Cabello FinalReport

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Dynamic Stress Analysis of the effect of an Air Blast Wave on a Stainless Steel

Plate

by

Brian Cabello

A Project Submitted to the Graduate

Faculty of Rensselaer Polytechnic Institute

in Partial Fulfillment of the

Requirements for the degree of

MASTER OF MECHANICAL ENGINEERING

Approved:

_________________________________________

Professor Ernesto Gutierrez-Miravete, Project Adviser

Rensselaer Polytechnic Institute

Hartford, Connecticut

May, 2011



© Copyright 2011

by

Brian Cabello

All Rights Reserved



i

CONTENTS

LIST OF SYMBOLS ..................................................................................................................... ii

LIST OF FIGURES ...................................................................................................................... iii

ABSTRACT .................................................................................................................................... 1

1 INTRODUCTION/BACKGROUND ....................................................................................... 2

2 PROBLEM DESCRIPTION .................................................................................................... 3

2.1 AIR BLAST WAVES ....................................................................................................... 3

2.1.1 DETONATIONS ...................................................................................................... 3

2.1.2 AIR BLAST WAVES ............................................................................................. 4

2.1.3 PRESSURE-TIME DISTRIBUTION ..................................................................... 5

2.1.4 MAXIMUM/MINIMUM PRESSURE..................................................................... 6

2.1.5 IMPULSE ................................................................................................................. 7

2.1.6 NEGATIVE PHASE ................................................................................................ 7

2.1.7 WAVE FORM PARAMETER ................................................................................. 8

2.1.8 SHOCK FRONT VELOCITY ................................................................................. 9

2.1.9 SPECIFIC HEAT RATIO ........................................................................................ 9

2.2 TNT-METHOD .................................................................................................................. 9

2.3 COMPUTATIONAL METHODS ................................................................................... 12

3 METHODOLOGY/APPROACH........................................................................................... 13

3.1 ABAQUS MODELING APPROACHES AND TECHNIQUES .................................... 14

3.2 ANALSIS TYPES ........................................................................................................... 15

3.3 MATERIAL MODEL ..................................................................................................... 15

4 DISCUSSION ........................................................................................................................ 16

5 CONCLUSIONS .................................................................................................................... 18

6 REFERENCES ....................................................................................................................... 19

7 APPENDIX ................................................................................................................ 20



ii

LIST OF SYMBOLS

Symbol Eng. Unit Description

r meter Radius of explosion

W, WTNT kg TNT equivalent of the explosion

WHC kg Mass of hydrocarbons

ta sec The arrival time

td sec The positive phase duration

tn, tneg sec The duration of the negative phase

pmax Pa The peak overpressure

p0 Pa The reference pressure

pmin Pa The maximum negative pressure

pr Pa The reflected overpressure

γ N/A Ratio of specific heat of air

d meter Distance of the centre of the charge

I Pa-sec Impulse of the air blast wave

u m/sec Shock front velocity

c0 m/sec Sound velocity

c p kJ/kg K Specific heat at constant pressure

cv kJ/kg K Specific heat at constant volume

V m

Total volume of the congested region

ε p

N/A Effective plastic strain

TH Kelvin Homologous Temperature



iii

LIST OF FIGURES

Figure 1.0 Blast wave propagation

Figure 2.0 Pressure-time curve for a free air blast wave

Figure 3.0 Model of Kingery [4] with scaled distancesFigure 4.0 Different parameters for the negative phase

Figure 5.0 Peak explosion pressure (side-on) vs. distance for TNT ground burst

Figure 6.0 Peak explosion pressure (side-on) vs. distance for TNT equivalent method

