U.S. Department

of Transportation

Federal Railroad

Administration

Deformation Behavior of Welded Steel Sandwich 

Panels under Quasi-Static Loading

Office of Railroad Policy

and Development

Washington, DC 20590

DOT/FRA/ORD-11/06 Final Report 

 March 2011 
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Technical Report

4. TITLE AND SUBTITLE

Deformation Behavior of Welded Steel Sandwich Panels Under Quasi-Static Loading

5. FUNDING NUMBERS

RR-28/HG-034

RR-28A1/HG-269

RR-93/HG-284

6. AUTHOR(S)

D. Y. Jeong, M. E. Carolan, A. B. Perlman, and Y. H. Tang

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)

U.S. Department of Transportation

Volpe National Transportation Systems Center

Research and Innovative Technology Administration

55 Broadway

Cambridge, MA 02142-1093

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ORGANIZATION REPORT 

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DOT-VNTSC-FRA-10-08

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Federal Railroad Administration

Office of Railroad Policy and Development

1200 New Jersey Avenue, SE

Washington, DC 20590

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AGENCY REPORT NUMBER

DOT/FRA/ORD-11/06

11. SUPPLEMENTARY NOTES

Program Manager: Francisco González III

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This document is available to the public through the FRA Web site at 

http://www.fra.dot.gov. 

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13. ABSTRACT (Maximum 200 words)

This report describes engineering studies that were conducted to examine the deformation behavior of flat, welded steel

sandwich panels under two quasi-static loading conditions: (1) uniaxial compression; and (2) bending with an indenter. 

Testing and analysis are conducted to study the force-displacement response of sandwich structures with different core 

geometries: (1) pipe or tubular cores with outer diameters equal to 2, 3, and 5 in; (2) a 2-inch square diamond core; and (3) a 

double corrugated core called an X-core with a 5-inch core height. Deformation and local collapse modes of sandwich 

panels under these loading conditions are also studied. Moreover, the work described in this report represents basic research

to investigate the concept of applying welded steel sandwich structures as a means to offer protection to railroad tank cars—

especially those carrying hazardous materials—in the event of an accident.

14. SUBJECT TERMS

Bending, compression, core geometry, face sheet, sandwich structure

15. NUMBER OF 

PAGES

71

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ABSTRACT

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ii

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 3 4 5

Inches

Centimeters 0 1 2 3 4 5 6 7 8 9 10 11 12 13

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
iii

Preface

The Federal Railroad Administration (FRA), Office of Railroad Policy and Development,

sponsored the work described in this report. Specifically, this work was conducted and managed 

by the Structures and Dynamics Division of the John A. Volpe National Transportation Systems 

Center (Volpe Center) through Project Plan Agreements and Inter-Agency Agreements with 

FRA. Moreover, this work is carried out through two Volpe Center projects: Tank Car 

Structural Integrity in FRA’s Rail Equipment Safety Research Program and Improved Tank Car 

Design Development in FRA’s Railroad Systems Safety Research Program. Mr. Kevin Kesler is 

the Chief of the Equipment and Operating Practices Research Division. Mr. Francisco 

González III is the project manager for research on railroad tank cars and hazardous materials.
iv

Contents

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

1 Introduction ........................................................................................................................ 4

2 Sandwich Panel Characteristics........................................................................................ 7

3 Behavior under Uniaxial Compression .......................................................................... 15

3.1 Quasi-Static Compression Tests..................................................................................... 18

3.2 Comparisons with Analysis............................................................................................ 21

4 Behavior under Bending with an Indenter .................................................................... 25

4.1 Local Collapse Modes.................................................................................................... 27

4.2 Quasi-Static Bend Tests................................................................................................. 31

5 Summary of Test and Analysis Results.......................................................................... 35

6 Discussion.......................................................................................................................... 37

7 References ......................................................................................................................... 41

Appendix A. Force-Crush Curves from Compression Tests................................................. 44

Appendix B. Estimation of Core Strength by Limit Analysis................................................ 47

Appendix C. Force-Displacement Curves from Bend Tests and Analyses........................... 58

Abbreviations and Symbols ....................................................................................................... 63
v

Illustrations

Figure 1. Percent of Damage by Location on Insulated Tank Cars in Mainline Accidents .......... 5

Figure 2. Strength Properties of Sandwich Panel Materials ........................................................ 12

Figure 3. Ductility of Sandwich Panel Materials......................................................................... 13

Figure 4. Schematic Force-Crush Response of Sandwich Panel with Idealized Elastic-Plastic 

Core Material to Uniaxial Compression....................................................................... 16

Figure 5. Generic Force-Crush Response of Pipe Core Sandwich Panel .................................... 16

Figure 6. Generic Force-Crush Response of Diamond Core Sandwich Panel ............................ 17

Figure 7. Generic Force-Crush Response of X-Core Sandwich Panel ........................................ 18

Figure 8. Variation of Compressive Strength with Relative Density .......................................... 20

Figure 9. Variation of Theoretical and Measured Strength of Pipe Cores with 

Relative Density ........................................................................................................... 22

Figure 10. Variation of Theoretical and Measured Strength of Diamond Core with 

Relative Density ........................................................................................................ 23

Figure 11. Measured and Calculated Force-Crush Curves for 5-Inch Pipe Core ........................ 23

Figure 12. Predicted and Observed Deformation of 5-Inch Pipe Core........................................ 24

Figure 13. Closeup View of Final Deformation Mode in 5-Inch Pipe Core................................ 24

Figure 14. Schematic of Bend Test Configuration ...................................................................... 25

Figure 15. Schematic Force-Displacement Curve in Bend Test.................................................. 26

Figure 16. Possible Local Collapse Modes in Bending ............................................................... 27

Figure 17. Collapse Mechanism Map for 3-Inch Pipe Core Panel in Bending............................ 28

Figure 18. Collapse Mechanism Map for Diamond Core Panel in Bending ............................... 29

Figure 19. Photograph of Face Sheet Fracture from Quasi-Static Bend Test.............................. 29

Figure 20. Bend Test with Bubble Wrap to Quantify Load Spreading ....................................... 30

Figure 21. Collapse Loads in Bend Tests .................................................................................... 33

Figure 22. Comparison of Compressive Strength for Different Cores........................................ 39
vi

Tables

Table 1. Recent Railroad Hazmat Accidents................................................................................. 4

Table 2. Details of Pipe Core Panels ............................................................................................. 8

Table 3. Details of Diamond Core Panels...................................................................................... 9

Table 4. Details of X-Core Panels ............................................................................................... 10

Table 5. Tensile Properties of Sandwich Panel Materials ........................................................... 11

Table 6. Tensile Property Specifications for TC-128B Steel ...................................................... 11

Table 7. Relative Densities for Different Cores........................................................................... 14

Table 8. Quasi-Static Compression Test Matrix.......................................................................... 19

Table 9. Quasi-Static Bend Test Matrix ...................................................................................... 32
1

Executive Summary

This report describes basic research (i.e., testing and analysis) conducted to examine the 

deformation behavior of flat-welded steel sandwich panels under two types of quasi-static 

loading: (1) uniaxial compression and (2) bending through an indenter. The objectives of these 

tests were to (1) confirm the analytical and computational (i.e., finite element) modeling of 

sandwich structures, (2) examine the fabrication issues associated with such structures (e.g., 

material selection and welding processes), and (3) observe the deformation behavior and local 

collapse mechanisms under the two different types of loading. In addition, the uniaxial 

compression tests were performed to rank or screen different core geometries. Five core 

geometries were examined in the compression tests: pipe or tubular cores with outer diameters 

equal to 2, 3, and 5 inches; a 2-inch square diamond core; and a double-corrugated core called an 

X-core with a 5-inch core height.

Previous research conducted by the Federal Railroad Administration’s (FRA) Office of Research 

and Development and the John A. Volpe National Transportation Systems Center (Volpe Center)

included applying semi-empirical and computational (i.e., finite element analysis) methods to 

estimate the puncture resistance of conventional railroad tank cars under generalized head and 

shell impact scenarios. Subsequent work identified sandwich structures as a potential technology 

to improve the puncture resistance of the commodity-carrying tank under impact loading 

conditions.

Test articles for the compression and bend tests were manufactured by Cellular Materials 

International, Inc. (CMI), located in Charlottesville, VA. CMI was under contract with FRA 

through the Small Business Innovation Research Program. Test articles include the flat welded 

steel sandwich panels, plates to support the panels in the text fixture, and the indenter for the 

bend tests. Test panels were fabricated with materials that were readily available in the desired 

quantities at the time of construction.

Quasi-static tests were carried out using a 5-million-pound capacity universal testing machine by 

the Center for Advanced Technology for Large Structural Systems at Lehigh University’s Fritz 

Engineering Laboratory in Bethlehem, PA. Uniaxial compression tests were conducted on 

14 sandwich panels on May 7, 2009. Panel dimensions were approximately 24 inches in length 

and 6 inches in width. Deformation and crushing of the different cores were measured and 

observed as uniform pressure was applied onto the face sheets over an area of about 1 square 

foot (1 ft2).

The compression tests showed excellent repeatability of structural (i.e., force-crush) response for 

panels with similar cores and welding. Compressive strength was estimated using plastic limit 

load analysis and shown to be in excellent agreement for the sandwich panels with pipe cores. 

Similar estimates for the diamond cores overpredict the measured strength. The 3-inch pipe core 

and the diamond core were selected as candidate cores for the next series of tests, because they 

possess attributes of moderate strength and moderate relative density. In addition, force-crush 

curves calculated from finite element analysis were in reasonable agreement with the measured 

curves for all cores.
2

Bend tests using a 12-inch by 12-inch indenter with 1-inch-radius rounded edges were conducted 

on 16 sandwich panels on December 9 and 10, 2009. The dimensions of the bend panels were 

approximately 48 in long by 18 in wide. The panels were simply supported over 4-inch-diameter 

rollers spanning 24 in between the centers of the rollers. The bend tests included three variables: 

(1) core type (diamond core and 3-inch pipe core); (2) core orientation relative to the supports 

(cores running either parallel or perpendicular to the rollers used to support the panels); and (3) 

face sheet type (solid plates on both sides, strips used as face sheets on both sides, and a 

combination of solid plates and strips). Force-displacement (i.e., head travel) curves were 

recorded during each test. Force levels were read off these curves, which were used to 

characterize the local collapse load of the sandwich panels in bending and to quantify the relative 

effects of the test variables. On the basis of this metric, the diamond core panels were shown to 

have slightly higher bending strength than the pipe core panels when solid face sheets were used. 

The difference in bending strength is almost negligible when strip face sheets were used. 

Bending strength of the panels was demonstrated to be strongly anisotropic. Panels with cores 

perpendicular to the supports had much higher bending strength than those with cores parallel to 

the supports. Panels with solid face sheets on both sides were shown to have higher bending 

strength than those with strips as face sheets.

Finite element analysis of the bend tests produced nearly identical shapes to the measured forcedisplacement curves. The computational modeling, however, overpredicts the force in the forcedisplacement curves compared with the test data in each case. The discrepancy is attributed to 

modeling perfect geometry, whereas imperfections likely exist in the actual test specimens. 

Sandwich structures are now being used in applications such as impact-attenuation devices in the 

event of run-off-the-highway accidents and as protection for ship hulls against impulsive loading 

from blasts and explosions. The potential merits of using sandwich structures investigation for 

railroad applications have been identified in this research project. However, fundamental 

research is needed to evaluate the performance characteristics of sandwich structures under 

controlled loading conditions that are representative of the intended application. 

