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1

Proceedings of the ASME/ASCE/IEEE 2011 Joint Rail Conference

JRC2011

March 16-18, 2011, Pueblo, Colorado, USA

JRC2011-56028

SEMI-ANALYTICAL APPROACH TO ESTIMATE RAILROAD TANK CAR SHELL 

PUNCTURE

David Y. Jeong

Yim H. Tang

A. Benjamin Perlman

Volpe National Transportation Systems Center

Research and Innovative Technology Administration

US Department of Transportation

Cambridge, Massachusetts, USA

ABSTRACT

This paper describes the development of engineeringbased equations to estimate the puncture resistance of railroad 

tank cars under a generalized shell or side impact scenario. 

Resistance to puncture is considered in terms of puncture 

velocity, which is defined as the impact velocity at which 

puncture is expected to occur. In this context, puncture velocity 

represents a theoretical threshold limit. A given object striking 

the side of a tank car at an impact speed below the threshold 

velocity is not expected to penetrate the commodity-carrying 

tank. This definition for puncture velocity is similar to that for 

ballistic limit velocity, which is used to measure a target’s 

ability to withstand projectile impact in military applications 

[1].

The term “semi-analytical” is used to characterize the 

current approach in developing equations for shell puncture in 

order to distinguish the present work from the semi-empirical 

approach used previously to develop equations corresponding 

to head puncture. While several tests have been conducted to 

study tank car head puncture, only a limited number of tests 

have been performed to study tank car shell puncture. The 

semi-analytical approach employs a combination of three 

tactics to deal with the paucity of test data. The first tactic

applies collision dynamics to derive an idealized relationship 

between impact speed and maximum force for a generalized 

tank car shell impact scenario. Specifically, the principle of 

conservation of energy is applied. The second tactic applies 

computational methods to simulate tank car shell impacts in 

greater detail. Specifically, finite element analysis is used to 

examine the force-deformation behavior of different tank car 

configurations under different loading conditions. Regression 

analyses are performed on the results of the detailed finite 

element results to develop best-fit curves to account for the 

effects of various factors such as shell thickness, tank diameter, 

internal pressure and indenter size. The third tactic is 

empirical, in which various factors are related to puncture force 

using empirical formulas that have been developed in research 

to examine impact resistance in pipeline applications.

Results from applying the semi-analytical method to 

estimate shell puncture velocity are presented. Similarities and 

differences between the current method for shell puncture and 

the semi-empirical method for head puncture are discussed. In 

addition, results from sensitivity studies are presented to show 

the relative effect of different factors on estimated puncture 

velocity. These studies indicate that indenter size and internal 

pressure have the most significant effect on shell puncture 

velocity. Conversely, these studies indicate that tank diameter 

and ram car weight have a relatively weak effect on shell 

puncture velocity.

INTRODUCTION

Studies on railroad tank car head puncture were conducted 

by industry and government in the 1970s [2] and 1980s [3]. 

Based on these studies, equations were developed to calculate 

puncture velocity, i.e. the velocity at which puncture of the 

commodity-carrying tank is expected to occur for an assumed 

set of conditions. The term “semi-empirical” was used to 

describe the method because the functional forms of the 

equations comprising the method have a theoretical basis but 

certain constants in the equations were calibrated with test data
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2

[4]. Subsequently the methodology was modified to account 

for the presence of jackets and head shields [5].

Figure 1 shows a schematic of the generalized tank car 

head impact scenario described in the Code of Federal 

Regulations (CFR), Title 49 § 179.16, Specifications for Tank 

Cars. A moving ram car strikes the head of a subject tank car 

below the centerline of the head. The impact location on the 

tank head is specified in Appendix A to Part 179 as a certain 

height above the top of the sill. The subject tank car with its 

brakes released is braced for the impact of the ram car by three 

fully-loaded backup cars with their brakes applied.

