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Final Report 

October 2007 

Development and Application of Methodology 

for Reliability Assessment of Tank Car 

Structures: Phase I 

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Transportation 

Federal Railroad 

Administration
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Development and Application of Methodology for Reliability Assessment of Tank Car 

Structures: Phase I 

6. AUTHOR(S)

Michael A. Sutton and Wei Zhao 

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University of South Carolina 

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Columbia, SC 29208 

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DOT/FRA/ORD-07/29 

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13. ABSTRACT (Maximum 200 words)

Safe packaging and transport of hazardous materials is of vital importance to tank car builders, users, Federal regulatory agencies, 

and the general public. Reliability-centered maintenance (RCM) can be used to effectively maintain the reliability of tank car

systems in guarding against lading loss due to equipment failures. Adequate and cost-efficient maintenance decisions depend on

the ability to understand the tank car’s characteristics and its responses under various service conditions, as well as the ability to 

analyze and predict the tank car’s performance and resistance deterioration as a function of usage. 

To support the informed decisionmaking and planning for the RCM through quantitative risk analysis and prediction, this project

has developed a methodology that applies modern reliability methods to perform reliability analysis on tank car structures. 

15. NUMBER OF PAGES

67 

14. SUBJECT TERMS

Reliability-centered maintenance, tank car, hazardous materials, reliability analysis, tank car 

structure 16. PRICE CODE

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i
METRIC/ENGLISH CONVERSION FACTORS 

ENGLISH TO METRIC METRIC TO ENGLISH

LENGTH (APPROXIMATE) LENGTH (APPROXIMATE) 

1 inch (in) = 2.5 centimeters (cm) 1 millimeter (mm) = 0.04 inch (in) 

1 foot (ft) = 30 centimeters (cm) 1 centimeter (cm) = 0.4 inch (in) 

1 yard (yd) = 0.9 meter (m) 1 meter (m) = 3.3 feet (ft) 

1 mile (mi) = 1.6 kilometers (km) 1 meter (m) = 1.1 yards (yd) 

1 kilometer (km) = 0.6 mile (mi)

AREA (APPROXIMATE) AREA (APPROXIMATE) 

1 square inch (sq in, in2) = 6.5 square centimeters (cm2) 1 square centimeter (cm2) = 0.16 square inch (sq in, in2) 

1 square foot (sq ft, ft2) = 0.09 square meter (m2) 1 square meter (m2) = 1.2 square yards (sq yd, yd2) 

1 square yard (sq yd, yd2) = 0.8 square meter (m2) 1 square kilometer (km2) = 0.4 square mile (sq mi, mi2) 

1 square mile (sq mi, mi2) = 2.6 square kilometers (km2) 10,000 square meters (m2) = 1 hectare (ha) = 2.5 acres 

1 acre = 0.4 hectare (he) = 4,000 square meters (m2) 

MASS - WEIGHT (APPROXIMATE) MASS - WEIGHT (APPROXIMATE) 

1 ounce (oz) = 28 grams (gm) 1 gram (gm) = 0.036 ounce (oz) 

1 pound (lb) = 0.45 kilogram (kg) 1 kilogram (kg) = 2.2 pounds (lb) 

1 short ton = 2,000 pounds (lb) = 0.9 tonne (t) 1 tonne (t) =

=

1,000 kilograms (kg) 

1.1 short tons

VOLUME (APPROXIMATE) VOLUME (APPROXIMATE) 

1 teaspoon (tsp) = 5 milliliters (ml) 1 milliliter (ml) = 0.03 fluid ounce (fl oz) 

1 tablespoon (tbsp) = 15 milliliters (ml) 1 liter (l) = 2.1 pints (pt) 

1 fluid ounce (fl oz) = 30 milliliters (ml) 1 liter (l) = 1.06 quarts (qt) 

1 cup (c) = 0.24 liter (l) 1 liter (l) = 0.26 gallon (gal) 

1 pint (pt) = 0.47 liter (l) 

 1 quart (qt) = 0.96 liter (l) 

1 gallon (gal) = 3.8 liters (l) 

1 cubic foot (cu ft, ft3) = 0.03 cubic meter (m3) 1 cubic meter (m3) = 36 cubic feet (cu ft, ft3) 

1 cubic yard (cu yd, yd3) = 0.76 cubic meter (m3) 1 cubic meter (m3) = 1.3 cubic yards (cu yd, yd3)

TEMPERATURE (EXACT) TEMPERATURE (EXACT) 

[(x-32)(5/9)] F = y C [(9/5) y + 32] C = x F

QUICK INCH - CENTIMETER LENGTH CONVERSION

0 1 2 34

Inches

Centimeters 01 3 26 1 4 5 7 1 1098 1 12

5

3

QUICK FAHRENHEIT - CELSIUS TEMPERATURE CONVERSION

 -40° -22° -4° 14° 32° 50° 68° 86° 104° 122° 140° 158° 176° 194° 212°

 

°F

 °C -40° -30° -20° -10° 0° 10° 20° 30° 40° 50° 60° 70° 80° 90° 100°

For more exact and or other conversion factors, see NIST Miscellaneous Publication 286, Units of Weights and Measures. Price 

$2.50 SD Catalog No. C13 10286 Updated 6/17/98

ii 
Table of Contents 

Illustrations ..................................................................................................................................... v 

Tables............................................................................................................................................ vii 

Acknowledgements........................................................................................................................ ix 

Executive Summary........................................................................................................................ 1 

1. Introduction .......................................................................................................................... 3 

1.1 Objective ..................................................................................................................... 3 

1.2 Scope of Work............................................................................................................. 3

1.3 Organization of the Report .......................................................................................... 4 

2. A Review of Some Important Aspects ................................................................................. 5 

2.1 Primary Mechanisms of Tank Car Structural Failures................................................ 5 

2.2 General Formulations Based on the Structural Reliability Theory ............................. 6 

2.3 RCM ............................................................................................................................ 7 

2.4 Risk Criteria ................................................................................................................ 9 

3. Reliability Analysis for Corrosion ..................................................................................... 13 

3.1 Background ............................................................................................................... 13

3.2 A Model for Tank Car General Corrosion ................................................................ 14 

3.3 Reliability Analysis Software and Solution Method................................................. 18 

3.4 Results and Discussions ............................................................................................ 18 

3.5 Summary of Corrosion Reliability Analysis ............................................................. 24 

4. Reliability Analysis for Fatigue ......................................................................................... 25 

4.1 Introduction ............................................................................................................... 25 

4.2 Key Elements Characterizing Tank Car Fatigue Crack Growth ............................... 26 

4.3 Fatigue Reliability Analysis ...................................................................................... 33 

4.4 Results and Discussion.............................................................................................. 36 

4.5 Concluding Remarks ................................................................................................. 42 

5. Conclusions and Future Work............................................................................................ 43 

5.1 Conclusions from the Corrosion and Fatigue Reliability Analysis........................... 43 

5.2 Future Work .............................................................................................................. 44

6. References .......................................................................................................................... 45 

Abbreviations and Acronyms ....................................................................................................... 49 

iii 
iv 
Illustrations 

Figure 1. Major failure mechanisms for SR-NARs .................................................................................. 5 

Figure 2. Event of SR-NARs is a union of individual failure modes ....................................................... 6 

Figure 3. The seven aspects of reliability RCM ....................................................................................... 8 

Figure 4. U.S. tank car NARs by hazard class, 1998 (AAR, 1999)........................................................ 13 

Figure 5. A schematic of tank plate cross section with corrosion damage............................................. 14 

Figure 6. PDF for initial tank thickness, T0 ............................................................................................ 16 

Figure 7. PDF for critical tank thickness, Tc........................................................................................... 17 

Figure 8. PDF for three different corrosion rates, Cr .............................................................................. 17 

Figure 9. Failure probability versus service time for μT0 = 7/16”........................................................... 18 

Figure 10. Failure probability versus service time for μT0 = 9/16”......................................................... 19 

Figure 11. Failure probability versus service time for μT0 = 11/16”....................................................... 19 

Figure 12. Failure probability versus service time for μT0 = 13/16”....................................................... 20 

Figure 13. Influence of μTc on the failure probability versus service time for nominal tank 

thickness μT0 = 11/16”.......................................................................................................... 21 

Figure 14. Corrosion failure probability as a function of service time for corrosion rate ...................... 22 

Figure 15. Corrosion failure probability as a function of service time for corrosion rate ...................... 23 

Figure 16. Corrosion failure probability as a function of service time for corrosion rate ...................... 24 

Figure 17. Half model of a non-pressure, general purpose, stub sill tank car ........................................ 27 

Figure 18. Downward VCF on an empty tank car: (a) location of high stress area and (b) hoop 

stress on tank’s bottom surface around the front sill pad area, psi....................................... 27 

Figure 19. Through-thickness distribution of the hoop stress at the FCL .............................................. 28 

Figure 20. Tank car downward VCF spectrum....................................................................................... 29 

Figure 21. Welding residual stress in transverse direction for a butt weld of tank car plates made 

of TB128-B (Sutton, et al., 2002)......................................................................................... 30 

Figure 22. Walker equation coefficient as a function of R-ratio for A516-70 ....................................... 31 

Figure 23. A 3-dof surface crack ............................................................................................................ 32 

Figure 24. A flow chart of the fatigue reliability analysis process ......................................................... 35 

Figure 25. Failure probability versus mileage for the five different values of the welding residual 

stresses.................................................................................................................................. 37 

Figure 26. Comparison of stress uncertainty factor as a random variable and as a deterministic 

parameter with three different values................................................................................... 38 

v 
Figure 27. Comparison of failure probabilities with and without correlation between initial crack 

depth and initial crack aspect ratio....................................................................................... 39 

Figure 28. Samples for a0 and (a/c)0 from Monte Carlo simulation (5000 samples), (a) Cc[a0, 

(a/c)0] = 0.............................................................................................................................. 40 

Figure 29. Samples for a0 and (a/c)0 from Monte Carlo simulation (5000 samples), (b) 

Cc[a0,(a/c)0] = 0.5 ................................................................................................................. 41 

Figure 30. Relative sensitivities of Pf to the basic random variables...................................................... 42 

vi 
Tables 

Table 1. Cases of SR-NARs reported for 1998 (AAR, 1999) .................................................................. 6 

Table 2. Suggested mishap severity categories....................................................................................... 10 

Table 3. Suggested mishap probability levels......................................................................................... 11 

Table 4. Example mishap risk assessment values................................................................................... 11 

Table 5. Example mishap risk categories and mishap risk acceptance level.......................................... 11 

Table 6. Probabilistic distributions for the basic random variables........................................................ 34 

vii 
viii 
Acknowledgements 

Mr. Jose Penã, Federal Railroad Administration (FRA), served as the FRA Technical 

Representative for the grant. His technical input on various tank car issues, helpful discussions, 

support, and encouragement were greatly appreciated. 