The crushing and the bending characteristics of sandwich structures with the different cores 

tested under quasi-static loading can be incorporated into detailed finite element analyses that 

may eventually include fluid-structure interaction to account for the lading as well as material 

failure to estimate tank puncture. Moreover, computational models can be used to extrapolate 

performance of sandwich structures with features not considered in the present work. 

Preliminary results from simulations of tank shells protected by sandwich panels were discussed

at a meeting about the Next Generation Tank Car project. The next logical step in continuing the 

basic research to investigate the proof-of-concept is to modify the loading condition from quasistatic to dynamic impact. Introducing dynamic loading will allow for examination of inertial 

effects on the deformation behavior of sandwich structures as protective panels. 

Sandwich structure design aspects such as weight and space considerations are beyond the scope 

of the work described in this report. However, such limitations have been discussed in previous 

work. The present work focuses on the basic mechanisms of deformation and failure associated 

with idealized loading conditions. As the potential for sandwich structures to adequately protect 
3

railroad tank cars becomes realized, industry interest in this technology may take hold. At this 

point, a cost-benefit analysis can be conducted; additional studies will be warranted to further 

examine the practical design aspects of this novel technology in railroad transportation 

applications.

For the past two decades, the Office of Research and Development (FRA) has sponsored 

research performed by the Volpe Center in safety matters related to the transportation of 

hazardous materials by railroad tank cars. This research is used to support government and 

industry efforts in resolving problems related to metal fatigue, fracture, and welding in the 

current fleet; structural behavior under derailment and collision scenarios; and improving 

standards and procedures for future railcar designs. Moreover, the Volpe Center has performed 

and managed research on the structural integrity of railroad tank cars under various loading 

conditions, ranging from normal operations to rare and extreme circumstances such as accident 

loading.
4

1. Introduction

Accidents involving the release of hazmat from rail cars during transport can result in fatalities, 

injuries, and evacuations of nearby towns, as well as property and sometimes environmental 

damage. Table 1 lists some of the recent railroad hazmat accidents investigated by the National 

Transportation Safety Board (NTSB). Since the train derailment that occurred in Minot, ND, on 

January 18, 2002, the safety performance of railroad tank cars during accidents has come under 

great scrutiny. For example, NTSB made a series of safety recommendations to FRA after the

NTSB’s investigation of the Minot derailment [1]. Subsequently, the government and industry 

initiated research activities to develop strategies to maintain structural integrity and improve the 

crashworthiness of railroad tank cars carrying hazmat during accidents. 

Table 1. Recent Railroad Hazmat Accidents

Location Date Type of Accident

Minot, ND January 18, 2002 Derailment

Macdona, TX June 28, 2004 Collision

Graniteville, SC January 6, 2005 Collision

Anding, MS July 10, 2005 Collision

Texarkana, AR October 15, 2005 Collision

New Brighton, PA October 20, 2006 Derailment

Shephardsville, KY January 16, 2007 Derailment

Oneida, NY March 12, 2007 Derailment

Rockford, IL June 19, 2009 Derailment

In some of the accidents listed in Table 1, the release of hazmat occurred from an impacting 

object (e.g., broken coupler) penetrating and puncturing the end (also called the head) or the side 

(also called the shell) of the commodity-carrying tank. The table also shows that these accidents 

were classified as either a derailment or a collision. Although accidents are rare events, caution 

must be exercised in making statistical inferences from small sample-size data. However, 

comprehensive studies of available accident data, which include FRA’s Railroad 

Accident/Incident Reporting System (RAIRS) [2], indicate that derailments and collisions 

account for more than 90 percent of all hazmat-related accidents [3]. In addition, accident data 

on damage to tank cars is maintained by the Railroad Tank Car Safety Research and Test Project, 

which is cosponsored by the Railway Supply Institute (RSI) and the Association of American 

Railroads (AAR). A recent report [4] provides information on the relative frequency of damage 

to various regions of the head and the shell, and the associated losses of lading from these 

regions, for various tank car configurations (i.e., cars with and without jackets). Figure 1 shows 

a graphical depiction of the aggregate findings for insulated tank cars involved in mainline 

accidents. The figure indicates that damage can occur anywhere on the commodity tank, 

although it is more likely to occur below the horizontal centerline and near the ends of the car.
5

Figure 1. Percent of Damage by Location on Insulated Tank Cars in 

Mainline Accidents [4] 

In 2004, the Volpe Center began conducting research to address the NTSB recommendations to 

FRA following the Minot, ND, derailment. This research included studies on derailment 

dynamics and the structural behavior of railroad tank cars during accidents [5]. Subsequent 

Volpe Center research identified sandwich structures technology as a potential technology to 

improve the crashworthiness of tank cars [6–8].

In 2006, the railroad industry began a research and development effort called the Next 

Generation Rail Tank Car (NGRTC) project to develop and implement tank car designs to 

improve safety performance. Dow Chemical Company, Union Pacific Railroad, and Union Tank 

Car were the industry sponsors of this project. In 2007, FRA and Transport Canada, two 

government regulatory agencies in North America, began collaborating with the NGRTC project 

through a Memoranda of Cooperation. Cooperative activities included full-scale shell impact 

testing of conventional tank cars designed to carry liquid chlorine. Separate activities included 

design development for improved crashworthiness. Government and industry collaboration in 

the NGRTC project dissolved in 2008. 

One of the concepts considered during the government and industry research to improve 

crashworthiness of railroad tank cars proposed treating the commodity-carrying tank car as a 

protected entity. Moreover, welded steel sandwich structures were examined as a means to 

protect the commodity-carrying tank against penetrations and punctures from impacting objects 

in the event of a derailment or collision. 

This report describes basic research, which entails testing and analysis conducted to examine the 

deformation behavior of flat-welded steel sandwich panels under quasi-static loading conditions. 

Specifically, two types of loads were applied: uniaxial compression and bending with an 

indenter. The objectives of these tests were to (1) confirm the analytical and computational (i.e., 

finite element) modeling of sandwich structures; (2) examine the fabrication issues associated 

with such structures (e.g., material selection and welding processes); and (3) observe the 

deformation behavior and local collapse mechanisms under the two different types of loading. In 
6

addition, uniaxial compression tests were performed to rank or screen different core geometries. 

The core geometries used in the compression tests were as follows: (1) pipe or tubular cores 

with outer diameters equal to 2, 3, and 5 in; (2) a 2-inch square diamond core; and (3) a doublecorrugated core called an X-core with a 5-inch core height. 

Test articles were manufactured by CMI, which is located in Charlottesville, VA, under contract 

with FRA through the Small Business Innovation Research Program. The compression and bend 

tests were conducted by the Center for Advanced Technology for Large Structural Systems 

(ATLSS) at Lehigh University’s Fritz Engineering Laboratory in Bethlehem, PA. The analyses 

described in this report and the test designs were performed by the Volpe Center. 

The organization of this report is as follows. Specific details of the sandwich panels used in the 

present work are described in Section 2. General descriptions and results on the behavior of 

sandwich panels subjected to uniaxial compression are given in Section 3. Force-crush 

measurements from individual quasi-static compression tests are provided in Appendix A. A 

detailed description of closed-form methods to interpret the compression test results is given in 

Appendix B. Generalizations on the behavior of sandwich panels subjected to bending with an 

indenter are given in Section 4. Force-displacement test data and comparisons with finite 

element results for the quasi-static bend tests are provided in Appendix C. Results from both 

quasi-static tests and the corresponding analyses are summarized in Section 5. Finally, 

suggestions for applying the results of the quasi-static tests and analyses to develop protective 

panels for railroad tank cars are discussed in Section 6.
7

2. Sandwich Panel Characteristics 

Sandwich structures are generally composed of two face sheets that are separated by a core. In 

the traditional design of sandwich structures, the separation of the face sheets by the core 

increases the moment of inertia of the panel, which produces a higher bending stiffness-toweight ratio than solid or monolithic plates. The face sheets carry almost all of the bending and 

in-plane loads while the core carries the shear load to prevent the face sheets from sliding past 

one another. 

The work described in this report represents basic research that was used to investigate the 

concept of applying welded steel sandwich panels as a means to offer protection to the 

commodity-carrying tank against puncture in the event of a collision. In this concept, sandwich 

panels provide protection through two mechanisms: load blunting and energy absorption. 

Blunting means that the impact load is distributed over a larger area of the tank with respect to 

the indenter footprint, which effectively increases the energy needed to puncture the commoditycarrying tank. Protective panels also absorb collision energy, which decreases the effective 

impact speed felt by the commodity-carrying tank during impact. Moreover, the load-blunting 

and energy-absorbing capabilities of sandwich panels depend on the core shape or geometry. 

The particular core geometries examined in this report are the following: (1) circular pipe or 

tubular cores with outer diameters equal to 2, 3, and 5 in; (2) a 2-inch square diamond core; and 

(3) a double corrugated core or X-core with a height of 5 in. 

Table 2 shows a schematic of the unit cell for a pipe core. The table also lists the nominal 

dimensions of the pipe cores and the materials used to construct the cores and the face sheets. 

Similarly, Table 3 lists the specific details for the diamond core and Table 4 provides 

characteristics for the X-core. All of the sandwich panels in this study have face sheets with 

equal thickness. 

In the present study, the forces produced from impact are decomposed into separate and 

idealized loads associated with crushing and bending of sandwich panels. By simplifying the 

application of loads, local collapse mechanisms can be observed and discriminated, which may 

help understand the failure mechanics of sandwich panels. The application of loads in a quasistatic manner allows for careful observation of the failure mechanisms while gathering data 

without the difficulties of dynamic (i.e., inertial) factors affecting instrumentation. Moreover, 

this basic research is intended to lay the foundation for improved puncture resistance for the 

commodity-carrying tank. 

The core geometries described in this report have been examined previously for other 

applications. For example, impact-attenuation devices are used to decelerate an errant vehicle in 

the event of run-off-the-road highway accidents. Also known as crash cushions, the devices 

consist of clusters of mild steel cylinders or tubes that dissipate the impact energy [9, 10]. Pipe 

or tubular cores have also been considered in automobile applications as car bumpers and 

guardrails [11]. Research sponsored by the Office of Naval Research has been conducted on 
8

sandwich panels with diamond cores and X-cores to examine their application in protecting ship 

hulls against impulsive loads created by blasts and explosions [12–14].

Table 2. Details of Pipe Core Panels

Core Characteristics

Material AISI 1010 steel

Pipe Outer Diameter, D 2, 3, and 5 in

Thickness, tc 0.125 in

Face Sheet Characteristics

Material Domex (100XF) plate

Thickness, tf 0.25 in for compression panels

0.118 in for bend panels
9

Table 3. Details of Diamond Core Panels

Core Characteristics

Material AISI 1010 steel

Height, hc 2.62 in

Thickness, tc 0.125 in

Leg Length, L 2 in

Corner Radius, r 0.25 in

Face Sheet Characteristics

Material Domex (100XF) plate

Thickness, tf 0.25 in for compression panels

0.118 in for bend panels
10

Table 4. Details of X-Core Panels

Core Characteristics

Material AISI 1010 steel

Height, hc 5 in

Thickness, tc 0.125 in

Leg Length, L 2.53 in

Weld Flat Length, l 0.56 in

Corner Radius, r 0.125 in

Width, W 5.4 in

Face Sheet Characteristics

Material Domex (100XF) plate

Thickness, tf 0.25 in for compression panels

In the traditional design of sandwich panels, the properties of interest in the face sheets are high 

stiffness to provide high flexural rigidity and high tensile and compressive strength. The 

properties of interest in the core are generally weaker than those in the face sheets. For each 

sandwich panel in the present study, the face sheet material is Domex (100XF) plate and the core 

material is American Iron and Steel Institute (AISI) 1010 steel. Domex is a high-strength 

structural steel that is generally used for applications such as agricultural machinery, crane 

booms, and railcar components. AISI 1010 steel is a low-strength steel that is typically used for 

automobile components (e.g., auto bodies and fenders), washers, nails, and rivets.