Figure 1: Generalized Tank Car Head Impact Scenario

Puncture velocities calculated using the semi-empirical 

equations were compared to head-impact test data from several 

sources. The comparisons indicated that the estimates for 

puncture velocity tend to be conservative [6-7]. In this context, 

conservative means that puncture is expected to occur at 

velocities greater than the calculated value. In other words, the 

actual puncture velocity is most likely to be higher than that 

calculated by the semi-empirical method.

Prior to 2007, no research had been conducted to examine 

the structural integrity and crashworthiness of railroad tank cars 

under shell impacts. In 2007, full-scale tests and analyses were 

performed to examine the structural response of tank cars in the 

generalized shell impact scenario shown schematically in 

Figure 2. A moving ram strikes the side of a tank car at its 

centerline as it is braced against a rigid wall. Dow Chemical 

Company provided the tank cars for these full-scale shell 

impact tests, which were conducted under the Next-Generation 

Rail Tank Car (NGRTC) project [8]. In addition, the test effort 

was supported by finite element and collision dynamics 

analyses conducted by the Volpe National Transportation 

Systems Center (Volpe Center) [9]. Subsequent Volpe Center 

research examined the force-indentation behavior and puncture 

of tank cars under the generalized shell impact scenario for 

different shell thicknesses and indenter sizes [10]. Results from 

this previous research are used in the work described in this 

paper.

Figure 2: Generalized Tank Car Shell Impact Scenario

This paper describes the development of an engineeringbased methodology to facilitate the estimation of puncture 

velocity in generalized tank car shell impacts. The 

methodology uses a combination of three tactics to deal with 

the paucity of data on tank car shell puncture. The first tactic is 

the application of engineering principles, specifically the 

conservation of energy, to derive an idealized relationship 

between impact speed and maximum impact force for a tank 

car shell impact scenario. The second tactic is the application 

of computational methods to simulate tank car shell impacts. 

Specifically, finite element analysis is applied to simulate 

different tank car configurations (i.e., shell thicknesses and tank 

diameters) under different loading conditions (i.e. indenter 

sizes). The third tactic is empirical, in which various factors 

are related to puncture force using empirical formulas that have 

been developed in research to examine impact resistance in 

pipeline applications.

SEMI-ANALYTICAL APPROACH

The term “semi-analytical” is used to characterize the 

equations comprising the present methodology because their 

derivation is based largely on collision dynamics and finite 

element analyses that have been confirmed by a limited number 

of shell impact tests. The semi-analytical method for shell 

impacts mimics the semi-empirical approach for head impacts 

to account for jacket thickness and standoff distance. 

Moreover, the equations presented in this paper provide the 

basis for an engineering framework to facilitate the estimation 

of tank car shell puncture velocity. Such a framework

complements the semi-empirical equations previously 

developed to calculate puncture velocity for head impacts.

Using Collision Dynamics

Theoretical collision dynamics are used to derive an 

idealized relationship between impact speed and maximum 

force in a generalized shell impact scenario. Specifically the 

principle of conservation of energy is applied, in which the 

kinetic energy of the moving ram car is set equal to the energy 

required to stop the ram car. Kinetic energy is defined as

2

0

1 KE mv 2

(1)

where m is the mass of the moving ram car and v0 is the impact 

velocity. In the present development, the energy to stop the 

ram car is assumed to be equal to the strain energy due to 

deformation, or the area under the force-indentation curve:

max

0

SE F x dx ( )

(2)

where max is the maximum indentation. This last assumption

implies that the maximum force and the maximum indentation 

occur simultaneously. In addition, the functional form of the 

force-indentation curve is assumed to be a power-law relation:
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3

n

F k

(3)

where is the indentation. Moreover, this functional form of 

the force-indentation curve facilitates integration of equation

(2) and provides a closed-form expression for strain energy. 