The authors want to thank Mr. James H. Rader, Staff Director, Hazardous Materials Program, 

FRA, for his support and informative conversations on tank car issues and for his effort in 

getting FRA tank car data and information. 

Timely technical support and help from Dr. Stephan Gollwitzer, RCP GmbH, regarding using 

STRUREL, were also greatly appreciated. 

Support and assistance from Dr. William T. Riddell, Volpe National Transportation Systems 

Center, are thankfully acknowledged. 

Financial support from FRA under Grant No. DTFRDV-00-G-60020 is gratefully acknowledged. 

ix 
x 
Executive Summary 

Safe packaging and transport of hazardous materials is of vital importance to tank car builders, 

users, Federal regulatory agencies, and the general public. Reliability-centered maintenance 

(RCM) can be used to effectively maintain the reliability of tank car systems in guarding against 

lading loss due to equipment failures. Adequate and cost-efficient maintenance decisions depend 

on the ability to understand the tank car’s characteristics and its responses under various service 

conditions, as well as the ability to analyze and predict the tank car’s performance and resistance 

deterioration as a function of usage. 

To support the informed decisionmaking and planning for the RCM through quantitative risk 

analysis and prediction, this project has developed a methodology that applies modern reliability 

methods to perform reliability analysis on tank car structures. 

Tank car failure with respect to lading loss under service conditions falls into two broad 

categories: 

• Tank car structure failures 

• Non-structural failures, such as loading/unloading devices, pressure release devices, and 

valves 

Tank car structure failures occur infrequently but likely with high consequences. It is often agerelated and due to structural deterioration, often in terms of corrosion and metal fatigue. This 

category is the focus of the project. Reliability analysis methodology has been developed for 

two common failure mechanisms: corrosion and fatigue. 

The following partially summarizes achievements as a result of this Phase I effort. 

1. A method for tank car reliability analysis against corrosion has been developed. The 

model includes basic features characterizing the tank car general corrosion process. 

Reliability analysis performed for a series of tank thickness, corrosion rates, and critical 

tank thickness predicts reasonable behaviors and trends of tank car corrosion failure. 

2. A methodology for tank car fatigue reliability analysis is being developed. It uses 

commercial reliability analysis software, STRUREL, for the reliability analysis while 

incorporating FORTRAN routines to perform fatigue crack growth analysis with the 

Walker crack growth law for a three degrees-of-freedom (dof) surface crack. 

3. The methodology is versatile in dealing with a variety of random variables (with or 

without correlation) and capable of considering various features involved in tank car 

fatigue crack growth analysis, such as spectrum loading, residual stresses, and an 

asymmetric, bi-variant stress field, thus offering a state-of-the-art fatigue reliability 

analysis tool for tank cars. 

4. It is clearly demonstrated that the reliability analysis methodology developed in this work 

can be used to quantify probability of failure for tank cars. Use of the information on the 

quantified risk allows better risk management through informed decisionmaking on 

inspection, maintenance, and repair. 

1 
A partial list of recommended future work includes the following: 

1. What to do in response to a quantified level of risk depends on what is considered an 

acceptable risk. Therefore, risk criteria or target reliability must be developed for 

effective and consistent risk management. 

2. To promote technology progress and improvement of tank car reliability, exploration 

should be made to find ways for the industry to share detailed data on lading loss 

instances with the research community, without concerns of litigation issues or negative 

impact on business as a result. This will allow specific reliability models to be developed 

to directly address various problems the tank car industry is facing. In this way, 

technology transfer can be done more effectively, which will effectively improve the 

reliability of railroad tank cars. 

3. Some failures due to non-structure related non-accident releases (NARs) can also be 

studied for possible development of reliability analysis models. Since this failure 

category, though often involving small quantities, accounts for the majority of the lading 

loss cases, it is very important to the reduction of the total failure rate. 

4. The reliability theory and solution methods used for the corrosion and fatigue reliability 

analysis are generally applicable to other systems and/or components of tank cars (e.g., 

linings, coatings, valves, pressure relieve devices, and brake systems). Therefore, 

reliability models can also be developed for such systems and components, if desired and 

if data is available. 

2 
1. Introduction 

1.1 Objective 

Safe packaging and transport of hazardous materials is of vital importance to tank car builders, 

users, Federal regulatory agencies, and the general public. RCM can be used to effectively 

maintain the reliability of tank car systems in guarding against lading loss due to equipment 

failures. Adequate and cost-efficient maintenance decisions depend on the ability to understand 

the tank car’s characteristics and its responses under various service conditions, as well as the 

ability to analyze and predict a tank car’s performance and resistance deterioration as a function 

of usage. 

To support the informed decisionmaking and planning for the RCM through quantitative risk 

analysis and prediction, the objective of this project is to develop a methodology that applies 

modern reliability methods to perform reliability analysis on tank car structures. 

1.2 Scope of Work 

It is generally recognized that a typical tank car is a complex system involving a variety of 

components, manufacturing procedures, and operational environments (lading type, car handing, 

maintenance and repair, and routes and climate). Thus, although the methodology and 

procedures developed are generally applicable to the entire tank car system, this project focuses 

on the tank’s reliability against lading loss. 

Tank car failure with respect to lading loss under service conditions falls into two broad 

categories: 

• Tank car structure failures 

• Non-structural failures, such as loading/unloading devices, pressure release devices, and 

valves 

Non-structural failures are often due to reasons such as loose fitting, venting, overloaded cargo, 

and missing parts. Some of the characteristics of this category include high occurrences, mostly 

in small quantities, and often human error-related. Thus, it has more bearing on design and 

training than maintenance. 

Tank car structure failures occur infrequently but likely with high consequences. It is often agerelated and due to structural deterioration, often in terms of corrosion and metal fatigue. This 

category is the focus of the project. Thus, this report will emphasize the primary tank structure 

and its immediate attachments (pads). It is realized that, although non-structural, linings and 

coatings are also important elements of the tank car system. The methodology and procedures 

developed will also serve as a good foundation for analyzing such elements. 

Phase I effort, as described in this report, focuses on the component reliability problem. 

Reliability analysis methodology is developed for two common failure mechanisms: corrosion 

and fatigue. Phase II of the project will analyze system reliability problems (multiple-site 

damage or multiple failure modes). 

3
1.3 Organization of the Report 

This chapter introduces the project objective, scope of work, and organization of the report. 

Chapter 2 briefly reviews a few important aspects including primary mechanisms of tank car 

structural failures, general formulation of structural reliability problems, reliability-centered 

maintenance, and risk criteria. 

Chapter 3 presents reliability methodology development and illustrative analyses for the 

corrosion failure mode of tank car tank. 

Chapter 4 describes reliability methodology development and illustrative analyses for fatigue 

failure mode of tank car tank structures. 

Chapter 5 summarizes major conclusions and recommendations for future study.

4
2. A Review of Some Important Aspects 

2.1 Primary Mechanisms of Tank Car Structural Failures 

Based on the experiences and the information known, major causes for structure-related nonaccident releases (SR-NARs) can be attributed to the following four primary failure mechanisms. 

They are fatigue, corrosion, corrosion-fatigue, and fracture, as schematically shown in Figure 1. 

Fatigue 

CorrosionFatigue 

Corrosion 

SR-NAR

Fracture 

Figure 1. Major failure mechanisms for SR-NARs

In the case of fatigue failure, release results from fatigue crack propagation and penetration 

through the tank wall. Environmental attack is negligible to crack propagation in this case. 

In the case of corrosion failure, release is due to uniform corrosion, which reduces tank wall 

thickness to such a degree that wall rupture occurs. 

In the case of corrosion-fatigue failure, release results from corrosion-assisted fatigue crack 

propagation and penetration through the wall thickness. In this case, corrosion occurs as pitting 

or similar localized damages. These damages serve as stress concentrations. Cracks initiate at 

these sites and propagate under combined action of the fatigue loading and the environmental 

attack. 

In the case of fracture failure, release is due to the tank’s separation into two pieces or a large 

through crack in the tank wall. Fracture may occur by cleavage or rupture, which may initiate 

from a manufacturing or a material defect, or from a damaged area attacked by one of the above 

mechanisms. 

5
Table 1 lists cases of SR-NARs reported for 1998 (Association of American Railroads (AAR), 

1998). 

