1.35) and YTR- YC = YCR + dT sin OCR - YC. (2.39) Therefore the position constraint is,
The fkst derivative of the constraint equation is where
The second derivative, after grouping acceleration terms onto the left hand side and
FRONT TRUCK PIVOT
y-- - RAIL--
Y curve radius, R centre at (X, , Y, )
UCK
curve radius, R centre at (Xc,Y, )
Figure 2.6: Car-rail constraint geometry: Constant radius curved track. Top: front truck pivot geometry. Bottom: Rear truck pivot geometry. velocity terms onto the right, is
The resulting Jacobian is then
Similarly for the rear truck, the resulting position constraint equation is,
(XCR- d~ COS OCR - XC)* + (YCR- dT sin BCR - Y~)~- R2 = O.
The velocity constraint is then,
wbere
The second derivative, after grouping acceleration terms onto the left hand side and velocity terms onto the right, is then
ZCRXRC+ YCRYRC - &R~T[YRC cos BCR - XRCsin BCR] = -@RdT[(xcR - XC) COS CR + (YCR- YC)sin BCR] -2&RdT( xCR sin BCR - yCRcos eCR)- GR- (2.50)
The resulting Jacobian for a rear tmck on cuved track is then 2.3.4 Coupler-to-Coupler Pivot Constraints
Figure 2.7 shows the geometry for the coupler-to-coupler pivot constraints at the front and rear of the car. The distance from the centre of mass to the coupler-tebody pivot point is dp and the coupler length is dc.
For two cars, the lead vehicle as body i and the following vehicle as body k, the position constraints for the cornmon point on the rear coupler and front coupler are,
L
Figure 2.7: Coupler-to-coupler constraint geometry. The second derivatives, required for the augmented form, &er grouping acceleration terms onto the left hand side and velocity terms onto the right, are
&ÿCRk - B;dpksin 8r - &dck sin ak - XcR - Jidp, sin Bi - adc, sin fli = 2 -2 *2 6, dfi COS ak + ~k2dc,COS a& + Bi dpi COS di + pi dc, COS Pi (2.56)
The resulting Jacobian is then
-1 O -dpi sin0i O -dc, sin pi 1 O -dpk sin 4 -dck sin ak [JI = [ O -1 dp, COS^^ O dc, COS pi O 1 dpkcosBk dck cosak O I (2.58)
2.3.5 Constraint Stabilization
In Sections 2.3.2, 2.3.3, and 2.3.4 the constraint equations used in the mode1 were presented. The constraints originate as algebraic equations of position that are incorporated into the equations of motion through differentiation into acceleration const raint equat ions.
The application of Lagrange's equations of motion wi t h added multipliers assumes that the position, velocity and acceleration constraints are ail satisfied in the solution. However, as only the acceleration form of the constraints are incorporated into the equations of motion when using the augmented method in a numerical solution, the satisfaction of the position and velocity constraints are not guaranteed. In a numerical solution the result is an increase in the error in the position and velocity of constrained bodies away from the common point of constraint. To overcome this increase in error Baumgilrte[52] proposed a method that incorporates the position and velocity constraints into the acceleration equation. A succinct explanat ion is found in Amirouche(45j and in Shabana[43]. The method uses the complete set of constraint conditions, defined by the position constraint equat ion, fi( for each constraint, i. The acceleration constraint, Equation 2.61 is replaced wit h a linear equation t hat includes the position and velocity constraints: where Ai and Bi are ôrbitrary constants. The added terms are placed on the right hand side of the constraint equations, and effectively alter the solution of the Lagrange multipliers such that the error in the position and velocity constraints is partidy corrected by slight changes in the constraint forces. The selection of the values used for the constants is not well defined. Amirouche[45] suggests that to achieve asymptotic stability the constants A and B be chosen such that, Ai 2 O (2.63) Note that the relative dues of Ai and Bi may be selected, but a single correct set for a given constraint is not possible. The appropriate values of Ai and Bi must be selected through t rial-and-error and analyst experience wit h a particdar problem. Also, the dues used for Ai and Bi must be altered if any parameter in the mode1 is altered. For specialized models the method does provide a decrease in constraint position and velocity error, and by using smaller than ided values for Ai and Bi the solution is less sensitive, such that redefining Ai and Bi for minor changes to the system is not required. 2.3.6 Assembled Equations of Motion The equations of motion as described by Equations 2.19 and 2.20 may now be assembled. St arting with the unconstrained assembly of sub-systems, the equat ions of motion for each sub-system present are arranged in order of placement in the train. This procedure is done sequentially from the lead sub-system toward the last sub-system to ensure that the complete mass matrix remains diagonal. The resulting system mass matrix remains constant throughout the solut ion. The sub-systems are then coupled using the required constraint equations. For each required constraint, the appropriate Jacobian is added as a row to the system [JI and the corresponding -[& added to the right hand side of Equation 2.20. The rernaining terms to be defined are the generalized forces {Q,) on the right hand side of Equation 2.19, discussed in the next section. 2.4 Generalized Forces The generalized non-conservative forces, {Q,), included in the sub-system model are the braking forces, coupler reaction forces, ground reaction forces, and collision forces. These forces are functions of the generalized coordinates and velocities, and are added to the right hand side of the equations of motion. These forces essentially determine the motion of the system and as such are important to the validity of the final rnodel. These forces must be as representative of the actual forces as possible. Models for each generalized force will be developed separately. 2.4.1 Brake Force Mode1 During a derailment the emergency braking for the train is either controlled bq the locomotive operator if the derailment is anticipated, or automatically when the brake line is broken at any point dong the train. In either situation the result is the rapid reduction of air pressure in the brake Line, originating from the opening. The application of the brakes at each car along the train is, therefore, delayed by the time it takes for the air line to depressurize and by the distance from the break in the line or from the locomotive. Under emergency conditions the brake signal propagation rate is 290 m/s [11][57][58].Once the brake Line pressure at a car has dropped to sufficiently low levels, a control valve opens to release pressurized air stored on each car into the brake cylinders. The pressurized air acts on the brake cylinder or cylinders, resulting in a force applied to the brake shoes at each wheel on the car. For service or emergency braking below rail adhesion lirnits (i-e.wheel lock-up) , the applied braking force is dependent on the type of brake shoe in use. AU modem freight cars are equipped with high friction composition brake shoes, replacing the cast iron brakes shoes which were in general use from the 1900's to the early 1970's. Although the Friction characteristics of cast iron and composition shoes are drastically different(591, the total braking force produced by each car is adjusted by the operator to be approximately the same, resulting in similar stopping distances and times. The significant difference between the materials is that to produce a similar stopping distance with identical cars the required brake shoe force for the composition shoe is less than one half the required force needed for a cast iron shoe[ôO]. An added benefit of the composition shoes, and the main feature that caused the railways to switch from cast iron shoes, is the reduction in wheel temperature during heavy braking. The reduct ion in thermal stresses, cracked w heels and premat ure w heel failure all offset any extra cost in retro-fitting freight cars with composition shoes. Ultimate brake force for any type of brake shoe is dependent on the traction available between the wheel and the track. Under normal operating conditions, adhesion levels for poor track conditions are expected to be the best available[59]. As such, the applied brake force for cars in general service with any type of brake shoe materid is adjusted such that, for dl operational weights from empty to full, the application of full service braking will not produce wheel skid under poor track conditions. Thus a typical freight train wiil seldom reach the adhesion Iimit even for poor track conditions. This level of tuning is not dways met and wheel skidding on the application of emergency braking sornetimes does occur. Thus, the brake mechanisms for freight cars are adjusted to produce wheel-rail braking forces just below adhesion limits for an empty car, which for a fidl car that weighs more produces braking forces well below the rail adhesion Lmits. Thus, the applied braking force is generdy not a function of freight car weight. The brake force model used in the derailment model delays the initial application of the brakes for each car consistent with the brake Line pressure propagation velocity, aod also ramps the applied force linearly up to full force in 10 seconds from the time of the brake application applied to each car. The 10 second ramp represents the tirne taken for the brake control system on each freight car to react to the brake line signal and apply full air pressure to the brakes[57]. The applied brake force is modelled as a total braking force generated by the brakes for each car, and is applied at the centre of mass of the car. The Nth car, representing the virtual cars of the train that will not derail, carries the total brâking force of ail the virtual cars, and applies the force under the appropriate time delay rules for each virtual car. For the model used in this research the applied braking forces are accepted as being typical of modem freight cars[57]. The brake force model is represented by a function presented by Yang and Manos [34], which was developed from experimental data. The function, miid for speeds below 80 krn/hr is given by where UT is the velocity of the train, in metres per second, and the brake force is given in Newtons. The resulting force-velocity cuve is shown in Figure 2.8. Above 80 km/hr (22.2 m/s), the brake forces are assumed to remain constant. The curve represents the typical brake force generated by a modem 90 tonne freight car using composition brake shoes, wit h brake mechanisms adjusted to produce adhesion Figure 2.8: Brake force model. limited braking at a car mass of approximately 36 tonne. The resulting brake force estimates follow the accepted industry brake force curves[11][57] [58]. Altering the applied brake force, to account for changes in brake types or to model higher or lower levels of braking other than typical forces, is done by simply altering the value of overall brake force for a given car. Control of the timing of the brakes is through adjustment of a second parameter. The overall brake force is then given by where Kb is the overall brake force adjustment parameter, and Kd is the timing parameter that accounts for the propagation and actuation delay of the brake signal for each car in the train. 2.4.2 Coupler Mode1 Figure 1.5 and Figure 1.6 in Section 1.2.3 show the main coupler types used in all North American freight cars. Figure 2.9 shows design and construction details of one of these coupler types. Apparent from the figures is the solid nature of the coupler body and that the link is not fixed in position but has limited yaw and pitch freedom of motion. Previous derailment models have assumed the coupler to be rigidly fixed to the car body, ignoring the yaw degree of freedom. To improve upon past models, the mode1 presented in this research includes the yaw motion of the coupler, which requires the modelling of the associated react ion forces generat ed between the coupler and the freight car body and between the mated coupler faces. TYo-~iionEIFCarg(rTyp.€mAnTE Figure 2.9: Design drawing of a typical E/F Type coupler[8]. 2.4.2.1 Striker Reaction Mode1 The yaw degree of &dom for all couplers is limited by the opening of the striker casting, as seen in Figure 2.9. The strïker opening is standardized for the different coupler bodies used in service, such that depending on the length of the freight car and the location of the truck pivots, a suitable coupler body must be used to ensure that during cornering adjacent cars do not force their couplers to rotate up to the striker plate limits. This would generate high lateral forces at the striker plates and potentially cause a wheel to be pushed over a rail. During a derailment, the action of the car bodies will engage the coupler shanks onto the striker plates, resulting in high lateral wheel-terail forces. To accurately model the coupler, the striker plate reaction must be suitably modelied to Limit coupler angles to within the striker opening. To model this system, a non-linear force element is placed at the contact point between the coupler shank and the striker plate. To model only the striker plate reaction, a force element would be applied only after the coupler has rotated up to the lirnit of the striker opening. However, other forces act on the coupler body before the striker limitation is reached. Another design feature of the coupler, yoke and draft gear assembly is a self-centering action under compressive loads[8](91 [61]. This manifes ts itself as a react ion moment at the coupler-yoke pivot joint under compressive loads. This reaction is highly complex and variable for different coupler-yoke combinations, but must be included in the model to correctly represent the loading conditions present under the compressive loading generated during braking and derailment. For the coupler model the moment at the coupler-yoke interface is transferred to act as a reaction force present at the striker plate which is a linear function of coupler angle. This reaction also serves to stabilize the numerical solution in the same way as the real system, by keeping the coupler bodies aligned under compressive loads as occurs during braking. 1 COUPLER BODY '\ \ '\ / COUPLER PIVOT FORCE ELEMENT Figure 2.10: Location of coupler striker force element, and associated dimensions used in the model. The coupler typically remains aligned with the freight car under compressive loading during braking or during shunting of the cars. The draft gear absorbs the shock loading which occurs during these standard operations. For this research, it is assumed that under emergency braking the draft gear is Mycompressed throughout the event. Any extension of the draft gear that may occw is assurned to have little impact on the kinematics of the motion, as the draft gear travel is less than 156 centimetres (6 inches) for most draft gear designs. To mode1 the self-centering action and the striker plate reaction a bi-linear force element is inserted between the coupler body and the freight car. The application of the force element to the model is shown in Figure 2.10. The force element characteris t ics are given by the bi-linear funct ion, where x is the displacement of the coupler body from the aligned position measured at the striker plate location, d, is the maximum striker opening distance, kl is the stiffness representing the self-alignrnent feature up to d., and k2 is the stifiess of the system past the striker opening. Table 2.1: Coupler parameters used in the model. Coupler Type E60 E67 E68 E69 F70 F79 F73 pivot-coupler length, 4 (m) 0.851 0.940 1.092 1.524 0.743 1.092 1.524 pivot-striker length dst (m) 0.546 0.635 0.788 1.22 0.438 0.788 1.22 striker opening, d, (m) 0.150 0.150 0.260 0.390 0.080 0.170 0.300 h (Io6 N/m) 2.18 2.54 3.15 4.88 1.75 3.15 4.88 k2 (log N/m) 50.0 , 50.0 50.0 50.0 50.0 50.0 50.0 The dimensions of the couplers and the values for kl and k2 used in the rnodel are surnmarized in Table 2.1. The dimensions given represent the seven types of couplers in generd use in freight service. The initid stifiess, kl, is assigned based on the experimental results of coupler mgling tests[6l]. The second stage stifiess. kz, represents the stifiess of the coupler shank in bending, and the striker plate stifiess. The resulting force is applied to the freight car as the equivaient force couple acting on the coupler and an equd and opposite moment acting on the car body. The resulting striker plate stiflness functions for seven common types of couplers in use are shown in Figure 2.1 1. 