Furthermore it is assumed that the value of k depends on the 

tank geometry (i.e. shell thickness and diameter), internal 

pressure and indenter size. Results from finite element analysis 

simulations are used to calculate specific values of k, which are 

discussed in the next section of this paper. Meanwhile, the 

value of the exponent n is assumed to be fixed. In severe 

impacts, plastic deformation of the tank typically occurs, in 

which case the value of the exponent in power-law relation for 

force-indentation is a positive number less than one (i.e. 0 < n <

1). In the development of semi-empirical equations for tank car 

head puncture [4], Hertz contact was assumed, which 

corresponds to a value of n equal to 3/2. The Hertz theory 

assumes elastic contact between impacting bodies. However, 

the force levels associated with collisions and train accidents 

are likely to create permanent plastic deformations for which 

the Hertz contact theory is not applicable. In the present 

development, n is equal to 1⁄2. The basis for this assumption is 

also discussed in the next section of this paper. 

Figure 3 shows a schematic of a force-indentation curve in 

which the shaded area under the curve represents the strain 

energy due to deformation. The shape of the force-indentation 

curve implies plastic deformation. In addition, increasing 

impact speed is implied in the force-indentation curve; i.e.

greater indentations and correspondingly higher force levels 

will occur as the impact speed is increased. When the impact 

speed reaches a critical value, the localized plastic strain in the 

tank will exceed the load-bearing strength of the material. At 

this point, failure by puncture of the commodity-carrying tank 

is expected to occur. However, the precise impact speed at 

which failure occurs is bounded by the inherent scatter 

associated with material properties. Therefore, three regions 

are labeled in the schematic to represent possible outcomes in 

terms of failure.

A closed-form expression for maximum force as a 

function of impact velocity can be derived by substituting 

equation (3) into (2) and integrating the strain energy equation:

1

2

max 0

1

2

n

n m n

F k v

k

(4)

In the current methodology, the impact speed reaches the 

critical value or puncture velocity when the maximum force 

reaches the puncture force of the tank. Empirical equations are 

used to relate the various factors to puncture force, which are 

described later in this paper.

Figure 3: Generic Force-Indentation Curve

Using Results from Finite Element Analysis

Dynamic, nonlinear finite element analysis (FEA) was 

conducted in previous work to examine the force-indentation 

behavior of tank cars in generalized shell impacts [9-10]. In 

this context, dynamic means the simulations were performed 

with explicit FEA codes. Nonlinear means that elastic-plastic 

stress-strain behavior was assumed in conjunction with large 

deformations. In addition, fluid-structure interaction was

included in the FEA analysis. In some case, the initiation and 

evolution of damage leading the catastrophic failure of 

commodity-carrying tank was included to calculate puncture 

velocity.

Three full-scale shell impact tests were conducted under 

the NGRTC project [8]. The tank cars used in these tests 

weighed 263,000 lb. In addition, they contained fluid with the 

approximate density of liquid chlorine with an outage of 10.6 

percent, and were pressurized to 100 psi.

Two of the full-scale tests were instrumented to capture 

the force-indentation behavior of the tank during impact. 

Figure 4 compares the measurements of force and indentation 

with calculations from finite element analysis of these two

tests. A relatively large indenter (17 inches in height by 23 

inches in width) was used in Test 1, in which the integrity of the 

tank was maintained. A smaller (6 inch by 6 inch) indenter was 

used in Test 2 which punctured the tank. The regression curve 

in the figure refers to the power-law relation in equation (3)

with the exponent n equal to 1⁄2. A closed-form solution to the 

rigid-plastic deformation of tubes under different loading and 

boundary conditions also corroborates the 1⁄2-power law 

exponent in force-indentation characteristic [11]. Moreover, 

the power-law regression curve provides a reasonable 

approximation of the force-indentation curve of the full-scale 

tests. For these reasons, the exponent is held constant and 

assumed to be to 1⁄2 in remainder of this paper.
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4

Figure 4: Force-Indentation Curves in Full-scale Shell Impact 

Tests

In the current methodology, the effect of different factors 

on the force-indentation behavior is embedded in the k-term in 

equation (3). That is, the value of k is depends on shell 

thickness, internal pressure, tank diameter, and indenter size. 