Table 1. Cases of SR-NARs reported for 1998 (AAR, 1999) 

Source of Leak Commodity Number of Cases 

Cracked Tank Shell Isobutane 1 

Cracked Tank Shell Acetone 1 

Cracked Tank Shell Corrosive Liquid N.O.S. 1 

Broken Tank Shell N.O.S. 1 

Broken Tank Shell Sodium Hydroxide 2 

Corroded Tank Shell Molten Sulfur 1 

Corroded Tank Shell Phosphoric Acid 2 

Rusted Tank Shell Phosphoric Acid 1 

Rusted Tank Shell Methanol 1 

From Report BOE 98-1, “Report of Railroad Tank car Leaks of Hazardous Materials by 

Commodity by Source of Leak for the Year 1998 (Non-Accident Releases)” 

2.2 General Formulations Based on the Structural Reliability Theory 

Recognizing the four failure mechanisms, the event of the SR-NARs occurs if any one or more 

of the four mechanisms causes a failure. Therefore, the event of the SR-NARs is a union of one 

or more individual failure events, as shown in Figure 2. 

Fatigue

(mode 1, fa)

Corrosion

(mode 2, co)

Corrosion-Fatigue

(mode 3, cf)

Fracture

(mode 4, fr)

Structure-Related

N.A.R

Figure 2. Event of SR-NARs is a union of individual failure modes

6
Mathematically, this is: 

()( ) )2(

)1(

frcfcofafff

frcfcofaf

EEEEPEP

EEEEE

UUU

UUU

=

=

where Ef stands for an event of failure leading to SR-NAR. Similarly, Efa stands for an event of 

failure due to fatigue, Eco due to corrosion, Ecf due to corrosion-fatigue, and Efr due to fracture. 

Pf is the probability of failure. In addition to the system reliability problems resulting from the 

multiple failures modes as shown, each failure mode can also pose system reliability problems 

due to multiple-site damages. For example, a number of fatigue critical locations (FCLs) may 

exist in a tank car structure, designated as Faj, j=1, 2, ... j0. The following notations are used: 

 Efaj Event of failure at Faj 

 gfaj(Xfaj) Performance function for Faj 

 Xfaj Vector of random variables for Faj 

Thus, the probability of failure at Faj can be expressed as: 

 Pfaj(Efaj) = P[gfaj(Xfaj)≤0] 

The event of the tank car failure due to fatigue, Efa, is the union of the events Efaj at each Faj: 

Efa = ∪j=1,j0 Efaj 

where ∪j=1,j0 Efaj ≡ Efa1∪ Efa2 ∪......∪ Efaj0. Therefore, the probability of failure due to fatigue is: 

 

P(Efa) = P(∪j=1,j0 Efaj). 

2.3 RCM 

United Airlines initially developed RCM during 1970s (Nowlan and Heap, 1978). Now it has 

become an industry standard (Society of Automotive Engineers, 1999). Basically, it is a logical 

and systematic approach for effective and cost-efficient maintenance of any physical assets. As 

shown in Figure 3, RCM has the following seven aspects (Moubray, 1997): 

1. Functions and Performance Standards–Identifying functions and the associated 

performance standards of the physical asset. 

2. Functional Failures–Determining in what ways it may fail to fulfill its functions. 

3. Failure Modes–Determining what causes each functional failure. 

4. Failure Effects–Determining what happens when each failure occurs. 

5. Failure Consequences–Assessing in what way does each failure matter. 

6. Proactive Tasks–Planning and performing proactive tasks that are technically feasible and 

worth doing to prevent failure or to reduce its consequences. 

7. Default Actions–If no proactive tasks are available, taking default actions, which include 

redesign, or letting it fail, then fixing it, if the failure consequences can be tolerated. 

7
Functions and 

Performance RCM 

Functional 

Failures 

Failure 

Effects 

Failure 

Modes 

Default 

Actions 

Proactive 

Tasks 

Failure 

Consequences 

1 

2 3

4

5

7 6

Figure 3. The seven aspects of reliability RCM

So, the first five aspects are about getting to know the assets, their operational environment, and 

failure consequences. Based on this understanding, proactive tasks are planned to prevent failure 

or to reduce its consequences if the proactive tasks are technically feasible and worth doing. 

Otherwise, default actions are taken to redesign, or to let it fail then fix it, if the failure 

consequences can be tolerated. 

Although the logic is simple, when it comes to a tank car structure that is subjected to 

deterioration due to complex loading and environmental conditions involving various 

uncertainties, the task to understand the tank car’s characteristics and its responses under various 

service conditions, as well as to analyze and predict the tank car’s performance and resistance 

deterioration as a function of usage, is highly challenging. 

To support the informed decisionmaking and planning for the RCM through quantitative risk 

analysis and prediction, the objective of this project is to develop a methodology that applies 

modern reliability methods to perform reliability analysis on tank car structures. Specifically, 

the structural reliability analysis can help identify the most probable failure modes, predict 

failure effects, and estimate failure consequences. The information on the quantified potential 

risks helps plan the proactive tasks to reduce or eliminate risks. Or if no suitable proactive tasks 

are available for a particular failure mode, then it helps take default actions. 

8
2.4 Risk Criteria 

Establishment of risk criteria is necessary for effective and cost-efficient management of risks by 

tank car owners, users, and regulatory agencies. Risk criteria are used to determine acceptable 

levels of risk or target reliability, with which an assessed risk (e.g., probability of an NAR under 

a given situation) from analysis can be compared. Thus, a decision can be made as to whether 

remedial actions need to be taken to reduce the risk. Determination of target reliability, or 

acceptable risk, is a complicated task that involves social and economical backgrounds and 

cultures, as people with different social and economical backgrounds and cultures may have 

different views of acceptable risks. (Some work involving the topic can be found in the 

following: Wilson and Crouch, 1987; Okrent, 1987; Rackwitz, 2002; Ellingwood, 2000; 

Ditlevsen, 1997; Ang and Leon, 1997.) 

Two approaches exist for establishing risk criteria. One is based on the minimization of the life 

cycle cost (Menzies, 1996; Mrazik and Krizma, 1997; Elms, 1997; Wen, 2000; Sarveswaran and 

Roberts, 1999; Singh and Koenke, 2000). The life cycle costs include all costs involved, such as 

the costs of manufacturing, maintenance and repair, loss of contents, injuries, the loss associated 

with fatalities, and environmental cleanup. This approach is very appealing and requires 

extensive data and a high degree of coordination between various aspects and stages of a life 

cycle. It is an active area of current research and should be a long-term goal for the tank car 

industry. 

The other approach can be well illustrated by the military standard practice for system safety, 

documented in MIL-STD-882D, February 2000 (Department of Defense, 2000). A risk is an 

exposure to possible loss or damage. Therefore, risk is measured by a combination of likelihood 

and consequences. To quantify a risk, it is necessary to quantify the severity of the consequences 

that a hazard (mishap) may cause and the level of probability with which the hazard may occur. 

The MIL-STD-882D classifies the mishap severity, as shown in Table 2. 

9
Table 2. Suggested mishap severity categories 

Description Category Environmental, Safety, and Health Result Criteria 

Catastrophic I Could result in death, permanent total disability, loss exceeding 

$1million, or irreversible severe environmental damage that 

violates law or regulation. 

Critical II Could result in permanent partial disability, injuries, or 

occupational illness that may result in hospitalization of at least 3 

personnel, loss exceeding $200,000 but less than $1million, or 

reversible environmental damage causing a violation of law or 

regulation. 

Marginal III Could result in injury or occupational illness resulting in one or 

more lost work day(s), loss exceeding $10,000 but less than 

$200,000, or mitigatible environmental damage without violation 

of law or regulation where restoration activities can be 

accomplished. 

Negligible IV Could result in injury or illness not resulting in a lost work day, 

loss exceeding $2,000 but less than $10,000, or minimal 

environmental damage not violating law or regulation. 

Table 3 shows the classification of the mishap probability levels suggested by the MIL-STD882D. The quantities shown in these tables are guidelines and need to be adjusted with respect 

to specific systems. 

With the classification of risk severity and probability, the qualitative risk assessment can be 

made by assigning a risk value to a hazard based on its mishap severity and probability. Table 4 

shows an example of a mishap risk assessment matrix. Such a matrix can be used to rank 

different hazards according to their associated risks. 

Such risk assessment values as in Table 4 can be used to group individual hazards into mishap 

categories, which can then be used to generate specific actions. Table 5 shows an example of 

mishap risk categories and mishap risk acceptance levels. 

10
Table 3. Suggested mishap probability levels 

Description Level Specific Individual Item Fleet or Inventory 

Continuously 

experienced. 

Frequent A Likely to occur often in the life of an item, with a 

probability of occurrence (POO) greater than 10-1

in that life. 

Probable B Will occur several times in the life of an item, 

with a POO less than 10-1 but greater than 10-2 in 

that life. 

Will occur frequently. 

Occasional C Likely to occur some time in the life of an item, 

with a POO less than 10-2 but greater than 10-3 in 

that life. 

Will occur several 

times. 

Remote D Unlikely but possible to occur in the life of an 

item, with a POO less than 10-3 but greater than 

10-6 in that life. 

Unlikely, but can 

reasonably be 

expected to occur. 

Improbable E So unlikely, it can be assumed occurrence may 

not be experienced, with a POO less than 10-6 in 

that life. 

Unlikely to occur, but 

possible. 