Figure 7.0 Plate Boundary Condition

Figure 8.0 Blast Source

Figure 9.0 Element Mesh

Figure 10.0 Displacement of Top/Bottom Center Surface for Various Weight Charge

Figure 11.0 0.5kg TNT Charge at 1.5 millsec






iv

KEYWORDS

Blast Wave

Detonations

Pressure Time Distribution

Impulse

Negative Phase

Shock Front Velocity

TNT Method

Abaqus

Finite Element Method

Dynamic stress analysis

CONWEP

Johnson-Cook Flow Stress Model



0

ABSTRACT

In this report I analyzed the effect of a blast loading on a Stainless Steel Plate. The

Dynamic analysis was performed using the Abaqus explicit finite element program. Solutions

were computed up to 1.5 milliseconds, where no further permanent deformation was observedfor all load values. The modeling of the blast was implemented using the blast modeling

software CONWEP (Conventional Weapons Effects), which is an empirically based loading

model within Abaqus. The report identified key areas needed for the modeling a CONWEP blast

load. The property of the blast load was specified using the incident wave interaction property

and the CONWEP charge property at the model level and the incident wave interaction at the

step level. The different weight charges used for the analysis were 0.5, 1, 2, and 3 kg TNT. The

Johnson-Cook flow stress model was used for the computation of the deformation due to the

impulse loading produced by the explosive detonation. Plots were created to show maximum

deflection of the plate under these loads. Calculations show that the Stainless Steel plate was

permanently deformed with 1, 2, 3 kg TNT charges but that it withstood elastically the 0.5 kg

TNT.



1

1 INTRODUCTION/BACKGROUND

A bomb explosion within or immediately nearby a building can cause catastrophic

damage on the building’s external and internal structural frames, collapsing of walls and blowing

out of large pieces of structure. Loss of life and injuries to occupants can result from many

causes, including direct blast-effects, structural collapse, debris impact, fire and smoke. Due to

the threat from such extreme loading conditions, efforts have been made during the past three

decades to develop methods of structural analysis and design to resist blast loads. The analysis

and design of structures subjected to blast loads require a detailed understanding of blast

phenomena and the dynamic response of various structure elements.

An explosion is a rapid release of stored energy characterized by a bright flash and an

audible blast. Part of the energy is released as thermal radiation (flash); and part is coupled into

the air as air blast and into the soil (ground) as ground shock, both as radially expanding shock

waves. The rapid expansion of hot gases resulting from the detonation of an explosive charge

gives rise to a compression wave called a shock wave, which propagates through the air. The

blast wave instantaneously increases to a value of pressure above the ambient atmospheric

pressure. This is referred to as the positive phase that decays as the shock wave expands outward

from the explosion source. After a short time, the pressure behind the front may drop below the

ambient pressure. A schematic representation of these processes is shown in Figure 1.0. During

such a negative phase, a partial vacuum is created and air is sucked in. This is also accompanied by high suction winds that carry the debris for long distances away from the explosion source.

Figure 1.0 Blast wave propagation [3]

As the shock wave travels outward from the charge, the pressure in the front of the wave,

called the peak pressure, steadily decreases. At great distances from the charge, the peak

pressure is infinitesimal, and the wave can be treated as a sound wave.



2

2 PROBLEM DESCRIPTION

This project will focus on the analysis of how different blast loading (shock wave) affects

a plate of High Ductility Stainless Steel Alloy (Al-6XN). The specimen will consist of a 0.305

m x 0.305 m x 0.061 m plate. The solid plate is modeled with three-dimensional continuum

elements and is subjected to CONWEP blast loading [12] using different charge masses (0.5, 1, 2

and 3 kg TNT).

2.1 Air Blast Waves

2.1.1 Detonations

Explosions can be distinguished in detonations and deflagrations. The difference

between detonations and deflagrations is the velocity of the reaction zone in the explosive.

Deflagrations have a slower reaction zone than the sound speed. Examples for deflagrations are

the burning of gas-air-mixtures and slow explosives like gun powder.

Detonations have a faster reaction zone than the sound speed. The most common explosives

react with detonations.