Moreover, regression or best-fit curves are developed from

dynamic, nonlinear finite element analysis results to account for 

these dependencies. The calculation of k is carried out by 

multiplying a series of factors that are functions of shell 

thickness, internal pressure and tank diameter. Thus, for a fixed 

indenter size:

2

0 1 2 1 2 ( , , ) ( , ) ( ) i i k h p D a h a h p D D a

(5)

In addition, the coefficients a0, a1 and a2 are assumed to depend 

on indenter size. Based on a closed-form solution for rigidplastic indentation of tubes [11], the following functional form 

is assumed for the dimensionless factor to account for internal 

pressure:

0

1

( , ) 1

( )

i

i

p

p D

p D

(6)

where piis the internal pressure and p0 depends on tank 

diameter. The following regression curve is derived from 

results of finite element analysis for shell impacts:

2

0

87.8 89.8 27.8

100 100

D D

p

(7)

Another dimensionless factor is used to account explicitly for 

the effect of tank diameter using a parabolic curve fit:

2

2

0.969 0.288

100 100

0.275

D D

(8)

Three different indenters were considered in previous 

finite element simulations of generalized tank car shell impacts 

[10]. The edges of these indenters are rounded. Referring to 

Figure 5, Table 1 lists the relevant dimensions for the footprint 

of these indenters. 

Figure 5: Schematic of Indenter Footprint

Table 1: Dimensions of Different Indenters

Height, H

(inches)

Width, W

(inches)

Edge radius, r

(inch)

1 6 6 1⁄2 

2 12 12 1

3 17 23 1

Regression curves for k as a function of shell thickness 

and indenter size were derived from the previous finite element 

results. Regression curves to account for the dependence of k

on internal pressure and tank diameter were derived from 

additional FEA results conducted for this paper. Table 2 lists 

the values of the coefficients used in equation (5) to account for 

different indenters.

Table 2: Coefficients for Different Indenters

a0 a1 a2

1 66.1 -13.0 42.8

2 69.2 -17.1 42.7

3 63.1 12.1 20.7

In the semi-empirical equations for tank car head 

puncture, the modification to account for the presence of 

jackets and head shields is summation of the individual 

thicknesses to calculate an equivalent or effective thickness of a 

single plate. The additive nature of this equivalent thickness 

calculation implies that the layering is commutative. In other 

words, changing the order of layering does not change the 

outcome in terms of puncture. Consequently, the puncture 

velocity associated with a thick jacket combined with a thin 

tank is the same for a thin jacket combined with a thick tank as 

long as the added thicknesses are equal. In the present 

methodology, an effective thickness is calculated for jacketed 

tank cars:

Indentation (inches)

0 5 10 15 20 25 30

Force (kips)

0

200

400

600

800

1000

1200

1400

1600

1800

TEST 1 MEASUREMENT (14.0 mph, LARGE PUNCH)

TEST 1 ANALYSIS (14.0 mph, LARGE PUNCH)

TEST 2 MEASUREMENT (15.1 mph, SMALL PUNCH)

TEST 2 ANALYSIS (15.1 mph, SMALL PUNCH)

Regression Curve
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5

eff S J h h h

(9)

where hS is the shell thickness and hJis the jacket thickness 

(typically 11-gauge or 0.119 inch). Moreover, the effective 

thickness is used wherever h appears in these equations.

Using Empirical Equations

The failure mechanics associated with puncture involve 

complex interactions between plastic bending, stretch, tearing, 

and mechanical properties of tank car steels. However, a 

simplifying assumption is applied in the development of both 

the semi-empirical equations for head puncture and the semianalytical equations for shell puncture. In both cases, puncture 

force is related to a shear failure criterion [4]. That is, failure 

by puncture was assumed to occur when the maximum stress 

exceeds the ultimate shear strength of the tank material. In the 

case of head puncture, the transverse shear component of stress 

was calculated for a flat circular plate subjected to a 

concentrated load offset from the center to represent a knuckle 

impact on a tank head. Moreover, this application of plate 

theory provided a means to relate puncture force to factors such 

as perimeter of the impressed area, tank head thickness and 

ultimate tensile strength of the tank head material.