Although the characteristic material properties of Domex and AISI are consistent with those used 

in traditional sandwich design for the face sheets and the core, selection of these materials to 

construct the sandwich panels was not based on design considerations. Rather, these materials 

were chosen because they were readily available in the desired quantities and thicknesses at the 

time of construction and scheduling of the tests.

Mechanical properties of the face sheet and core materials were measured from tensile tests 

conducted on coupons from the same batch of material from which the test panels were 

constructed. These tensile tests were conducted by Thielsch Engineering, Inc., in Cranston, RI 

(under contract to CMI), in accordance with standards developed by the American Society for 

Testing and Materials (ASTM) [15]. The mechanical properties measured in these tests were 

ultimate tensile strength, 0.2 percent offset yield strength, and elongation over a 1-inch gage 

length. Moreover, five replicate coupon tests were conducted on each of the five different 
11

materials used to construct the sandwich panels in this study. Table 5 lists the results from the 

tensile tests on the different sandwich panel materials. The values listed in the table represent 

the average of five tests; the numbers in parentheses are the calculated standard deviation. In 

subsequent analyses, the compressive and tensile properties of these materials were assumed 

equal. 

Table 5. Tensile Properties of Sandwich Panel Materials 

Material Ultimate Tensile 

Strength

(ksi)

Yield Strength

(ksi)

Elongation

(%)

Domex plate 113.2 (0.8) 101.2 (1.1) 17.8 (0.3)

AISI 1010 steel 2-in square 71.4 (1.0) 60.5 (2.0) 21.9 (0.4)

AISI 1010 steel 2-in pipe 62.6 (0.4) 52.3 (1.3) 32.2 (0.5)

AISI 1010 steel 3-in pipe 53.6 (0.4) 43.0 (0.6) 35.5 (0.9)

AISI 1010 steel 5-in pipe 51.8 (0.3) 37.1 (1.2) 38.7 (0.9)

Notes: 

Values are averages from five tests. 

Numbers in parentheses are standard deviations. 

Elongation measured over 1-inch gage length. 

Pressurized tank cars manufactured since 1989 were built with normalized TC-128B steel.1 For 

the sake of comparison, Table 6 lists the minimum requirements for strength and ductility of TC128B steel. 

Table 6. Tensile Property Specifications for TC-128B Steel 

Property Allowable*

Ultimate Tensile Strength 81–101 ksi

Yield Strength Greater than 50 ksi

Elongation Greater than 22%

* AAR Manual of Standards and Recommended Practices and Specifications for 

Tank Cars, M-1002, Appendix M. 

Figure 2 shows the variation of the average yield strength and ultimate tensile strength for the 

different sandwich panel materials. The error bars in the figure represent the scatter associated 

with ±2 standard deviations from the mean. The dashed lines represent the nominal properties of 

52.9 kilopounds per square inch (ksi) for the ultimate tensile strength and 44.2 ksi for the yield 

 

1

 Prior to 1989, nonnormalized steel was predominantly used. Normalization is a heat treating process that produces 

a finer and more abundant pearlitic microstructure, which is intended to make the steel stronger and harder 

compared to non-normalized steel. 
12

strength of AISI 1010 from www.matweb.com. Apparently, the differences between the nominal 

and actual strength properties can be attributed to cold-forming the shapes of the cores. 

Domex plate 1010 steel 2" square 1010 steel 2" pipe 1010 steel 3" pipe 1010 steel 5" pipe Strength (ksi)

20

40

60

80

100

120

Ultimate Tensile Strength

Yield Strength

Error bars represent ±2 standard deviations

Mean and standard deviation based on results from 5 tensile tests

52.9 ksi

44.2 ksi

Figure 2. Strength Properties of Sandwich Panel Materials 

Similarly, Figure 3 shows the variation and scatter for percent elongation over 1 in for the 

sandwich panel materials. The figure also shows the nominal value of 20 percent elongation for 

AISI 1010 from www.matweb.com, which is based on a gage length of about 2 in (50 mm). 

Elongation is a measure of ductility, which quantifies the ability of the material to deform 

plastically without fracturing. These figures suggest that for these five materials, strength and 

ductility are inversely related. Therefore, the material with the highest strength properties (i.e., 

Domex plate) has the lowest ductility; the material with highest ductility (i.e., AISI 1010 steel 

5-inch pipe) has the lowest strength properties. 
13

Domex plate 1010 steel 2" square 1010 steel 2" pipe 1010 steel 3" pipe 1010 steel 5" pipe Elongation Over 1 inch (%)

15

20

25

30

35

40

45

Error bars represent ±2 standard deviations

Mean and standard deviation based on results from 5 tensile tests

20%

Figure 3. Ductility of Sandwich Panel Materials 

The structural characteristics of sandwich structures are often described in terms of relative 

density. That is, stiffness and strength properties of sandwich panels are commonly reported in 

relation to core relative density. Core relative density is defined as the ratio of the average 

density of the core to the density of the material, or equivalently, the volume fraction of the core 

occupied by the material. The following equations are used to calculate the relative density for 

the different cores examined in this report: 

Pipe Core:

 

2

*

c c

t D t

D









(refer to Table 2) (1)

Diamond Core:

 

2

4 2

*

c c c

c

Lt t r t

h





 



(refer to Table 3) (2)

X-core:

 

4

* 2 2

3

c

c

c

t

r t L

h W





 

       

(refer to Table 4) (3)

Table 7 lists the relative densities for the different cores using these equations and the details of 

the unit cell dimensions for the individual cores. By comparison, cellular solids such as 

honeycombs and foams generally have relative densities less than 0.3 [16].2

 

2

 Special ultra-low-density foams can be made with relative density as low as 0.001 [16]. 
14

Table 7. Relative Densities for Different Cores 

Core Geometry Relative Density, *

X-Core 0.075

5-in Pipe Core 0.077

3-in Pipe Core 0.125

Diamond Core 0.131

2-in Pipe Core 0.184

A first-order approximation to calculate relative density of the pipe cores can be calculated 

using: 

*

c

t

D



 

(4) 

This equation neglects the higher-order terms and tends to overestimate the relative densities 

calculated by equation (1) and listed in Table 7. The overestimation becomes larger as the outer 

diameter of the pipe core decreases. For example, the first-order approximation overestimates 

the relative density of the 5-, 3-, and 2-inch pipe cores by 4.0, 4.8, and 6.5 percent, respectively. 

The geometries of the diamond core and the X-core are similar except for the weld flats. If the 

weld flats in the X-core are ignored, then the geometries are identical. Under this idealized 

assumption, a first-order approximation to calculate the relative density of both cores can be 

made from this equation: 

2

*

sin

c

t

W





 

    

(5) 

In this equation,  is equal to 45 degrees for the diamond core and 60 degrees for the X-core. In 

the case of the diamond core, the first-order approximation overestimates the relative density 

calculated with equation (2) by only 3 percent. The first-order approximation for the X-core, 

which neglects the presence of the weld flats, underestimates the relative density calculated with 

equation (3) by more than 20 percent. 

In the Metal Foams Design Guide [17], core stiffness and strength are normalized by the relative 

density and material properties as a way to examine and compare different core types or 

geometries. This normalization practice is adopted in the next section of the report to compare 

the core strength under uniaxial compression. 
15

3. Behavior under Uniaxial Compression 

Tests and analyses were conducted to examine the deformation behavior of flat sandwich panels 

as a uniform compressive load is applied. The sandwich panels used for test specimens were 

manufactured by CMI. The crush tests were conducted by ATLSS at Lehigh University’s Fritz 

Engineering Laboratory. The analyses and the test design were performed by the Volpe Center. 

Two types of loads are associated with forces impacting sandwich panels: bending and pressure 

loading. Sandwich panel testing under uniaxial compression is conducted before bend testing 

because uniform pressure is more basic and simpler to carry out. Testing under this idealized 

load case examines two components of the sandwich panels: (1) mechanical response or 

deformation behavior of the core; and (2) strength of the welds. The ultimate mode of failure in 

sandwich panels subjected to uniaxial compression is the crushing of the core. 

Force-crush curves are measured in these tests to characterize the deformation behavior of 

sandwich panels subjected to uniaxial compression. For sandwich panels with an idealized 

elastic-perfectly-plastic core material, the force-crush response to uniaxial compression 

comprises three regimes (see Figure 4): (1) an initial rise in force, (2) a force plateau, and (3) a 

steep rise in force [16]. Each regime is associated with a different mechanism of deformation. 

For example, the initial rise in force is representative of linear-elastic material behavior. The 

force plateau corresponds to plastic deformation or buckling of the core. The steep rise in force 

is due to densification in which the core is solidly packed together. Two regimes in this 

idealized force-crush curve are also observed in the compression tests for the cores examined in 

this report: (1) the initial rise in force representative of linear elasticity; and (2) the densification 

regime. The shape of the intermediate regime in the force-crush curve depends on the particular 

core. 

For example, Figure 5 shows a trace of a measured force-crush curve from one of the pipe core 

sandwich panels. The shape of the force-crush characteristic is generally the same for all pipe 

cores but the force levels associated with the different regimes of the curve depend strongly on 

the pipe outer diameter. 

The initial force rise (i.e., linear-elasticity regime) is followed by a force plateau characteristic of 

plastic deformation. The plateau is then followed by another rise in force until a local maximum 

is reached, which is marked in the figure by the solid circle. This local maximum occurs when 

the deformed shape of the circular cores resembles a rectangle. The vertical legs of the 

rectangular shape behave as columns, which provide the additional compressive strength above 

the plateau. The level of the peak force depends on the core height or the pipe diameter. The 

drop in force after the local maximum signifies plastic collapse or buckling of the core. The 

force continues to drop until the onset of core densification. 
16

Figure 4. Schematic Force-Crush Response of Sandwich Panel with Idealized ElasticPlastic Core Material to Uniaxial Compression 

Figure 5. Generic Force-Crush Response of Pipe Core Sandwich Panel 
17

Figure 6 shows a trace from a measured force-crush curve from a diamond core sandwich panel. 

In this case, the linear-elasticity regime is immediately followed by a rapid drop in force, which 

represents plastic collapse or buckling of the core. The termination of linear-elasticity regime, 

which is denoted by the solid circle, coincides with the peak compressive force sustained by the 

panel. Similarly, Figure 7 shows a trace from a force-crush characteristic measured from one of 

the X-core sandwich panels. The overall shape of the force-crush characteristic for the X-core is 

quite similar to that of the diamond core. The similarity in the force-crush response is logical 

because the diamond core geometry is virtually the same as the X-core without the weld flats. 

Differences in these two characteristics are evident in the portion of the curves between the peak 

force and the onset of densification. The diamond core characteristic shows a relatively smooth 

curve in this regime while the X-core is somewhat jagged. The difference may be attributed to 

the manner in which the cores were made. The diamond core is a square tube, which is a single 

part that merely deforms as the core is compressed. The X-core is made from four individual 

parts that are cold-worked to form the weld flats. The bumps in the jagged curve for the X-core 

represent deformation of the four welded parts and intermittent tearing of the welds holding them 

together. 

