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Proceedings of IMECE2006 

2006 ASME International Mechanical Engineering Congress and Exposition 

November 5-10, 2006, Chicago, Illinois, USA 

 IMECE2006-13212 

ENGINEERING ANALYSES FOR RAILROAD TANK CAR HEAD PUNCTURE 

RESISTANCE 

D.Y. Jeong 

U.S. Department of Transportation 

Volpe National Transportation Systems Center 

Cambridge, Massachusetts, USA 

jeong@volpe.dot.gov

Y.H. Tang 

U.S. Department of Transportation 

Volpe National Transportation Systems Center 

Cambridge, Massachusetts, USA 

tangy@volpe.dot.gov

H. Yu 

Chenga Advanced Solutions & Engineering, LLC 

Cambridge, Massachusetts, USA 

yu@volpe.dot.gov

A.B. Perlman 

Tufts University 

Medford, Massachusetts, USA 

bperlman@tufts.edu

ABSTRACT 

This paper describes engineering analyses to estimate the 

forces, deformations, and puncture resistance of railroad tank 

cars. Different approaches to examine puncture of the tank car 

head are described. One approach is semi-empirical equations 

to estimate the velocity at which puncture is expected to occur. 

Other approaches apply elastic-plastic finite element analysis. 

The results from these approaches are compared with 

experimental data from impact tests, and are shown to provide 

reasonable estimates of impact forces. 

INTRODUCTION 

Studies on tank car head puncture were conducted by the 

Railway Progress Institute – Association of American Railroads 

(RPI-AAR)1

 Tank Car Safety Research and Test Project in the 

1970s [1]. However, recent train derailments involving the 

subsequent release of hazardous material have prompted a 

renewed focus on the structural integrity of tank cars involved 

in accidents [2,3,4]. 

This paper describes engineering analyses to estimate the 

impact forces, deformations, and puncture resistance of railroad 

tank cars. In the present context, puncture resistance is defined 

in terms of the impact velocity at which puncture of the tank is 

expected to occur. These analyses include: (1) semi-empirical 

(or semi-analytical) equations, (2) static finite element analysis 

coupled with dynamic lumped mass modeling, and (3) dynamic 

 1

 In 2003, the Railway Progress Institute (RPI) consolidated with the 

Railway Supply Association to form the Railway Supply Institute (RSI). 

finite element analysis. Some of the work presented in this 

paper was documented in previous government reports 

prepared for the Federal Railroad Administration (FRA) by the 

Volpe National Transportation Systems Center (Volpe Center) 

[5,6], but has not been presented in any previous technical 

papers. 

SEMI-EMPIRICAL APPROACH 

Semi-empirical equations to calculate puncture velocity 

were originally developed by the RPI-AAR Tank Car Safety 

Research and Test Project for bare tank heads [7], and later 

modified by the DuPont Company to account for shield 

protection and thermal insulation. 

In deriving these equations, the energy transmitted by 

wave propagation is considered small compared to the initial 

kinetic energy and the energy dissipated during deformation, 

and is therefore neglected. In such cases, local indentations or 

penetrations are strongly coupled to the overall deformation of 

the structure. Moreover, the process is considered as 

isothermal so that temperature and other thermodynamic effects 

are also neglected. 

The maximum force due to a coupler impacting the head of 

a tank is related to the indentation or dent depth by: 

3 0.6

6 3/2 15

( ) 35 10

2 15

h p Fd d

a

+ = × ⎛ ⎞⎛ ⎞ ⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠ (1) 
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 2 

where F is the maximum impact force (in units of kips), d is the 

dent depth (in inches), h is the shell thickness (in inch), a is the 

tank head radius (in inches), and p is the internal pressure (in 

psi). The exponent of 3/2 for d indicates that a Hertzian 

relationship between the contact force and the indentation was 

assumed in the formulation. Moreover, the Hertz contact 

assumption implies that the problems of elastic contact and 

elastic impact are treated identically in this formulation. The 

assumption of Hertz contact may be valid for low-velocity 

impacts, but may be questionable for impacts involving large 

plastic deformations or those resulting in puncture or other 

types of failure. 