The failure mechanics associated with puncture have been 

studied in other applications such as buried onshore oil and 

natural gas transmission pipelines. The integrity of pipelines 

may be threatened by puncture during digging operations using 

excavators and backhoes. Results from testing and analysis 

indicate that puncture force for line pipe is strongly dependent 

on the size of the excavator tooth (analogous to indenter size in 

tank car impacts), pipe wall thickness and material properties 

[12-15]. In addition, puncture force in this application is 

insensitive to pipe diameter and internal pressure.

Some of the research on impact strength in the pipeline 

application has been empirical. For example, the following 

empirical equation for puncture force is assumed in the present 

development for tank car shell puncture, which is similar in 

mathematical form to one that was considered in the studies of 

pipeline puncture and tank car head puncture:

1

a b

F C P h p U

(10)

where P is the perimeter of the indenter footprint, h is the shell 

thickness and U is the ultimate tensile strength of the shell 

material. The perimeter of the indenter footprint is

P H W r 2 4

(11)

In equation (10), a, b and C1 are constants, which are 

determined using results from detailed finite element analyses. 

These analyses accounted for the initiation and evolution of 

material damage, and were confirmed by the data from the fullscale shell impact tests [10]. Based on the FEA results a, b and 

C1 are equal to 0.6, 0.8 and 4.19 respectively.

Puncture velocity is calculated by setting the maximum

force in equation (4) equal to puncture force in equation (10):

0.907 1.202

0 2

1

17.6 ( , , )

U

i

v C P h

k h p D m

(12)

where C2 is equal to 9.912, and the values for exponents n, a

and b have been substituted. In addition, the factor of 17.6 has 

been included to convert the units of velocity from inches per 

second to miles per hour.

Equation (12) represents the puncture velocity for tank

cars without a jacket. For jacketed tank cars, a multiplying 

factor is applied, which accounts for the presence of the jacket 

and for the gap or standoff distance, which is usually 4 inches. 

Such a factor was developed of the semi-empirical equations 

for head puncture. In the tank car head impact tests, a small but 

measureable reduction in impact velocity was observed when a 

coupler hits a head shield and then when it hits the commoditycarrying tank because the head shield is generally placed with a 

gap distance between it and the tank. The reduction in impact 

velocity was estimated in the semi-empirical method for head 

puncture by applying the principle of conservation of energy. A 

similar gap factor is derived in the semi-analytical method for 

shell puncture for jacketed tank cars:

p G 0

v v K

(13)

where vp is the velocity to puncture the commodity-carrying 

tank, v0 is the impact velocity when the indenter strikes the 

jacket and KG is the so-called gap factor. From the 

conservation of energy,

2 2

0

1 1

2 2 mv mv F

p p

(14)

where Fp is the puncture force and is the gap or standoff 

distance between the jacket and the commodity-carrying tank. 

The gap factor is derived as

2

0

1

2

17.6

p

G

F

K

m v

(15)

The physical interpretation of this so-called gap factor is that 

the puncture velocity for a jacketed tank car is higher than that 

of a non-jacketed car. Moreover, this factor would account for 

the ability of the jacket to blunt the load of an impacting object. 

Using this factor, the puncture velocity of jacketed tank car is 

generally about 10 percent higher than that of a non-jacketed 

tank car.
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6

RESULTS

Figure 6 shows results from applying the semi-analytical 

method to estimate shell puncture velocity as a function of shell 

thickness and indenter size. These results are based on the 

following assumptions: ram car weight of 286 kips, tank 

diameter of 100 inches, internal pressure of 100 psi, and 

ultimate tensile strength of 81 ksi for TC-128B steel. In 

addition, these results include the presence of an 11-gauge (i.e. 

0.119 inch thick) jacket. 