To compare different explosive the TNT equivalent can be used. The TNT equivalent is

a method for quantifying the energy released in the detonation of an explosive substance, by

comparing it to that of an equal quantity of TNT. It is known that 1 kg TNT releases the energy

of 4.520x106 J.

The effects of an explosion can be distinguished in three ranges:

Contact detonation: The explosive is in contact with the loaded material. The load-time

function depends on the loaded material, which, in most cases, is destroyed.

Occurrences are the blasting of concrete (demolition etc.) or terrorist attacks where the

explosive is located directly on the structure.

Near zone of the explosion: In most cases the material is also directly damaged like in the

contact zone.

Far zone. The blast wave resulting from the detonation dominates the effects on humansand structures.

The size of all these zones depends on the quantity of the explosive charge.



3

Additional parameters for a detonation, depending on the size of the explosive, can be defined.

For example, the radius in which debris from the explosion (not from the blast wave) are

possible is give by Kinney [5] as

3

1

45W r (1)

Where, r is expressed in m and W is the TNT equivalent of the explosive in kg.

2.1.2 Air Blast Waves

The pressure that arrives at a certain point depends on the distance and on the size of the

explosive.

Figure 2.0 Pressure-time curve for a free air blast wave [5]

The main characteristics of the development of this pressure wave are the following:

The arrival time ta of the shock wave to the point under consideration. This includes the

time of the detonation wave to propagate through the explosive charge.

The peak overpressure pmax – The pressure attains its maximum very fast (extremely short

rise-time), and then starts decreasing until it reaches the reference pressure p0 (in most

cases the normal atmospheric pressure).

The positive phase duration td, which is the time for reaching the reference pressure.

After this point the pressure drops below the reference pressure until the maximum

negative pressure pmin. The duration of the negative phase is denoted as tn.

The incident overpressure impulse, which is the integral of the overpressure curve over

the positive phase td.



4

The idealized (free air blast) form of the pressure wave of Figure 5.0 can be greatly altered

by the morphology of the medium encountered along its propagation. For instance, peak

pressure can be increased up to 8 times if the wave is reflected on a rigid obstacle. The effects of

the reflection depend on the geometry, the size and the angle of incidence. By setting γ =1.4

(ratio of specific heats of air), it can be shown that the reflected overpressure pr is

max0

maxmax

7

472

p p

p p p p o

r (2)

All parameters of the pressure time curve are normally written in terms of a scaled distance

3 W

d Z (3)

where W is the mass of the explosive charge and d the distance of the centre of the charge.

2.1.3 Pressure-Time Distribution

There are several pressure-time-curves for different kinds of explosions. The effects of

nuclear explosions here should be disregarded.

The pressure at a known point can be described by the modified Friedlander equation (from

Baker [6]) and depends on the time t from the arrival of the pressure wave at the time (t = t0-ta)

d t

bt

d t

t p pt p )1()( max0 (4)

The other parameters involved are the atmospheric pressure p0, the maximum

overpressure pmax and the duration of the positive pressure td. The parameter b describes the

decay of the curve. It can be calculated with a known minimum pressure after the positive

phase. Alternatively, the parameter b can be calculated with the knowledge of the impulse.

All parameters for the pressure-time curve can be taken from different diagrams and equations

(Baker [6], Kinney [5], Kingery [4]) See Figure 6.0.



5

Figure 3.0 Model of Kingery [4] with scaled distances

2.1.4 Maximum/Minimum Pressure

Kingery [4] developed in 1984 curves for the description of the different air blast

parameters by using a rich body of experimental data, which had been properly homogenized.



6

The parameters are presented in double logarithmic diagrams with the scaled distance Z as

abscissa, but are also available as polynomial equations. These diagrams and equations enjoy the

greatest overall acceptance and are widely used as reference by most researchers. The

parameters are also implemented in different computer programs that can be used for the

calculation of air blast wave values. e.g. they are implemented in CONWEP. The same curves

are also used for an easy air blast load model in ABAQUS.