Table 4. Example mishap risk assessment values 

Severity Probability 

Catastrophic Critical Marginal Negligible 

Frequent 1 3 7 13 

Probable 2 5 9 16 

Occasional 4 6 11 18 

Remote 8 10 14 19 

Improbable 12 15 17 20 

Table 5. Example mishap risk categories and mishap risk acceptance level 

Mishap Risk 

Assessment Value Mishap Risk Category Mishap Risk Acceptance Level 

1-5 High Component Acquisition Executive 

6-9 Serious Program Executive Officer 

10-17 Medium Program Manager 

18-20 Low As directed 

The above approach provides a good framework for establishing risk criteria for tank cars. 

11
12
3. Reliability Analysis for Corrosion 

3.1 Background 

As shown in Figure 4 (AAR, 1999), corrosive materials share a significant portion of tank car 

NAR cases. Although detailed corrosion mechanisms were not reported in the “Annual Report 

of Hazardous Materials Transported by Rail,” nearly half of the cases listed in the AAR report’s 

Table 1 are related to corrosion (classified as corroded tank shell and rusted tank shell) (AAR, 

1999). As to specific forms of corrosion, tank car specification (AAR, 1996) uses the following 

categories to distinguish them: random pits, grouped pits, adjacent to weld, general corrosion, 

bathtub ring, and blister. 

Class 8: 

Corrosive 

Materials

34%

Class 9

4%

Comb. Liq.

6%

Other

3%

Class 3: 

Flammable 

Liquids

32%

Division 2.3

2%

Division 2.2

12%

Division 2.1

7%

Figure 4. U.S. tank car NARs by hazard class, 1998 (AAR, 1999)

Extensive efforts in understanding and characterizing corrosion behaviors of various material 

and environment systems can be seen in the literature. For example, Kondo (1989) studied 

fatigue crack initiation life based on pit growth of NiCrMoV low alloy steels in hot water with 

dissolved oxygen. A residual life prediction method was proposed that considered transition 

from pit growth to fatigue crack growth. Bayoumi et al. (1995) reported the effects of pitting on 

fatigue behavior of plain carbon steels based on numerical simulation using the finite element 

13
method to determine pit stress concentration while adopting an empirical stress-life relationship 

to predict fatigue life. Enright and Frangopol (1998) predicted service life of deteriorating 

concrete bridges using time-variant reliability methods in which both loads and resistance are 

time dependent. Effects of variability in various parameters were studied using the Monte Carlo 

method. Harlow and Wei (1998) proposed a probability model for the growth of corrosion pits 

in aluminum alloys induced by constituent particles in aqueous environments. They show that 

pitting was initiated at constituent particles, which were either anodic or cathodic relative to the 

matrix and involved complex electrochemical processes. Rokhlin et al. (1999) studied the effect 

of pitting corrosion on fatigue crack initiation and fatigue life in Al 2024-T3 aluminum alloy. 

They performed experiments using a single artificial pit of different depths. An empirical 

relationship between pit depth and fatigue life was established. Zhang and Mahadevan (2001) 

assessed corrosion fatigue life using the reliability method. In-service inspection data was used 

to update prior reliability analysis. It also showed how the uncertainty in non-destructive 

inspection techniques affects the prediction. 

As well documented in the literature (Van Der Sluys, et al., 1997; ASTM, 1995; ASTM STP 

1238, 1995; Ross, 1977; ASM International, 1996), corrosion behavior is very sensitive to 

material features and environmental variables. Different corrosion behaviors occur for different 

combinations of material/environment systems and operating conditions. Lacking specific tank 

car corrosion data, no attempt is made in this work to model and analyze specific cases of tank 

car corrosion. Instead, a simplified tank car corrosion model is developed for use in reliability 

analysis. The corrosion model includes basic features for characterizing tank car corrosion 

behavior. It considers tank wall thinning (material loss) due to general corrosion. 

3.2 A Model for Tank Car General Corrosion 

Figure 5 shows a schematic of through-thickness tank plate cross section, which is used to 

develop a simple general corrosion model. 

T0 

Tc 

T

 corroded 

Figure 5. A schematic of tank plate cross section with corrosion damage

The following notations are used: 

• T0: As built or newly repaired tank wall thickness 

• Tc: Critical tank wall thickness, at or below which rupture and leakage occur 

• T: Time in service (years) 

• T: Tank wall thickness at time t 

14
• Cr: Corrosion rate, thickness loss per year (inch/year) 

• N(μ, σ): Normal distribution with mean value of μ, and standard deviation of σ 

• LN(μ, σ): Lognormal distribution with mean value of μ, and standard deviation of σ

Since leakage or rupture will occur when tank thickness reduces to a critical thickness, the failure 

criterion is, therefore, defined as: 

T ≤ Tc 

Tank thickness at time t is equal to initial thickness minus thickness loss: 

T= T0 - Crt 

Thus, a performance function for reliability against corrosion is: 

g = T - Tc= (T0 - Crt) - Tc 

In a general situation, T0, Tc, and Cr are all random variables. For example, T0 varies within 

tolerances of the rolling process that produced the steel plate; Tc will vary due to variations in 

tank radius, lading density, tank pressure, and material properties, as well as uncertainties 

involved in measurement; Cr is a primary random variable because the corrosion process is 

influenced by numerous factors related to materials and operating conditions. T0 may be 

considered as a constant because the rolling process is capable of limiting the thickness 

tolerances to very small values per American Society for Testing and Materials (ASTM) 

standard (ASTM, 2002). Nevertheless, T0 is treated as a random variable to generally include 

other factors that may contribute to the variability of the plate thickness, such as a case where T0

represents a measured thickness after a certain period of service, instead of starting from a 

pristine condition. 

The following lists the assumed random variables and distribution parameters: 

• Critical Thickness: Tc~N(μTc, 0.01”), μTc = 0.1” and 0.2” 

• Tank Initial Thickness: T0~N(μT0, 5%), μT0 = 7/16”, 9/16”, 11/16”, and 13/16 

• Corrosion Rate: Cr~LN(μCr, 0.01), μCr = 0.01, 0.02, and 0.03 [inch/year] 

The standard deviation for T0 is given as 5 percent of μT0 (coefficient of variation = 5 percent). 

The parameters used in the example cases are not based on an analysis of actual tank car data. 

Hence, the results are used to highlight the type of results that can be achieved using the 

reliability method being developed. Once tank car data is available, prediction regarding trends 

that may be expected in the tank car fleet can be analyzed using the same methodology. Figures 

6, 7, and 8 show the probability density functions (PDF) for the random variables considered. 

Figure 6 shows the PDF for the four different initial thicknesses considered. Since the same 

coefficient of variation of 5 percent is assumed for the four different thicknesses, the probability 

density functions show more scatter for thicker plates. 

15
0

2

4

6

8

10

12

14

16

18

20

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Tank thickness T0 (inch)

PDF for T 0

T0~N(13/16", 5%)

T0~N(11/16", 5%)

T0~N(9/16", 5%)

T0~N(7/16", 5%)

Figure 6. PDF for initial tank thickness, T0

Figure 7 shows the PDF for the two different critical tank thicknesses considered. 

16
0

5

10

15

20

25

30

35

40

45

0.0 0.1 0.2 0.3 0.4 0

Critical tank thickness, Tc (inch)

PDF for Tc

.5

Tc~N(0.1", 0.01")

Tc~N(0.2", 0.01")

Figure 7. PDF for critical tank thickness, Tc

Figure 8 shows the PDF for the three different corrosion rates considered. 

0

10

20

30

40

50

60

70

80

90

100

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

Cr, Corrosion Rate (inch/year)

f(Cr)

Cr~LN(0.03, 0.01)

Cr~LN(0.02, 0.01)

Cr~LN(0.01, 0.01)

Figure 8. PDF for three different corrosion rates, Cr

17
3.3 Reliability Analysis Software and Solution Method 

Commercial structural reliability analysis software, STRUREL (2000), is employed for the 

corrosion reliability analysis. First order reliability method (FORM) (Rackwitz and Fiessler, 

1978; Chen and Lind, 1983; Wu and Wirsching, 1987; Der Kiureghian and Liu, 1986) has been 

widely used due to its computational efficiency. As a useful by-product, the method also 

provides importance measures (ranking) of random variables relative to the failure probability. 

The specific FORM algorithm offered in STRUREL is based on (Rackwitz and Fiessler, 1978), 

which is now a standard approach. This FORM is used to locate the most probable point (MPP), 

or β-point, on the failure surface (g = 0). The method produces exact failure probability in cases 

where failure surface is a (hyper) plane (in standard and independent normal space). Accurate 

results can be obtained in cases where the curvature of failure surface at the MPP is small or 

moderate, as is the case for the corrosion model. 

3.4 Results and Discussions 

Figure 9 gives reliability analysis results for the above corrosion model. 

T0~N

Figure 9. Failure probability versus service time for μT0 = 7/16” 

(7/16", 5%) 

Tc=N(0.2", 0.01") 

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 2 4 6 8 10 12 14 16 18 20

Time (ye rs in service)

Pf

Cr~LN(0.01, 0.01)

Cr~LN(0.02, 0.01)

Cr~LN(0.03, 0.01)

a

18
Figure 10 shows the failure probability, Pf, versus service time for the four different values of 

initial tank thickness with μTc = 0.2”. Figures 9, 10, 11, and 12 are, respectively, for μT0 = 7/16”, 

9/16”, 11/16”, and 13/16”. 

Figure 10. Failure probability versus service time for μT0 = 9/16”

Figure 11. Failure probability versus service time for μT0 = 11/16”

T0~N(9/16", 5%) 

Tc~N(0.2", 0.01") 

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 2 4 6 8 10 12 14 16 18 20

Time (years in service)

Pf

Cr~LN(0.01, 0.01)

Cr~LN(0.02, 0.01)

Cr~LN(0.03, 0.01)

T0~N(11/16", 5%) 

Tc~N(0.2", 0.01") 

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 2 4 6 8 10 12 14 16 18 20

Time (years in service)

Pf

Cr~LN(0.01, 0.01)

Cr~LN(0.02, 0.01)

Cr~LN(0.03, 0.01)

19
T0~N

Figure 12. Failure probability versus service time for μT0 = 13/16”

The failure probability increases as service time increase. The higher the corrosion rate is, the 

higher the failure probability. The thicker the tank plate is, the lower of the failure probability. 