2.4.2.2 Coupler-to-Coupler Reaction Mode1 A second reaction occurs between the coupler heads in contact between mated coupler faces. Mated couplers do not dow unrestricted rotation; approximately 17' of relative rotation is available before the slack in the mated coupler faces is taken up. A non-linear reaction moment is the result, in which the initial relative rotation 0.00 O. 10 0.20 0.30 0.40 0.50 displacement at striker piate, m Figure 2.11: Coupler striker force model, as applied to seven common coupler types. produces no reaction followed by a rapidly increasing reaction moment until failure of the component. Figure 2.12 shows the applied moment functions as used in the model for the three possible Type-E and Type-F coupler combinations, labelled as Mee, Mff, and Mef. These functions are based on AAR coupler knuckle reaction equations, which were used by Yang and Manos[34] to model coupler moment reactions. However, Yang and Manos did not dow fôilure of the couplers and altered the equations to provide an increased band of non-reactive anguiar motion to account for the coupler body yaw motion which their model ignored[34]. As used in this research, the reaction moment functions are applied as they were determined by the AAR and reported by Yang and Manos[34] prior to modification. The reaction moment functions as used in the model are given by, MEE = -203.08 + 1061.7s - 455.87n2, (0.22 5 n 5 0.9), (2.69) MFF = -762.77 + 3536.2~- 2777.1n2, (0.28 5 n 5 0.71), (2.70) where n is the relative angle between mated couplers in radians, and MEE,MFF, and MEF represent the reaction moment between Type E to E, Type F to F, and Type E to F couplers in kN-m. In the coupler mode1 the coupled joints are set to fail, by removal of the coupler constraints, when the relative coupler angle reaches the maximum angle given by Equations 2.69, 2.70, and 2.71. 2.4.3 Ground Reaction Mode1 Once a freight car has derailed the wheels, truck bodies or the bottom of the car body are in contact with the ground producing a reaction force. Different freight car types will react in different ways with the variety of ground surfaces present near a rail Line. To completely mode1 all possible combinations of surfaces and freight cars would be the ideal solution; however, a compromise between exactness and useiùhess rnust be made. The most basic model for the body-ground interaction is the classic Coulomb friction model in which the laterai reaction force is a linear function of the normal force and opposes the direction of the relative velocity of the contacting bodies. Both the Yang and Manos[34] and the Andenon[36] models used Coulomb friction to estimate the ground reaction force, with both models using a value of 1.O to 1.5 relative angle between couplers, degrees -----.Mee ME --- I Md I Figure 2.12: Coupler reaction moment. for the coefficient of friction. This value resulted in approximately the correct number of derailing cars for the calibration cases used by these authors. However, t hese derailment models were applied to a limited number of test cases and the applicability of this value for a coefficient of friction is not known for a wider range of train speeds or lengths. As well the Yang and Manos mode1 did not allow uncoupling of the cars which may have affected the friction dues required to match the number of derailed cars produced by the mode1 with the reported statistics. Findy Yang and Manos[34] did report an excessively high number of derailed cars at speeds higher than 96 krn/hr (60 milelhr) for the base case train6 used in their study, causing numerical problems. This result brings into question the applicability 661 cars, each with a mas of 72 600 kg, base speed of 65 h/hr, ground hiction 1 .O. of a Coulomb friction ground reaction mode1 that is constant with velocity. In this research, several sources of information were used to develop the ground reaction model, which is applicable over a wide range of velocities and train lengths. A review of the soil mechanics literature suggested that the ground reaction force produced by a body sliding over the ground is velocity dependent. A basic physical analysis of the system was also completed, with the results of the basic analysis of the system compared to pubiished derailment statistics, helping to narrow the range of applicable forms for a constant coefficient model, and gaining significant insight into the relative importance of the variables involved. In this section, the basic physical analysis is presented fist, followed by a review of the soil mechanics literature, and concluding with a description of the ground reaction rnodel used in the derailment model. 2.4.3.1 Basic Analysis of the Derailing Train System: Case 1. An analysis of the derailing train with the system reduced to its basic components is presented as an aid in the study of the relationship between ground reaction force, train braking force, train length, initial velocity and the resulting number of derailing cars. The goal of the analysis is to nanow the range of an appropriate ground reaction model, to assist in the final determination of the model, and to shed light on the importance the miables involved. To simplify the derailing train, it is assumed that the train is replaced by a solid bar of length 1 and mass m. The bar is moving with an initial forward velocity ui on a surface with a constant friction coefficient of pi, which represents the brâking friction coefficient for the train on the rails. Initially the bar is just about to slide onto a surface with a friction coefficient of pa, which represents the ground. A free body diagram of the system after some time has passed is shown in Figure 2.13. The forces on the system consist of the vertical reaction force distributed dong the length of the bar and the corresponding distributed lateral friction reaction forces, Fiand Fa. The magnitudes of these friction forces are given by FI = ~1N1 (2.72) and w here and The sum of the forces in the direction of travel of the bar is then given by ma = -mg [pi (q)+ 1i2 (j)] x.v uniform bar l 1 Figure 2.13: Basic derailing train system: case 1. The acceleration of the bar, after rearranging, is then given by The solution to an equation of the fom of (2.77) is given as[62], The result is an equation describing the velocity, v, of the bar at a given displacement, x: 2 9 v2 = V; - -[2xfp1 + x2(p2- pl)]. 1 To find the distance required to bring the bar to a complete stop, the final velocity is equated to zero, resulting in, 9 vi2 = -[221/~1 xs2(p2 - PL)]* (2.80) 1 + produces a quadratic equation as a function of stopping distance x, The solution for x, is then Two Limiting cases exist, for pl = O and = O. For pi = 0, the case with no braking, the derailed distance becomes which must be positive, resolving the value of the (I)in Equation 2.82 for cases in which is larger than pl, including pl = O. For pz = 0, the case with no ground friction, the derailed distance becomes I AS pl + oo the result should be x, = O, thus the (f)is negative for cases in which p 1 is larger t han pz, including pz = 0. Thus the solution has two forms, lui xS = - + (Pl < ~2) (2.85) ~2 - PI (~2-d~(~a-~i)g' and Interestingly, for pl = p2 the above analysis produces the unrealistic result of x, = oo for any two surfaces with equd friction. However, by taking the Limit of Equation 2.81 as (p2 - pi) approaches zero, i.e. Lim S(x,) = O b2-~1)+- does provide the expected result of An analysis assuming a single surface with friction p produces the same result as the limiting case above. Applying this simple mode1 to the case of a train derailment will require the use of Equation 2.85 as the ground fkiction is assumed higher than the braking friction. To bring the analysis closer to the realm of a train, we replace the uniform bar with a series of Nt bars, each with length 2,. Therefore with Nt the number of cars initially on the tradc and 1, the average length of the cars. Similarly X, = Ndlc (2.90) O 10 20 30 40 50 60 70 80 90 100 train length, number of cars Figure 2.14: Results of basic rigid bar analysis: case 1. with Nd being the number of derailed cars. Equation 2.85 then becomes Using 2, =15m, pl = 0.05, and pz = 0.5 as estimates for average car length, braking friction and ground friction and using v; =64 km/hr, Equation 2.91 produces the curve shown in Figure 2.14. The shape of the curve is similar to the statistical results shown in Figure 1.2, presented in Section 1.1. However, the number of derailing cars is lower than expected, even with the low value of 0.5 for the ground friction coefficient. The lower than expected number of derailing cars may be caused by the assumption in the analysis that the cars remain coupled and aligned during the derailment, i.e. the bar remained ngid and complete. To improve this estimate, a change to the mode1 assumptions may be made in which the uncoupling of the derailed cars is accounted for. The basic analysis so far assumes a rigid bar passes from surface 1 to 2, andogous to derailed cars rernaining coupled and aligned after leaving the rails. In reality the derailed cars will become unaiigned and uncoupled and are typically pushed aside as the couplers break. 2.4.3.2 Basic Analysis of the Derailing nain System: Case 2. To account for the uncoupling and laterd movernent of the cars away from the track during a derailment, it is now assumed that the rigid bar travels onto surface 2 up to a specified distance, beyond which the mass is removed from the system7. This distance represents the length of the derailed train that remains aligned with the train still on the tracks. The removd of the mass represents the uncoupling of cars and their removal from the system. Figure 2.15 shows the system. The limiting distance is shown as d. While x < d the acceleration of the system is given by Equation 2.77. Once the forward end of the bar ha travelled through distance d the mass in the system is no longer constant. For this phase of the motion Newton's 2nd law in the form is applied. For the system, the resulting equation of motion dong the linear direction of the bar is then where m. is the total initial mass of the bar, I is the initial length, d is the fixed distance the bas travels over surface 2, x is the distance the left end of the bar has travelled since encomtering the transition between surfaces 1 and 2, and v is the instantanmus velocity of the bar. After differentiating and removing common terms, the equation of motion for x 2 d 7The point on surface 2 where mass is removed may be thought of as a ledge which the overhanging portion of the bar, no longer solid in this analysis, simply fails over. After substituting a = 6 and rearranging, the complete solution for the system acceleration is given by and Equation 2.96 is a function of both the position of the end of the train, x, and the velocity of the bar, v2, i.e. it is a 2nd order non-linear differential equation aod thus a closed form solution is not easiIy found. However, a numerical solution of Equation 2.95 and Equation 2.96 can be used to provide some insight into the physical significance of the paramet ers invoived. A numerical solution for a system with 1, = 15.0m, pl = 0.05, and p2 = 1.0 with an initial train velocity of vi =64 km/hr (40 mph) produced the results shown in Figure 2.16. The values of x, l and d have been substituted with I = /,Nt, d = 1,N, 1 X. V- uniform bar ic-d* I 1 ;///'//'////,'//'//>'//////'/////'/////Y/ ,'/ surface 1 surface 2 mg. x Figure 2.15: Basic derailing train system: case 2. 72 and x = 1, Nd in the presentation of the results. The variation in the number of cars to derail for a train length of 10 to 100 cars is show for the case of 3 cars remaining aligned during the derailment (d = 45 m). The values selected for the constant parameters are not unreasonable estimates for each, and the resulting number of derailed cars for a given train length is reasonably close to the statistical data for the 40 mph result presented as Figure 1.2, in Section 1.1. The generd shape of the curve is similu, with a decreasing slope with increasing train length observed in both curves. An interesting, and quite distulbing, result of the above analysis is that for (x 2 d) a limiting velocity exists for a given set of parameters (pi,pz, 1, d). When the limiting velocity is exceeded the acceleration is positive, i.e. the train will increase in velocity and not stop before completely passing onto surface 2. A complete derailment would occur. This result was observed during testing of the numerical solution of Equation 2.95 and Equation 2.96 when low values of pi and p2 were used to test the effects of low braking and ground reaction forces. The cause of this result, a complete derailment, is seen in Equation 2.96, in which it O IO 20 30 40 50 60 70 80 90 100 train length, number of cars Figure 2.16: Results of basic rigid bar analysis: case 2. is observed that should the v2 term in the numerator be sufliciently large, the acceleration will be positive, resulting in an increase in velocity. To guarantee negative acceleration for (x 2 d) requires sufficiently large dues for the products (1 - d)pl or dpz. Thus the system must possess the correct combination of being long enough, remain aligned with the track sufnciently, have suficient wheel-terail traction for braking, and have a high resistance to sliding over the ground. Further andysis of this process requires the addition of more details to the simple model. 2.4.3.3 Basic Analysis of the Derailing Train System: Case 3. The analysis of Case 2 can be taken one step further, by introducing the known equation for brake application tirne and the braking forces presented in Section 2.4.1. The braking coefficient of friction as used in Equation 2.95 and Equation 2.96 is replaced with a velocity and time dependent function based on Equation 2.65. The braking coefficient of friction function is where Kt = O ,t i td Kt = O.l(t -td) ,td < t 5 (td+ 10) Kt = 1.O ,t > (td + 10). and and ~ A numerical solution of Equation 2.95 and Equation 2.96, with Equation 2.97 controlling the brake force produced the results shown in Figure 2.17 for a system with 1, = 15.0m. td=O seconds, pb = 0.1, pz = 1.0, and an initial train velocity of v; =64 krn/hr (40 mph). The variation in the number of cars to derail for a train length of 10 to 100 cars is shown for the case of d = 45 m, representing 3 cars remaining digned during the derailment. The resulting curve shown in Fi yre 2.17 is now improved over the results for the Case 2 analysis in Section 2.4.3.2. The general shape of the cuve and the number of derailed cars for an initial train length are both comparable to the statistical data presented as Figure 1.2, in Section 1.1 for the 40 mph data. Not shown by these results is that when the solution for Case 3 is performed with an initial speed of 105 krn/hr all the cars derail for initial train lengths up to t O0 cars. The possibility of this result was discussed in the analysis of Equation 2.96 in Section 2.4.3.2, and also suggested by the results of the Yang and Manos[34]* as discussed in Section 2.4.3. For the above solution (Case 3), the friction coefficient value, p2 =1.0, and the brake force given by Equation 2.97 is not great enough to provide sufficient drag force. Experimentation with the numerical model has shown that pz > 2.0 is needed to bring the number of derailed cars to below 40 for a train lengt h of 100 cars. Thus a velocity dependent ground reaction model is required to allow the basic analysis to be applicable over a wide range of train speeds. This resdt leads to the conclusion that a velocity dependent ground reaction model is required to produce acceptable results from a cornplete derailment model. The final basic physical analysis presented in Case 3 above has demonstrated the sensitivity of the system to the constant coefficient ground reaction model. The sensitivity of the derailment process to variations in the ground reaction force and the brake force show the importance of improving the ground reaction force modei beyond a si~pleconstant coefficient type. To improve the model further requires BTheCase 3 analysis is quite similar to the Yang and Manos model, in that braking is realistically accounted for and a constant coefficient of friction of 1.0 is used. The assumption that 3 cars remain aligned with the track while deraiiing is not uareasonable, as the Yang and Manos model did not allow uncoupling, so the similarity of the results is not unexpected. O 10 20 30 40 50 60 70 80 90 100 train length, number of cars Figure 2.17: Results of basic rigid bar analysis: case 3. the analysis of the process of pushing or dragging a freight car body over the ground, the cause of the underlying reaction force. 