Figure 6. Generic Force-Crush Response of Diamond Core Sandwich Panel 
18

Figure 7. Generic Force-Crush Response of X-Core Sandwich Panel 

3.1 Quasi-Static Compression Tests 

The test objectives are: (1) to confirm the analytical modeling of sandwich panels; (2) to 

examine the fabrication issues associated with sandwich panels such as material selection and 

weld type; (3) to observe the mechanical response or deformation behavior of sandwich panels 

under uniform pressure; and (4) to rank or prioritize the different cores. 

A universal testing machine with a 5-million-pound capacity was used to carry out the uniaxial 

compression tests. In these tests, quasi-static means that the targeted rate of head travel was 

0.2 in per minute. 

The matrix for the quasi-static compression tests is given in Table 8, which includes two test 

variables: (1) core type, and (2) weld type. Generally, the test panel dimensions are 

24–27 inches in length by 6 inches in depth. The table includes the weight of each panel, which 

varied between 33 and 40 pounds depending on the core. Other details such as dimensions and 

materials used for the face sheets and the cores are described in Section 2. The force-crush 

curves measured during the 14 quasi-static compression tests are shown in Appendix A. The 

peak loads, as depicted in the schematic force-crush characteristics for each core, are determined 

from these curves and are also listed in Table 8. 
19

Table 8. Quasi-Static Compression Test Matrix 

Test 

Order

Serial 

Number

Core Type Panel

Length

Panel

Depth

(in)

Panel

Weight

(lb)

Weld Type Peak Load

(kip)

1 0047-01 5-in pipe core

5 cores

25" 6 36.5 Slot/skip(Manual) 38

2 0047-03 3-in pipe core

8 cores

24" 6 33.9 Slot/skip(Manual) 128

3 0047-05 2-in pipe core

12 cores

24" 6 34.3 Slot/skip(Manual) 273

4 0047-07 2-in pipe core

12 cores

24" 6 34.2 Slot/skip & stitch between core elements (manual) 615

5 0047-14 Diamond core

9 cores

24" 6 33.4 Slot/skip & stitch between core elements (robotic) 420

6 0047-09 X-core

5 hats

27" 6 40.0 Spot(manual) 52

7 0047-11 X-core

5 hats

27" 6 40.0 Slot & spot(manual) 53

8 0047-02 5-in pipe core

5 cores

25" 6 36.4 Slot/skip(robotic) 37

9 0047-04 3-in pipe core

8 cores

24" 6 33.8 Slot/skip(robotic) 126

10 0047-06 2-in pipe core

12 cores

24" 6 34.3 Slot/skip(robotic) 458

11 0047-08 2-in pipe core

12 cores

24" 6 34.3 Slot/skip & stitch between core elements (robotic) 628

12 0047-13 Diamond core

9 cores

24" 6 33.1 Slot & skip(robotic) 398

13 0047-10 X-core

5 hats

27" 6 40.0 Spot(manual) 62

14 0047-12 X-core

5 hats

27" 6 40.0 Slot & spot(robotic) 75
20

A normalization procedure adopted from the Metal Foams Design Guide [17] is used in this 

report to compare the compressive strength of different cores. Moreover, compressive strength 

is characterized by a nondimensional quantity, peak/o*. Peak compressive stress, peak is the 

peak load (from Table 8) divided by the loaded cross-sectional area of the panel (generally 

2 2 approximately 144 square inch (in ) or 1 ft ). Nondimensional compressive strength is peak 

compressive stress divided by yield strength of the core material, o (from Table 5) and relative 

density, * (from Table 7).

Figure 8 shows a log-log plot of nondimensional compressive strength as a function of relative 

density. In this plot, attributes of an optimal design would be low relative density (implying light 

weight) and high strength. The figure shows that the nondimensional compressive strength of 

the cores examined in this report is correlated to relative density. Cores with the lowest relative 

densities, namely, the 5-inch pipe core and the X-core, have the lowest compressive strengths. 

Conversely, cores with the highest relative densities (the diamond core and the 2-inch pipe core) 

have the highest compressive strengths. Because the correlation does not trend toward an 

optimal design, a trade-off between strength and core density (i.e., weight) must be made to 

select the most favorable cores from the five that were tested. For example, the 2-inch pipe core 

exhibits the highest strength, but sandwich panels made with such cores may be too heavy and 

might require more welding than is practical for the intended application. The diamond core and 

the 3-inch pipe core appear to be reasonable alternatives that may provide adequate structural 

performance within weight constraints and are used in the bend tests (which are described in the 

next section of this report). 

Relative Density, *

0.01 0.1 1 Compressive Strength, peak

/o

0.01

0.1

1

5" Pipe Core

3" Pipe Core

2" Pipe Core

X-Core

Diamond Core

Figure 8. Variation of Compressive Strength with Relative Density 
21

Figure 8 shows a variation in strength for the 2-inch pipe cores. The highest level of 

compressive strength for the 2-inch pipe core corresponds to the panel in which the core 

elements are welded to each other at the 3- and 9-o’clock positions. Compressive strength is 

reduced when the cores are not welded to each other at these horizontal locations. Similarly, the 

figure shows a variation in strength for the X-cores, although this variation is less than that for 

the 2-inch pipe cores. In the case of the X-cores, the highest strength is from the panel that was 

slot and spot welded robotically. Whereas, the lowest strength is from a panel that was spot 

welded manually. 

3.2 Comparisons with Analysis 

Analytical and computational methods are applied to examine the strength and deformation 

behavior of sandwich panels under compression. Analytical methods are based on the 

assumption of elastic-perfectly-plastic material behavior. Calculations based on finite element 

analysis account for strain hardening effects. Results from these methods are used to design the 

quasi-static tests and to interpret test results. 

Analytical methods based on plastic limit load analysis [18, 19] are applied to estimate the 

strength (and stiffness) of sandwich structures. Limit analysis is well established for estimating 

collapse loads under the assumption of elastic-perfectly-plastic material behavior. The following 

equation is derived from limit analysis to estimate the compressive strength of the pipe core 

sandwich panels: 

2

2.56 4 1 1 *

* *

peak

o





   

 

      

(6) 

where o is the yield strength of the core material and * is the relative density. Specific details 

to derive this equation are described in Appendix B. 

Figure 9 compares the nondimensional compressive strength measured in the pipe cores to the 

theoretical estimate based on limit analysis. The figure shows excellent agreement between the 

measured strength and the closed-form equation. 

A similar equation can be derived from limit load analysis to estimate the compressive strength 

of the diamond core sandwich panels: 

0.5

*

peak

o



 



(7) 
22

Relative Density, *

0.01 0.1 1 Compressive Strength, peak

/o

0.01

0.1

1

5" Pipe Test Data

3" Pipe Test Data

2" Pipe Test Data

Theoretical

Figure 9. Variation of Theoretical and Measured Strength of Pipe Cores 

with Relative Density 

Figure 10 compares the nondimensional compressive strength from the measurements of the 

diamond core sandwich panels with the theoretical estimate. In this case, however, the analytical 

equation overpredicts the measured data by more than 40 percent. The discrepancy is due to the 

differences in the assumed and actual core geometries. The analytical equation is based on the 

idealized assumption that the diamond core is a perfect square, whereas the actual core has 

rounded corners (see Table 3). Because the legs of the diamond are somewhat curved and not 

perfectly straight, plastic collapse of the core initiates at a lower force level than would be 

expected under the idealized geometry. 

Finite element calculations are performed using the general-purpose finite element package 

ABAQUS. The face sheets and the cores of the sandwich panels are modeled using linear shell 

elements. Finite element analysis produces qualitatively similar force-crush characteristics as 

those measured in the quasi-static compression tests. However, the force levels in the finite 

element results are generally higher than the test data at the same levels of displacements. For 

example, Figure 11 shows a comparison between the measured and the posttest calculated forcecrush curves for the 5-inch pipe core. In the present context, ―posttest‖ means that the materials 

properties used in the finite element calculations were those measured from the tensile tests that 

were reported in Section 2. The comparison shown in the figure is typical of the qualitative 

agreement in the shape of the measured and calculated curves for all cores, but is also 

representative of higher force levels calculated by the finite element analysis. The 

overestimation is due to neglect of manufacturing imperfections and the assumption of perfect 

geometry. The sensitivity of finite element predictions to imperfections was discovered in 

previous work on diamond lattices [12]. 
23

Relative Density, *

0.01 0.1 1 Compressive Strength, peak

/o

0.01

0.1

1

2" Square Test Data

Theoretical

Figure 10. Variation of Theoretical and Measured Strength of Diamond Core with 

Relative Density 

Displacement (inches)

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

Force (kips)

0

10

20

30

40

50

60

0047-01 (Manual Weld)

0047-02 (Robotic Weld)

Post-Test FEA

Figure 11. Measured and Calculated Force-Crush Curves for 5-Inch Pipe Core 

Deformation patterns for different cores from finite element analysis resemble the observed 

deformation behavior. For example, Figure 12 shows a comparison of the deformation predicted 

from the finite element analysis at four levels of crush and video clips from the quasi-static 
24

compression test at the same levels of crush. The predicted deformation patterns are similar to 

the observed modes. The predicted deformations are based on finite element analyses performed 

prior to conducting the tests and before the actual mechanical properties of the materials were 

measured. 

Figure 12. Predicted and Observed Deformation of 5-Inch Pipe Core 

Figure 13 compares a close-up view of the predicted (left) and observed (right) deformation in a 

single pipe core at 4 in of crush. The test for this particular panel was terminated at this point. 

Moreover, the details of the deformation pattern at this level of crush are remarkably similar. 

Figure 13. Closeup View of Final Deformation Mode in 5-Inch Pipe Core 
25

4. Behavior under Bending with an Indenter 

Similar tests and analyses were conducted to examine the deformation behavior of welded steel 

sandwich panels under bending by using an indenter to transmit load. The quasi-static bend tests 

were also conducted at Lehigh University’s Fritz Engineering Laboratory by using the 5-millionpound capacity universal machine that was used in the compression tests. In the bend tests, local 

collapse modes of the sandwich panels were examined in addition to the force-displacement 

behavior. 

Results from the uniaxial compression tests were used in designing the bend tests. Of the five 

cores examined previously in the uniaxial compression tests, two were selected for bend testing: 

the 3-inch pipe core and the 2-inch square diamond core. The face sheets were made using the 

same material (Domex plate) as the compression tests, but the thickness was reduced from 

0.25 to 0.118 in, which reduced the overall panel weight. 

Figure 14 shows a schematic of the bend test configuration. Bending load was applied to the 

sandwich panels using a 12- by 12-inch indenter with rounded (1-inch radius) edges. The panels 

were simply supported over solid, 4-inch diameter round bars with a center-to-center span of 

24 in. The round bars were welded onto a virtually rigid (i.e., very stiff) platen. The overall 

length of the sandwich panels used in the bend tests was typically 48 in. Therefore, each end of 

the sandwich panels has about a 12-inch overhang. After about 4 in of indenter travel, the 

sandwich panel contacts the rigid platen. Additional crushing of the core after this point was 

similar to that observed in the previous compression tests. The sequence of loading, such as 

panel bending followed by crushing of the core, was similar to that in an idealized tank car 

impact scenario where the panel must bend and crush to protect the commodity-carrying tank. 

24”

12”

R 1”

4”

48”

Panel depth: 18”

Block depth: 12”

4”

Figure 14. Schematic of Bend Test Configuration 
26

Figure 15 shows the general shape of the expected force-displacement response in the bend tests. 