Although these trends are well within the expectation without analysis, the analysis does quantify 

the amount of the risk involved in each case. Furthermore, the failure probability may increase 

much more rapidly after a certain period of service time, say ta. Based on this information, 

inspection and repair can be scheduled around time ta, so that a desired level of reliability can be 

achieved with a minimal frequency of inspection and repair. 

Figure 13 shows the influence of the critical tank thickness, μTc, on the failure probability versus 

service time for initial tank thickness μT0 = 11/16”. The smaller the critical tank thickness is the 

lower the failure probability, as expected, because a smaller critical thickness means a larger 

material loss can be tolerated. 

(13/16", 5%) 

Tc~N(0.2", 0.01") 

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 2 4 6 8 10 12 14 16 18 20

Time (years in service)

P f

Cr~LN(0.01, 0.01)

Cr~LN(0.02, 0.01)

Cr~LN(0.03, 0.01)

20
T0~N(11/16", 5%)

Open Symbols: Tc~N(0.2", 0.01")

Solid Symbols: Tc~N(0.1", 0.01") 

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 2 4 6 8 10 12 14 16 18

Time (years in service)

Pf

20

Cr~LN(0.01,0.01)

Cr~LN(0.02,0.01)

Cr~LN(0.03,0.01)

Cr~LN(0.01,0.01)

Cr~LN(0.02,0.01)

Cr~LN(0.03,0.01)

Figure 13. Influence of μTc on the failure probability versus service time for nominal tank 

thickness μT0 = 11/16” 

Figures 14, 15, and 16 compare the effect of different μT0 on the failure probability versus 

service time, for the three different corrosion rates: μCr = 0.01”/year, 0.02”/year, and 0.03”/year. 

21
T0~N(11/16", 5%)

Open Symbols: Tc~N(0.2", 0.01")

Solid Symbols: Tc~N(0.1", 0.01") 

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 2 4 6 8 10 12 14 16 18 2

Time (years in service)

Pf

0

Cr~LN(0.01,0.01)

Cr~LN(0.02,0.01)

Cr~LN(0.03,0.01)

Cr~LN(0.01,0.01)

Cr~LN(0.02,0.01)

Cr~LN(0.03,0.01)

Figure 14. Corrosion failure probability as a function of service time for corrosion rate 

Cr = 0.01”/year 

22
Cr~LN(0.02, 0.01)

Tc~N(0.2", 0.01") 

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 2 4 6 8 10 12 14 16 18 2

Time (years in service)

Pf

0

T0~N(7/16", 5%)

T0~N(9/16", 5%)

T0~N(11/16", 5%)

T0~N(13/16", 5%)

Figure 15. Corrosion failure probability as a function of service time for corrosion rate 

Cr = 0.02”/year 

23
Cr~LN(0.03, 0.01)

Tc~N(0.2", 0.01") 

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 2 4 6 8 10 12 14 16 18 20

Time (years in service)

Pf

T0~N(7/16", 5%)

T0~N(9/16", 5%)

T0~N(11/16", 5%)

T0~N(13/16", 5%)

Figure 16. Corrosion failure probability as a function of service time for corrosion rate 

Cr = 0.03”/year 

Figures 14, 15, and 16 provide a convenient form for design in determining how a thicker tank 

can reduce failure probability. The information can be useful to arrive at an optimal design by 

minimizing the life cycle cost when incorporating costs of manufacturing, maintenance and 

repair, and risk of failure. 

3.5 Summary of Corrosion Reliability Analysis 

The following summarizes the points regarding the above corrosion analysis. 

A method for tank car reliability analysis against corrosion has been developed. Lacking 

specific tank car corrosion data, a simplified corrosion model for the case of general corrosion is 

developed. The model includes basic features characterizing the tank car’s general corrosion 

process. 

Reliability analysis using assumed random variables and distribution parameters has been 

performed for a series of tank thickness, corrosion rates, and critical tank thickness. The results 

show reasonable behaviors and trends for the effects of tank thickness, corrosion rates, and 

critical tank thickness. 

The model and method can be used to analyze specific tank car corrosion problems once the 

relevant data is available. 

Models for other forms of corrosion can also be developed once relevant tank car data is 

available. 

24
4. Reliability Analysis for Fatigue 

4.1 Introduction 

After a series of tank car fractures occurred under normal service conditions in the aging U.S. 

tank car fleet, fatigue cracking has emerged as a primary safety concern. To effectively deal 

with the problem, damage tolerance analysis (DTA) has been performed on the tank car stub sill 

underframe (Cardinal, et al., 1998) and on the tank structure itself (Zhao, et al., 2000) to 

determine inspection intervals. It is widely realized that the DTA process, due to complexities 

involved, is subject to uncertainty arising from incomplete knowledge and/or intrinsic variability 

of parameters, such as material properties, defect location and geometry, load spectrum, service 

history or usage projection, and inspection and manufacturing variability. 

Built on the probability and statistics concepts, reliability methods based on limit-state 

formulation provide a useful tool to quantitatively assess uncertainties involved in the DTA 

process, which, in turn assists in better risk management. Indeed, a rich literature documents 

development and applications of reliability methods to various problems involving DTA or 

fatigue crack growth analysis. For example, Sutharshana et al. (1992) presented a probabilistic 

fracture mechanics approach for predicting the fatigue life distribution using Monte Carlo 

simulations and used it to perform probabilistic fatigue crack growth analysis on a welded tube 

in a space shuttle main engine. Manning et al. (1992) proposed a reliability-centered 

maintenance analysis method based on a simple stochastic fatigue crack growth approach. To 

account for uncertainties in various parameters, they applied a single random factor (dispersion 

parameter) to a deterministic fatigue crack growth law. The method was demonstrated using a 

cutout in a fuselage bulkhead. 

Harkness et al. (1992) combined FORM with a finite element method (FEM) to perform fatigue 

crack growth reliability analysis for two-dimensional cracks obeying the Paris law (1963). 

Relative to the PDF for initial crack length, they noted that only the portion corresponding to the 

largest cracks has a significant effect on reliability. Millwater et al. (1994) integrated a fatigue 

crack growth life assessment methodology with finite element analysis and advanced 

probabilistic algorithms. The weight function method was implemented for efficient 

computation of stress intensity factors (SIF). An application example of a blade attachment 

steeple in a steam turbine also demonstrated the accuracy and efficiency of the proposed 

probabilistic life assessment method. Tryon et al. (1996) performed a detailed study of fatigue 

crack growth reliability of a disk rim in a high pressure turbine using FORM and Monte Carlo 

methods. For computational efficiency, they used a response surface approach to construct a 

fatigue performance function. The Paris law was adopted to describe fatigue crack propagation. 

Both component and system reliability problems were analyzed. Berens (1996) used a 

probabilistic fracture mechanics approach to perform risk analysis of aging military aircraft. 

Berens considered four types of problems: (a) cracking at a single critical location, (b) material 

thinning effects due to corrosion, (c) multiple element damage, and (d) expected costs of 

inspections and repairs. Cardinal and Enright (2001) performed sensitivity studies on tank car 

stub sill DTA to various parameters involved by interfacing the fatigue crack growth software, 

NASGRO (Forman, et al., 2000), with probabilistic analysis software, NESSUS (Riha et al., 

2000). Among other things, the results showed that the vertical coupler force (VCF) scale factor 

25
had the most significant influence on crack growth life, indicating the importance of accurate 

determination of the VCF spectrum and the associated tank car stresses. 

In this work, a methodology for assessing structural reliability of railroad tank cars undergoing 

fatigue crack growth is being developed. Significant statistical parameters affecting fatigue 

crack growth life predictions are treated as random variables. A surface crack with 3-dof is 

employed in fatigue analysis to account for the asymmetric stress field that may exist at a fatigue 

critical location in tank car structures. A three-dimensional weight function method (3D-WFM) 

developed based on the work of Zhao, et al. (1997a) and Zhao, et al. (1997b), was used to 

determine the SIF for the 3-dof surface crack. A fatigue life prediction program using the 

Walker equation (1970) to account for stress ratio effects was developed for the 3-dof surface 

crack model. The component reliability problem is formulated as a limit-state function and 

solved using FORM along with the importance sampling method within the commercial software 

for reliability analysis, STRUREL (2000). Without losing generality, the methodology is 

developed in reference to a fatigue critical location at a weld between the tank head and the front 

sill pad. 

4.2 Key Elements Characterizing Tank Car Fatigue Crack Growth 

To facilitate understanding of structural issues related to the reliability analysis, the following 

section presents a brief account of the structural features of a typical railroad tank car, followed 

by a description of a few important elements characterizing tank car fatigue crack growth. These 

elements include stress distributions, load spectrum, welding residual stresses, fatigue crack 

growth law, and life prediction algorithm. 

4.2.1 Structural Features of Railroad Tank Cars 

The majority of tank cars in service are stub sill tank cars that utilize a stub (short) sill at each 

end of the tank, with the tank acting as a structural component to transfer in-train loads. Figure 

17 shows a half model of a non-pressure, general purpose, stub sill tank car (cut along the 

symmetry plane). The figure does not show running gears (trucks) supporting the tank car at 

each end since they are not a subject of the current analysis. The supporting actions of the trucks 

are simulated by springs at each end. The primary fatigue loads for tank cars are those 

transferred through couplers and are classified as the VCF and longitudinal coupler force (LCF). 

