MSC - Pressure evolution inside complex corridor system induced by blast action.pdf

7/30/2019 MSC - Pressure evolution inside complex corridor system induced by blast action.pdf

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Poznan University of Technology

Faculty of Civil and Environmental Engineering

Masters Thesis

Pawe Mamrak

Pressure evolution inside complex corridor

system induced by blast action

Supervisors: Marcin Wierszycki, PhD,

Piotr Sielicki, MSc.

Pozna 2011


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Contents

Contents ............................................................................................ 3

1. Introduction ............................................................................. 52. Phenomenon of explosion ...................................................... 7

2.1. Definition and classification of explosives ..........................................72.2. Blast phenomenon ...............................................................................7

2.3.

Blast-loading classification ................................................................ 12

2.4. Calculation methods of blast-loading ............................................... 193. Finite Element Analysis techniques ..................................... 224. Tools and methods .................................................................23

4.1. Software .............................................................................................. 23

4.2.

Creation of CEL model in Abaqus .................................................... 26

4.3. Assumptions and restrictions in the script ...................................... 294.4. Simplified block scheme of the script ...............................................304.5. Structure of the script ........................................................................ 344.6. Problems and difficulties .................................................................. 404.7. Further development ......................................................................... 41

5. Analyses ................................................................................. 425.1. Benchmarking analysis of blast wave propagation ......................... 42

5.1.1. Description .......................................................................................................... 425.1.2. Analytical solution .............................................................................................. 435.1.3. Numerical solution ............................................................................................. 435.1.4. Discussion of the results .................................................................................... 44


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5.2. Analysis of the cubicle ...................................................................... 465.2.1. Description .......................................................................................................... 465.2.2. Analytical solution .............................................................................................. 475.2.3. Numerical solution ............................................................................................. 565.2.4. Discussion of the results .................................................................................... 60

5.3. Shock wave propagation analysis in corridor .................................. 655.3.1. Description .......................................................................................................... 655.3.2. Numerical solution ............................................................................................. 65

5.4. Shock wave propagation analysis in tunnel shelter ........................ 745.4.1.

Description .......................................................................................................... 74

5.4.2. Numerical solution ............................................................................................. 755.4.3. Discussion of the results .................................................................................... 86

6. Summary ................................................................................ 88Bibliography .................................................................................... 91

Appendix A .................................................................................... 93Appendix B ................................................................................... 102


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1. IntroductionPreface

In recent years one can observe fast development of design methods in the field of civil

engineering. The threat of terrorism (but not only) pointed peoples attention to the

problem of resistance of the buildings to different, extreme kinds of accidental loading.

Among them the most dangerous seem to be explosions, which can seriously damage or

even destroy the structure of the object, forcing it to collapse. Also buildings, where

explosive materials are manufactured and stored should be evaluated against structure's

resistance to the effects of accidental explosions, as concentration of explosives increases

this possibility.

Nowadays computers are sufficiently powerful to successfully deal with these topics.

It is possible to simulate the explosion taking place in a given space, as well as behaviour of

the structure subjected to dynamic loading induced by the explosions (mainly air pressure).

Both analyses are relatively new issue; therefore they are tersely described in national

standards. In case of blast simulations the principal interest is value of loading subjecting

the building. Many engineering methods have been introduced in the past, computing the

load on a basis of weight of the explosive and distance to the structure. Many applications,

e.g. ConWep for years has been serving people as useful tool to predict the influence of the

blast wave. This approach, although widely used all over the world, e.g. to estimatepressure acting on a structure of tall buildings, is sometimes insufficient, as it does not take

into account individual circumstances.

In this thesis author presents a method of blast wave propagation analysis in Abaqus

Explicit. In order to perform this kind of analyses, so called coupled Eulerian-Lagrangian

analysis has been applied. The method allows detailed simulation of blast phenomena,

requires however slightly different approach in modelling elements comparing to standard,

Lagrangian analysis technique. The differences are described in next chapters.

In order to automate the modelling process a special script has been prepared.

Objectives

Within the thesis four groups of simulations are conducted. In the first two, one seeks for

appropriate mesh size on the basis of comparison between the analytical and numerical

results. The parameters describing the shock wave are studied. In third simulation, the

main objective is to obtain information on propagation of shock wave for different


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conditions (e.g. arrangement and dimensions of corridors, location of explosives). The

fourth simulation concerns analysing explosion in tunnel shelter. The results are compared

to values obtained in experiments by Ishikawa and Beppu [27].

Structure of the thesis

The thesis consists of six chapters, followed by bibliography and one appendix. In the

Chapter 1 the motivation and objectives are briefly discussed. Chapter 2 presents the

theoretical background of explosion phenomenon. The types of explosions are described,

followed by explanation of the crucial shock wave parameters. At the end, the historical

and the current calculation methods of blast wave parameters are briefly described.