2.4.3.4 Soi1 Mechanics Literature The process of pushing or dragging the complex irregular structure of a freight car body over surfaces that range from loose grave1 to soft clay or sand is essentially a soi1 mechanics problem. The Literature associated with agricdtural tools, earth moving equipment, and basic soil medianics was investigated to gain insight into the processes taking place during the derailment. The literature regarding soil mechanics suggested that a mode1 more complex than classic Coulomb friction is applicable. Research regarding the resistance of soils to agricultural and earth moving tools is fairly extensive. However, no studies were found involving tests on dragging large objects over soil, other than standard earth noving equipment . No experiments have been done on the movement of freight cars or other similar bodies pushed over trackside soils. The most applicable papers were associated with bulldozer and other soil cutting studies, wheels rolling or being dragged through soils, and basic soil mechanics theory. Qinsen and Shuren(631 presented a mathematical model of the soil cutting process of bulldozer blades, with experimental verification of the results. These results show that for velocities below 2 m/s draft force does not vary, whereas above a transition speed near 3 m/s the draft force begins to increase with velocity due to the inertid effects of soil being displaced by the blade. Cutting force also increased with cutting depth. A ratio of the laterd to vertical force on the blade of about 5 resulted from the experiments. Boccafogli et al. [64] presented the results of soil bin experiments on cutting blades in soil, with cornparisons made to mathematical models. The results show cutting forces increasing with cutting velocity. The normal force required to produce a given cutting force is not presented. Dechao and Yusu(651 developed a similar dynarnic model for soil cutting resistance, verifying the model with soil bin tests. Their results also show an increase in cutting force with velocity. The authors also present the ratio of the lateral to vertical load under various cutting conditions for speeds up to 2.5 m/s. The range of lateral to vertical ratios presented is from approximately 1 to 3.5 for various cutting angles, cutting widths, and soi1 types. All the results show an increase in draft force with veloci t y. Research regarding wheels rolling or sliding over soil includes [66][67] [68] [69][70] [71][72][73] [74]. Although the initial phase of a derailment may see the wheels rolling over the rail bed or the surrounding grade, the mas of the freight cars would very quickly push the wheels deep into the soil, creating high drag forces and very LkeIy forcing the wheel-sets off the trucks or the trucks off the car bodies. In eit her case, research regarding the motions of wheels ouer soils becornes inapplicable. General theory of soil mechanics was also surveyed. Yusu and Dechaa [75] presented the results of theoretical and experimentd analysis of metal sliding on soil for a speed range of 0.4 to 1.7 m/s. Measured friction coefficients ranged from 0.3 to over 2.0 for various soil types and normal pressures. A similar study by Dechao and Yusu [76]showed the rate dependence of soil shear strength. The above literature al1 suggests a ground reaction mode1 that is a function of velocity and normal force. The estimates of coefficient of fiction values summaxized above may be taken as the maximum values of the ground reaction for freight cars, as the agricultural tools and bulldozer blades are designed to till and move soil. These results suggest a range of 1 to 3.5 for a Coulomb type coefficient for a body digging into or scraping the soi1 surface. However, the applicability of these results for speeds approaching those of a derailment, 20 m/s or more, is not known. As a freight car body has large flat areas, such as tank bodies or flat frames, dong with smdler protmsions such as valve stems and Banges, it can be estimated that the lower end of this range would be appropriate for a velocity dependent ground reaction model, with the reaction force increasing with velocity. 2.4.3.5 A Velocity Dependent Ground Reaction Force hnction. The ideal ground reaction model for this research would, of course, be based on experimentd data of rneasurements of freight cars forced over various soils. Unfortunately no such data exist. Based on the basic analysis of the system and the results presented in soil mechanics literature it has been concluded that a velocity and normal force dependent function is applicable. The type of function, its shape and characteristics are, however, unknown. An estirnate of the type of function must be made, and the characteristics of it inferred from the results produced by the use of the model in the complete derailment model. A suitable function is suggested by Shilling for use as a model for the friction of joints and sliders in robots(771. The versatility of the function makes it useful as a general purpose ground reaction force model. The function has the general foxm where K, : effective coefficient of friction c : static coulomb coefficient cd : dynamic coulomb coefficient c, : velocity coefficient v : velocity at point of interest 6 : a smd positive number. This general purpose friction function is usefd as a ground reaction model if the values of c,, cd, C, and e are set to values that result in the derailment model producing results that correlate with real derailments, the statistical data presented in Section 1.1, and produces reaction forces consistent with the basic physical analysis rtnd the known properties of soils. A trial and error approach was adopted to arrive at the values used in the final ground reaction force model. Several test cases were run to determine suitable values for the constants in Equation 2.98, followed by a series of tests comparing the results of the derailment model with the statistical data aMilable[3][4]. The final assignment of values was not done until the entire derailment model was complete and tested, using data for the Mississauga derailment[l] as a final calibration test. Cornparisons in number of cars derailed and the Iength and width of the haI accident scene guided the selection of the parameters. The final model takes the form shown in Figure 2.18. The reaction force is a linear function of normal forces, but is non-iinear with velocity. The coeEcients used Equation 2.98 in the ground reaction mode1 are: c, = 0.8, cd = 0.7, c, = 3.0, and e = 0.01. The results of the complete derailment model using the above gound reaction force function compare very well with the statistical data for the 64 krnfhr train speed veiocity (Ws) Figure 2.18: Ground reaction force model. data. However for slower speeds and shorter trains the number of derailed cars resulting from the final model deviate from the published statistical data. Several different ground reaction models, which reduced the ground friction coefficient to 0.3 as velocity approached zero, were tested in an effort to match the data at lower speeds. The resulting number of cars to derail could be matched to the statistical data, however the out corne of the model typically produced unrealist icaily long accident scenes. A ground reaction force 0.3 times the local normd force is also unrealist ically low . An explanation for the under-prediction of the number of derailed cars may be found in the action of the brakes under an emergency. When the point of derailment is behind the locomotive position and the train speed is slow, the brake pipe is not immediately broken and the crew do not immediately identify that a derailment has occurred. The train simply continues on until several cars have derôiled, which are then noticed by the crew or result in the failure of a coupler and mpture of the brake pipe. With no delay in the application of the brakes, the derailment model wilI underestimate the number of cars derailing at slower speeds. However, when a delay in the brake application is included the model reproduces the statistical data to wit hin 2 derailing cars. Figure 2.19 shows a cornparison of the results, in which the solid Lines are the statistical baseline taken fiom [3] and [4] and the data points show the number of derailed cars produced by the cornplete derailment model. The data points represent the outcome of the model for a given train length from 20 to 90 cars. Brake delay of 30 seconds and 10 seconds for the 24 km/hr and 48 km/hr case were required to match the results. This corresponds to 200 and 133 metres of travel past the derailment position before the application of the brakes. These delays are not unreasonable. 2.4.3.6 Application of the Reaction Force. The ground reaction force is applied to each derailed car depending on the location of the truck pivots with respect to the break in the rail, the statu of the front and rear couplers of the car, and the velocity of the car. While one tmck is still on the track wit h the other derailed (either front or rear), the gound reaction force is applied solely at the derailed truck. If both trucks are derailed, but either coupler is intact, the car is assumed to be upright and in contact with the ground at the truck locations only, and the reaction force continues to be applied at the trucks. The upper sketch within Figure 2.20 shows the application of the reaction forces for t his situation. When both the front and rear couplers have failed, and either the front or rear truck is off the track the reaction is calculated over ten discrete points evenly distributed dong the length of the car body. The assumption is that for the time when the car has one truck engaged in the track, the other truck is still attached to the car and reacting with the ground until the point that both couplers have failed. With both couplers failed it is assumed that the car has either roiled onto its side or has lost both trucks and is therefore contact ing the ground dong its entire length. The lower sketch within Figure 2.20 shows the application of the reaction forces for this situation. The total ground reaction is found by fmt solving for the local velocity of the point of interest between the car body and the ground. The local velocities determine the local reaction coefficient and the direction of the applied force at that point. The 10 20 30 40 50 60 70 80 number of cars in train Figure 2.19: Number of derailed cars for a given train speed and length. 82 Figure 2.20: Application of ground reaction forces. reaction force at a point i on the car is then given by {Fg)i= (K9)iNi where ( KJi is given by Equation 2.98 and Ni is the proportion of the total normal force of the car, mg, at the point of interest. For either truck location in contact with the ground Ni = mg/2. For the body in contact with the ground Ni = mgllO. The total applied force to the body is then calculated as the vector sum of the reactions at the discrete points. 2.4.4 Car Collision There were four main requirements for a collision model suit able for the derailment model presented in this thesis. These were that the method work with multiple collisions between several bodies, be valid for a wide range of collision speeds, mode1 the time of the collision and the forces realisticdy, and be numerically robust . The traditional method of modelling collision between rigid bodies has been to use the classical impact theory based on the impulse-momenturn laws [78][79]. This method was used by Anderson[36] to model the collisions between freight cars, the only other derailment mode1 to account for the collision between the freight cars. Other research regarding the application of the classical impact theory to areas of mechanics and dynamics of machines include[80][8 11 [82]. However, classical impact theory was not used in this research to model coilisions between freight cars. Classical impact theory was rejected for several reasons. Firstly, the theory assumes prior knotvledge of the coefficient of restitution for a given impact between two bodies[78], with the dueselected usuaily based on previous experimental evidence or an estimate of the expected energy loss. Secondly, the theory assumes that the impact occurs over an infinitesimal period of time such that the displacements of the coUding bodies are not changed after the impact[78]. For collisions between relatively solid bodies this may hold tme, however for colliding railroad tank cars experimental collision tests have shown that the contact time lasts up to 0.3 seconds and that the positions of the tank cars are not constaot[83][84]. Thirdly, the method is unable to describe the transient forces or deformations which take place during the collision[78]. Findy, the application of the classical theory to bodies other than point masses is not straightforwôrd[85] [86][8ï] [88] [89] [go]. The representation of collision and contact between rigid bodies is best done with the application of a force function at the point of contact. As used in the derailment model, force elements are applied corner-to-wall or corner-to-corner between colliding rail cors, with all the rail cars assumed to be rectangular. The force elements are inserted when the gap from the corner of one car to the outer limits of another car is less than a bufTer or maximum gap distance, ds. This method dows all four corners of a given car to be in contact with the walls or corners of any other car in the model, placing no restrictions of the occurrence of a collision. To detect a coilision, the positions of d the cars in the train relative to each other are moni tored, and if two cars are wit hin a minimum distance t hen a detailed cornparison of the positions of the comers of one car against the wds and corners of the other car is done. Should a corner be within a limiting distance of the opposing wd the contact force is applied in a direction perpendicular to the contacting wall, and an opposing force is applied to the corner of the other car. For corner-tecorner contact the force is applied dong the iine comecting the comers, equal and opposite for both cars in the direction of the comecting line. Figure 2.21 shows three cars in contact with each other, showing the application of the contact force for corner-tewd and corner-tecorner contact. Figure 2.21 also shows how the wd-tewall contact between stationary and adjacent, or "stackedn, cars is modelled using the force elements at one corner on each wd. The insertion of a non-linear elastic element between two bodies about to collide has been studied by de la Fuente and Felippa[91] to model the collision of simple objects, such as falling balls or rods contacting hard %at surfaces. Dias and Pereira[92] used a similar method to study the impact of automobiles, in which the structure was modelled as a system of rigid links and flat plates connected by non-linear spring elements. These authors have shown that the insertion of an elastic element replaces the deformation of materid over a finite surface area with the displacement of a single force element. Although the bodies are assumed rigid, deformation of the bodies during impact is accounted for by the displacement of the force element. The force function used in the derailment model is based on the function used by de la Fuente and Felippa[91]. The variation in force with gap distame between the where Fd(xd)is the displacement dependent contact force, xd is the gap distance at the imminent point of contact between the colliding bodies, db is the maximum gap or buffer distance beyond which no force is appiied, Fm, is the peak force which BUFFER DISTANCE, db C--- CORNER-TO-WALL ,' COLLISION i .. FORCE ELEMENT CORNER-TO-WALL COLWSION FORCE ELEMENT , i CORNER-TO-CORNER COLLISION FORCE ELEMENT Figure 2.21: Collision mode1 geometry. occurs at xd = 0, and d, is the control distance (the perpendicular distance at which Fd = Fmor/2)AS used by de la Fuente and Felippa[91] Equation 2.100 is constant with approach velocity, thus a given function will be appropriate for ody a single pair of masses approaching at a given velocity. To accommodate different approach velocities and body masses de la Fuente and Felippa[91] would alter the peak contact force Fm., and the control distance d, based on the system momentum. The system rnomentum method of collision force control was not used in this research. As the derailment process involves a variety of bodies with difierent masses and approach speeds, the contact force is altered to be variable with approach velocity. The equation is a quadratic function of velocity which modifies the instantaneous collision force given by Equation 2.100 to produce a collision force funct ion, FC = Fd(~vi(vi(+ C2Vi + ~3) (2.101) where ci, CZ, and CJ are coefficients controlling the degree of velocity dependency, v; is the instantaneous relative velocity at the point of contact of the two cars, Fd is the displacement dependent contact force given by Equation 2.100, and Fc is the collision force applied to the cars, in equal and opposite directions