As shown in Figure 17, the VCF is either in the upward direction or in the downward direction. 

The LCF is either in tension (draft) or in compression (buff). 

26
Draft 

Draft 

Downward 

VCF 

Top nozzle 

assembly 

Bottom outlet 

and skid 

Upward 

VCF 

Symmetry 

plane 

Springs 

simulating 

truck support 

Figure 17. Half model of a non-pressure, general purpose, stub sill tank car 

4.2.2 Stress Distribution at a Fatigue Critical Location 

DTA of the above tank car has identified a fatigue critical location (FCL), as shown in Figure 18 

(Zhao et al., 2000). The FCL is located outside the tank along the weld toe between the front sill 

pad and the tank head, and stress is induced by a downward VCF load. 

RSYS=333 

DMX =1.703 

SMN =-10975 

SMNB=-52492 

SMX =35540 

SMXB=64438 

-10975 

-5806 

-638.121

4530 

9699 

14867 

20035 

25204 

30372 

35540 

High hoop stress location: 

Weld toe between front sill 

pad and the tank head 

(a) (b) 

Downward 

VCF 

...

Figure 18. Downward VCF on an empty tank car: (a) location of high stress area and (b) 

hoop stress on tank’s bottom surface around the front sill pad area, psi 

27
Since the process of fatigue crack growth and fracture in tank car structures has been shown to 

initiate primarily at welds, especially those welds that connect the tank to the underframe 

structure (the stub sill region), the FCL shown in Figure 18 is a typical one. A surface crack 

located at this FCL is generally subjected to an asymmetric stress distribution, as shown in 

Figure 19 (X and Y in inch, with Y = 0 at the outer surface), as well as welding residual stresses, 

thus covering the practical complexities that may be experienced by tank cars. Therefore, 

without losing generality, the tank car fatigue reliability analysis methodology is developed in 

reference to this FCL. 

Y 

X 

tank outer 

surface 

hoop stress 

(psi) 

 

Figure 19. Through-thickness distribution of the hoop stress at the FCL 

4.2.3 Tank Car Fatigue Load Spectrum 

The load spectrum causing most fatigue damage at the FCL is the one for the downward VCF 

load. The downward VCF spectrum has two parts: one for a loaded tank car and the other for an 

empty tank car. The current practice is to assume that the ratio of unloaded (empty tank) mileage 

to the loaded mileage is RUL = 1.06. Figure 20 shows the load step information for the 

downward VCF spectrum, where the load steps for the unloaded tank car are arranged to follow 

the loaded portion. Figure 20 does not show the number of cycles at each load step. Sutton and 

Zhao (2001) have shown, among other things, that load ordering effects are negligible if the 

spectrum is repeated many (e.g., > 10) times during a life prediction where load sequence effects 

are neglected. 

28
Downward VCF, RUL=1.06

0

10

20

30

40

50

60

1 21 41 61 81 101 121 141

Load Steps

Load, kips

For loaded car For empty car 

Figure 20. Tank car downward VCF spectrum 

Mean loads corresponding to lading and gravity effects at the FCL are added to the above 

spectrum to obtain a load spectrum specific to the FCL considered. 

4.2.4 Welding Residual Stresses 

Tank cars are built-up structures of shells and beams through fusion welding. Although postweld heat treatment is performed, welding residual stresses may not be negligible in fatigue 

crack growth analysis. Therefore, welding residual stresses are considered in developing the 

fatigue reliability analysis methodology. This work uses neutron diffraction measurements of 

welding residual stresses for a butt weld of tank car plates made of TC128-B steel (Sutton, et al., 

2002). 

For the FCL considered, the responsible residual stress component is in the direction transverse 

to the weld bead, which is about the same or very close to the second principal stress direction. 

Figure 21 shows the second principal welding residual stress distributions at various thickness 

positions (Sutton et al., 2002). The figure shows that the welding residual stress distribution is 

very complicated. Though the actual residual stress distribution can be implemented in the SIF 

calculation using the enhanced 3D-WFM, this work made no such attempt for the methodology 

development. Rather, the residual stress effect on life prediction is assessed by assuming a 

relevant constant value as described later. 

29
-150

-100

-50

0

50

100

150

200

250

0 5 10 15 20 25 30 35 40 45

Transverse Distance (mm)

2nd Principal Stress (MPa)

S2_1.9mm

S2_3.7mm

S2_9.1mm

S2_12.7mm

S2_14.5mm

Weld center 

line 

Figure 21. Welding residual stress in transverse direction for a butt weld of tank car plates 

made of TB128-B (Sutton, et al., 2002) 

4.2.5 Fatigue Crack Growth Law 

Commonly used fatigue crack growth laws vary from Paris equation (Paris and Erdogan, 1963) 

to those applicable to the entire fatigue crack growth process from the threshold regime to the 

rapid fracture regime, such as the NASGRO equation (Forman, et al., 2000) and the FASTRAN 

equation (Newman, 1992). Since the available fatigue crack growth data for the tank car steel 

A516-70 (Suresh and Ritchie, 1982) is too limited to determine parameters needed for 

description of the entire fatigue crack growth process, the Walker equation (Walker, 1970) is 

adopted for this work. The Walker equation is simple, yet it incorporates the stress ratio 

(R-ratio) effect, a desirable feature for spectrum loading, as is the case for tank cars. No load 

sequence effects (retardation and acceleration) are considered since neither load sequence 

information for the tank car load spectrum nor material response data is available. The Walker 

equation can be expressed as follows: 

[ ]

)3(

)1(

)(

)( )3(

RRfor b

q R

C RC

a m KRC

dN

da

W co

W

<

−

=

Δ=

30
)3(

)1(

)( RRfor c

q R

C RC co

co

W ≥

−

=

where Rco is the cutoff value of the R-ratio above which the R-ratio effect diminishes in Paris 

regime. Based on the limited fatigue crack growth data for A516-70 (Suresh and Ritchie, 1982), 

along with data for structural and low alloy steels (Hudak, et al., 1985), the parameters in Eq. 

(3a, 3b, and 3c) are estimated as follows: C = 1.36x10-10, q = 2.09, m = 3.15, and Rco = 0.5, with 

crack growth rate, da/dN, expressed in inch/cycle. Figure 22 shows the resulting Cw, where the 

low R (R = 0.2) and high R (R = 0.5) values for C of Paris equation used for tank car stub sill 

DTA (Cardinal et al., 1998) are shown as triangles. The advantage of using the Walker equation 

is the ability to use the actual R-ratios of the tank car spectrum in fatigue life prediction, instead 

of assuming a fixed R-ratio for high R (Cardinal et al., 1998) or for a dominating R-ratio (Zhao, 

et al., 2000). 

0.0E+00

1.0E-10

2.0E-10

3.0E-10

4.0E-10

5.0E-10

6.0E-10

7.0E-10

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

R

Cw (R)

Rcutoff=0.5

SwRI_Data

Figure 22. Walker equation coefficient as a function of R-ratio for A516-70

31
4.2.6 Life Prediction Algorithm for the 3-dof Surface Crack 

Figure 23 defines geometrical parameters for the 3-dof surface crack considered in this work. 

The 3-dof surface crack is more general because it does not require symmetry of geometry and 

load with respect to the crack’s a-axis, as does a 2-dof surface crack. Therefore, it allows a 

better representation for cracks involving asymmetric load and/or geometry. 

y 

a x 

c 

e 

w

t

w/2

B A 

Figure 23. A 3-dof surface crack

The capability of the 3D-WFM (Zhao, et al., 1997a; Zhao, et al., 1997b) has been enhanced to 

analyze off-center surface and embedded cracks1

. The enhanced 3D-WFM is adapted for use in 

this work to determine SIFs for the 3-dof surface crack. 

The fatigue life prediction for the 3-dof surface crack under downward VCF spectrum is 

performed as follows. First, the SIF range for load step i of the spectrum is determined as: 

[ ]),(,,,,,, )4( 111 Δ=Δ iiiiii −−− φ yxStwecafSK

where ΔSi is the stress range at the FCL; S(x,y) is the stress distribution at the FCL due to a unit 

downward VCF load; φ is a parametric angle of the crack with φ = 0° at point A in Figure 23, 

φ = 90° at point B, and φ = 180° at point C; fi is SIF due to the unit downward VCF load. Then, 

the crack growth increments are calculated according to Eq. (5): 

 

1

 The work was done in cooperation with Dr. G. Graham Chell and Mr. Yi-Der Lee, Southwest Research Institute 

(SwRI), while Dr. Zhao worked as a subcontractor to SwRI through funding of Pratt & Whitney (P&W). SwRI’s 

contribution to the enhanced 3D-WFM and P&W’s kind permission for the authors to use the enhanced 3D-WFM 

are gratefully acknowledged. 

32
[ ]

[ ]

[ ]

)5(

180 )5(

)0()( )5(

)90()( )5(

0

0

0

Δe Δc Δc d

ΔN c m Δc ΔK()C(R )

N b m KRCc

N a m KRCa

i i i

i i i i

i i i i

i i i i

A B

B

A

−=

=

Δ=Δ Δ

Δ=Δ Δ

where Ri is the stress ratio for load step i and ΔNi is the number of cycles at load step i. Subincrementation of ΔNi will be performed if a large crack growth increment may occur during ΔNi

load cycles. Lastly, the crack size and eccentricity are updated as follows: 

)6(

)6(

)6(

)6(

1

1

1

1

Δeee d

Δccc c

Δccc b

Δaaa a

ii i

ii i

ii i

ii i

B B B

A A A

+=

+=

+=

+=

−

−

−

−

The procedure starts from initial crack size and continues until either crack penetrates tank or 

final fracture occurs. 