The semi-empirical equation for indentation or dent depth 

is a linear function of impact velocity: 

1

2 0.5

5 1/16 2

( ) 8.8 10 1 0.23

40

a p W v

d v

h g

α − =× − ⎛ ⎞⎡ ⎤ ⎛⎞ ⎛⎞ ⎜⎟ ⎜⎟ ⎜ ⎟⎢ ⎥ ⎝⎠ ⎝⎠ ⎝ ⎠⎣ ⎦

 (2) 

where d is the indentation relative to its undeformed condition 

(in inches), v is the impact velocity (in miles per hour, mph), 

W1 is the weight of the impacting car (in kips), and g is the 

acceleration due to gravity (386 in/s2

). Also, α is the ratio 

between the weights of the tank car and the impacting car or 

W2/W1. 

An equation to calculate the maximum coupler force as a 

function of impact velocity can be derived by combining 

equations (1) and (2): 

( ) 1

3 / 32 3/2 F( ) 0.00383 ( ) v Wv p = α λ (3) 

In this equation, a dimensionless function of internal pressure is 

defined as 

3/2 0.5 0.6 15

( ) 1 0.23

40 15

p p λ p+ = −

⎡ ⎤ ⎛⎞ ⎛ ⎞ ⎢ ⎥ ⎜⎟ ⎜ ⎟ ⎝⎠ ⎝ ⎠ ⎢ ⎥ ⎣ ⎦

 (4) 

The numerical value of λ is always greater than or equal to 1. 

For example, λ equals 1.0 when the internal pressure is zero; λ

is equal to 1.72 when the internal pressure is 100 psi. 

Failure is assumed to occur when the maximum stress is 

greater than or is equal to the ultimate shear strength. 

Moreover, the failure criterion is based on the transverse shear 

component of stress, which is calculated for a flat circular plate 

with clamped edges subjected to a concentrated load offset 

from the center to represent a “knuckle” impact. An analytical 

solution for transverse shear is calculated from the first five 

terms of an infinite series solution [9]. Mathematically, the 

failure criterion is expressed as: 

1.81 u

F

ah

≥ τ (5) 

where τu is the ultimate shear strength of the head material (in 

ksi). In general, mechanical properties for a given material are 

reported in terms of yield strength, ultimate tensile strength, 

and percent elongation. Assuming that triaxial stresses are 

related to uniaxial properties by the von Mises equivalent 

stress, the ultimate shear strength is equal to 57.7% of the 

ultimate tensile strength. 

Figure 1 compares impact forces measured in the RPIAAR tank car head study [1] with those calculated from 

equation (3). The data are segregated by the internal pressure 

level in the tank. The straight line shown in the plot represents 

perfect correlation. Data points to the right of the straight line 

represent cases where the calculated impact forces 

overestimated the measured test results. A quantitative measure 

of the overall scatter observed in the plot is estimated by 

calculating the correlation coefficient, which is equal to 0.920. 

The correlation coefficient for a perfect correlation is equal to 

one. The correlation coefficient corresponding to the data for 

only non-pressurized tank cars is equal to 0.988, and for only 

pressurized cars the correlation coefficient is 0.773. These 

values for the correlation coefficient indicate that the semiempirical equation for impact force is more accurate for nonpressurized tanks than it is for pressurized tanks. 

0 100 200 300 400 500 600 700

Calculated Maximum Impact Force (kips)

0

100

200

300

400

500

600

700

Measured Maximum Impact Force (kips)

p = 0 psi

p = 10 psi

p = 20 psi

p = 30 psi

p = 40 psi

Fig. 1: Comparison between calculated and measured 

impact forces. 

Figure 2 compares measured indentation or dent depth 

from the RPI-AAR tests [1] with those calculated with equation 

(2). Similarly, this plot segregates the data into non-pressurized 

tank cars and pressurized tank cars. The correlation coefficient 

for all data is 0.777. If only non-pressurized tank cars are 

considered, the correlation coefficient is 0.809; if only 

pressurized tank cars are considered, it is 0.757. As in the case 
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 3 

of calculating impact forces, the semi-empirical equation for 

dent depth is relatively more accurate for non-pressurized tank 

cars than for pressurized tank cars. 