at the point of contact. The final values for the parameters used in Equation 2.100 are; Fm, = 3.0 MN, ds = 0.8 meter, and d, = 0.08 meter. The resulting variation in force with gap distance is shown in Figure 2.22. The values used for the coefficients cl, cz, and c3 in Equation 2.101 are; ci = 0.025, cz = 0.1, and c3 = 1. The variation of Fc with velocity and separation distance using the above values is shown in Figure 2.23. To calibrate the collision force element model given by Equation 2.100 and Equation 2.101, cornparisons were made between published tank car collision test data[83][84] and the resdts of the model. The experiments measured the forces produced when a 158 260 kg tank car with a fixed-in-place coupler body collided with a group of four cars with a total mass of 326 673 kg[83]. The tests were performed at 15.0, 20.4 and 25.0 km/hr collision speeds, with the contact occurring between the rigicüy fixed coupler head of the moving car and the tank head of the first stationary car in the group of four. AU the cars were fiLIed with water[83]. The experimental results show that the collision lasts 0.2 to 0.3 seconds with peak 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 perpendicuiar approach distance, m Figure 2.22: Contact force variation wit h gap distance. forces ranging from 2225 MN to 4450 MN (500 to 1000 kips)[83]. Although tank cars appear to be rigid, they are in fact relatively weak structures given their mass and large dimensions. Researchers with the AAR tank car research group[83] state uthat the heads of loaded tônk cars in combination with their Iiquid backup axe good energy absorbers and through their soft spring action reduce impact forces during tank-car collisions to moderate levelsn. Speeds higher than 25.0 km/hr resulted in ruptured tank heads, even with head shields in place, in aIl tests done under these condit ions. Experiments performed wit h 115th scale mode1 tank cars[84] confirm the results of the earlier experiments[83]. Using the data provided from the collision tests of tank cars[83][84]as a guide for the duration of the collision and the forces produced, values for the parameters used in Equations 2.100 and 2.101 were determined. However, the test data only provide a reference for the collision of three or four cars up to 25 km/hr, while actual derailments could possibly produce higher speed collisions between a greater number of cus. For this reason the parameters that matched the test data were approach vefocity, m/s perpendicular distance, rn Figure 2.23: Collision contact force variation with approach velocity and gap distance. used in the validation test caseg and adjusted until the validation case produced acceptable results without numericd problems. The changes made to the parameters resulted in the collision model with a larger Fm, and smaller d, (i.e. a stifier model) than is required to reproduce the tank car collision experiments. Thus the fui$ model overpredicts the collision force for lower energy collisions in order to model high energy collisions. 'The validation case is discussed in detail in Chapter 3, Section 3.1. 2.5 Computer Program Implementat ion The equations describing the complete train system, the constraints, and the applied forces are incorporated into a cornputer program which solves the equations of motion to produce the position and velocity history of each car in the system. The program is written in FORTRAN, and is modularized such that the individual force models and constraint equations may be tested independently of each other. Extensive testing of the program during development was performed to ensure the validity of the theory and the correct translation into computer code. Figure 2.24 outlines the main flow of the program. The program starts with reading the input data and setting the initial conditions. The train is always set such that ail cars and couplers are aligned on straight track running parailel to the global X-axis, with an initial forward velocity as assigned by the input. To nui on curved track the mode1 is initiaily staxted on straight track, with ail the cars on the rails, and the train about to enter the curved portion of the track. This method ensures consistent initial conditions for the system, regardless of curve radius1'. The conditions for the coupler and wheel constraints are assigned for each car, such t hat any car may be set as uncoupled or derailed. Thus the derailment may originate from any car on the train, using a srnall lateral force appiied to the selected car to initiate the derailment. As used in this thesis, the origin of the derailment is the first car. The couplers of the initial derailed car are not required to be uncoupled as a car derails, as the coupler constraints are removed independently of the rail constraints. The exception to the above is the last car. The finai, or Nth rigid body in the mode1 assumes the role of applying the braking force and accounting for the mass of non-derailing, or virtual, cars and is not expected to derail or uncouple. Should the final car reach the point of derailing, the number of cars should be increased and the 'O~heresuits for curved trdmodels are not presented in this research. 90 l~eodinput dota ( 1Set initial conditions set maximum e 7 time step I form inertia motrix h brake forces coupler forces ground reoction forces collision forces tform constroint equations . no print output data <. t-, Y- Y- Figure 2.24: Flow chart of the derailment mode1 cornputer program. model should be nui again. An initial break in the rad may be defined at any desired location dong the X-axis, independent of the car-rail c0nstra.int status defined for each truck on each car. This X-axis coordinate represents the location where the car-rail constraints are removed when a car passes over the location. This point can be moved during the simulation to a new location, such as the position of a violated car-rail lateral c0nstra.int to rnodel rail roil over. With the initial conditions set, the maximum dowable time step is calculated. The collision model, described in Section 2.4.4, assumes that the positions of the bodies will be monitored such that a.adjacent cars corne within a set bder distance, a force function will be applied. To ensure that a collision is not missed the displacement of dlthe bodies in the model must be limited to be less than the buffer distaoce defined in the collision model. The maximum displacement of the bodies is controlled by the time step size. The maximum time step is calculated based on the maximum velocity of the corners on my car in the solution. The integration time step is Limited such that the largest displacement resulting from a solution step will not be greater than 20% of the buffer distance (db)defined in the collision model. The program then completes the equations of motion for integration and integrates to the next step in the solution. A variable time step Runge-Kutta Fehlburg integration routine[93] is used, with modifications made to allow for variable error control and for the control of the constraint removal between successful integration steps. The alterations to the constraint equations must be made outside of the Runge-Kut ta integration routine or integration failure will result . The car-tecoupler and tmck-to-rail constraints are both removable on an independent basis, modelling failure of a coupler or the derailment of a truck from the rails. The coupler limitation is based on the relative displacement of the couplers at the mated faces. Displacement, and not force, is used as it provides a convenient reference, and the limitations are known[34] (351. Exceeding the dowed displacement produces the removd of the constraint. Two truck-to-rail constraint remod criteria are in place: the position of the truck pivot point on the rail and the magnitude of the ratio of the lateral constraint force to the vertical force at the truck. The In the first instance, if a truck passes the global position on the rail set as the break in the rail, the tmck-terail constraint is removed. In the second instance, the failure criteria is based on the well established derailment quotient, or L/V ratio, used by the rail industry to quantify the resistance of wheel-rail profile combinations to derailment due to lateral loading. Numerous studies into the theoretical aspects of the ratio and experimental studies to substantiate the theoretical L/V limits show a maximum achievable ratio in the range of 1.4 to 1.8 for a single wheel on rai1[24] [25][26][27]. The value required to derail a complete wheelset is slightly higher due to the lateral load contribution of the second wheel, thus the failure value of 2.0 was chosen to provide slightly excessive lateral load capability. As well, the position on the rail where the constraint was removed may be set as the new break in the rail. Moving the rail break position models the failure of the rail due to excessive lateral loading. Several additions are made to the model in order to reduce numericd errors. Firstly, Baumgaxte constra.int stabilization is applied to the coupler-to-coupler cons traint equations, as described in Section 2.3.5. Secondly, the rotational inertia defined for the coupler bodies is set at 20 000 kgm2, a value approximately 10 times higher than the true coupler inertia but still 100 times lower than the inertia of the rail car bodies. This increase in coupler inertia reduces constraint errors, reduces integration routine failure, and decreases the running time of the program. The effects of the added inertia on the motion of the car bodies is minimal. Thirdly, acting in pardiel with the stifhess element at the striker plate location a smdl amount of velocity dependent damping (4 000 Ns/m) is introduced to model friction present in the system and to damp out the free vibrations of the coupler bodies once uncoupled. 2.5.1 Input Data The data input into the program are separated into data files for the initial conditions, integration and computer control, train consist data, car type data, and coupler type data. For the complete model, the input data include: - initial speed - position of break in rail - ground reaction parameters - collision parameters The consist of the train is defined by specifying the input data for each car, which includes: - car type - mass and inertia - front and rear wheel constraint status - rear coupler status - brake type - initial applied forces For each car type used in the model, the input data includes: - wheelbase - pivot-tepivot distance - body length - coupler type For each coupler type the input data includes: - coupler mas and inertia - coupler lengt h - pivot-to-striker length - striker opening distance 2.5.2 Output Data The data output by the program are separated into data files for the generalized coordinates and the forces. The position data for each car is used to draw the figures of the derailment outcornes seen in this thesis. For each rail car, the generalized coordinate output files include: - the - position data, for each generalized coordinate - velocity data, for each generalized coordinate - acceleration data, for each generalized coordinate - coupler-coupler constraint position error For each rail car, the force output files include the following data: - time - braking force - ground reaction force on each truck pivot - ground reaction force over surface of car body - total applied ground reaction force and moment at CG of rail car - coupler striker reaction force, front and rear coupler - coupler-tecoupler reaction moment, rear coupler - collision force reaction at each corner - perpendicular distance of collision - perpendicular relative approach velocity - vector sum of the collision forces acting at the CG of the rail car - sum of the collision forces acting simultaneously upon the matching car - the peak collision force acting at the corners Chapter 3 Results This chapter presents the resdts of the validation and sensitivity tests performed on the derailment model. The purpose of the validation test is to compare the results of the model against the outcome of an actual derailment, critically comparing the results of the model against the actual event. The sensitivity test consists of a variation of the input parameters from the inputs defined for a set base case derailment, and tests the ability of the model to remain valid through a wide range of inputs. The results also show the physical effects of the test parameters on the outcome of a derailment. This Chapter begins with the results of the validation test case, followed by results of the variation of parameters tests. 3.1 Mode1 Validation: Modelling of the Mississauga Derailment To reproduce the main features of an actual derailment is possibly the best method to test the validity of the complete computer model. To accomplish this, a well known and documented derailment was modelled. The cornparison case selected is the November 10, 1979 derailment of a 106 car, 3 locomotive freight train in Mississauga, Ontario. The east-bound train was travelling at 90 km/hr, had a total length of 2020 metres, and a mass of 8210 tomes, d of which were typical of freight trains of the tirne. The derailment resdted in 24 cms leaving the track, 19 of which were tank cars carrying dangerous goods[l][94]. The cause of the derailment was the failure of a plain journal bearing on the right-rear wheelset of a toluene tank car, the 33rd freight car from the front. At the time of the accident toluene was not considered a dangerous commodity and as such was not required to have rouer bearings or shelf couplers. The train travelled for m estimated 10 kilometres with the journal bearing box visibly on fie, and traveiled another 3 kilometres after the right rear-wheel dropped off before fdyderailing. As the freight train passed over a level crossingl at 1l:56 pm the toluene car finally fully deraiied and uncoupled from the 32nd, causing the brake line to be torn open. The failure of a rear wheel on car 33 very probably caused vertical loads on the front and rear couplen of that car, which being of non-shelf type simply slipped upwards and uncoupled from the rear coupler of car 32 and possibly from the front of car 34. The lead locomotives and cars 1 to 32 continued travelling 1.9 km east under emergency braking before stopping. The remaining cars (33 to 106) now acted as a derailing 74 car train with car 33 as the lead derailing car. As the cars derailed, several propane tank cars ruptured and caught fie. One propane tank car exploded during the derailment, as witnessed by the crew of the train, and Mr. and Mrs. Ronald Dabor who were in their car at the rail crossing as the derailment occurred. The Dabors were forced to drive their car in reverse to 0ee from the crossing to avoid being hit by derailing cars in front of and approaching them. Mr. Dabor lost control of his car and the couple then fled on foot. The first tank car explosion occurred seconds later, throwing them to the ground and shattering windows in the buildings adjacent to the scene. Figure 3.1 shows a schematic of the final positions of the cars[l]. The remains of l~avizadjust North of ~undFforreaders farniliar with the area. HYDRO EYf- \ Figure 3.1 : Outcome of the Mississauga derailment [l]. cars 7, 8, 9, 10, 11, 12, 13, 15 and 16 show extensive damage, with car 7 being the ruptured chlorine car. The damage to the cars was very severe as nine propane tank cars ruptured resulting in t hree Boiling Liquid Evaporating Vapour Explosions (BLEVES)~and six tank fires. Car 8, a propane car, appears to be out of position, resting next to cars 3 and 4, but in fact had ruptured and resuited in a BLEVE during the derailment, travelling 45 metres east. This was the explosion t hat knocked the Dabors to the ground. Cars 12 and 13, also propane cars, are not shown on the figure as they were ejected from the accident site after each resulted in 2BLEVEs aze extreme cases of sudden depressurizing of liquefied petroleum gases (LPGs) result- ing in rapid conversion of the LPG frorn a iiquid to a gaseous state, combineci with the combustion of the gas. Table 3.1: Derailed freight car list . Mississauga derailment. [l]. 