4.3 Fatigue Reliability Analysis 

4.3.1 Performance Function 

A performance function for fatigue reliability can be defined as: 

)7( 0 −= NNg DP

where NP is the predicted fatigue life, ND is the demanded fatigue life during which a tank car is 

expected to perform without leakage or fracture due to fatigue crack growth (i.e., failure in this 

case is defined as leakage or fracture due to fatigue crack growth). Therefore, two failure criteria 

exist: 

)( )8(

)8(

max K forK fracture b

leakageforta a

c L

L

≥

≥

φ

For practicality, the leakage criterion is implemented as a ≥0.985t. Failure occurs when 

predicted fatigue life fails to meet the demanded life (i.e., (NP - ND)<0). The probability of 

failure, Pf, is thus expressed: 

]0)[()0( )9( f 0 =<= − NNPgPP DP <

33
4.3.2 Basic Random Variables and Deterministic Parameters 

This study considered six basic random variables. They are: 

1. Initial crack depth, a0 

2. Initial crack aspect ratio, (a/c)0 

3. Coefficient of fatigue crack growth law, C (Equ.(3)) 

4. Stress uncertainty factor, Suf, which represents the combined influence of uncertainty in 

amplitudes for the load spectrum and stress analysis 

5. Cutoff value of R-ratio effect, Rco 

6. Initial crack eccentricity, e0

The statistic nature of the first four basic random variables is well recognized, and they are 

routinely considered in probabilistic analysis of fatigue crack growth, (e.g., Sutharshana, et al., 

1992; Harkness, et al., 1992; Millwater, et al., 1994; Tyron, et al., 1996; Cardinal and Enright, 

2001). 

The Rco is assumed to be a random variable based on the observation that the R-ratio effect in the 

near-threshold to Paris regimes is primarily attributed to various crack closure mechanisms, 

which, in turn, may be sensitive to material microstructure, mechanical properties, and 

environment, as well as the statistical nature of measuring the crack opening load. 

The e0 is assumed to be a random variable since a crack initiation site is dependent on features, 

such as welding details and material microstructures, which apparently are statistic in nature, in 

addition to being in a high-stress area as determined by structural finite element analysis. 

No sufficient data is available to establish probabilistic distribution types and parameters for the 

tank car material considered. Based on experiences and engineering judgment, Table 6 lists the 

distribution types and parameters assumed for this methodology development work. 

Table 6. Probabilistic distributions for the basic random variables 

Name Symbol Probabilistic Density Function Parameters 

Initial crack depth a0 U(0.0125", 0.2") Distribution interval 

Initial crack aspect ratio (a/c)0 U(0.1, 1) Distribution interval 

Crack growth coefficient C LN(1.36e-10, 0.272e-10) Mean, standard deviation 

Stress uncertainty factor Suf N(1, 0.08) Mean, standard deviation 

R-ratio cutoff value Rco N(0.5, 0.03) Mean, standard deviation 

Initial crack eccentricity e0 U(0", 3") Distribution interval 

34
In Table 6, U, LN, and N stand for uniform (rectangular), lognormal, and normal distributions, 

respectively. The dimensions for C correspond to crack growth rate expressed in inch/cycle. 

In addition, the effects of welding residual stresses, Sr, are considered through deterministic 

parameter studies. 

The fracture toughness, Kc, is another legitimate random variable but intentionally treated as a 

deterministic parameter of Kc = 250 ksi√in (roughly a mean value for data from a range of steels 

and conditions). This Kc value essentially suppresses the fracture failure (leak before break), 

thus avoids dealing with system reliability problems (multiple failure modes in this case), which 

will be covered in the next phase of the project. 

4.3.3 Reliability Analysis Software and Solution Methods 

Reliability analysis software. Commercial structural reliability analysis software, STRUREL 

(2000), is employed for the fatigue reliability analysis. To do so, FORTRAN subroutines and a 

static library have been developed to interface with STRUREL. Figure 24 shows a flow chart for 

the fatigue reliability analysis process. The fatigue life calculation (in the dashed rectangle) is 

carried out from initial crack size to failure (leakage) or to a pre-determined run-out life, for each 

realization (sample) of the set of random variables. 

STRUREL reliability analysis 

engine with various solution 

methods and pre- and postprocessing capabilities 

Performance function definition 

(interface routine with 

STRUREL) 

Fatigue life calculation of 3-dof 

surface crack under spectrum 

loading 

Determine stress intensity factor 

of 3-dof surface crack using 3DWFM 

STRUREL input file for random 

variables, correlations, 

deterministic parameters, and 

analysis options 

Inner loop performed for each 

set of samples from initial crack 

to failure

Figure 24. A flow chart of the fatigue reliability analysis process

Solution methods. FORM (Rackwitz and Fiessler, 1978; Chen and Lind, 1983; Wu and 

Wirsching, 1987; Der Kiureghian and Liu, 1986) has been widely used due to its computational 

efficiency. As a useful byproduct, the method also provides importance measures (ranking) of 

random variables relative to the failure probability. The specific FORM algorithm offered in 

STRUREL is based on Rackwitz and Fiessler (1978), which is now a standard approach. This 

35
FORM is used to locate the most probable point (MPP), or β-point, on the failure surface (g0=0). 

The method produces exact failure probability in cases where failure surface is a (hyper-) plane 

(in standard and independent normal space). Accurate results can be obtained in cases where the 

curvature of failure surface at the MPP is small or moderate. Preliminary tank car reliability 

analyses, however, have clearly demonstrated that the failure surface is highly nonlinear. To 

reduce the nonlinearity, the following equivalent performance function (logarithmic 

transformation of Eq.(7) is used: 

)()( )10( P −= NLnNLng D

Furthermore, after locating the β-point using FORM, importance sampling (Hohenbichler and 

Rackwitz, 1988) is performed based on the β-point information to improve the accuracy of the 

FORM solution. This approach allows arbitrarily accurate solutions to be obtained as more and 

more importance samples are used. This work used 75~150 importance samples in most of the 

cases, which generally gives a failure probability with a coefficient of variation less than 15 

percent, which is considered reasonable for illustrating the methodology. 

4.4 Results and Discussion 

To illustrate the fatigue reliability analysis methodology, probabilistic analyses are performed 

using the random variables given in Table 6, along with parametric studies for the deterministic 

parameter, Sr. Five different values are considered for residual stresses, Sr. They are Sr = 0 (no 

residual stresses), 3.24, 6.47, 10.9, and 15.33 ksi (22.3, 44.5, 75, and 105.5 MPa, respectively). 

The residual stress effect is considered by applying a mean load of Qr = 3.8, 7.6, 12.8, and 18 kip 

in the load spectrum. This mean load is determined such that it corresponds to a tensile residual 

stress given above. 

Figure 25 shows the probability of failure, Pf, as a function of mileage traveled for the four cases 

considered. Instead of using the number of load cycles, the usage is expressed as a number of 

passes through the tank car spectrum that represents 10,000 miles of service. Figure 25 clearly 

shows the following: 

• In all the cases, Pf increases as mileage increases. 

• Welding residual stress can significantly increase Pf.

• The residual stress effect on Pf tends to saturate above Sr = 10.9 ksi (75 MPa), which 

corresponds to a load level of Qr = 12.8 kip in the downward VCF spectrum. 

The saturation behavior is related to the R-ratio cutoff effect of the material. The saturation 

behavior observed from the analysis has a practical implication that post-weld heat-treatment for 

reducing welding residual stresses can be effective in increasing fatigue crack growth resistance 

only if it reduces the residual stresses to well below the saturation level. The residual stress 

saturation level depends on load spectrum and stationary (non-alternating) stress distribution, as 

well as materials’ R-ratio cutoff values. 

36
0.00E+00

2.00E-03

4.00E-03

6.00E-03

8.00E-03

1.00E-02

1.20E-02

25 30 35 40 45 50 55 60 65 70 75 80 85

Number of passes through 10,000 mile spectrum

Qr=0

Qs=3.8

Qr=7.6

Qr=12.8

Qr=18

f P

Figure 25. Failure probability versus mileage for the five different values of the welding 

residual stresses

The information provided in Figure 25 should be useful in managing risk. For example, 

inspections and repairs can be scheduled at a certain mileage interval to satisfy a given target 

reliability (reliability = 1-Pf). 

In DTA (Cardinal, et al., 1998; Zhao, et al., 2000; Zhao and Sutton, 2001), all parameters are 

deterministic. To be conservative, a common practice is to use upper bound values for loads and 

lower bound values for material properties. Figure 26 compares the effect of assuming Suf as (a) 

a random variable of N(μ, σ) with mean μ = 1 and standard deviation σ = 0.08 and (b) a 

deterministic parameter with three different values equal to μ, μ+2σ, and μ-2σ. In this case, it 

can be seen that assuming an upper bound value (μ+2σ) for Suf is overly conservative, while 

using mean value of Suf will be unconservative. In other words, adopting a reliability approach 

may reduce the conservativeness while still maintaining adequate reliability, thus reducing the 

life cycle cost. 

Figure 26 also shows the results obtained using Monte Carlo simulation (5000~9000 samples) 

for Suf = 1.16. Good agreement observed between Monte Carlo and FORM, plus importance 

sampling, provides additional verification that the FORM, plus importance sampling, produced 

correct results in the case of the highly nonlinear performance function analyzed. 