0 5 10 15 20

Calculated Indentation (inches)

0

5

10

15

20

Measured Indentation (inches)

p = 0 psi

p = 20 psi

p = 30 psi

p = 40 psi

Fig. 2: Comparison between calculated and measured 

indentations. 

An expression to calculate puncture velocity (i.e., the 

velocity at which puncture of the tank may be expected to 

occur) can be derived by substituting the equation for 

maximum coupler force into the failure criterion: 

1

2/3

1/16

27.6

( ) p

u ah

v

W p

τ

α λ = ⎡ ⎤

⎢ ⎥ ⎣ ⎦ (6) 

In this equation, vp is the puncture velocity in miles per hour 

(mph). 

Test results indicate that there is a small reduction in 

impact velocity between head shield and tank car head impacts, 

suggesting that the head shield has a negligible energy 

absorption capability. Apparently, the primary benefit of a head 

shield is to increase the overall material thickness. In 

calculating the puncture velocity, the apparent increase in 

material thickness is represented by an effective thickness 

parameter defined as 

1/1.33 1.33 1.33 1.33

eff hs j h hhh = ++ ⎡ ⎤ ⎣ ⎦ (7) 

where hh is the tank car head thickness, hs is the head shield 

thickness, and hj is the jacket thickness. The exponent of 1.33 

is an empirical constant. 

A small but measurable reduction in impact velocity has 

been observed when a coupler hits a head shield and then when 

it hits the tank car head because, in general, the head shield is 

placed with a gap distance between it and the tank head. 

Applying the principle of energy conservation, a so-called gap 

factor can be derived: 

( )2

1

1

2 (

1

17.6

) G

pb

pb

K

F v g

v W

=

∆ −

 (8) 

where ∆ is the gap distance between the head shield and the 

tank head, vpb is the puncture velocity for a bare tank car head 

(in mph) with effective thickness as defined in equation (7), or 

1

2/3

1/16

27.6

( ) pb

u eff ah

v

W p

τ

α λ = ⎡ ⎤

⎢ ⎥ ⎣ ⎦ (9) 

Then, the puncture velocity for a tank car head with shield 

protection and/or an insulating jacket can be calculated from: 

p G pb v Kv = ⋅ (10) 

where KG is the gap factor defined by equation (8) and vpb is 

defined by equation (9). 

A secondary benefit of the head shield is that it prevents 

puncture by blunting the corner edges of the coupler making 

puncture less likely. This effect has not been taken into account 

explicitly in these equations. 

The semi-empirical equations for puncture velocity were 

evaluated by comparing results from these equations with test 

data from various sources. The first source is the RPI-AAR 

Tank Car Head Study [1] which included 42 full-scale tests (34 

on bare head tank cars, 6 with a head shield, and 2 with an 

insulating jacket) and 3 tests on DOT112A340W tank cars (2 

on bare heads and 1 with a head shield). The second source is 

9 full-scale impact tests conducted on chlorine tank cars (3 with 

head protection and 6 with steel jackets) [9]. The third source 

of test data is 6 impact experiments that were conducted on 

aluminum DOT111A60ALW1 tank cars (3 with a bare head 

and 3 with a head shield) [10]. 

Table 1 summarizes the comparison between the calculated 

puncture velocity and the results from the various impact tests. 

The data from the three different sources represent a total of 65 

test cases. The semi-empirical equation for puncture velocity 

agreed with the test result in 51 out of these 65 cases. Of the 

14 cases where the predicted and actual outcomes disagreed, 

the semi-empirical equation overestimated the puncture 

velocity in 6 cases. In other words, the semi-empirical 

equation predicted that no puncture would occur in 6 cases 

where puncture occurred in the test. But in 2 of the 6 cases in 

which the semi-empirical equations overestimated the actual 

puncture velocity, the difference was less than 1 mph. 
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Table 1: Comparison of Impact Test Data with SemiEmpirical Equation for Puncture Velocity 

Source Cases Disagree Over 

RPI-AAR tests [1] 45 10 4 

FRA tests on chlorine 

tank cars [9] 14 3 1 

FRA tests on 

aluminum tank cars 

[10] 

6 1 1 

TOTALS 65 14 6 

ENGINEERING APPROACH 

Alternative methods to examine the impact forces and 

deformations of tank car heads involve elastic-plastic finite 

element (FE) analysis. 