1 Number 1 Car Tvpe 1 Contents 1 1 1 ltankDOT111 1 toluene 1 1 24 1 tank DOT 111 1 toluene 1 a BLEVE shortly after the cornpletion of the derailment. A portion of the tank from car 12 travelled 134 metres southeast, and a portion of the tank from car 13 travelled 677 metres northeast. Table 3.1 lists the freight car type and contents of the derailed cars. The contents of one of the two box cars was reported to be insulation, the other is unknown; both are assumed to be near empty. Ten of the eleven propane cars involved in the derailment were darnaged, with three resulting in BLEVEs, six ruptured or venting continuously until totd loss of contents, and one damaged without losing any contents. The ruptured and venting propane resulted in a continuous fire which lasted for several days following the accident. The ruptued chlorine tank car surrounded by burning propane created the extreme risk of the formation of a chlorine gas cloud. This resulted in the evacuation of 8000 people from the surrounding neighborhoods within two hours of the accident, and within 24 hours approximately 250 000 people from the surrounding communities were evacuated. The risk associated with the possible formation of a lethal chlorine gas cloud was far more severe than the fie, or even the possible explosion of ot her propane tank cars [l][94]. Miraculously no fat ali ties resulted, and the only serious injury during the eight day crisis was a journalist who broke a leg while jumping over a fence. The Mississauga derailment was selected as the validation case for the following reasons: 24 cars derailed - a number greater than the average number of cms t hat derail during a main track derailment 0 90 km/h train speed is near the top speed allowed on main track freight lines 74 cars behind the point of derailment represents the upper range of presently allowabie train lengt hs the geography of the surrounding area is sufficiently flat to be modelled with a plan= model the incident is well known and documented. See Grange[l] and Burton et al. [95l The first three items above are important as the scale of the accident allowed the model to be validated at the limits of its usefulness. Subsequent derailment models will more than likely be less taxing numericdly than the severe derailment that occurred at Mississauga. The fourth item is important as the planar aspect of the model required a validation case that had occurred on relatively flat terrain. 3.1.1 Input Data Table 3.2 summarizes the input parameters for each car in the train for the computer mode1 of the Mississauga derailment. The dimensions used are representative for fieight cars of the type, and the mass assigned to each car represents the upper range of the gross dowed mass for the freight cars. The estimated length for each car is taken from the pictures and sketches of the accident scene, and from the tank car type on record[l]. Exact dimensions for each car were not available, even at the time of the accident. The moment of inertia values are calculated assuming a uniform solid cylinder for the freight car bodies. Al1 cars were assigned a width of 2.6 metres, 0.4 metres less than the typical width of 3.0 metres to account for a portion of the b&er distance required by the collision model in the detection and application of collision forces. The initial speed was assigned as 90 kilometres per hour, the estimated speed of the train at the point of deraiIment[l]. The front and rear trucks of the fist car were set as derailed. The point of derailment is at the origin, coordinates (0,0),as is the initial position of the centre of mass of the first car. The brake initiation was set with no delay. To mirnic the derailment forces and to initiate motion in the numerical model a lateral force of 50 000 N was applied for 1.0 second to the first car at the front truck position. This lateral force represents less than 9% of the normal force initidy acting on the front truck. More significantly the impulse due to the force acting over 1.0 second represents less than 0.04% of the initial system momentum, or less than 2% of the initial momentum of the first car. Thus the init id applied force has Little consequence on the out come. Table 3.2: Input data for Mississauga model. car mass inertia body coupler truck coupler lengt h pivot centre type 1o3 1O6 spacing spacing (kg) (kg-m2) (ml (ml (ml 1 120 1.97 13.58 13.26 11.50 F-79 2 30 0.60 14.48 13.30 12.42 F-79 3 120 1.49 12.44 12.10 9.86 F-79 4 120 2.49 12.44 12.10 9.86 F-79 5 120 1.49 12.44 12.10 9.86 F-79 6 120 1.49 12.44 12.10 9.86 F-79 7 80 1.00 12.44 12.10 9.86 F-79 8 120 3.32 1 18.04 1 16.88 15.96 F-73 ---8 ---8 - 9 120 1.97 13.58 13.26 11.50 F-79 10 120 1.97 13.58 13.26 11.50 F-79 11 120 1.97 13.58 13.26 11.50 F-79 12 120 3.32 18-04 16.88 15.96 F-73 13 120 3.32 18.04 16.88 15.96 F-73 14 120 3.32 18.04 16.88 15.96 F-73 15 120 1.97 13.58 13.26 11.50 F-79 16 30 0.60 14.48 13.30 12.42 F-79 17 120 3.32 18.04 16.88 15.96 F- 73 18 120 3.32 18.04 16.88 15.96 F-73 19 120 3.32 18.04 16.88 15.96 F-73 20 120 3.32 18.04 16.88 15.96 F-73 I 21 120 3.32 18.04 16.88 15.96 F-73 22 120 3.32 18.04 16-88 15.96 F-73 23 120 3.32 18-04 16.88 15.96 F-73 24 120 3.32 18.04 16.88 15.96 F-73 25 3070 N/A represents cars 25 to 74 3.1.2 Result s and Discussion Figure 3.2 is a drawing of the final positions of the cars as produced by the computer model of the Mississauga derailment. A comparison with the sketch of the real accident scene, shown in Figure 3.1 reveals several similarities. The number of cars to derail is the same, at 24 and the grouping of the cars is similar but not an exact reproduction. However, the left-right placement of the cars on either side of the rail is similar between the model and the actual event for cars 5, 6, 7, 9, 10, 11, 19, 21, 22 and 23. The front grouping of cars 1 to 3 in the model is more compact than in the actual event, possibly due to the initial conditions applied to cars 1 and 2 for the model being different from the actual event. The orientation of the central group of cars is similar in both cases, being bunched in a tight grouping perpendicular to the track. The final group of cars to derail in the model is oriented in a chevron pattern sirnllar to that seen in the actual event. The accident scene dimensions produced by the model are 109 metres by 41 -7 metres compared to 132 metres by 39 metres for the actual event, with the corresponding axeas being 4545 and 5148 square rnetres respectively. The accident scene dimensions are taken as spanning across the width of the accident scene and lengthwise from the end of car 24 to the front of car 1, excluding fragments ejected from explosions. The difference in orientation of cars 1 and 2 between the model and the actual event accounts for much of the difference in the length dimension. The damage to the freight cars in the actual event is evident from Figure 3.1, with cars 7, 8, 9, 10, 11, 12, 13, 15, and 16 being shown as severely damaged. The published records confirm that cars 8, 12, and 13 exploded during or shortly after the derailment, indicating severe damage to the cas. The damage sustained by the cars in the derailment model can be inferred from the peak collision force on each car during the entire derailment. Figure 3.3 shows a graphical comparison of the peak collision force nonnalized to the largest peak due. Table 3.3 summasizes the peak collision forces for each car and the opposing 1 MISSISSAUGA MODEL 1 Figure 3.2: Mississauga derailment model: final outcorne. car, the ranking of the collision from highest to lowest, and the time of occurrence. For the derailment model, within the hst 9 seconds the five most severe collisions have occurred, involving cars 6, 7, 8, 9, 10, 11 and 12. Figure 3.3 shows a grouping of the peak collision forces from cars 4 to 12, with the coilision of car 11 into 9 producing the peak. Also evident are the high collision forces experienced by cars 14, 15, 18, and 20. The top five peak collisions produced by the model were car 11 into 9 at 8.02 seconds, collision force of 23.09 MN car 8 into 6 at 5.94 seconds, coliision force of 16.1 1 MN car 8 into 7 at 5.74 seconds, collision force of 15.70 MN car 11 into 10 at 8.61 seconds, collision force of 15.36 MN car 12 into 11 at 8.62 seconds, collision force of 13.77 MN Figures 3.4 through 3.7 show the positions of the cars at the times of these coIlisions, in the order of occurrence. Figure 3.3: Mississauga derailment model: normalized peak collision forces for each Car. Figure 3.4 shows the positions of the cars just prior to the collision of car 8 (propane) into 7 (chlorine) . The coUision is between the front corner of car 8 and the rear of car 7, which produces a peak force of 15.70 MN. At this time cars 1 through 6 have stopped and car 7 has almost stopped and is fully abutted against car 6. Car 8 is traveling at 21 m/s, 84% of the initial speed of the train. A collision at this speed would certainly result in the rupture of the tank on car 7 and damage or rupture to the tank on car 8. Figure 3.5 shows the positions of the cars 0.2 seconds later, just prior to the collision of car 8 into car 6 (caustic soda). Car 7 appears to overlap into car 6, which indicates a large amount of deformation. Car 8 is now travelling at 18.6 m/s and the resulting peak collision force between car 8 and 6 is 16.11 MN. This collision clearly involves the coupler head of car 8 into the tank of car 6, and would resdt in the rupture of the tank on car 6. Car 8 has now sustained two collisions directly to the head of the tank, with car 6 having sustained a severe blow to the side of the tank. Table 3.3: Peak collision forces. Computer mode1 of the Mississauga derailment. Car Peak Force Ranking time Numben (MN) (seconds) Figure 3.6 shows the positions of the cars 2.10 seconds later, just prior to the collision of car 11 (styrene) into 9 (styrene). Car 11 is travelling at 18.0 m/s, 72% of the initial speed, prior to the collision with the stationaxy car 9. The resulting collision force of 23.09 MN is the highest for the entire derailment. Figure 3.7 shows the positions of the cars 0.6 seconds later, just prior to the collision of car 11 into 10 (styrene) and car 12 (propane) into 11. The position of the cars is sirnilar to the placement seen in Figures 3.4 to 3.6. Car 11 has stopped against the grouping of cars 2 through 10, while car 12 has not rotated enough away from the track to avoid a collision with car 11 or 10. Car 12 is travelling at 17.3 m/s just prior to the collision with car 11, 69% of the initial speed. The peak collision forces produced are 15.36 MN between cars 10 and 11 and 13.77 MN between cars 11 and 12, occurring at 8.61 and 8.62 seconds. The peak collision forces produced by the model are considerably higher than any force measured in experiments presented in the literature[83][84], in which collisions between tank cars with relative velocities of up to 7 m/s produced forces on the order of 5 MN in magnitude. The results of the computer model clearly show relative velocities much higher than those used in the experiments, and as such the peak forces cm only be an estimate of the forces that were produced by the actual event. However, even if a force of 10 MN produced by the model, twice the known force that will rupture a tank car, is accepted as nipturing a tank car the resuits of the model show 13 mptured tank cars. It is notable that the cars sustaining or producing high collision forces, and thus heavy damage, are also arnong the longest and heaviest cars in the consist. For example, both cars 8 and 12 are significantly longer and heavier than the cars immediately preceding them. Car 8 is 1.5 times more massive, 1.45 times longer resulting in 3.32 times the moment of inertia of car 7, while Car 12 is 1.33 times longer and 1.69 times the moment of inertia of car 11. The collisions of cars 8 into 7 and car 12 into 11 both appear to be the result of the shorter less massive car rotating to a perpendicular orientation with the track as it derails, followed by a longer more massive car not rotating as quickly and colliding into the side of the car or cars stopped in front of it. I MlSSlSSAUGA MODEL 5.70 seconds 20 IO 5 O rnetre I I Figure 3.4: Mississauga derailment model, collision of car 8 into 7 at 5.70 seconds. MlSSiSSAUGA MODEL 5.90 seconds -5 O rnetre Figure 3.5: Mississauga derailment model, collision of car 8 into 6 at 5.90 seconds. 108 MISSiSSAUGA MOOEL 8.00 seconds 20- 10 5 O metre Figure 3.6: Mississauga derailment model, collision of car 11 into 9 at 8.00 seconds. 8.60 seconds 20- 10 5 O metre Figure 3.7: Mississauga derailment model, collision of car 11 into 10 and 12 into 11 at 8.60 seconds. 3.1.3 Conclusions: Mode1 Validation Although it does not, and camot be expected to, reproduce the exact placement of the cars, the model does reproduce several significant rneasures: 0 the number of derailed cars produced by the model matches the actual event, at 24 cars 0 the accident scene dimensions produced by the model compare well with the actual event 0 the orientation of the cars produced by the model has notable sirnilarities to the real event 0 the damage sustained by the cars in the model compares well with the actual event The accident scene dimensions produced by the model compare well with the actual event , being 109 metres by 41.7 metres, compared to 132 metres by 39 metres for the actual event. This represents an error of -17.4% and -7.0% from the actual site dimensions. The accident scene area produced by the model is -1 1.7% in error of the actual site area. Again, the dimension results were used in the final adjustment of the ground reaction model. The orientation of the cars seen in the resdts of the model has notable sirnilarities to the car placement in the actual event. Cars 5, 6, 7, 9, 10, 11, 19, 21, 22, 23, and 24 are al1 positioned on the same side of the track in the rnodel results as in the real event. Notably, results of the model show the final placement and the positions relative to the break in the rail of the last three cars to derail, cars 22, 23 and 24, to be very similar to the actud event. Both the actual event and the model show the central group of cars, between cars 6 to 17, to be the most darnaged. As weu, the light damage to the lead group of three cars compares weil with the little to no damage experienced by these cars in the actual event. The three peak collisions produced by the model were of car 8 into 7 at 5.74 seconds, car 8 into 6 at 5.94 seconds, and car 11 into 9 at 8.02 seconds. This result concurs with the real event where car 8 undergoes a BLEVE, car 7 is severely damaged, and cars 6, 9 and 11 are damaged. Finally, of significance is that the results of the mode1 show cars 6, 7, 8, 9, 10, 11, 12, 14, 15, 18, and 20 as sustaining peak collision forces above 10 MN. This result compares well with the outcome of the actual event where cars 8, 12 and 13 exploded, cars 7, 9, and 11 are tom open, and cars 10, 15 and 16 are severely damaged. Although the model does not predict a large collision force for car 13, the forces acting on car 12 may be expected to damage car 13 in manners not accounted for by the derailment model, such as a failure of the coupler in the vertical plane. As well, the effects of the early explosion of car 8 on the integrity of the other cars is not accounted for in the computer model. These results confirm that the model produces results capable of reproducing the important rneasures of an actual event. The parameter variation, or sensitivity test of the computer model is presented next. 3.2 Variation of Parameters The parameter variation consists of individuaily altering selected input parameters about a set base condition, from which changes in the outcome of the model are then interpreted as the direct result of the single change to the input. The parameter variation test perforrns two functions: to test the ability of the model to remain valid, or numericdy stable, throughout a wide range of inputs, and to show the effects that the test parameters have on the outcome of the derailment model. The first role of the parameter variation testing is in effect a sensitivity test of the computer model. The second role is a test of the physical results produced by the model, with the outcome measures used to interpret the effects of the parameter variations on the physical system. If the model does not produce results that correlate with known physical laws, then a flaw in the model is present. The second function cannot be completed if the sensitivity test has failed. The parameter variation is performed on a set of input data defining a model that dows for significant wiations to be made without exceeding physical realities. For this reason, the set of input data defining the base system produces a mode1 similar to an average size f~eighttrain operating in North America. To remain consistent, the base system was defined using the computer model as finalized for the Mississauga derailment validation case discussed in Section 3.1. Changes were made only to the type and number of cars, the coupler type, and the initial train speed. All other specifications remained the sarne as used in the Mississauga model, which include the ground reaction model, the brake model, the collision model, and input specific to the computer implementation. Six parameters were selected for variation: train speed, car mas, train length (number of cars in the train), brake force, derailment quotient (L/V ratio), and ground reaction force. These pammeters were individudy varied *20% and f50% from the base system values, keeping ail other parameters fixed. Of these variations, the changes in car mas and train length also represent equal changes to the initial kinetic energy of the system, allowing cornparisons between these variations. The outcome measures used for cornparison with the base system are: number of derailed cars, No duration of the event, b peak coilision force, FC cars involved in the peak collision accident scene length, -XD accident scene widt h, YD accident scene area, AD- As well the general appearance of the final positions of the cars will be compared. This section fht presents the results of the base case model. Foiiowing is the presentation of the results of each variation of train speed, car mas, lengt h, brake force, L/V ratio, and ground reaction force. For each variation the outcorne measures axe presented, dong with drawings of the final positions of the cars. Following the results for the individual variations is a summary of the complete results, where the outcome measures for all the variations are compared. 