2.1.5 Impulse

The impulse of the air blast wave has a big influence on the response of the structures.

The impulse is defined here as the area under the pressure time curve with the unit of

pressure*sec. The impulse can be calculated with [5].

3 42

4

)55.1/(1

)23.0/(1067.0

Z Z

Z I (5)

2.1.6 Negative Phase

Detonations produce an overpressure peak, and afterwards the pressure decreases and

drops below the reference pressure (generally the atmospheric pressure). The influence of the

so-called negative phase depends on the scaled distance. For scaled distances Z larger than 20

and especially for Z larger than 50 the influence of the negative phase can not always be

neglected. The size of the positive impulse and of the negative impulse is then nearly the same.

If the structure can react successfully to the positive pressure but is more sensitive to negative

pressure, failure of parts of the structure can result from this negative pressure phase [8].

However, in several cases the negative phase is neglected e.g. in the air blast function of the

CONWEP-Code. Smith [7] presents the following equation to calculate the value of the negative

pressure.

5

min 1035.0

Z p Pa for Z > 1.6 (6)

The duration time of the negative pressure pmin can be calculated with

31

00125.0 W t n [sec] (7)

Another possibility to get these parameters in a diagram (see Figure 7.0) in Krauthammer [xx].

By using this diagram the limitation of equation (7) can be overcome by assuming

5

min 1035.0

Z p Pa for Z > 3.5 (8)





8

2. Using the impulse of the positive phase. Then, as it will be shown, the minimal pressure

in the negative phase is not accurate. An additional equation for the negative phase

should be used to avoid a smaller underpressure than the atmospheric pressure.

2.1.8 Shock Front Velocity

The arrival time of the shock front at different points can be used to calculate the velocity

of the shock front. With the knowledge of this velocity the pressure can be obtained with the

Rankine Hugoniot relationship.

Kingergy [4] calculates also the shock front velocity depending on the pressure as

21

)2

11( max

0

o p

pcu (11)

The parameter γ (ratio of specific heats of air) depends also on the overpressure and can be taken

from a table in [9]; c0 is the sound velocity in air (331 m/sec); pmax is the peak overpressure and

p0 is the atmospheric pressure (101.3 kPa).

2.1.9 Specific Heat Ratio

The specific heat ratio γ is defined as

v

p

c

c (12)

with c p being the specific heat at constant pressure and cv the specific heat at constant volume.

Both the specific heat ratio and the speed of sound depend on the temperature, the pressure, thehumidity, and the CO2 concentration. Kingery [4] defines the variation of the specific heat ratio

with a range of 1.402 to 1.176.

2.2 TNT-Method

The diagram for TNT detonations (Figure 8.0) have been used for estimations of blasts

from gas explosions, even though there are differences between the blasts from a gas explosion

and a TNT-detonations (Shepherd et al., 1991, van den Berg, 1985). In a gas explosion the local

pressure may reach values as high as a few bars. The blast pressure for TNT explosions is much

higher close to the charge. Such near-field data are therefore irrelevant for gas explosions and it

is recommended not to use TNT-data indicating pressures higher than 1 bar to estimate gas

explosion blasts.



9

Figure 5.0 Peak explosion pressure (side-on) vs. distance for TNT ground burst [4]

The so-called TNT equivalence method has been widely used for gas explosions. The TNT

equivalence method applies pressure-distance curves for TNT explosions to gas explosions and

the equivalent TNT charge is estimated from the energy content in the exploding gas cloud.

For typical hydrocarbons, such as methane, propane, butane etc., the heat of combustion is

10 times higher than the heat of reaction of TNT. The relation between the mass of

hydrocarbons WHC and the equivalent TNT charge WTNT is then

HC TNT W W **10 (13)

where h is a yield factor (η = 3%-5%), based on experience, see Gugan, 1978.