The open circles in Figure 6 represent results from the 

FEA simulations that include material failure. The solid

symbols in the figure were calculated from applying the semianalytical method described in this paper. Each solid symbol

has error bars that represent ±1.5 mph variation from the 

calculated puncture velocity. The variation in puncture velocity 

attempts to quantify the region of possible failure shown 

schematically in Figure 3. The magnitude of this variation (i.e. 

1.5 mph) corresponds to a variation of one standard deviation 

from the average measurement of ultimate tensile strength 

conducted on samples of TC-128B during a test program to 

characterize steels obtained from tank cars retired from the fleet 

[16]. The variation in puncture velocity associated with the 

scatter in material properties is described further in the 

Discussion section.

Figure 6: Estimated Shell Puncture Velocity as a Function of Shell 

Thickness and Indenter Size

SENSITIVITY STUDIES

Sensitivity studies are performed to show the relative 

effect of different factors that enter into the semi-analytical 

equations developed to estimate tank car shell puncture. 

Baseline values are assumed for each of the different factors, 

and then varied one factor at a time while holding the other 

factors equal to their baseline values to calculate the difference 

in estimated puncture velocity. Table 3 lists the different 

factors that are considered in the sensitivity studies along with 

their corresponding baseline values.

Table 3: Factors and Assumed Baseline Values

Factor Baseline

Value

Indenter size 12” × 12”

Shell thickness 0.777 inch

Ultimate tensile strength 81 ksi

Ram car weight 286 kips

Tank diameter 100 inches

Internal pressure 100 psi

Jacket thickness 0.119 inch

Figure 7 illustrates the relative sensitivity of the factors 

considered in these studies. The figure shows that the puncture 

velocity is 19.9 mph when the baseline values for each of the 

factors are assumed. The factors are arranged to illustrate the 

reduction on puncture velocity from the baseline. In addition, 

the figure shows the range of assumed values for each factor 

and the corresponding change in estimated puncture velocity

from the baseline. For example, changing the indenter from the 

12-inch by 12-inch baseline to the 6-inch by 6-inch indenter, 

changes the puncture velocity from 19.9 mph baseline to 11.3 

mph. Decreasing the shell thickness from 0.777 inch to 1⁄2 inch 

decreases the puncture velocity from the baseline to 15.3 mph. 

Conversely, increasing the shell thickness to 0.975 inch 

increases the puncture velocity to 21.9 mph. The effect of 

material properties is represented by ultimate tensile strength 

which has a similar effect as shell thickness. That is, the 

estimated puncture velocity increases or decreases as the 

ultimate tensile strength increases or decreases. The assumed 

range of values for the ultimate tensile strength corresponds to 

±2 standard deviation from the mean based tensile test 

measurements on TC-128B [16]. The figure also shows that 

puncture velocity decreases when the 11-gauge jacket is 

removed. Tank diameter is shown to have a relatively weak 

effect on puncture velocity. The difference between a 90-inch 

diameter tank and 120-inch diameter tank is slightly more than 

0.5 mph. The ram car weight of 286,000 lb is considered as a 

worst-case assumption because real-life accidents do not 

always involve cars with the heaviest gross rail weight. 

However, the difference in puncture velocity for a ram car 

weighing 286 kips and 263 kips is shown to be relatively small. 

The effect of internal pressure is shown to be significant. The 

tank structure becomes more compliant when the internal 

pressure is reduced, which requires more energy and therefore 

greater velocity to cause puncture than a stiffer tank. 

Moreover, the length of the bars in Figure 7 symbolizes the 

relative effect of each factor on puncture velocity. Therefore, 

internal pressure and indenter size are shown to have the most 

significant effect on shell puncture velocity. Conversely, ram 

car weight and diameter have a relatively weak influence on 

shell puncture velocity, at least for the range of values 

considered.