The schematic exhibits three phases of panel deformation. The first phase of the curve is an 

initial rise in force, representative of elastic bending. The second phase initiates when a local 

collapse mode is engaged. The particular local collapse mechanism depends on several factors 

such as span length, face sheet and core characteristics (e.g., dimensions, material properties, and 

geometry), and indenter characteristics (e.g., size and shape). These local collapse mechanisms 

are discussed in the next section of this report. The schematic characterizes the local collapse 

phase as a force plateau, but in some cases, the response may actually exhibit a softening 

behavior (i.e., a drop in force). Softening occurs when the relative distance between the face 

sheets decreases as the core compresses, which effectively reduces the bending inertia and 

decreases the moment and, therefore, the load carried by the panel. The third phase of 

deformation starts when the indenter travel is approximately equal to the roller support diameter, 

and the bottom of the sandwich panel comes into contact with the rigid platen. During the 

compression phase, the deformation of the sandwich panel is similar to that observed in the 

uniaxial compression tests. 

Figure 15. Schematic Force-Displacement Curve in Bend Test 

One of the factors affecting the force level that characterizes the local collapse phase (denoted as 

FC in Figure 15) is bending direction. The bending strength of the panel depends on the 

orientation of the cores with respect to the axis of bending. Moreover, the dependence of 

strength on core orientation means that bending behavior is anisotropic. Referring to the test 

configuration shown in Figure 14, the cores in the bend test panels were aligned either 

perpendicular or parallel to the supports. Panel bending strength is expected to be higher when 

the cores are perpendicular to the supports compared to when the cores are parallel to the 

supports. 
27

4.1 Local Collapse Modes 

The collapse mode of sandwich panels under uniaxial compression is crushing of the core. 

However, sandwich panels subjected to bending with an indenter can collapse locally in several 

ways. Three local collapse modes of sandwich panels in bending, which are shown 

schematically in Figure 16, are (1) face sheet yield or fracture, (2) local indentation, and (3) core 

shear. 

Figure 16. Possible Local Collapse Modes in Bending 

For a panel undergoing bending in the test configuration, the stress state in the top face sheet is 

compressive while that in the bottom face sheet is tensile. As the load transmitted through the 

indenter increases, plastic hinges may form at the edges of the indenter when the deformations 

become plastic. At sufficiently large deformations, the stress state in the top face sheet may also 

become tensile. Moreover, local failure of the face sheets may occur if the compressive or 

tensile stresses are high enough to cause yielding or fracture. 

Local indentation may occur when the face sheet comes in contact with the indenter and acts as a 

plate on an elastic-plastic support provided by the core. In this case, the top face sheet bends 

independently of the opposite (or bottom) face sheet. If the stress applied to the core exceeds its 

compressive strength, the core will collapse. 

In short-span panels under three-point bending, the core is mainly subjected to shear stresses and 

carries most of the transverse force. Core failure may occur if the shear stress exceeds the shear 

strength of the core material. In long-span panels, normal stresses in the core can be of the same 

order of magnitude or even higher than the shear stresses. In this case, the core is subjected to 

biaxial stresses and core failure may occur if the effective transverse shear stress exceeds the 

shear strength of the core material. 
28

Previous research has been conducted to examine the concept of a collapse mechanism map that 

describes how various factors affect the initiation of different local collapse modes [17, 20–22]. 

These factors include characteristics of the core and the face sheets (i.e., material properties, 

geometry, and dimensions), span length, and indenter size. For example, the Metal Foams 

Design Guide [17] provides analytical equations to calculate the collapse loads associated with 

the three modes shown in Figure 16 based on the assumptions of perfectly-plastic material 

behavior and three-point bending applied through a flat-bottomed indenter. The collapse loads 

for the three modes are plotted on a diagram with nondimensional axes of core height divided by 

span length and core thickness divided by core height. 

Figure 17 shows a diagram developed for a 3-inch pipe core panel with characteristics provided 

in

Table 2 and assuming a 12-inch indenter with a simply-supported span length of 24 in. The solid 

circle in the diagram indicates that failure of the 3-inch pipe core panel is likely to occur from 

local collapse due to excessive shear stresses in the core. 

Core Height/Span, hc/L

0.001 0.01 0.1 1 Face Sheet Thickness/Core Height, t

f/hc

0.001

0.01

0.1

1

Face Sheet Yield or Fracture

Core

Shear

Indentation

Figure 17. Collapse Mechanism Map for 3-Inch Pipe Core Panel in Bending 

Figure 18 shows a similarly constructed collapse mechanism map for the diamond core panel. In 

this case, the core height (see Table 3) is slightly smaller than the pipe core (2.62 vs. 3 in). The 

change in core dimensions relative to span length and face sheet thickness alters the local 

collapse mode from core shear to face sheet yield or fracture. Results from the compression tests 

are used to construct the collapse mechanism maps for the two different cores. In both cases, the 

collapse mechanism maps indicate that face sheet yield or fracture is likely to occur in panels 

with relatively thin face sheets (i.e., small values of tf/hc) and in relatively long panels (i.e., small 

values of hc/L). Core shear is generally expected to occur in the panels in the present bend tests. 

The collapse mechanism map shown in Figure 18; however, suggests that face sheet yield or 

fracture may also be expected to occur in some cases with the diamond core panels. 
29

Core Height/Span, hc/L

0.001 0.01 0.1 1 Face Sheet Thickness/Core Height, t

f/hc

0.001

0.01

0.1

1

Face Sheet Yield or Fracture

Core

Shear

Indentation

Figure 18. Collapse Mechanism Map for Diamond Core Panel in Bending 

Interestingly, fracture of the bottom face sheet occurred during the quasi-static bend tests with 

both diamond core panels with solid face sheets and cores oriented perpendicular to the supports. 

Figure 19 shows the extent of cracking between welds in the bottom face sheet of one panel. 

Moreover, the face sheet fracture corresponds to the local failure mode indicated by the collapse 

mechanism map for the diamond core panels in Figure 18. 

Figure 19. Photograph of Face Sheet Fracture from Quasi-Static Bend Test 
30

In traditional sandwich design, panels that are efficient in absorbing energy are also stiff. For 

panels with the attributes of high strength and stiffness, in-plane tension in the face sheets 

increases the likelihood of local failure by yielding, tearing, or fracturing. Finite element 

simulations have shown a rapid decrease in panel effectiveness once local failure of the face 

sheets occurs. To mitigate the likelihood of engaging this local failure mode, the concept of strip 

face sheets was introduced as a variable in the test matrix for the bend tests. Simulations also 

confirm that using strips relieves in-plane tension in the face sheets, and may help distribute the 

applied load over a larger area. 

A quick and inexpensive method was conceived to quantify load spreading during some of the 

quasi-static bend tests. A sheet of plastic bubble wrap was taped to the rigid platen between the 

roller supports before conducting the test. As bending loads were applied through the indenter, 

the bottom face sheet eventually came into contact with the platen. At this point, the protruding 

air-filled hemispheres in the bubble wrap were squeezed and ―popped.‖ After the test was 

completed, the boundaries of popped bubbles on the sheet that outlined the contact area, such as 

the area of load spreading, were examined. 

The bubble wrap method was used to measure load spreading in two bend tests. In both tests, the 

sandwich panels were made with solid face sheets and cores running parallel to the supports. 

One test involved a pipe core sandwich in which the boundaries of the contact area were 

measured as 16 inches in the direction of the supports by 14.5 inches in the direction 

perpendicular to the supports. The other test involved a diamond core panel, where the 

corresponding measurements were 16 by 13.25 in. Sixteen inches was the total width for the 

particular brand of bubble wrap that was used in these tests. Therefore, the actual amount of 

load spreading along the direction of the parallel cores may be even greater. On the basis of 

these two measurements, the sandwich panels distributed load over an area that is conservatively 

estimated to be 50–60 percent greater than the indenter footprint (12 by 12 in). Figure 20 shows 

the bend test configuration with the bubble wrap as the pipe core sandwich panel is bending. 

Figure 20. Bend Test with Bubble Wrap to Quantify Load Spreading 
31

4.2 Quasi-Static Bend Tests 

The objectives of the quasi-static bend tests were as follows: (1) to confirm the analytical and 

computational modeling of sandwich panels and (2) to observe the mechanical response of the 

sandwich panels under bending. The observation of mechanical response includes measuring the 

force-displacement behavior and indentifying the local collapse modes which were previously 

described in Section 4.1. 

Table 9 shows the test matrix for the quasi-static bend tests in which three test variables are 

evident: (1) core geometry (3-inch pipe core and 2-inch square diamond core), (2) core 

orientation relative to the rollers used to support the panel (cores running either parallel or 

perpendicular to the supports), and (3) face sheet type (solid plates on both sides, strips used as 

face sheets on both sides, and a combination of the two). In the present bend tests, quasi-static 

means that the indenter was lowered onto each of the sandwich panels at a rate of approximately 

one-quarter inch per minute. 

The term ―hybrid‖ is used in this report to refer to the face sheet type where a solid plate is 

attached to one side of the sandwich panel and strips are used on the opposite side. In the test 

matrix, Serial Number 0047-29 is called Hybrid A in which strips are placed on top and a solid 

face plate is on the bottom in the test fixture. Serial Number 0047-30 is called Hybrid B in 

which the placement of the face sheets with respect to the indenter is reversed (i.e., solid plate on 

top, strips on bottom). Both hybrid panels are constructed with diamond cores that are aligned 

perpendicular to the supports. 

Appendix C is a compendium of the 16 force-displacement (i.e., distance of head travel) curves 

measured during the bend tests. These curves are used to estimate the collapse load as shown in 

Figure 15. The local collapse loads are plotted in Figure 21 in ascending order for each panel. 

Moreover, the collapse loads are used as a measure to quantify the relative effect of the test 

variables. In some cases, the collapse load is distinct, and is read directly from the forcedisplacement curve as the local maximum value within the local collapse phase. In other cases, 

particularly for panels with cores aligned parallel to the supports, no local maximum is evident. 

In these cases, the collapse load is considered as the force level at which the compression phase 

appears to initiate. Generally, this occurs after about 4 in of displacement, when the bottom of 

the panel comes into contact with the very stiff platen. 

Figure 21 clearly indicates that the bending strength of the panels used in this test series is 

strongly anisotropic. Panels with cores perpendicular to the supports are much stronger in 

bending than those with cores parallel to the supports. The collapse load for the perpendicular 

core panels is about five (for pipes) to six (diamonds) times that of the corresponding parallel 

core panels. 