The physics of impact may be captured with a dynamic, 

elastic-plastic FE model. But the development of a model that 

includes impact dynamics as well as nonlinear material 

behavior may be labor intensive, and the model execution may 

require high computational times. Moreover, the combination 

of finite difference in time for dynamics and nonlinear material 

behavior makes it difficult in practice to obtain good model 

performance without assuming extremely small time steps. 

In linear problems, numerical stabilities in FE analysis are 

self-evident because the solution grows without limit. In 

nonlinear problems, with elastic-plastic or other energydissipating materials, extra energy introduced into the system 

by numerical instabilities may be dissipated by plastic work 

which makes it possible for the instability to be arrested. An 

arrested instability may be difficult to detect because the 

solution, which may be in error by 10% to 100% or more, may 

appear to be reasonable [11]. 

For these reasons, an engineering method was developed 

which couples static elastic-plastic finite element analysis with 

a dynamic lumped-mass model, which is shown schematically 

in Figure 3. The spring characteristic representing the stiffness 

of the tank car head as it deforms is assumed to be a nonlinear 

function of impact load. The impact load-deformation behavior 

is calculated from static elastic-plastic finite element analysis. 

The lumped-mass model uses the non-linear load-deflection 

curve as calculated from the static FE analysis to calculate the 

impact force as a function of closing velocity. Moreover, the 

lumped-mass model provides a means to verify the results from 

a dynamic finite element model. 

m1 m2

k(x)

vo

z1 z2

x =z1-z2

Fig. 3: Dynamic lumped-mass model for tank car impacts. 

For mathematical convenience, the nonlinear spring 

characteristic in the dynamic lumped-mass model may be 

defined by a piecewise linear curve. The piecewise linear 

representation allows for an analytical solution to the equations 

of motion to the lumped-mass system [12]. The analytical 

equations for this particular application were derived and 

described in previous work [6]. Alternatively, the equations of 

motion may be solved numerically. 

The finite element analyses described in this paper were 

carried out using the commercial codes LS-DYNA [13] and 

ABAQUS [14]. Elastic-plastic (E-P) material behavior is 

modeled using the Ramberg-Osgood equation for strain as a 

function of stress: 

n

E K

σ σ

ε = + ⎛ ⎞ ⎜ ⎟ ⎝ ⎠ (11) 

where ε is the strain, σ is the stress, and E is the modulus of 

elasticity (30,000 ksi assumed for tank car steel). Also, n and K

are material constants. Table 2 lists the assumed values of 

these constants for two different tank car steels. The current 

tank car fleet has been built predominantly using TC-128B. 

The cars used in impact tests conducted in the 1970s [1] were 

made from AAR M-115. The table also lists the yield and 

ultimate tensile strengths assumed for these tank car steels. 

Table 2: Assumed Ramberg-Osgood (R-O) Material Constants 

and Mechanical Properties for Two Tank Car Steels 

 AAR M-115 TC-128B 

Hardening exponent, n 6.65 9.41 

R-O Constant, K 73.1 ksi 96.8 

Yield strength 30 ksi 50 ksi 

Tensile strength 60 ksi 81 ksi 
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Figures 4 and 5 show the finite element (FE) meshes used 

to represent the ellipsoidal tank car head. The quartersymmetric mesh is used for cases where the load is applied at 

the center of the head. The half-symmetric mesh is used for 

cases with off-center loading. Moreover, three-dimensional 

solid elements are used to construct these meshes. Two layers 

of solid elements are used to model the thickness of the tank 

car head. 

Fig 4: Quarter-symmetric mesh for typical tank car head. 

Fig 5: Half-symmetric mesh for typical tank car head. 