In the original TNT equivalence method, the mass of hydrocarbon WHC was based on the

total mass released and the yield factor η. In order to estimate consequences of gas explosions,

the geometrical conditions (i.e. confinement and obstructions) have to be taken into account. In

the original TNT equivalence method, the geometrical conditions are not taken into account. The

results from this type of analysis have therefore hardly any relevance and should in general not

be used.

The drawbacks of the TNT equivalence method can be listed as follows:

a non-unique yield factor is necessary

representation of weak gas explosions is a problem

positive phase duration only

gas explosion processes are not represented well

choice of "blast centre" is problematic (no well-defined and sensible method exists)



10

In order to take the geometrical effects into account in the TNT equivalence method, Harris

and Wickens (1989) proposed to use a yield factor of 20% (η = 0.2) and the mass of

hydrocarbon, WHC, contained in Stoichiometric proportions in any severely congested region of

the plant. For natural gas the equivalent mass of TNT can be estimated from (assuming

atmospheric pressure initially)

][16.0 kg V W TNT (14)

where V [m3] is the smaller of either the total volume of the congested region or the volume of

the gas cloud. Equation 14 will also hold for most hydrocarbons, since the energy content per

volume Stoichiometric mixture is approximately the same (~3.5MJ/m3).

Figure 9.0 shows the results from a TNT equivalent analysis, as suggested by Harris and

Wickens, in comparison with CMR's experimental results from 50 m3 tests.

Figure 6.0 Peak explosion pressure (side-on) vs. distance for TNT equivalent method [4]

As we can see from this figure there is fairly good agreement between the predicted values

and the experimental values as long as the explosion pressure in the cloud is in a few bars range.

Weak gas explosions (less than 0.5 bar) are not represented satisfactorily. This indicates that the

TNT equivalence method can be useful as a rough approximation if one uses a yield factor of

20% and appropriate values for WHC or V. However, for explosion pressures below 1 bar, the

TNT equivalence method will overestimate the blast. More sophisticated methods must therefore

be applied for such cases.



11

2.3 Computational Methods

Computational methods in the area of blast-effects mitigation are generally divided into

those used for prediction of blast loads on the structure and those for calculation of structural

response to the loads. Computational programs for blast prediction and structural response use

both first-principle and semi-empirical methods. Programs using the first principle method can

be categorized into uncouple and couple analyses. The uncouple analysis calculates blast loads

as if the structure were rigid and then applying these loads to a responding model of the

structure. The shortcoming of this procedure is that when the blast field is obtained with a rigid

model of the structure, the loads on the structure are often over-predicted, particularly if

significant motion or failure of the structure occurs during the loading period. For a coupled

analysis, the blast module is linked with the structural response module. In this type of analysis

the computational fluid mechanics (CFD) model for blast-load prediction is solved

simultaneously with the computational solid mechanics (CSM) model for structural response.

To account for the motion of the structure while the blast calculation proceeds, the pressures that

arise due to motion and failure of the structure can be predicted more accurately by using

ABAQUS, AUTODYN, DYNA3D AND LS-DYNA.



12

3 METHODOLOGY/APPROACH

The boundary conditions for the analysis will have all degrees of freedom including the

rotational degrees of freedom fixed at the top and bottom side of the plate (Figure 2.0).

Figure 7.0 Plate Boundary Condition

The 0.5, 1, 2 and 3 kg TNT blast source (RP-1) is kept at a standoff distance of 0.20 m from

the front surface and center of the plate (Figure 3.0). The plate surface uses a 31 x 31 C3D8R

element with five layers of elements through the thickness of the plate (Figure 4.0).

FixedBoundaryCondition



13

Figure 8.0 Blast Source

Figure 9.0 Element Mesh

3.1 Abaqus modeling approaches and simulation techniques

This project demonstrates the usage of CONWEP blast loading using Abaqus. The solid

plate is modeled using three-dimensional continuum elements and subjected to CONWEP blast

loads due to 0.5, 1, 2, and 3 kg of TNT.