For solid face sheets, the collapse loads for the diamond core panels were slightly higher on 

average (about 15 percent) than those for the pipe core, independent of the core orientation. For 

strip face sheets, the collapse loads for both cores were nearly the same. 
32

Table 9. Quasi-Static Bend Test Matrix 

Test 

Order

Serial 

Number

Core

Type

Core Orientation 

Relative to Supports

Face Sheet

Type

Panel

Length

Panel

Depth

Panel

Weight

(lb)

Weld

Type

1 0047-19 3-in Pipe Parallel Solid 16 cells

48” 18” 139.5 Manual

2 0047-15 Diamond Parallel Solid 18 cells

47.2” 18” 139.5 Manual

3 0047-26 3-in Pipe Parallel Strip 16 cells

48” 18” 111.0 Robotic

4 0047-24 Diamond Parallel Strip 47.2” 7 cells

18.34” 111.5 Robotic

5 0047-21 3-in Pipe Perpendicular Solid 48” 6 cells

18” 141.0 Manual 

6 0047-17 Diamond Perpendicular Solid 47.2” 7 cells

18.34” 140.0 Manual

7 0047-25 3-in Pipe Perpendicular Strip 48” 6 cells

18” 110.5 Robotic

8 0047-23 Diamond Perpendicular Strip 47.2” 7 cells

18.34” 109.5 Robotic

9 0047-29 Diamond Perpendicular Solid & strip 47.2” 7 cells

18.34” 126.0 Manual

10 0047-20 3-in Pipe Parallel Solid 16 cells

48” 18” 139.5 Robotic

11 0047-16 Diamond Parallel Solid 18 cells

47.2” 18” 139.5 Robotic

12 0047-28 Diamond Parallel Strip 18 cells

47.2” 18” 109.0 Manual

13 0047-22 3-in Pipe Perpendicular Solid 48” 6 cells

18” 142.5 Robotic

14 0047-18 Diamond Perpendicular Solid 47.2” 7 cells

18.34” 137.0 Robotic

15 0047-27 Diamond Perpendicular Strip 47.2” 7 cells

18.34” 110.0 Manual

16 0047-30 Diamond Perpendicular Solid & strip 47.2” 7 cells

18.34” 126.0 Manual
33

Figure 21. Collapse Loads in Bend Tests 
34

Panels with solid face sheets are stronger in bending than those with strips. The difference in the 

collapse loads is between 35 and 49 percent depending on the particular configuration. The 

differences are slightly greater for the diamond cores (42–49 percent) compared to the pipe cores 

(35–40 percent). 

The force-displacement behavior for the two hybrid panels is nearly the same before the 

compression phase. The collapse load for Hybrid A is approximately 10 percent higher than that 

for Hybrid B. 

The two panels exhibiting the largest collapse loads are the replicate diamond core panels with 

solid face sheets and cores oriented perpendicular to the supports (i.e., strong-bending direction). 

In the test matrix (Table 9), these panels correspond to Serial Numbers 0047-17 and 0047-18. 

Cracks between welds were observed in the bottom face sheets of both panels. 

The general-purpose finite element package ABAQUS/Explicit was used to carry out 

calculations to predict the force-displacement response of the sandwich panels in the bend tests. 

The pretest predictions from the finite element analysis are included in the compendium of forcedisplacement curves given in Appendix C. The predictions are generally higher than the 

measured responses. Again, the discrepancy in overpredicting strength is attributed to the 

neglect of manufacturing imperfections in the analyses, which was seen in the comparisons in 

the compression tests and in the previous work on diamond lattices [12]. 
35

5. Summary of Test and Analysis Results 

Observations and results from the tests and analyses of sandwich panels subjected to quasi-static 

uniaxial compression are summarized as follows: 

1. Tests on panels with similar cores and weld types exhibit excellent repeatability. 

2. Compressive strength is correlated to relative density. Compressive strength increases as 

relative density increases. The 2-inch pipe cores provide the highest compressive strength 

but also have the highest relative density. 

3. Strength estimates for the pipe cores based on plastic limit load analysis are in excellent 

agreement with measured compressive strength. Similar analytical estimates for the diamond 

cores are more than 40 percent higher than the measured data. The discrepancy may be 

attributed to the idealized assumptions regarding the geometry (i.e., perfectly square diamond 

core) and material (i.e., elastic-perfectly-plastic) behavior. 

4. Finite element analysis produces force-crush curves that are in good agreement with those 

measured in the tests. The finite element analysis results generally overestimate strength, 

which may be attributed to the neglect of manufacturing imperfections. 

5. Deformation modes simulated by finite element analysis resemble those observed in the tests. 

Observations and results from the tests and analyses of sandwich panels subjected to quasi-static 

bending using an indenter are summarized as follows: 

1. Tests on panels with similar cores and weld types exhibit excellent repeatability. 

2. Bending strength of the sandwich panels is strongly anisotropic. Panels with cores 

perpendicular to the supports are much stronger in bending than those with cores parallel to 

the supports. Collapse loads for perpendicular core panels are five (for pipes) to six 

(diamonds) times those for the parallel core panels. 

3. Replacing solid face sheets with strip face sheets reduces panel bending strength by 

approximately 35–49 percent. 

4. Cracking in the bottom face sheet was observed in the two diamond core panels with cores 

perpendicular to the supports using solid face sheets. Cracking was not obvious in the 

diamond core panels with the same orientation using strip face sheets. On the basis of these 

results, it appears that the use of strip face sheets may mitigate the onset of face sheet yield or 

fracture. 

5. The force-displacement behavior for both hybrid panels is nearly the same before the 

compression phase. The collapse load for Hybrid A (i.e., strip face sheets on top, solid plate 
36

on bottom) is approximately 10 percent higher than that for Hybrid B (solid plate on top, 

strip face sheets on bottom). 

6. Force-displacement (i.e., indenter travel) curves calculated by finite element analysis 

overpredict the force level compared to the test data in every case, but are within reasonable 

agreement. The discrepancy is attributed to modeling perfect geometry, whereas 

imperfections likely exist in actual test panels. The sensitivity of finite element analysis to 

imperfections was also found in previous testing and computational analysis of sandwich 

panels with stainless steel diamond lattices [12].
37

6. Discussion 

Previous research identified sandwich structures as a potential technology that could improve the 

crashworthiness of commodity-carrying tank cars [6–8]. One of the concepts currently under 

development treats the pressurized commodity-carrying tank as a protected entity using 

sandwich structures as protective panels in the event of an accident. Two critical advantages for 

using sandwich structures in traditional design applications are: (1) sandwich structures are 

known to provide greater flexural stiffness and strength than those of a solid plate of equal mass; 

and (2) material arrangement in the core can add functionality such as thermal insulation and 

energy absorption.

The energy-absorption capability makes sandwich structures attractive as a means to protect 

railroad tank cars during crashes. In theory, protective sandwich panels under impact would 

absorb the kinetic energy of the moving object. Energy-absorption requirements for generalized 

tank car head and shell impact scenarios have been estimated by calculating kinetic energy [25]. 

The kinetic energy associated with the impact of two fully loaded tank cars (each weighing 

286,000 lb) with a closing speed of 25 miles per hour (mph) is 6 million ft-lb. For a closing 

speed of 30 mph, the kinetic energy is 8.6 million ft-lb. However, absorbing energy levels of 

this magnitude is a formidable challenge because the design of such a system is constrained by 

the maximum allowable weight and clearance standards for rail cars. The effect of weight and 

space budgets on improved designs is discussed in previous work [7]. 

Finite element simulations to model sandwiches as protective panels during impact are ongoing. 

These models incorporate results from the engineering studies described in this report on 

mechanical behavior under compressive and bending loads. Force-displacement curves from 

these simulations, however, do not differ significantly from those of a conventional tank car 

impact without protective panels. Such results indicate that the levels of energy absorbed by 

sandwich panels are relatively small compared to the kinetic energies corresponding to impact 

closing speeds of 25 and 30 mph. Moreover, the results suggest that, due to weight and space 

limitations, the structural rigidity of the commodity-carrying tank must be integrated with the 

sandwich panels in order to absorb more kinetic energy than the protective panels alone. 

The simulation results further suggest that the primary benefit of using sandwich structures in the 

protection concept is gained from their ability to distribute load over a larger area of the tank 

with respect to the indenter footprint (i.e., load blunting). The bubble wrap method described in 

Section 4 indicates that sandwich panels can distribute the load over an area that is at least 

50–60 percent greater than the indenter footprint. These estimates of the contact area are 

conservative because an indenter with a footprint of 12 by 12 in is assumed. Because the edges 

of the indenter are rounded with a 1-inch radius, the footprint may be considered to be 

10 by 10 in. In this case, the increased contact area becomes 116–132 percent larger than the 

indenter footprint. Whereas, requirements for energy absorption are readily determined from a 

calculation of kinetic energy; it is unclear how to determine an appropriate target for load 

spreading. It is also unclear whether or not a contact area that is 1.5 or 2.3 times that of the 

indenter footprint is sufficient to prevent puncture in a tank covered by a sandwich panel as an 
38

effective shield. Additional research may be needed to better understand the mechanics of load 

spreading offered by protective panels. 

Finite element analysis reveals the following sequence of events that typically occurs when a 

sandwich panel cannot protect the commodity-carrying tank against puncture from an indenter. 

The face sheet initially impacted by the indenter fails by fracture. Failure of this outer face sheet 

is followed by core failure, which in turn is followed by failure of the inner face sheet, and then 

puncture of the tank. Moreover, once failure in the outer face sheet occurs, global failure of the 

sandwich and the commodity-carrying tank is imminent. 

The finite element analysis also indicated that failure of the face sheets is inherently due to 

excessive in-plane tension. Therefore, design features that can reduce or mitigate the effect of 

in-plane tension in the face sheets will help to maintain the integrity of the sandwich structure 

under impact and to enable its load-spreading functionality. For example, the detrimental effect 

of in-plane tension was demonstrated in the quasi-static bend tests using diamond core panels 

with solid face sheets and cores oriented in the strong bending direction (i.e., perpendicular to the 

supports). But when the sandwich design was modified by using strips rather than solid face 

sheets, local failure of the bottom face sheet by fracture was preempted. In-plane tension is also 

influenced by the boundary conditions of the panel or the manner in which the panel is 

supported. Another concept to mitigate this effect is to employ relatively small panels or tiles as 

a protective layer rather than a larger continuous cover. Additional research is needed to explore 

this concept. 

All sandwich panels used in the compression and bend tests described in this report were 

constructed with equal thickness face sheets. Finite element models were also used to examine 

the concept of using unequal thickness face sheets. Preliminary results indicate that thickening 

the outer face sheet delays the time to initiate failure, but the failure rate actually increases 

rapidly through the protective panels once it initiates in the core. Thickening the inner or bottom 

face sheet allows more time for the core to completely fail. Therefore, it appears that increasing 

the thickness of the bottom face sheet is more effective in protecting the tank than increasing the 

thickness of the top face sheet. This inference is supported by the fact that the sandwich panels 

that failed in the quasi-static bend tests exhibited cracking in the bottom face sheet. 

Structural behavior and functionality of sandwich panels depends strongly on the core 

arrangement and the relative density. Figure 22 compares the nondimensional compressive 

strength of the cores examined in the present study with data extracted from other studies on

stainless steel corrugated cores [12], stainless steel diamond lattices [12], aluminum alloy metal 

foams [17], stainless steel square honeycombs [23], and stainless steel pyramidal truss cores 

[24]. The dashed lines through the extracted data represent best-fit curves whereas the solid line 

through the pipe core data is the theoretical estimate based on limit analysis from equation (6). 

The figure indicates that the cores examined in this report (namely the pipe, diamond, and Xcores) exhibited lower compressive strength than the square honeycomb, pyramidal truss, and 

corrugated cores. Conversely, the cores in this report exhibited greater strength than aluminum 

alloy foam. 
39

Relative Density, *

0.01 0.1 1 Compressive Strength, peak

/o

0.01

0.1

1

10

5" Pipe Core

3" Pipe Core

2" Pipe Core

X-Core

Diamond Core

Square honeycombs

Foam

Pyramidal Pipe cores

core

Corrugated cores

Diamond lattice

Figure 22. Comparison of Compressive Strength for Different Cores 

Clearly, several options exist for selecting core shape or geometry for optimal design. In 

traditional sandwich design, high strength and stiffness are attributes of an optimal design. In the 

concept of a protection system against puncture from an indenter; however, it is unclear whether 

these attributes are desirable. Stiffer panels imply more strength, but finite element simulations 

suggest that such panels are more vulnerable to face sheet tearing due to excessive in-plane 

tension as described previously. 