In order to ascertain confidence in the model development, 

results from the elastic-plastic (EP) finite element (FE) model 

are compared with a closed-form solution for the plastic 

collapse of a rigid-plastic ellipsoidal shell subjected to a 

concentrated load applied at the center [15]. Figure 6 compares 

the closed-form (analytical) solution with finite element results 

using ABAQUS for AAR M-115 tank car steel. In these 

results, the ellipsoidal tank head has an aspect ratio of 2 to 1, 

diameter of 87.5 inches, thickness of 0.5 inches, and no internal 

pressure. In the FE model, the applied load is distributed over 

a 1 square-foot area at the center of the cap. In addition, FE 

results are shown for two material models regarding stressstrain behavior: elastic-perfectly plastic (EPP) and elasticplastic behavior modeled with the Ramberg-Osgood equation. 

In the analytical solution, the flow stress was assumed to be 

equal to the yield strength of the material. The solution 

algorithm in the finite element analysis uses Riks method [16], 

which is efficient when the static load-deflection curve is 

expected to exhibit snap-through or snap-back behavior. 

0 10 20 30 40 50

Center Deflection (inches)

0

200

400

600

800

1000

Force (kips)

Analytical (Rigid-Plastic, Flow stress=30 ksi)

Finite element (Elastic-Perfectly Plastic)

Finite element (Ramberg-Osgood)

AAR M-115

Fig. 6: Analytical and FE results for an ellipsoidal cap 

loaded at the center (AAR M-115 tank car steel). 

Figure 7 shows similar results for TC-128B tank car steel. 

Given the differences in the assumed material behavior and 

applied loading, these comparisons show reasonable 

agreement. 

0 10 20 30 40 50

Center Deflection (inches)

0

200

400

600

800

1000

Force (kips)

Analytical (Rigid-Plastic, Flow stress=50 ksi)

Finite element (Elastic-Perfectly Plastic)

Finite element (Ramberg-Osgood)

TC128B

Fig. 7: Analytical and FE results for an ellipsoidal cap 

loaded at the center (TC-128B tank car steel). 

Figure 8 shows the effect of load location on the static 

elastic-plastic (i.e., Ramberg-Osgood) finite element results for 

an ellipsoidal end cap assuming AAR M-115 tank car steel. In 

the off-center case, the load is applied at a distance equal to 

one-third of the radius down from center. The figure shows a 

slight stiffening effect for the off-center loading case. 
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0 10 20 30 40 50

Maximum Deflection (inches)

0

200

400

600

800

Force (kips)

Load applied at center

Off-center loading

AAR M-115

Fig. 8: Effect of load location on static elastic-plastic analysis 

(AAR M-115 tank car steel). 

The load-deflection results from the static elastic-plastic 

finite element analysis for off-center loading are then used as 

input to the lumped-mass model to calculate impact force as a 

function of approach velocity or closing speed. Figure 9 shows 

the results from three different approaches: (1) the semiempirical method, (2) the lumped-mass model, and (3) dynamic 

elastic-plastic finite element analysis. In these results, a ram 

car that weighs 128.9 kips is assumed to strike an empty tank 

car weighing 48.5 kips. The semi-empirical method calculates 

impact forces that are lower than the lumped-mass model and 

dynamic FE analysis for relatively low closing speeds (roughly 

less than 15 mph). Conversely at relatively high closing 

speeds, the semi-empirical method calculates impact forces 

greater than the lumped-mass and the dynamic FE analyses. 

0 5 10 15 20 25

Approach Velocity (mph)

0

100

200

300

400

500

600

700

800

Impact Force (kips)

Semi-Empirical Equation

Lumped Mass Model

Dynamic Finite Element Analysis

RPI-AAR Test Data

AAR M-115

W1 = 128.9 kips

W2 = 48.5 kips

D = 87.5 inches

h = 0.5 inch

Fig. 9: Impact force as a function of approach velocity. 

CONCLUDING REMARKS 

The semi-empirical or semi-analytical approach to 

calculate impact forces, indentations, and puncture velocity 

provides reasonable agreement with experimental data from 

three different sources. The agreement between predictions 

from this approach and experimental data becomes worse when 

head shield protection or an insulating jacket is present and 

when the tank is internally pressurized. 

Alternative engineering approaches to examine puncture 

resistance were described which apply nonlinear finite element 

analysis which minimizes the need for empiricism. The results 

from this work indicate that the semi-empirical method and the 

engineering approaches provide similar results for impact 

forces. 