Shell Thickness (inch)

0.4 0.5 0.6 0.7 0.8 0.9 1.0

Puncture Velocity (mph)

5

10

15

20

25

30

35

40

6" by 6"

12" by 12"

17" by 23"

FEA Result
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7

Figure 7: Relative Effect of Various Factors on Shell Puncture Velocity

DISCUSSION

Similar sensitivity studies can be conducted to examine 

the relative effect of different factors on head puncture velocity 

using the semi-empirical method. The original semi-empirical 

method was modified to account explicitly for indenter size. 

Figure 8 shows results from such studies, considering the same 

factors as in the shell puncture studies. The range of assumed 

values for the factors is also the same except for an additional 

case of a 1⁄2-inch head shield for protection. The head shield 

case is shown to provide an improvement in puncture resistance 

over the baseline case of an 11-gauge jacket. That is, using a 

1⁄2-inch head shield increases the puncture velocity by 5 mph 

compared to a bare head, and by about 2 mph compared to a 

jacketed tank head. In comparing the relative effect of the 

various factors on head and shell puncture velocity, similarities 

and differences are evident. In both cases, for example, 

indenter size is shown to have a relatively strong effect on 

puncture velocity while the effect of ram car weight is 

relatively weak. However, internal pressure is shown to have a

significant effect on shell puncture velocity, but its effect on 

head puncture velocity is not as strong. The remaining factors 

have relatively moderate effects on puncture velocity in both 

head and shell cases.

In the semi-empirical equations for head puncture and the 

semi-analytical equations for shell puncture, the force at which 

puncture is expected to occur is directly related to the ultimate 

tensile strength of the tank material. However, measurements 

of mechanical properties such as ultimate tensile strength 

exhibit inherent scatter, which can lead to uncertainty in 

predictions of puncture. For example, standard tensile tests on 

material extracted from tank cars retired from the fleet indicate 

that the standard deviation for the ultimate tensile strength of 

TC-128B steel is between 6 and 7 ksi [16]. The scatter 

associated with ±one standard deviation variance corresponds 

to ±1.5 mph in the estimation of shell puncture velocity.

The effect of strain rate has not been studied explicitly in 

the present work. However, measurements of the strength 

properties for TC-128B steel at different strain rates were 

conducted under the NGRTC project [8]. Figure 9 shows 

measurements of yield strength and ultimate tensile strength as 

a function of strain rate, and a power-law regression formula 

for each strength property. The figure also shows the scatter 

band corresponding to ±2 standard deviations in these strength 

properties from their average values at nominally low strain 

rate (i.e. the strain rate at which tensile test measurements are 

normally conducted). The regression curves indicate that yield 

strength and ultimate tensile strength tend to increase with 

increasing strain rate. But the figure also shows that the 

strength properties measured at relatively high strain rate are 

within the scatter bands of the strength properties conducted at 

nominally low strain rate.

Evaluations of the semi-empirical method to estimate tank 

car head puncture velocity revealed that the method tends to be 

conservative [6-7]. That is, the velocity at which head puncture 

is expected to occur is likely to be greater than the speed 

predicted by the semi-empirical approach. The observed trend 

of the semi-empirical method was based on comparisons with 

results of head puncture test data from several sources. To date, 

however, only one full-scale tank car shell puncture test has 

been conducted. Therefore, a similar conclusion cannot be 

drawn to assess whether the semi-analytical approach will 

consistently provide conservative (or non-conservative) 

predictions for shell puncture. On one hand, estimates of 

puncture velocity based on finite element analysis are expected

to be less conservative as those based on the semi-empirical 

method. But on the other hand, the conservation of energy 

Puncture Velocity (mph)

0 10 20 30 40 50

Internal pressure

Ram car weight 

Tank diameter

Jacket thickness

Ultimate tensile strength

Shell thickness

Indenter size

Baseline = 19.9 mph

0.500" 0.975"

67 ksi 95 ksi

286 kips 263 kips

100 psi 0 psi

120 inches 90 inches

6" by 6" 17" by 23"

None 11-gauge
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8

assumes that the entire kinetic energy is dissipated through

plastic deformation of the tank, which is a conservative 

assumption. In addition, the gap factor developed to account 

for the standoff distance between a jacket and the commoditycarrying tank is also based on energy conservation. However, 

the benefit of the jacket in terms of protection is its ability to 

blunt load since its energy absorption capability is negligible. 