3.2.1 Base System The train system used in the base mode1 is defined by the input data specified by: number of cars 60 initial speed 60 km/hr car mass, per car 68 O00 kg car length 14.1 m coupler pivot-twpivot 13.3 m wheel base 11.5 m coupler type F-73 The values selected for train length, speed and car mass are al1 close to the average for a freight train in North America. As well, a train length of 60 cars allows a substantial variation in the number of cars, from 30 up to 90. The speed of 60 km/hr and the 68 000 kg mass were selected for similar reasons. The car dimensions represent a typical tank car. Figure 3.8 is a drawing of the hal position of the cars as produced by the computer model of the base train system. The final position of the cars is very symmetrical, not unexpected for a train with identical car mass and dimensions for al1 freight cars. The peak collision occurs when the fiont of car 5 deflects ofE the rem of car 4 and collides with the side wall of car 3, producing a peak force of 9.47 MN (2120 kips). BASE CASE 2 1.40 seconds Figure 3.8: Variation of parameters: final car positions for the base system model. The outcorne mesures, which will be compared to the results of each vaiiation case, ue: nurnber of derailed cars duration of derailment (s) peak collision force (MN) cars involved in peak collision accident scene length (m) accident scene width (m) accident scene area (ma) 3.2.2 Variation of Initial Thin Speed The parameter variation of f20% and f50% from the base system value of 60 km/hr resulted in initial train speeds of 30, 48, 72, and 90 km/hr for the model. Al1 other inputs remained fixed. The model was numericdy stable for all speeds. The result ing outcome measures are summarized in Table 3.4. Figures 3.9 t hrough 3.12 show the final position of the cars for the initiai speeds of 30, 48, 72: and 90 km/hr. Table 3.4: Results of Mnation of initial train speed. r out corne rneasure -50% -20% base +20% +50% number of derailed cars 5 10 14 18 24 duration of derailment (s) 12.7 17.9 21.4 23.7 25.9 peak collision force (MN) 0.78 3.16 9.47 9.48 19.0 cars involved in peak collision 3,2 3,4 5,3 12J0 14,12 accident scene length (m) 58.9 58.7 65.8 77.8 93.5 accident scene width (m) 21.1 25.4 26.2 29.3 37.0 accident scene area (m2) 1243 1491 1721 2277 3460 For an initial speed of 30 and 48 km/hr the symmetrical pattern of car placement similar to the base system outcome is seen, but with fewer derailed cars. For 72 km/hr the chevron pattern is seen on only the last few cars to derail, reflecting the increased number of collisions taking place in the initial phase of the derailment. The front group of cars have been pushed into being perpenàicular with the track from repeated collisions with derailing cars. For 90 km/hr the chevron pattern of car placement is almost completeiy replaced by a perpendicular orientation to the track centreline. Nearly all the cars have coilided with nearby cars and the cars have taken final resting positions with nearly complete wd-to-wa.ll contact. The +50% increase in train speed resulted in a 200% increase in accident scene area and peak coilision force, not unexpected for a 225% increase in initid system energy. As well, the increase in train speed results in the cars involved in the peak collision being farther back from the front of the train, dong with an increase in the magnitude of the peak force. The number of derailed cars produced by the variation in train speed also compares weU with the statisticd data presented as Figure 1.1, in Chapter 1 at Iower speeds. At higher speeds the variation results deviate from the statistical data, which may be due to the compilation of the statistical data fmm numerous sources and the use of a single length of train for the parameter variation tests. The parameter variation results clearly show that train speed has a very strong effect on the derailment outcome, with increasing speeds producing an approximately quadratic increase in the peak collision force and accident scene area. -50% initial veiocity (30 km/hr) 12.70 seconds ?O 10 5 O metre I Figure 3.9: Outcome with variation of -50% initial speed. 116 -20% initial velocity (48 krn/hr) 17 90 seconds 20- 10 5 O metre I Figure 3.10: Outcome with variation of -20% initial speed. +20% ~nitiol velocity (72 km/hr) 23.7 seconds 20 10 5 O metre ilI Figure 3.11: Outcome with variation of +20% initial speed. 117 +50X initial velocity (90 km/hr) 25.90 seconds - 2 4 7 m, , 20& 10 O metre Figure 3.12: Outcome with variation of +50% initial speed. 3.2.3 Variation of Car Mass The parameter variation of f20% and f50% from the base system car mass of 68 000 kg resulted in car mass of 34 000, 54 400, 81 600, and 102 000 kilograms. All other inputs remained fixed. The mode1 was numerically stable for all dues of car mas. The outcome measures results are summarized in Table 3.5. Figures 3.13 through 3.16 show the final position of the cars for models with car masses of 34 000, 54 400, 81 600, and 102 000 kg. The number of derailed cars, duration of the derailment, and the peak collision force ail increase with an increase in car mass. The accident scene dimensions and area increase with both an increase and decrease in car mas. The -50% variation shows a general scat tering of the cars, reflecting the lower ground reaction force resulting from motion over the ground. The increase in accident scene width and length from the base case also reflect this. As well, the reduced ground forces have reduced the coupler reaction force, such that cars 7 and 8 remained coupled after derailing and Table 3.5: Results of variation of car mas. outcome mesure -50% -20% base number of derailed cars 11 13 14 duration of derailment (s) 14.9 18.6 21.4 peak collision force (MN) 4.17 4.76 9.47 cars involved in peak collision 4,7 5,3 5,3 accident scene length (m) 83.9 72.2 65.8 accident scene width (m) 33.8 26.3 26.2 accident scene area (m2) 2835 1900 1721 coiliding with cars 4 and 5. The -20% variation shows results similar to the base case, with a symmetrical chevron position of the cars, and with the derailing cars al1 uncoupling. The +20% variation resulted in an increase in the accident scene dimensions and a slight increase in the severity of the collisions, but the final car positions are similar to the base results. The +50% variation shows an increased scattering of the cars with a wider accident scene due to the increase in the severity of the collisions. A +50% increase in car mass resulted in a 64% increase in the peak collision force during the derailment and a 51% increase in accident scene area, each very nearly following the 50% increase in system energy. However, the changes in peak force and accident scene area are greater than that resulting from the same increase in train length, which produces the same initial train mus. As the systems have identical initial kinetic energy the difference partially reflects the added braking force resulting from increasing total mass with additionai cars as opposed to heavier cars, and the fact that the collision of more massive cars at a given speed will produce higher collision forces. The results show that the cius involved in the peak collision do not drastically change with increasing car mass, but sirnply experience larger collision forces as the car mass increases. It should be noted that these results reflect the application of brake forces that are not a function of car mas, but are set to not produce wheel lockup at light weight conditions. -50% car mass 14.90 seconds 20- 10 5 O metre Figure 3.13: Outcome with variation of -50% car mass. -20% car mass 18.60 seconds Figure 3.14: Outcome with variation of -20% car mass. 120 +20% car rnass 22.10 seconds Figure 3.15: Outcome with variation of +20% car mass. +50% car mass 24-10 seconds Figure 3.16: Outcome with variation of +50% car mas. 3.2.4 Variation of Thin Length The paameter variation of f20% and I50% from the base system value of 60 cars resulted in initial train lengths of 30, 48, 72, and 90 cars for the model. Al1 other inputs remained fixed. The model was nurnericdy stable for all train lengths. The outcome measures results are summarized in Table 3.6. Figures 3.17 through 3.20 show the final position of the cars for initiai train lengths of 30, 48, 72, and 90 cars. Table 3.6: Results of vitriation of train length. out corne rneasure number of derailed cars duration of derailment (s) peak collision force (MN) cars involved in peak collision accident scene length (m) accident scene width (m) accident scene area (mZ) The final car positions for models with 30, 48, and 72 car trains appear very similar to the chevron pattern produced by the base model, but with differing number of derailed cars. The ha1 car positions for the 90 car model show an increase in the scattering, reflecting the result of an increased number and severity of collisions. The peak collision force and the accident area results clearly show this, with a 38% increase in accident scene area and 47% increase in peak collision force for a 50% increase in train length. Interestingly, the cars involved in the peak collision do not change with increasing train length until the length has increased by 50%. As well, the time history results (not presented) clearly show that the derailments for all variations of train lengths are dl nearly identical to the base system for the initial 7 seconds of the derailment, after which deviations in outcome occur. The result for the +50% train length does produce a collision with car 5 into 3, however a later coilision between cas 10 into 8 is more severe than the collision of 5 into 3 and replaces it as the peak instance. Train length does display an effect on the outcome of a derailment, with increasing severity with longer trains. However the increase in the number of cars to derail with increasing length is not as pronounced in cornparison to the results produced by changes in speed. This is clearly the result of the difference in the initial kinetic energy of the systems, with changes in system energy proportional to changes in length (Le. mass) but proportional to the square of the change in speed. Additionally, the percentage of initial train length to derail decreases with increasing train length. This result reflects the effect of the increased braking force of the addi tiond cars on the rails, off-sett ing to some extent the eRect of the increase in total train mass. The number of derailed cars resulting from the variations in train length also compare well with statisticd data presented in Chapter 1, Figure 1.2 and with the results of the basic system analysis presented in Chapter 2, Section 2.4.3.3. The number of cars derailing resulting from the model is at most in error by 2 cars in cornparison with the 40 mph (64 km/hr) data, the nearest speed to the base case of 60 krn/hr, seen in Figure 1.2 and in Figure 2.17. - - 3The very simple one degree of freedom model appears quite capable of determining the ex- pected number of cars to derail for a given train length and speed, agreeing with the conclusion of Johnson[39] that simple models of complex systems can be very revealing. -50% train size (30 cars) 1 4.20 seconds 20 10 5 O metre 111 Figure 3.17: Outcome with variation of -50% train length. -20% train size (48 cars) 19.00 seconds Figure 3.18: Outcome with variation of -20% train length. 124 +20% train size (72 cars) 23.50 seconds Figure 3.19: Outcome with variation of +20% train length. +50% train size (90 cars) 24.20 seconds Figure 3.20: Outcome with variation of +50% train length. 125 3.2.5 Variation of Brake Force The parameter variation of f20% and f50% from the base system brake force model was incorporated into the computer model as a variation in overall applied brake force to each car. Al1 other inputs remained fixed. The model was numerically stable for al1 brake force values. The outcome measures results are summarized in Table 3.7. Figures 3.21 through 3.24 show the final position of the cars for a variation in the brake force of -5096, -20%, +20%, and +50% of the base case. Table 3.7: Resdts of variation of brake force. ' outcome measuse -50% -20% base +20% +50% number of derailed cars l? 15 14 13 12 durat ion of deraihent (s ) 26.4 23.2 21.4 19.9 18.1 peak collision force (MN) 9.93 9.30 9.47 8.58 5.97 cars involved in peak collision 5,3 5,3 5,3 5,3 5,3 accident scene length (m) 77.5 67.6 65.8 61.9 60.3 accident scene width (m) 29.9 26.4 26.2 27.1 26.8 accident scene area (m2) 2319 1781 1721 1675 1616 The final chevron pattern for the positions of the cars in each case is similar to the base case. Braking does not appear to have a strong effect on the outcome of a derailment, which is not unexpected given the operation of freight train brake systems as reviewed in Section 2.4.1 of Chapter 2. With 50% more brake force the reduction in peak collision force was 5.3%, the nurnber of derâiled cars was 14.2% less, and the reduction in the accident scene area was 6%. The cars involved in the peak collision were unchanged for ail variations, reflecting the result that the derailment is relatively unchanged from the base system in the first 7 seconds for âny change in brake force. This is the direct resdt of the time delays inherent in the brake systems, including the speed of the brake signal at 290 m/s and the linear ramping to full brake force of 10 seconds once the si& is received by a car. -50% brakinq force 26.40 seconds 20- 10 5 O metre Figure 3.21: Outcome with variation of -50% braking force. -20% braking force 23.20 seconds 20- 10 5 O metre ------ Figure 3.22: Outcome with variation of -20% braking force. 127 +20% bruking force 19-90 seconds 20- 10 5 O metre Figure 3.23: Outcome with variation of +20% braking force. +50% brakinq force 18.10 seconds 20- 10 5 O metre Figure 3.24: Outcome with variation of +50% braking force. 128 3.2.6 Variation of the Derailment Quotient (L/V). The parameter variation of f20% and f50% from the base system value of 2.0 for the derailment quotient (L/V) resulted in ratios of 1.0, 1.6, 2.4 and 3.0 used in the model. AU other inputs remained fixed. The model was nurnericdy stable for al1 variations. The outcome rneasures resuits are summarized in Table 3.8. The outcome remained near constant with changes in L/V, with the exception of accident scene length and axea which increased slightly with lower L/V values. Figures 3.25 through 3.28 show the fmd position of the cars for a model using derailment quotients of 1.0, 1.6, 2.4, and 3.0 respectively. Table 3.8: Results of variation of L/V. outcome measure -50% -20% 1 base +20% +50% number of derailed cars 15 15 14 14 14 durat ion of derailment (s) 21.2 21.2 21.4 21.5 21.5 peak collision force (MN) 10.3 10.3 9.47 10.3 10.3 cars involved in peak collision 5,3 5,3 5,3 5,3 5,3 accident scene length (m) 67.0 66.6 accident scene width (m)-. 26.2 26.0 accident scene area (m2) 11 1755 / 1734 The variation of the derailment quotient produces almost no change in the outcome of the derailment, with an L/V ratio of 1.0 producing only a 2% change in the accident scene length compared to the base case. -50% derailment quotient (L/V ratio = 1) 21 -20 seconds 20 10 5 O metre 111 Figure 3.25: Outcorne with variation of -50% derailment quotient. -20% derailment quotient (L/V ratio = 1.6) 2 1 -20 seconds 20- 10 5 O metre Figure 3.26: Outcome with miation of -20% derailment quotient. 130 -20% derailment quotient (L/V ratio = 2.4) 2 1.50 seconds 20- 10 5 0 metre Figure 3.27: Outcome with variation of +20% derailment quotient. +50% deroilment quotient (L/V ratio = 3) 2 1 -50 seconds 20- 10 5 O metre Figure 3.28: Outcome with variation of +50% derailment quotient. 