Blast Source



14

Summary of analysis cases

Case 1 Solid plate modeled with C3D8R continuum elements under 0.5 kg TNT blast load.




3.2 Analysis Types

Dynamic analysis using Abaqus is performed for all cases. Solutions are computed up to

1.5 milliseconds, where no further permanent deformation is observed for all load values.

3.3 Material Model

The mechanical properties of the steel alloy as described in Nahshon [11] are specified as

follows: Young’s modulus of 1.61 x 105 MPa, Poisson’s ratio of 0.35, density of 7.85 x 10

-9

metric tons/mm3, and coefficient of expansion of 452 x 106 Nmm/metric tons,K.

A Johnson-Cook model [13] is used to model the elastic – plastic behavior with the

following coefficients and constants: A=400 MPa, B=1500 MPa, C=0.045 , n=0.4, m=1.2, and

έ p0=0.001 s-1

. The Johnson-Cook model is a phenomenological model, i.e. it is not based on

traditional plasticity theory that reproduces several important material responses observed in

impact and penetration of metals. The three key material responses are strain hardening, strain-

rate effects, and thermal softening.

])(1)[ln1]()([),,(

* m

p

n

p p pY T C B AT

(15)

Where p is the equivalent plastic strain, p is the plastic strain-rate, and A, B, C, n, m are

material constants. The normalized strain-rate and temperature in equation (15) are defined as

0

*

p

p

p

(16)

)(

)(

0

0*

T T

T T T

m

(17)

Where 0 p is a user defined plastic strain-rate, 0T is a reference temperature, and mT is a

reference melt temperature. For conditions where *T < 0, we assume that m = 1.



15

4 DISCUSSION

The center displacement after 1.5 milliseconds was monitored to compare each case with

experimental results. The animation of the deformed plate over the entire time period of 1.5

milliseconds shows large deformation at the center. The plate stabilizes after few oscillations.

The top surface center deflection and the bottom surface center deflection were quite close to

each other. The two values were plotted (Figure 10.0) for each case to compare the midsection

deflection of the solid plate.

Figure 10.0 Displacement of Top/Bottom Center Surface for Various Weight Charge



16

Figure 11.0, 0.5kg TNT Charge at 1.5 millsec Figure 12.0, 1.0kg TNT Charge at 1.5 millsec

Figure 13.0, 2.0kg TNT Charge at 1.5 millsec Figure 14.0, 3kg TNT Charge at 1.5 millsec



17

5 CONCLUSION

This report describes the results of dynamic stress analysis of the effect of a blast loading

on a Stainless Steel Plate using the finite element method. The solid plate was a high ductility

stainless steel alloy (Al-6XN) comprised of 49% Fe, 24% Ni, 21% Cr, and 6% Mo by weight.

Solutions were computed up to 1.5 milliseconds, where no further permanent deformation was

observed for all load values. The mechanical behavior of the material of the plate was

represented by Johnson-Cook Flow Stress Model and the blast wave modeling was represented

by the CONWEP model. All calculation were carried out using Abaqus explicit. The different

weight charges used for the analysis were 0.5, 1, 2, and 3 kg TNT and the maximum

displacement at various charges weights were 7.14 mm, 14.64 mm, 27.47 mm, and 55.86 mm

respectively. Results show that the plate was permanently deformed for the 1, 2, and 3 kg TNT

charges and that it deformed elastically for the 0.5 kg TNT charge with a maximum deflection of

0.4 mm at 1.5 millisec.





19

7 APPENDIX

7.1 Abaqus Input Files (Hyperlink)

..\02 Abaqus Modeling\Input Files\Result_05TNT.inp

..\02 Abaqus Modeling\Input Files\Results_1TNT.inp



http://localhost/var/www/apps/conversion/tmp/02%20Abaqus%20Modeling/Input%20Files/Result_05TNT.inp


http://localhost/var/www/apps/conversion/tmp/02%20Abaqus%20Modeling/Input%20Files/Results_1TNT.inp










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