Therefore, applying the gap factor for jacketed cars might tend 

to overestimate the actual puncture velocity.

Development of a performance standard for tank car 

puncture is a continuing topic of interest for both industry and 

government. Moreover, an open question is whether such a 

standard can be based on the ability to predict puncture velocity 

in lieu of actual testing. The semi-empirical method for head 

puncture and the semi-analytical methodology described in this 

paper for shell puncture may represent the initial steps in 

developing a rational basis for such a performance standard.

CONCLUDING REMARKS

This paper describes the development of a semi-analytical 

methodology to facilitate the calculation of puncture velocity 

for railroad tank cars involved in a generalized shell impact 

scenario. The approach in the development combines three 

tactics: (1) applying collision dynamics to derive an idealized, 

closed-form expression between impact speed and maximum

force, (2) employing results from detailed finite element 

analyses to characterize the effects of various factors on impact 

force, and (3) using empirical relations to relate these factors to 

puncture. Moreover, this methodology provides a framework 

to estimate shell puncture velocity in light of limited test data 

on tank car shell impacts.

The test data to validate the semi-analytical methodology 

is limited two full-scale shell impact tests, only one of which 

resulted in failure by puncture. Therefore, the application of 

the methodology to predict puncture under the generalized shell 

impact scenario should be carefully considered. Despite the 

paucity of test data, the current methodology is well suited to 

conduct comparative studies to evaluate the performance of 

different tank car configurations, in terms of puncture velocity,

in which factors such as tank thickness, tank diameter and 

internal pressure are varied. Therefore, the methodology is 

applied to conduct sensitivity studies to examine the relative 

effect of different factors on shell puncture velocity. Results 

from sensitivity studies indicate that indenter size and internal 

pressure have the most significant effect on shell puncture 

velocity. Conversely, tank diameter and ram car weight are 

shown to have a relatively weak effect on shell puncture 

velocity.

Moreover, the current methodology developed for shell 

puncture provides a complementary framework to the semiempirical method previously developed to estimate puncture 

velocity for head impacts.

Figure 8: Relative Effect of Various Factors on Head Puncture Velocity

Puncture Velocity (mph)

6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Internal pressure

Ram car weight 

Tank diameter

Ultimate tensile strength

Head thickness

Head protection

Indenter size

Baseline = 12.9 mph

0.500" 0.975"

67 ksi 95 ksi

286 kips 263 kips

100 psi 0 psi

90 inches 120 inches

None 0.5-inch head shield

6" by 6" 17" by 23"
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9

Figure 9: Effect of Strain Rate on Strength Properties of TC-128B and Scatter at Nominally Low Strain Rate

NOMENCLATURE

a exponent

b exponent

C1 constant

C2 constant

D tank diameter

F impact force

Fp puncture force

H indenter height

h thickness

heff effective thickness

hJjacket thickness

hS shell thickness

KE kinetic energy

KG dimensionless gap factor

k constant in power-law relation

m ram car mass

n exponent in power-law relation 

P perimeter of indenter footprint

piinternal pressure

r edge radius

SE strain energy

v0 impact velocity

vp puncture velocity

W indenter width

gap or standoff distance

indentation

1 dimensionless factor for internal pressure

2 dimensionless factor for tank diameter

U ultimate tensile strength

ACKNOWLEDGMENTS

The work described in this paper was sponsored by the 

Federal Railroad Administration (FRA), Office of Research and 

Development. Mr. Kevin Kesler is the Chief of the Equipment 

and Operating Practices Division. Mr. Francisco Gonzalez is 

the project manager for research on railroad tank cars and 

hazardous materials. Discussions with Mr. Karl Alexy of the 

FRA Office of Safety are greatly appreciated.

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