The materials used to construct the sandwich panels for the compression and bend tests were not 

varied. The selection of these materials was based on current availability of the desired 

quantities and thicknesses when tests were being planned. Moreover, the specific details of the 

sandwich panels as described in Section 2 do not represent optimal designs for structural 

performance. Preliminary finite element simulations of generalized tank car impacts suggest that 

an optimal design might employ materials with high-strength properties in the face sheets and 

high ductility in the cores. 

The basic research described in this report examined certain fabrication issues. For example, in 

previous work sponsored by the Office of Naval Research [12], diamond lattice materials were 

manufactured as cores of sandwich panels by slotting together stainless steel sheets and then 

brazing together the assembly. In the present study, off-the-shelf (OTS) tubes (circular tubes for 

the pipe cores and square tubes for the diamond cores) were welded to the face sheets to 

construct sandwich panels. Figure 22 also shows that the nondimensional compressive strength 
40

of the diamond cores in the present study is slightly lower than the regression curve for the 

diamond lattices assembled with the slotting and brazing technique. The different construction 

methods may explain the variation in compressive strength, but this is not significant. 

The quasi-static compression and bend tests included replicate panels to examine potential 

differences due to welding procedures (e.g., manual versus robotic). The results indicate that the 

variability in structural performance is almost negligible in the replicate panels. Therefore, the 

combination of using OTS tubing for circular pipes and square diamonds and joining them 

together with the various welding procedures provides repeatability in structural performance. 

Results from the quasi-static tests and analyses described in this report indicate that overall 

performance of sandwich panels is strongly sensitive to imperfections. The imperfection 

sensitivity of sandwich structures confirms similar results from previous work [12, 26]. In the 

present study, imperfections were considered in terms of imperfect geometry. However, flaws or 

defects may develop during the fabrication process that can degrade performance. The potential 

for such imperfections may introduce an issue of quality control if construction of sandwich 

structures for the current application becomes viable. 

Obviously, the quasi-static tests described in this report do not account for dynamic effects on 

deformation behavior. Two separate effects should be distinguished in the context of dynamic 

response of structures: strain rate and impact velocity. Strain rate effect is a material property in 

which the effective yield strength increases as the strain rate is raised. The latter effect is 

material inertia that leads to enhanced stresses during impact, which may affect the mechanism 

of local collapse. Understanding the development of local collapse mechanisms that evolve to 

global failure is critical in assessing the structural performance of sandwich structures as 

protective panels, and is a major objective of the present work. The next logical step for ongoing 

research would be to design and implement a dynamic impact test to examine inertial effects. 

Moreover, additional research is needed to reach the ultimate goal of demonstrating the 

performance of a candidate protective system against puncture from an indenter carrying 

sufficient weight. 
41

7. References 

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Freight Train 292-16 and Subsequent Release of Anhydrous Ammonia Near Minot, 

North Dakota January 18, 2002,‖ Railroad Accident Report NTSB/RAR-04/01. 

[2] Nayak, P. R., Rosenfield, D. B., and Hagopian, J. H. (1983). ―Event Probabilities and 

Impact Zones for Hazardous Materials Accidents on Railroads,‖ Final Report, 

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[3] Anderson, R. T., and Barkan, C.P. (2004). Railroad accident rates for use in 

transportation risk analysis. Transportation Research Record, 1863, 88–98. 

[4] Treichel, T. T. (2007). ―Specific Locations of Damage and Lading Losses on Tank 

Heads and Shells,‖ RSI-AAR Railroad Tank Car Safety Research and Test Project Report 

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[5] Jeong, D. Y., Tang, Y. H, Yu, H., Lyons, M. L., Gordon, J. E., Orringer, O., and Perlman, 

A. B. (2009). ―Engineering Studies on Structural Integrity of Railroad Tank Cars Under 

Accident Loading Conditions,‖ Final Report, DOT/FRA/ORD-09/18. 

[6] Tyrell, D. C., Jeong, D. Y., Jacobsen, K., and Martinez, E. (2007). ―Improved Tank Car 

Safety Research,‖ Proceedings of the 2007 ASME Rail Transportation Division Fall 

Technical Conference, RTDF2007-46013. 

[7] Tyrell, D. C., Jacobsen, K., Talamini, B., and Carolan, M. (2007). ―Developing 

Strategies for Maintaining Tank Car Integrity During Train Accidents,‖ Proceedings of 

the 2007 ASME Rail Transportation Division Fall Technical Conference, RTDF2007-

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[10] Carney, J. F., III. (1989). ―The Connecticut Narrow Hazard Crash Cushion,‖ Design and 

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[11] Alghamdi, A. A. A. (2001). Collapsible impact energy absorbers: an overview. ThinWalled Structures, 39, 189–213.
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[12] Côté, F., Deshpande, V. S., Fleck, N. A., and Evans, A. G. (2006). The compressive and 

shear responses of corrugated and diamond lattice materials. International Journal of 

Solids and Structures, 43, 6220–6242.

[13] Wei, Z., Deshpande, V. S., Evans, A. G., Dharmasena, K. P., Queheillalt, D. T., Wadley, 

H. N. G., Murty, Y. V., Elzey, R. K., Dudt, P., Chen, Y., Knight, D., and Kiddy, K.

(2008). The resistance of metallic plates to localized impulse. Journal of the Mechanics 

and Physics of Solids, 56, 2074–2091.

[14] Wadley, H. N., Fleck, N. A., and Evans, A. G. (2003). Fabrication and structural 

performance of periodic cellular metal sandwich structures. Composites Science and 

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[15] ASTM Standard A370. (2003). ―Standard Test Methods and Definitions for Mechanical 

Testing of Steel Products,‖ West Conshohocken, PA: ASTM International, 

www.astm.org. 

[16] Gibson, L. J., and Ashby, M. F. (1997). Cellular Solids: Structure and Properties, 2nd 

edition, Cambridge University Press. 

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H.N.G. (2000). Metal Foams: A Design Guide, Washington, DC: ButterworthHeinemann. 

[18] Chakrabarty, J. (1988). Theory of Plasticity, 2nd edition, London: McGraw-Hill. 

[19] Lubliner, J. (2008). Plasticity Theory, New York: Dover Publications Inc. 

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sandwich beams. Materials Science and Engineering, 95, 37–53.

[21] Petras, A., and Sutcliffe, M. P. (1999). Failure mode maps for honeycomb sandwich 

panels. Composite Structures, 44, 237–252.

[22] Tagarielli, V. L., and Fleck, N.A. (2005). A comparison of the structural response of 

clamped and simply supported sandwich beams with aluminum faces and a metal foam 

core. Journal of Applied Mechanics, 72, 408–417. 

[23] Côté, F., Deshpande, V. S., Fleck, N. A., and Evans, A. G. (2004). The out-of-plane 

compressive behavior of metallic honeycombs. Material Science Engineering A, 380, 

272–280.

[24] Zok, F. W., Waltner, S. A., Wei, Z., Rathbun, H. J., McKeeking, R. M., and Evans, A. G. 

(2004). A protocol for characterizing the structural performance of metallic sandwich 

panels: application to pyramidal truss core. International Journal of Solids and 

Structures, 41, 6249–6271.
43

[25] Jeong, D. Y., Tyrell, D. C., Carolan, M. E., and Perlman, A. B. (2009). ―Improved Tank 

Car Design Development: Ongoing Studies on Sandwich Structures,‖ Proceedings of the 

2009 ASME Joint Rail Conference, JRC2009-63025.

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sandwich structures. International Journal of Solids and Structures, 44, 4690–4706.
44

Appendix A. Force-Crush Curves from Compression Tests 

Displacement (inches)

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5

Force (kips)

0

10

20

30

40

50

0047-01 (Manual Weld)

0047-02 (Robotic Weld)

Figure A1. Measured Force-Crush Curves for 5-Inch Pipe Cores 

Displacement (inches)

0.0 0.5 1.0 1.5 2.0 2.5 3.0

Force (kips)

0

50

100

150

0047-03 (Manual Weld)

0047-04 (Robotic Weld)

Figure A2. Measured Force-Crush Curves for 3-Inch Pipe Cores 
45

Displacement (inches)

0.0 0.5 1.0 1.5 2.0

Force (kips)

0

200

400

600

800

1000

0047-05 (Manual Slot & Skip Weld)

0047-06 (Robotic Slot & Skip Weld)

0047-07 (Manual Slot/Skip & Stitch Between Core

0047-08 (Robotic Slot/Skip & Stitch Between Cores

Figure A3. Measured Force-Crush Curves for 2-Inch Pipe Cores 

Displacement (inches)

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5

Force (kips)

0

50

100

150

200

250

300

350

0047-09 (Manual Spot Weld)

0047-10 (Manual Spot Weld)

0047-11 (Manual Slot & Spot Weld)

0047-12 (Robotic Slot & Spot Weld)

Figure A4. Measured Force-Crush Curves for X-Cores 
46

Displacement (inches)

0.0 0.5 1.0 1.5 2.0 2.5

Force (kips)

0

200

400

600

800

1000

0047-13 (Robotic Slot & Skip Weld)

0047-14 (Robotic Slot/Skip & Stitch Between Cores

Figure A5. Measured Force-Crush Curves for Diamond Cores 

Figure A6. Measured Force-Crush Curves for All Cores 
47

Appendix B. Estimation of Core Strength by Limit Analysis

Analytical methods are applied to estimate the plastic collapse loads for the pipe and diamond 

cores under a compressive load. Moreover, these analytical methods are based on the theory of 

plastic limit analysis [18, 19]. Estimates of plastic collapse load are compared with the 

measurements obtained from the quasi-static compression tests that were conducted at Lehigh 

University. 

B.1 Pipe Core Estimates 

Figure B1 shows the geometry of a flat sandwich panel with two pipes with diameter D in its 

core. 

Figure B1. Sandwich Panel with Pipe Core 

Under compressive loading, the upper half of the center pipe in the sandwich cell shown in 

Figure B2 is assumed to behave in a similar manner as a semicircular arch with pinned supports 

subjected to an equivalent concentrated load (Figure B3):

P D b     2

(B-1) 

where 2 is the compressive stress, D is the outer diameter of the pipe, and b is the depth of the 

panel. 

Figure B2. Sandwich Cell under Compression 
48

Figure B3. Semicircular Arch with Pinned Ends

The vertical reaction load at the pinned supports, V is found from equilibrium, while the 

horizontal reaction load, H is determined from applying Castigliano’s theorem:

2

P

V 

(B-2)

P

H





(B-3)

As the semicircular arch deforms plastically, hinges are assumed to form at the locations shown 

schematically in Figure B4. One hinge is assumed to develop directly under the centrally applied 

load. Hinges are also assumed to form at the pinned supports. In addition, hinges are assumed 

to develop at the location of maximum negative (hogging) bending moment, which is 

characterized in Figure B4 by the rotational angle o.