3.2.7 Variation of Ground Reaction Force The ground reaction was varied f20% and f50% from the base system by scaling the calculated reaction force by the appropriate amount. AU other inputs remained fixed. The mode1 was numerically stable for ad variations. The outcome measures are summarized in Table 3.9. Figures 3.29 through 3.32 shows the final position of the cars for a variation in the ground reaction force of -50%, -20%, +20%, and +50% of the base case reaction force. Table 3.9: Results of variation of ground reaction force. 1 outcome measure 11 -50% 1 -20% base +20% +50% 1 number of derailed cars II 16 1 14 14 13 13 duration of derailment (s) 23.7 21.4 21.4 20.5 18.7 peak collision force (MN) 6.36 6.30 9.47 10.3 8.5 cars involved in peak collision 10,8 8,6 5,3 5,3 593 accident scene length (m) 102.5 67.3 65.8 72.8 71 accident scene width (m). - 41.0 29.6 1 accident scene area lm2) 11 4202 1 1992 The ground reaction force 50% lower than the base case results in a signifieant'y different derailment and final placement of the cars compared to the base case. The lower ground force produces lower coupler reaction forces upon derailment, resulting in cars 14, 15, and 16 remaining coupled at the end of the derailment. The miations of -20% and +20% ground reaction force produce results similar to the base case, with the final chevron appearance of the cas being similar. The increased ground force resulted in a shorter but wider accident scene and an increased number of collisions, compared to a decrease in ground reaction force. Within the f20% wiation limits the ground reaction displays a slight effect on the outcome of a derailment. The *50% variation shows a more marked effect on the system, however the peak collision force increased at most by 8.8% (for the +20% variation) showing that the ground reaction force has a slight effect on collision severity. However, the cars involved in the peak collision are further back for decreased ground reaction, but unchanged from the base for increased ground force. It should be noted that a f50% change in the ground reaction force represents a significant change in a physical parameter which rnay not be seen in reality. This is unlike the other parameters tested in this section which in reality may ~iywithin the boundaries of the dues tested. -50% ground reaction force 23.70 seconds 16 1 20- 10 5 O metre - - Figure 3.29: Outcome with variation of -50% ground reaction force. 133 -20% ground reaction force 2 1.40 seconds 20 10 5 O rnetre .111 Figure 3.30: Outcome with wiation of -20% ground reaction force. +20% ground reaction force 2 1 .20 seconds 20- 10 5 O metre Figure 3.31: Outcome with variation of +20% ground reaction force. +50% ground reaction force 18.70 seconds Figure 3.32: Outcome with variation of +50% ground reaction force. 3.2.8 Cornparison of the Parameter Variation Result s To perform an overall comparison of the parameter variations, a compilation and summary of the results was performed. The outcome measures were normalized against the outcome for the base model, and the normalized outcome measures were then grouped for comparison. The number of derailed cars, peak collision force, and accident scene area were selected for overall comparison between the variation tests, as these outcomes are of main concern to the public and the rail industry. Figures 3.33, 3.34, and 3.35 show the comparison charts for these three outcome measures. Figure 3.33 shows the effects of the six parameter variations on the normalized number of derailed cars. Speed has the most pronounced effect, with train length having the next strongest effect. Car mass has only a slight positive effect on the number of derailed cars. hcreasing brake force, ground reaction force, and derailment quotient d produce a slight decrease in the number of derailed cars. The number of derailed cars is most sensitive to changes in train speed. % Variation of Parameter +BRAKE FORCE +NUMBER OF CARS +î+ GROUND REACTION FORCE +wuATlo +INITIAL SPEED +CAR MASS Figure 3.33: Cornparison of changes in number of derailed cars. Figure 3.34 shows the effects of the six parameter variations on the peak collision force for a derailment. Increasing the initial train speed, car mas, and number of cars al produce increases in the maximum collision force, with the greatest increase due to increasing train speed. Increasing ground reaction force shows a slight increase in peak collision force. Increasing the brake force and L/V ratio both show littie effect on the maximum coUision forces. Collision force is most sensitive to changes in train speed. Figure 3.35 shows the effects of the six parameter variations on the area of the accident scene. Train speed clearly shows a strong positive effect on the area of the accident scene. The train length also shows a positive effect but to a lesser extent than speed. Ground reaction force and car mass result in an increase in accident scene area when each parameter is increased or decreased, with a decrease in ground reaction force producing the largest Mnation in accident area of al1 the variations. The increase in accident scene area with increasing mass and ground reaction forces is mostly due to increasing width of the accident scene, caused by increased nurnber and severity of collisions. Increasing derailment quotient and brake force both produce a slight reduction in accident scene area. To further quantify the effects of the variations on the outcome of a derailment, a composite outcome measure called derailnent severity is introduced in this research. This measure combines the t hree quantitative measures, number of derailed cars, peak collision force, and accident scene area, into a single value. This combination of outcome measures dows for an overd cornparison of the results to be made. Derailment severity is defined as the average of the normdized outcome measures presented in Figures 3.33, 3.34, and 3.35. Derailment severity is given by where ND,, FD,, and ADo are the number of derailed cars, peak collision force, and area of accident scene for the base system model. -60 -50 -40 -30 -20 -10 O 10 20 30 40 50 60 % Variation of Parameter +BRAKE FORCE +MJMBER OF CARS +GROUND REACTiON FORCE +UV RATIO +INITIAL SPEED +CAR MASS ------ Figure 3.34: Cornparison of changes in peak collision force. 138 Nonndized Accident Scene Area (Variation I Base) Figure 3.36 shows the effects of the six parameter variations on the derailment severity. The parameters producing the greatest changes in severity are variations in train speed, car mas, and train length. The composite measure results in a closer similarity between the effects of car mass and car length, while highlighting the result that the braise force and L/V Mnations have little effect on the derailment severity. As weU positive variations in the ground reaction show little change in der ailment Severit y, while decreasing ground react ion force does produce an increase in severity mostly due to increasing accident area. The sensitivity of derailment severity to train speed is clearly seen, and reflects the result that alI outcome measures are most sensitive to train speed. 3.2.9 Conclusion: Variation of Parameters The variation of parameters test successfdly performed the two intended functions: to test the ability of the model to remain numericdy stable throughout a wide range of inputs, and to expose the physical effects that the test parameters have on the outcome of the derailment model. The model was numericdy stable through aU the tests, with no changes made to the model other than the variations of the test parameters from the base condition. As well the results of the model do not contradict and are consistent with basic physics. Train speed has the greatest effect on the outcome of a derailment, as is expected given the v2 proportionality of the initial system energy. On a peak collision force basis, car mass has the second most pronounced effect, followed by train length. This result is not unexpected, given that collisions occur between a pair of cars for a given velocity thus the increased mass of each car will produce higher collision forces. On a number of derailed cars cornparison, train length has the second most pronounced effect on outcome, and car mass has the third most pronounced effect. This result is also not unexpected, given that increased car mass also increases the % Variation of Parameter +BRAKE FORCE -rt NUMBER OF CARS *GROUND REACTION FORCE +W RATIO +iNITiAL SPEED +CAR MASS Figure 3.36: Cornparison of changes in the accident severity measure. energy loss when a car has derailed and is sliding on the ground. As well the increased inertia keeps the cars aligned with the rail for a longer distance, increasing the braking effect of the derailed cars on the underailed train. With increased train length, the system energy loss in the fist few seconds of the derailment has not cbanged compared to the base case, but the overall system energy for the longer train is higher. With a longer train the system energy resides on the rails behind the point of derailment, and more cars must derail to remove energy fiom the system. The effect of increasing the mass of a train by additional cars is lessened slightly because of the additional braking force created by the added cars, and through lower collision forces resulting from less massive cars. These results show that train speed, car mass, and train length all produce the largest changes in outcome for a given change in input. The relatively small effect changes in braking have on the outcome of a derailment is at first surprising, however given the standard operation of freight train braking systems the results are not unexpected. For the base system of 60 cars, full application of emergency brake force of a non-derailing train (all the cars remain on the rail) would take from 15 to 20 seconds, with the train needing over 60 seconds to stop. For moût of the variation tests the peak collision force occurred with the collision of car 5 into car 3, at about 6 to 7 seconds into the event. From these results it appears that the most damaging collisions of a derailment occur before the brakes of the train have any effect on the train speed. In the initial stages of a derailment, the work done on the ground by the derailing cars is the main energy loss of the system, not the brakes. 3.3 Mississauga Derailment: Brake Application Tests The results of Section 3.2 show that train braking has little effect on the outcome of a deraitment. This result may be due to the tirne delay in the fdapplication of the brakes, or to the effective brake force appiied at each car. This of course leads to the question of what if the brakes codd be applied instantly, and with greater force. To observe the results of the instantaneous application of the brakes and the effect of improved brake traction, the Mississauga derailment model was tested with changes made to the braking model. Three tests were performed. For the first test the application of brakes was made instontaneous, with all other hinctions and inputs unchanged- The velocity dependent brake force function as described in Chapter 2, Section 2.4.1 is used intact, but with no delay due to brake signal propagation and the actuation of the brake valves- For the second test the applied brôke force was altered to represent traction Limited braking force, wit h the standard propagation and actuation time delays in place. The typical brake force used in the validation model, described in Chapter 2, Section 2.4.1, applied the same force function to each car regardless of normal force available at the wheei-rail interface. By applying a traction Limited brake force, each car has an applied brake force that is a function of the available normal force at the rail. In this marner increased car mass results in an increase in brake force. The applied brake force for each car is then given by a simple Coulomb friction model, with where Ptrodion = 0.3 was selected as as estimate of the typical friction coefficient between dry steel wheels and rails[l 11. The final test was the combination of the first two cases: an instantaneous application of traction limited brake force. The remaining input parameters are as described in Section 3.1 of Chapter 3. 3.3.1 Resdts and Discussion of Brake Application Tests The important outcome measures fiom the resdts of the brake tests performed on the Mississauga model are sumrnarized in Table 3.10. Table 3.10: Results of brake application tests to the Mississauga model. 1 II Mississauga Mode1 Validation Test 1 Test 2 Test 3 number of derailed cars 24 21 18 12 durat ion of derailment (s) 25.3 22.7 16.2 11.5 peak collision force (MN) 23.09 18.4 18.3 9.15 cars involved in peak collision 11 into 9 7 into 6 7 into 6 8 into 7 The instant application of the brakes, Test 1, has reduced the number of derailed cars by 3, the peak collision force has been reduced by 20.3% and the duration by 10.2%. The cars involved in the peak collision have changed from car 11 into 9 for the validation model to car 7 into 6, reflecting the decrease in train speed late into the derailment, preventing the later more severe collisions from occurring. The quicker application time has clearly reduced the accident severity, however the applied brake force appears to not be effective in removing enough energy from the system to prevent severe collisions from occurring in the early stage of a derailment. Standard timing of the traction limited brake force, Test 2, has reduced the number of derailed cars by 6, a 25% decrease. The peak collision force has been reduced by 20.7%, and the duration of the event by 36.0%. The increased brake force clearly reduces the accident severity, as the later collision between car 11 and 9 has been prevented from being as severe as in the validation model. However the time delay in the application of the brakes does not prevent the severe collisions which occur in the initial seconds of the derailment. Test 3 shows the most dramatic results. The instant application of traction lirnited braking reduced the number of derailed cars by 12, a 50% reduction. The duration of the event was reduced by 54.5%, and the peak collision force was reduced by 60.4% with the peak collision being car 8 into 7. The final position of the cars for Test 3 of the Mississauga brake tests is shown in Figure 3.37. The accident scene looks much less severe, as the width has been reduced to 26.3 metres, the length to 85.0 metres, and the area to 2236 square metres. These values represent reductions of 22.0%, 36.9%, and 50.8% from the results of the validation test model. The results of these tests clearly show that thorough investigations into the effects of braking on the severity of derailrnents are required. As well, the result that car 8 remains involved in a severe collision even with significant improvements in braking leads to other questions regarding the effects of the placement of long and short cars relative to each other in the consist, f MlSSlSSAUGA MODEL 1 1.50 seconds instant brake application traction Iirnited brake force .