37
0.00E+00

5.00E-03

1.00E-02

1.50E-02

2.00E-02

2.50E-02

3.00E-02

25 30 35 40 45 50 55 60 65 70 75 80 85

Number of passes through 10,000 mile spectrum

Pf

Suf=0.84

Suf=1

Suf=1.16

Suf~N(1, 0.08)

Suf=1.16, MC_5~9k

Figure 26. Comparison of stress uncertainty factor as a random variable and as a 

deterministic parameter with three different values 

In the above analyses, all the random variables are assumed to be independent. In reality, the 

initial crack depth, a0, and the initial crack aspect ratio, (a/c)0, are probably correlated, such that 

a crack with a larger surface length will be shallower (a larger c corresponds to a smaller a). For 

example, a crack initiating from weld porosity may have a smaller surface length and a larger 

aspect ratio, while a crack initiating from a weld undercut may have a larger surface length but a 

smaller aspect ratio. Thus, a0 and (a/c)0 may be positively correlated. As an illustrative 

example, the study considered two correlated cases between a0 and (a/c)0, with a correlation 

coefficient of 0.25 and 0.5, respectively (i.e., Cc[a0,(a/c)0] = 0.25 and 0.5). Figure 27 compares 

results with and without correlations. All three cases have residual stresses. The solid symbols 

are for the case without correlation. The open ones are for the cases with correlation. It is clear 

that correlation between a0 and (a/c)0 has a significant effect on the failure probability, even in a 

weakly correlated case (Cc = 0.25, triangles). 

38
0.00E+00

2.00E-03

4.00E-03

6.00E-03

8.00E-03

1.00E-02

1.20E-02

25 30 35 40 45 50 55 60 65 70 75 80 85

Number of passes through 10,000 mile spectrum

Qs=12.8, Cc=0

Qs=12.8, Cc=0.5

Qs=12.8, Cc=0.25

f P

Figure 27. Comparison of failure probabilities with and without correlation between initial 

crack depth and initial crack aspect ratio

The positive correlation reduces Pf significantly because the chances to have simultaneously long 

and deep surface cracks are reduced. Figures 28 and 29 show this where samples for a0 and 

(a/c)0 from Monte Carlo simulation (5000 samples) are compared between correlated and 

uncorrelated cases. The correlation effect boils down to the fact that SIF is a function of both 

crack depth and crack aspect ratio. 

39
cc00_116_128_5000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

a0

(a/c)0

Figure 28. Samples for a0 and (a/c)0 from Monte Carlo simulation (5000 samples), (a) 

Cc[a0, (a/c)0] = 0 

40
cc50_116_128_80_5000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.05 0.1 0.15 0.2

a0

(a/c)0 Figure 29. Samples for a0 and (a/c)0 from Monte Carlo simulation (5000 samples), (b) 

Cc[a0,(a/c)0] = 0.5 

Though considered as deterministic parameters in the above analysis, Sr and other parameters 

can be easily treated as random variables. Different probability density distributions, or 

distribution parameters, can also be considered for the variables listed in Table 6, so that the 

effects of distribution types and parameters on the failure probability can be studied. In this 

regard, the sensitivity measures obtained as a byproduct of FORM can provide guidance in 

determining the relative importance of random variables on failure probability. Figure 30 shows 

such an example, which is for Qr = 7.6 kip. In the order of decreasing importance (sensitivity) of 

the six random variables to the failure probability, they are (a/c)0, a0, Suf, C, Rco, and e0. In fact, 

based on the information that the sensitivity of Pf to e0 is nearly zero, e0 may well be treated as a 

constant, instead of as a random variable. The information is of practical interest in design 

because the effect of changing design variables on Pf can be evaluated. 

41
-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

25 30 35 40 45 50 55 60 65 70 75 80 85

Number of passes through the 10,000 mile spectrum

Relative sensitivity

a0

(a/c)0

e0

C

RCO

Suf

Figure 30. Relative sensitivities of Pf to the basic random variables 

Finally, the Walker equation is implemented for a 3-dof surface crack. Other crack growth laws 

and crack configurations can be added in parallel to the existing models. For example, by 

utilizing the weight function methods (Zhao, et al., 1997a; Zhao, et al., 1997b; Wu and Carlsson, 

1991), various 2D and 3D cracks can be efficiently analyzed. 

4.5 Concluding Remarks 

A methodology for tank car fatigue reliability analysis is being developed that uses a commercial 

reliability analysis software, STRUREL, for the reliability analysis, while incorporating 

FORTRAN routines to perform fatigue crack growth analysis with the Walker crack growth law 

for a 3-dof surface crack. 

Feasibility of using the commercial reliability software for tank car fatigue reliability analysis is 

demonstrated. The methodology is versatile in dealing with a variety of random variables (with 

or without correlation) and capable of considering various features involved in tank car fatigue 

crack growth analysis, such as spectrum loading, residual stresses, and asymmetric, bi-variant 

stress field. 

Results from the illustrative examples show that Pf increases as mileage increases, welding 

residual stresses can significantly increase Pf, uncertainty in the stress amplitude has a significant 

effect on Pf, and the correlation between initial crack depth and initial crack aspect ratio is an 

important parameter. 

42
5. Conclusions and Future Work 

5.1 Conclusions from the Corrosion and Fatigue Reliability Analysis 

A method for tank car reliability analysis against corrosion has been developed. Lacking specific 

tank car corrosion data, a simplified corrosion model for the case of general corrosion is 

developed. The model includes basic features characterizing tank car general corrosion process. 

Reliability analysis with assumed random variables and distribution parameters has been 

performed for a series of tank thickness, corrosion rates, and critical tank thickness, using the 

commercial reliability analysis software, STRUREL. The results show reasonable behaviors and 

trends for the effects of tank thickness, corrosion rates, and critical tank thickness. 

The model and the method can be used to analyze specific tank car corrosion problems once the 

relevant data is available. Models for other forms of corrosion can also be developed once 

relevant tank car data is available. 

A methodology for tank car fatigue reliability analysis is being developed that uses the 

commercial reliability analysis software, STRUREL, for the reliability analysis, while 

incorporating FORTRAN routines to perform fatigue crack growth analysis with the Walker 

crack growth law for a 3-dof surface crack. 

The FORTRAN routines developed for the tank car fatigue crack growth analysis can be 

extended to include various other crack configurations and crack growth laws as more tank car 

material data becomes available. 

Feasibility of using the commercial reliability software for tank car fatigue reliability analysis is 

demonstrated. The methodology is versatile in dealing with a variety of random variables (with 

or without correlation) and capable of considering various features involved in tank car fatigue 

crack growth analysis, such as spectrum loading, residual stresses, and asymmetric, bi-variant 

stress field. 

Results from the illustrative examples of fatigue reliability analysis show that probability of 

failure, Pf, increases as mileage increases; welding residual stresses can significantly increase Pf;

uncertainty in the stress amplitude also has a significant effect on Pf; and the correlation between 

initial crack depth and initial crack aspect ratio is an important parameter. 

The observed saturation behavior of the welding residual stress effect on fatigue crack growth 

life suggests that post-weld heat-treatment for reducing welding residual stresses can be effective 

in increasing fatigue crack growth life, but only if it reduces the residual stresses to well below 

the saturation level. 

It is clearly demonstrated that the reliability analysis methodology developed in this work can be 

used to quantify probability of failure for tank cars. Use of the information on the quantified risk 

allows better risk management through informed decisionmaking on inspection, maintenance, 

and repair. 

43
5.2 Future Work 

Tank car corrosion failure mechanisms are manifold. It is of practical interest to develop 

reliability models for other forms of corrosion failure, such as pitting. 

The possibility may exist that at certain stages and locations, fatigue crack growth will be 

accelerated by wall thinning effect due to tank car general corrosion. The current fatigue 

reliability model can be extended to incorporate the general corrosion effect. 

System reliability models must be developed to consider multiple failure modes and multiple site 

damage. In fact, the two failure criteria, fracture and leakage, encountered in the fatigue 

reliability analysis, pose a system reliability problem in general. 

What to do in response to a quantified level of risk depends on what is considered an acceptable 

risk. Therefore, risk criteria, or target reliability, must be developed for effective and consistent 

risk management. 

To promote technology progress and improvement of tank car reliability, exploration should be 

made to find ways for industry to share detailed data on lading loss instances with the research 

community, without concerns of litigation issues, or negative impact on business as a result. 

This will allow specific reliability models to be developed to directly address various problems 

that the tank car industry is facing. In this way, technology transfer can be done more 

effectively, which, in turn, will effectively improve the reliability of railroad tank cars. 

Reliability analysis method accounting for inspection and repair (both have their own 

uncertainties) needs to be developed, so that reliability models can be updated based on 

inspection and repair information, once available. 

Some failures due to non-structure related NARs can also be studied for possible development of 

reliability analysis models. Since this failure category, though often involving small quantities, 

accounts for the majority of the lading loss cases, it is very important to the reduction of the total 

failure rate. 

The reliability theory and solution methods used for the corrosion and fatigue reliability analysis 

are generally applicable to other systems and/or components of tank cars (e.g., linings, coatings, 

valves, pressure relieve devices, brake system). Therefore, reliability models can also be 

developed for such systems and components, if desired and data is available.

44
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Abbreviations and Acronyms 

3D-WFM three-dimensional weight function method 

dof degree-of-freedom 

DTA damage tolerance analysis 

FCL fatigue critical location 

FEM finite element method 

FORM first order reliability method 

MPP most probable point 

PDF probability density function 

POO probability of occurrence 

NAR non-accident release 

RCM reliability-centered maintenance 

SIF stress intensity factor 

SR-NAR structure-related non-accident releases 

VCF vertical coupler force 

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