Figure B4. Arch Mechanism at Plastic Collapse
49

Considering only the right half of the semicircular arch shown in Figure B3, the bending moment 

as a function of the rotational angle,  is given by

 

1 1 ( ) (1 cos ) sin 1 cos sin 0

2 2

M VR HR PR      



             

(B-4)

where R is the radius of the semicircular arch. The derivative of this equation with respect to  is

1 1 sin cos

2

M

PR  

 

  

      

. (B-5)

The angle at which the maximum hogging moment occurs is determined by setting this 

derivative equal to zero, from which 

1 2

tan 32.5oo



  

     

. (B-6)

The maximum hogging moment is

   

2

4 0.093

2

o

PR M PR   



     . (B-7)

The absolute value of this moment is used in the remainder of this derivation. The plastic 

collapse load is determined from equating this maximum moment to the fully plastic moment

1 2

4

M t b o o 

(B-8)

where o is the yield strength of the core material and t is the thickness of the pipe. Therefore, 

the compressive strength of the pipe core may be expressed in terms of its material yield strength 

and geometric parameters as follows

 

2 2

2 2

4 5.39

4

o

t t

D D

 

 



   

           

. (B-9)

Alternatively, a similar equation for plastic collapse of the pipe core may be derived from 

assuming a different type of collapse mechanism. In this method, the collapse mechanism is 

assumed to consist of a polygon inscribing the circular pipe. Figure B5 shows a schematic of the 

collapse mechanism in the shape of an octagon. 
50

Figure B5. Collapse Mechanism for Circular Pipe

Moreover, each side of the polygon is assumed to behave as a fixed-end beam, which the 

collapse load is calculated from

8 o

o

M

P

L



(B-10)

where L is the equivalent length of the beams. For a polygon of equal-length sides, the 

equivalent beam length is calculated from

L Dsin

n





(B-11)

where n is the number of sides to the polygon (which is also equal to the number of hinges). 

Therefore, an approximate expression for the collapse load of the pipe core can be derived as:

2

2

2

o sin

t

D

n



 

 

    

(B-12)

For n equal to 8,

2

2 5.23

o

t

D





 

    

(B-13)

which is about 3 percent lower than equation (B-9), derived from the semicircular arch 

mechanism. 

The plastic collapse loads associated with the sandwich panels used in the quasi-static 

compression tests are calculated from: 
51

2

2

sin

o

C

t

P A

D

n





 

     

(B-14)

where A is the loaded cross-sectional area of the panel.

Table B1 lists the outer diameter, the measured tensile yield strength, and the loaded crosssectional area for the tested sandwich panels with pipe cores. Yield strengths are the averages of 

tensile test measurements on five different samples of AISI 1010 steel from the same batch used 

to construct the test articles. The core thickness was the same for each panel, and was equal to 

0.125 in. Table B1 also compares the calculated plastic collapse load using equation (B-14) and 

assuming eight links (i.e., n=8) with measurements for the initial collapse load. Figure B6

compares the calculated and measured collapse loads corresponding to the initial peak for the 

three pipe diameters.

Table B1. Plastic Collapse Estimates and Initial Peak Observed in Tests on Pipe Cores

D

(inch)

o

(ksi)

A

(in2)

PC (kip)

Calculated

(n = 8)

Test

2 52.3 144 154 190

3 43.0 144 56 60

5 37.1 150 18 17

Calculated Collapse Load (kips)

0 50 100 150 200 250

Initial Collapse Load in Test Data (kips)

0

50

100

150

200

250

D = 5"

D = 3"

D = 2"

Figure B6. Comparison of Calculated and Measured Initial Peak Loads
52

On the basis of videos and finite element simulations of the quasi-static compression tests, the 

initial plastic collapse load appears to occur when the pipe deforms into a shape resembling a 

square, as shown in Figure B7.

Figure B7. Collapse Mechanism with Eight Hinges

The equation derived from the equivalent beam method is also used to estimate the maximum 

peak in the observed force-crush characteristic by assuming n equal to 16:

2

2 10.25

o

t

D





 

    

(B-15)

The corresponding estimates and test measurements are listed in Table B2, and compared in

Figure B8.

In Section 3.2, nondimensional compressive strength is calculated by dividing the normalized

peak stress by the relative density of the core. For the pipe core, the relative density is related to 

the core thickness and the pipe outer diameter by

* 1 t t

D D

 

 

     

(B-16)

from which

1 4 1 1 *

2

t

D





 

      

. (B-17)

Combining equations (B-15) and (B-17) gives

2

2

2.56 4 1 1 *

* *

o





   

 

      

(B-18)

Results in Section 3.2 for the pipe cores are based on this equation.
53

Table B2. Plastic Collapse Estimates and Maximum Collapse Load Observed in Tests on 

Pipe Cores

D

(inches)

o

(ksi)

A

(in2)

PC (kip)

Calculated

(n = 16)

Test

2 52.3 144 302 273

3 43.0 144 110 128

5 37.1 150 36 37

Calculated Collapse Load (kips)

0 50 100 150 200 250 300 350

Peak Collapse Load in Test Data (kips)

0

50

100

150

200

250

300

350

D = 5"

D = 3"

D = 2"

Figure B8. Comparison of Calculated and Measured Maximum Peak Loads

Similarly, from the videos and finite element simulations of the tests, the deformed pipe at the 

maximum plastic collapse load resembles a polygon that consists of 16 hinges, as shown in

Figure B9. The figure shows that the sides of the polygon buckle inwards. In the videos and 

simulations, outward buckling occurs to the neighboring polygons or deformed pipes.

Figure B9. Collapse Mechanism with 16 Hinges
54

Figure B10 shows two video clips taken from the quasi-static compression test conducted on a 

sandwich panel with the 5-inch pipe core. The first clip shows the deformation of the cores at 

approximately 0.8-inch crush (or equivalently about 16 percent strain). In the second clip, the 

crush and the strain are about twice that of the first. Moreover, the idealized deformed shapes of 

the circular pipes are overlaid to show the resemblance to the collapse mechanisms assumed in 

the approximate limit analysis method.

Figure B10. Annotated Video Clips from Compression Test on 5-Inch Pipe Core

B.2 Diamond Core Estimates

Similar limit analysis is applied to examine the collapse load for the idealized X-core without 

weld flats, shown in Figure B11, which is equivalent to the diamond core.

Figure B11. Sandwich Panel with Idealized X-Core

Under compressive loading, the upper half of the center diamond in Figure B11 is assumed to 

behave in a similar manner as a triangular truss with pinned supports subjected to an equivalent 

concentrated load (Figure B12):
55 P B b     2

(B-19)

where 2 is the compressive stress, B is the cell width, and b is the depth of the panel.

Figure B12. Triangular Truss with Pinned Supports

A free body diagram of the right-hand member of the triangular truss is shown in Figure B13, 

from which the compressive force in this member is equal to P/2sin . 

Figure B13. Free Body Diagram of Truss Member

Compressive yielding of this member occurs when the axial stress is equal to the yield strength

2

2 sin 2 sin o

P B

tb t





 

 

(B-20)
56

where o is the yield strength of the core material and t is the thickness of the core. 

Therefore, the collapse strength of the idealized X-core or diamond core is given by

2 2 sin

o

t

B







 

    

(B-21)

The relative density of the diamond core is estimated from

2

*

sin

t

B





 

    

(B-22)

Assuming  equal to 45 degrees for the diamond core geometry and combining equations (B-21)

and (B-22) gives

2 0.5

*

o



 



(B-23)

Results in Section 3.2 for the diamond core are based on equation (B-23).

The plastic collapse load associated with sandwich panels with the diamond cores used in the 

quasi-static compression tests are calculated from: 

2 sin C o

t

P A

B

 

 

     

(B-24)

where A is the loaded cross-sectional area of the panel.

Table B3 lists the characteristics of the test articles for the diamond cores. The table also lists 

the estimated plastic collapse load calculated from the limit analysis, which overestimates the 

collapse load observed in the two tests conducted on panels with the diamond core. The value of 

yield strength assumed in calculated collapse load is derived from the average of five tensile tests 

measurements on AISI 1010 steel of the same vintage used in the compression tests. 

Interestingly, if the nominal value of 44.2 ksi is assumed for the yield strength, the calculated 

collapse load is equal to 429 kilopounds (kip), which is in excellent agreement with the test 

results.
57

Table B3. Panel Characteristics and Collapse Loads for Tests on Diamond Cores

Characteristic Value

Yield strength, o 60.5 ksi

Area, A 144 in2

Core height, h 2.62 in

Cell width, B 2.62 in

Angle,  45o

Calculated Collapse Load 588 kips

Measured Collapse Load (0047-13) 398 kips

Measured Collapse Load (0047-14) 420 kips
58

Appendix C. Force-Displacement Curves from Bend Tests and

Analyses

Displacement (inches)

0 1 2 3 4 5 6 Force (kips)

0

100

200

300

400

0047-15 (Manual Weld)

0047-16 (Robotic Weld)

Pre-Test FEA

Figure C1. Diamond Cores Parallel to Supports with Solid Face Sheets

Displacement (inches)

0 1 2 3 4 5 6 Force (kips)

0

100

200

300

400

0047-17 (Manual Weld)

0047-18 (Robotic Weld)

Pre-Test FEA

Figure C2. Diamond Cores Perpendicular to Supports with Solid Face Sheets
59

Displacement (inches)

0 1 2 3 4 5 6 Force (kips)

0

100

200

300

400

0047-24 (Robotic Weld)

0047-28 (Manual Weld)

Pre-Test FEA

Figure C3. Diamond Cores Parallel to Supports with Strip Face Sheets

Displacement (inches)

0 1 2 3 4 5 6 Force (kips)

0

100

200

300

400

0047-23 (Robotic Weld)

0047-27 (Manual Weld)

Pre-Test FEA

Figure C4. Diamond Cores Perpendicular to Supports with Strip Face Sheets
60

Displacement (inches)

0 1 2 3 4 5 6 Force (kips)

0

100

200

300

400

0047-29 Hybrid A

Pre-Test FEA

Figure C5. Diamond Cores Perpendicular to Supports with Hybrid A Face Sheets

Displacement (inches)

0 1 2 3 4 5 6 Force (kips)

0

100

200

300

400

0047-30 Hybrid B

Pre-Test FEA

Figure C6. Diamond Cores Perpendicular to Supports with Hybrid B Face Sheets
61

Displacement (inches)

0 1 2 3 4 5 6 Force (kips)

0

100

200

300

400

0047-19 (Manual Weld)

0047-20 (Robotic Weld)

Pre-Test FEA

Figure C7. Pipe Cores Parallel to Supports with Solid Face Sheets

Displacement (inches)

0 1 2 3 4 5 6 Force (kips)

0

100

200

300

400

0047-21 (Manual Weld)

0047-22 (Robotic Weld)

Pre-Test FEA

Figure C8. Pipe Cores Perpendicular to Supports with Solid Face Sheets
62

Displacement (inches)

0 1 2 3 4 5 6 Force (kips)

0

100

200

300

400

0047-26 (Robotic Weld)

Pre-Test FEA

Figure C9. Pipe Cores Parallel to Supports with Strip Face Sheets

Displacement (inches)

0 1 2 3 4 5 6 Force (kips)

0

100

200

300

400

0047-25 (Robotic Weld)

Pre-Test FEA

Figure C10. Pipe Cores Perpendicular to Supports with Strip Face Sheets
63

Abbreviations and Symbols

AAR Association of American Railroads

AISI American Iron and Steel Institute

ASTM American Society for Testing and Materials

ATLSS Advanced Technology for Large Structural Systems

CMI Cellular Materials International, Inc.

FRA Federal Railroad Administration

ft2square feet

in2square inch

kip kilopound

ksi kilopound per square inch

NTSB National Transportation Safety Board

OTS off-the-shelf

RAIRS Railroad Accident/Incident Reporting System

RSI Railway Supply Institute

A area

D pipe outer diameter

hc core height

l, L length

Mo fully plastic moment

r radius

tc core thickness

tf face sheet thickness

 angle

* relative density

o yield stress

peak peak stress