------ 8 Figure 3.37: Outcome of brake application test 3. Mississauga derailment mode1 with instant brake application and traction limited brake force. Chapter 4 Summary and Conclusions This chapter presents a summary of the research presented in this thesis, a discussion of the result s wit h conclusions, followed by recommendat ions for related research work. 4.1 Summary This thesis has focused on improving the study of the motions and forces on rail cars during a derailment through the development of a comprehensive state-of-the-art computer derailment model. There is a growing need for a cornprehensive rnodel such as this. Between 1985 and 1994 in Canada an average of 128.6 freight trains per year derailed on a rnainline track. This accounts for 12% of aiI accidents reported by the rail companies. The cause of these deraihents is divided approximately into 40% track, 40% equipment, 20% operations related causes. Of high concern are those trains which are carrying dangerous goods such as chlorine, propane, and toluene, of which in Canada from 1985 to 1994 an average of 37.5 derailed per year. These accidents are serious events, as a typical freight train in Canada has 76 cars, each with a mass of about 120 000 kilograms, and travels at up to 100 km/hr. As a cornparison, the kinetic energy stored in a train of this configuration is the same as the energy needed to Iift a 280 000 kg Boeing 777 to 1280 metres, or to accelerate the same aircraft to 570 km/hrl. The consequences of such a derailment were seen in 1979 near Mississauga, Ontario, wheri a 106 car train carrying dangerous goods derailed, resulting in 24 derailed cars, several explosions and an evacuation of over 250 000 people. Improvernents in rail car design, track maintenance, and operating procedures have resulted in reductions in the total number of derailments occurring per year over the past 2 decades, but in the same period of time urban areas have expanded into regions closer in contact with the rail lines, increasing the risk associated with any type of derailment. The study of the derailment process by a computer model has not been ignored by past researchers. Previous computer models of a train derailment were published by Yang and Manos[34] in 1972, and Anderson[36] in 1994. The Yang and Manos[34] model consists of a planar chah of Linked bodies, representing the train. The number of derailed cors produced by the model matches the statistical data well, but the accident scene has the appearance of linked sausages. Major limitations of the model are the lack of uncoupling of the cars and the exclusion of collision between cars. The Anderson[36] model addressed these Limitations through the use of individual car masses connected by stiff springs representing the rail cars and couplers, with collision modelled by classical impact theory. The Anderson[36] mode1 produced acceptable results, however the uncoupling of the stiff springs acting as couplers caused numerical problems, and the method was not pursued further. Other basic analyses have appeared in the Literature but none were serious attempts at modelling the entire derailment event. The requirements for a derailment model are based on many factors, with the most important being the known operating procedures and equipment in use by the rail 'or to have a 60 kg mass escape Earth orbit! industry, the outcomes of typical derailments, and the previous derailment models and their known limitations. The requirements for the derailment model produced by this research were: 0 to model a freight train with mas, length and speed typical of trains in use a t O include coupler kinemat ics and react ion forces 0 to include uncoupling and deraihg based on failure loads and displacements 0 to model ground reaction forces to mode1 collisions between cars to permit the initiation of a derailment to occur where needed 0 to reproduce the outcome of a known derailment Lagrange's equations with added multipliers was the method used to develop the equations of motion for the special purpose mode1 presented in this thesis. The use of this method dows the mode1 to overcome the limitations of previous derailment models and to meet the main requirements set out for a comprehensive derailment model. The method is extremely well suited to handle the difficult problem of the uncoupling of cars, the modeling of coupler kinematics, and the control of the point of derailment. The main disadvantages of the method are the increased number of equations needed to define the system, and the possibility of numerical erron brought about through the use of the constraint equations. The unique difficulties involved in the use of the solution method were manageable and were more than offset by the advantages. For the plana derailment rnodel, a simplification in the constraints allowed the number of degrees of kdomper rail cas to be five, with three for each car and one rotational degree of freedom for each coupler. Force models are used to apply reaction loads at the coupler knuckle interface, the coupler-tecar interface, the car-tegound interface, and the car-to-rail interface. The braking of the train is controlled though a model of train brake functioning, including brake signal delay, signal propagation delay, and the linear ramping of brake actuation. The ground reaction model is a velocity dependent function, applied at either truck pivot or along the length of the rail car, depending on the derailment status of each car. The collision model applies non-linear force elements between cars when imminent contact is detected. The collision force is a function of relative approach velocity and distance, being self adjusting for each unique collision event which may occur during a derailment. To control numerical errors, Baumgarte stabilization is applied to the coupler constraints. To validate the rnodel a cornparison was made between the model results and the accident scene of a weLl known and documented event. The selected event was the derailment in November of 1979 near Mississauga Ontario which involved the rear 74 cars of a 106 car train. Initially travelling at 90 kilometres per hour, the train derailed at the 33rd car as it passed over a level crossing. The damage resulting was extensive, wit h 24 derailed cars 3 of which exploded during or shortly after the accident. This event was selected for the validation as it represents the upper extreme of derailment outcornes, exceeding the average derailment in Canada in number of cars derailed, initial train speed, and Length. The car length, mas, and coupler types for the validation model were defined to be as close a match to the real event as possible. The validation rnodel results match the actual event very well. The number of derailed cars is the same, at 24. Of more significance is that the most severely damaged cars produced by the model were cars 6, 7, 8, 9, and 11. These results compare well with the results of the actual event in which car 7 was torn open on the side, car 8 exploded during the derailment, and cars 6, 9, and 11 sustained severe damage. The accident scene length, width and area produced by the model were -17.4'35, -7.O%, and -11.7% in error of the actual event dimensions. As well, the overd positioning of the cars agrees well, with left-right placement relative to the rail matching in 11 of the 24 derailed cars. With these results, the validation of the model was concluded as being successfd. To complete the testing of the model a parameter variation test was performed. As a test of the model, both the numericd sensitivity of the cornputer program and the physicd sensitivity of the model were under scrutiny. The goal was to test the sensitivity of the model to input data to expose any numerical instabilities, and to test the physical system and expose the parameters of most influence on the outcome of a derailment. A base train system was defined, consisting of 60 cars, each 14.1 metres long, with a mass of 68 000 kilograms, and with an initial train speed of 60 kilometers per hour. These initiai values allowed for a large variation of the input parameters without becoming physicdy unrealistic. The other aspects of the model remained as defined for the Mississauga derailment. The parameters selected for variation were the initial train speed, car rnass, number of cars, braking force, ground reaction force, and derailment quotient. Each parameter was varied f20% and f50% from the base system value. The outcome measures compared were the number of derailed cars, duration of the event, peak collision force, cars involved in peak collision, and accident scene dimensions. A composite measure, called accident severity, was defined to quantify the overall outcome of a derailment based on the number of derailed cars, the peak collision force, and the accident scene area. The parameter variation study resulted in 24 separate models, in addition to the base model. The results of this study were extensive, however the model was numericaily stable for ail 24 variations. This result allowed for cornparisons to be made between the outcornes of each variation on a physical basis. The variation *in train speed had the most pronounced effect on the derailment outcorne. In cornparison to the base systern, a 50% increase in initial train speed resulted in a 71.4% increase in the number of derailed cars, a 101% increase in accident area, and a 101% increase in peak collision force. As a 50% increase in speed produces a 125% increase in initial kinetic energy of the system, the 101% increase in peak collision force and accident area are physicdy consistent results, not unexpected for the given change in the system. These results also comply with the statistical data for derailments in North America, with the number of derailed cars increasing with train length in a similar manner. The variation in car mass had the second most pronounced effect on the outcome in terms of the accident severity rneasure. In comparison to the base system, a 50% increase in car mass resulted in a 7.1% increase in the number of derailed cars, a 50.7% increase in accident area, and a 63.7% increase in peak collision force. As a 50% increase in car mass produces a 50% increase in initial kinetic energy of the system, the 63.7% innease in peak collision force and 50.7% increase in accident area are physically consistent results, not unexpected for the given change in the system. The variation in train length had the third most pronounced effect on the outcome in terrns of the accident severity rneasure. In comparison to the base system, a 50% increase in train lengt h resulted in a 21.4% increase in the number of derailed cars, a 37.6% increase in accident area, and a 46.8% increase in peak collision force. As a 50% increase in train length produces a 50% increase in initiai kinetic energy of the system, the 46.8% increase in peak collision force and 37.6% increase in accident area are physicdy consistent results, not unexpected for the given change in the system. The results of the train length variation also compare very well with the results of the statistical data. The variation in braking force had relatively little effect on overall accident severity in comparison to the eflects of speed, mass and train length. A 50% increase in brake force resulted in a 14.3% reduction in the number of derailed cars, a -5.3% change in peak collision force, and a -6.1% change in the accident scene area. The variation of the ground reaction force produced results consistent with the physical changes made to the system. A 50% reduction in the ground reaction force ~roduceda 144% increase in accident scene area and a 14.2% increase in the number of derailed cars. reflecting the increased scatter due to less energy dissipation in the system. Changing the ground reaction force had little effect on the peak collision force, reflecting the fact that the collisions are dependent on the relative velocity and mass of the cars involved, not the reactions these cars are having with the ground. The variation in derailment quotient produced very little effect on the outcome. with a reduction in the L/V ratio producing the greatest changes. A 50% reduction in L/V ratio increased the accident scene length by 1.8% and resulted in 1 more car derailing compared to the base system results. The cornparison of the overd results of the variation tests reveais severd interesting features. The sensitivity of the system to train speed is clearly in line with the basic physics of the system, and shows that methods of reducing freight shipment delivery times other than through increasing train speed need to be given priority. Also of note is that a greater number of cars derailed for a train with more cars compared to a shorter train of the same mas. This is due to the greater proportion of the system energy rernaining on the rails behind the point of derailment when a train is longer compared to a shorter train of equal mass. The added braking force of the longer train helps to partially negate some of the effects the extra mas has on the outcome, but the effect is not enough to overcome the added mas. However, increasing total train mass through increased number of cars does produce a less pronounced increase in peak collision force than an increase in individual car mas, a direct result of the fact that lighter cars produce lower collision forces compared to heavier cars for a given collision velocity. The insensitivity of the mode1 to changes in the applied brake force highlights the fact that the delay in brake signal propagation and actuation results in little to no brake force application during the crucial initial few seconds of a derailment. The Mississauga validation and the parameter variation tests show that the most severe collisions occur during the initial 5 to 10 seconds of the event. As an initial study, the effects of improvements in train braking was tested using the Mississauga model, which was altered and run for three test cases: O instant application of the brakes to all cars, using the accepted brake force curve O improved brake traction up to the estimated Limits of adhesion for a wheel on a rail (p = 0.3), with stôndard brake timing and control instant brake application for al1 cars, with maximum traction. The resuits of these three tests were compared to the results of the Mississauga validation model. The results show that instant brake application with standard brake apparatus brings about a 20.3% decrease in peak collision force, with 3 fewer cors derailing. Adhesion Limited braking with standard time delays improves on this slightly, with 6 fewer cars derailing and a 20.7% decrease in peak collision force. Instant application of the brakes with adhesion limited braking produces the most drastic reductions in the accident severity, with 12 fewer cars derailing (a 50% reduction), a 60.4% reduction in peak collision force, and 50.8% reduction in accident scene area. These results clearly show that improvements in safety can be made with changes to the brake systems in use on freight trains. 4.1.1 Recommendations for Future Research Given the results of the research presented in this thesis, several recommendations rnay be made. 1. Studies be undertaken to use the derailment model to determine the effect of brake system performance on derailment severity. The initial study of the Mississauga derailment clearly shows that improvements to train braking result in reductions in accident severity. However, more work is required to highlight the best and most efficient method of brake system improvement. An in-depth study of all aspects of braking on the derailment outcome would be required. 2- Studies be undertaken to use the derailment model to determine the effect of the placement of empty and fdl cars on derailment outcome. For trains of equal kinetic energy, the parameter variation tests cleady show differences in the results between more cars in a train vs. more massive cars. Variations in car mass dong a train may have an effect on derailment severity and warrant futher study. 3. Studies be undertaken to use the derailment model to determine the effect of the placement of long and short cars in the train. The Mississauga validation mode1 clearly showed car number 8 colliding into the side of cars 6 and 7. Car 8 was the first long propane car to derail after the first 7 shorter cars in front had derailed and rotated perpendicular to the tracks, resulting in car 8 being unable to rotate away from cars 7 and 6. Car 8 is 45% longer than car 6 and 7, with 3.32 times the rotational inertia of car 7 and 2.23 times the rotational inertia of car 6. This drastic variation in both length and inertia displayed a pronounced effect on the outcome, and requirea further study. 4. Studies be undertaken to use the derailment model to determine the effect of changes in the surroundhg terrain on the results of a derailment. Examples include the placement of bems, retaining walls, and energy absorbent soils near the rail lines. The present model wodd require modifications to include a ground reaction model mapping of the terrain near the track, and the addition of a coilision model for objects fixed in the reference frame. 5. Studies be undertaken to use the derailment model to determine the effect of total system kinetic energy on the outcome of a derailment. The results could be used to recommend an industry standard kinetic energy Limit level that could be used to set limits on train speed for a given number of cars of a given mass. The energy level limit could be based on the rail line proximity to popuiated areas, the type of freight being haded, the trackside terrain and grade, and the past safety record of the rail Company. 6. The modification of the mode1 to include the addition of roll and pitch motions of the rail cars be undertaken. These motions have important consequences on the outcome of a deradment, as tank car roll over typically results in a spill of the contents and the pitch motion of the freight cars on the rails effects the uncoupling of the cars. References [1] Gran e, Honorable Mr. Justice S. (1981) Report of the Misszssauga Railway Acci 8;ent Inquiry. Canadian Govemment Publishing Centre, Hull, Quebec, Canada. [2] Havey, M.C., Dickie, A., AUen, D., (1980) Deruilment: The Mississauga Miracle. 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