Research is ongoing to apply various fracture initiation 

criteria to the dynamic finite element analyses to estimate the 

force and corresponding velocity at which puncture is expected 

to occur. Work is also ongoing to examine the effect of fluidstructure interaction on the dynamic response of tank car head 

impacts. 

ACKNOWLEDGMENTS 

The work described in this paper was sponsored by the 

Federal Railroad Administration, Office of Research & 

Development. Ms. Claire Orth is the Chief of the Equipment 

and Operating Practices Division. Mr. Francisco Gonzalez is 

the project manager for research on railroad tank cars. 

Technical discussions with Dr. Oscar Orringer and Mr. 

Frank DiMasi of the Volpe Center have contributed to this 

research and are acknowledged. 

REFERENCES 

[1] Phillips, E.A., Olsen, L., 1972: “Final Phase 05 Report on 

Tank Car Head Study,” RPI-AAR Tank Car Safety 

Research and Test Project, RA-05-17. 

[2] National Transportation Safety Board, “Derailment of 

Canadian Pacific Railway Freight Train 292-16 and 

Subsequent Release of Anhydrous Ammonia Near Minot, 

North Dakota January 18, 2002,” Railroad Accident Report 

NTSB/RAR-04/01, March 2004. 

[3] National Transportation Safety Board, “Collision of 

Norfolk Southern Freight Train 192 With Standing Norfolk 

Southern Local Train P22 With Subsequent Hazardous 

Materials Release at Graniteville, South Carolina January 

6, 2005,” Railroad Accident Report NTSB/RAR-05/04, 

November 2005. 

[4] National Transportation Safety Board, “Collision of Union 

Pacific Railroad Train MHOTU-23 With BNSF Railway 

Company Train MEAP-TUL-126-D With Subsequent 

Derailment and Hazardous Materials Release, Macdona, 
This material is declared a work of the U.S. Government and is not subject to copyright protection in the United States. Approved for 

public release; distribution is unlimited. 

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Texas, June 28, 2004” Railroad Accident Report 

NTSB/RAR-06/03, July 2006. 

[5] Jeong, D.Y., Tang, Y.H., Perlman, A.B., 2001: “Evaluation 

of Semi-Empirical Analyses for Railroad Tank Car 

Puncture Velocity, Part I: Correlations with Experimental 

Data,” Final Report, DOT/FRA/ORD-01/21.I. 

[6] Jeong, D.Y., Tang, Y.H., Perlman, A.B., 2001: “Evaluation 

of Semi-Empirical Analyses for Railroad Tank Car 

Puncture Velocity, Part II: Correlations with Engineering 

Analyses,” Final Report, DOT/FRA/ORD-01/21.II. 

[7] Shang, J.C., Everett, J.E., 1972: “Impact Vulnerability of 

Tank Car Heads,” Shock and Vibration Bulletin 42, 197-

210. 

[8] Timoshenko, S., Woinowsky-Krieger, S., 1959: Theory of 

Plates and Shells, 2nd Edition, New York: McGraw-Hill 

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[9] Coltman, M., Hazel, M., 1992: “Chlorine Tank Car 

Puncture Resistance Evaluation,” Final Report, 

DOT/FRA/ORD-92/11. 

[10] Larson, W.G., 1992: “Aluminum/Cold Temperature Tank 

Car Puncture Resistance Tests: Data Report,” Final Report, 

DOT/FRA/ORD-92/29. 

[11] Cook, R.D., Malkus, D.S., Plesha, M.E., 1989: Concepts 

and Applications of Finite Element Analysis, 3rd Edition, 

New York: John Wiley& Sons. 

[12] Timoshenko, S., Young, D.H., Weaver, W., 1974: Vibration 

Problems in Engineering, 4th Edition, New York: McGrawHill Book Company. 

[13] LS-DYNA3D User’s Manual, Version 940, Livermore 

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[14] ABAQUS User’s Manual, Hibbit, Karlsson & Sorensen 

Inc., Pawtucket, Rhode Island. 

[15] de Oliveria, J.G., Weirzbicki, T., 1982: “Crushing Analysis 

of Rotationally Symmetric Plastic Shells,” Journal of 

Strain Analysis for Engineering Design 17, 229-23. 

[16] Riks, E., 1979: “An incremental approach to the solution 

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