Chapter 3 outlines the basic kinematical descriptions of the continuum that are used whiledealing with mechanics of fluids and solids. These issues are described in more detail in

Appendix A. In the Chapter 4 the tools that were used for computations are described, that

is: Abaqus/Explicit, Abaqus Scripting Interface, Python programming language. It is

followed by a short overview of creating CEL model in the mentioned tools. Later, the

script that was prepared for the purpose of easy and fast modelling of complex corridor

systems is described. The issues such as assumptions that have been made in the script, its

structure, as well as problems that have been encountered are discussed. Lastly authors

thoughts on further development of the script are given. Chapter 5 concerns the numericalsimulations that have been conducted. The first simulation is a comparison of results

obtained using Unified Facilities Criteria (UFC) methods [7] and Abaqus/Explicit. The

second analysis deals with problem of blast wave propagation in a simple cubicle, where

numerical results are compared to those calculated with use of analytical algorithms. The

third simulation concerns behaviour of blast wave reflecting in a corridor broken by some

angle. The fourth simulation is a numerical reconstruction of experiments performed by

Japanese scientists in a tunnel shelter [27]. Chapter 6 is an ultimate summary of the

information gathered in previous chapter. Appendix A discusses different Finite Element

techniques, presents derivation of important equations of Eulerian, Lagrangian, Arbitrary

Lagrangian-Eulerian and Coupled Eulerian-Lagrangian descriptions of continuum. Later

the universal conservation laws in the non-conservative and conservative forms are

provided. Appendix B contains UML class diagram of Python script.


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2.Phenomenon of explosion2.1. Definition and classification of explosivesExplosion is a rapid increase in volume and release of energy in an extreme manner [4].

One can distinguish two types of explosive materials depending on the speed of

propagation of the shock wave, namely "high explosives", if the detonation is supersonic

and low explosives when the burning process is slower. Then this process is known as

deflagration.

Explosives, according to physical state can be classified as solids, liquids or gases.

Solid explosives are mainly high explosives for which blast effects are best known. They can

also be classified on the basis of their sensitivity to ignition as secondary or primary

explosives. The latter are ones that can be easily detonated by simple ignition from a spark,

flame or impact. Secondary explosives when detonated create blast (shock) waves which

can result in widespread damage to the surroundings.

The most common are explosions caused by chemical reactions. The first such a

material was invented in 9th century in China and is known as black powder. In the

Industrial Revolution of the 18th and 19th centuries, when a very significant and rapid

development of chemistry was observed, new explosive materials have been invented. One

can mention here nitro-glycerine, nitrocellulose, smokeless powder, and probably the most

famous one, invented by Alfred Nobel, dynamite. Since World War II one of the most

popular materials has become trinitrotoluene, called shortly TNT. It is a high explosive

solid of yellow colour, an example of secondary explosive, which nature is a chemical event.

The numerous applications of TNT caused that it is considered to be the standard measure

of strength of bombs and other explosives. The method of quantifying the energy released

in explosions is called TNT equivalent. The ton of TNT is a unit of energy equal to the

amount of energy released in the detonation of one ton of TNT (which is approximately

4.184 gigajoules). TNT equivalency is usually based on experimentally determined factors

or the ratio of its heat of detonation to that of TNT [30].

2.2. Blast phenomenonIn the moment of ignition, the explosive charge rapidly releases energy in the forms such as

heat, sound and very dense and high pressure wave. From the viewpoint of influence on the

structure the most important is the latter. The detailed descriptions of detonation process

and blast wave propagation can be found in papers [6] and [7].


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In this thesis author focuses on the most common chemically induced explosions. In

the process of detonation a very rapid and stable chemical reaction takes place, which

proceeds through the explosive material at a supersonic speed, called the detonation

velocity. Detonation velocities range from 6700 to 8500 meters per second for most high

explosives. The detonation wave rapidly converts the solid or liquid explosive into a very

hot (about 3000- 4000C [6]), dense, high-pressure gas, and the volume of this gas which

had been the explosive material is then the source of strong blast waves in air. Pressures

immediately behind the detonation front range from 19000MPa to 33800MPa [7]. Only

about one-third of the total chemical energy available in most high explosives is released in

the detonation process. The remaining two-thirds are released more slowly in explosions in

air as the detonation products mix with air and burn. This afterburning process has only a

slight effect on blast wave properties because it is much slower than detonation. The blast

effects of an explosion are in the form of a shock wave composed of a high-intensity shock

front which expands outward from the surface of the explosive into the surrounding air.

Figure 2-1. Blast wave pressure-time history [6].


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The shape of the blast wave is very characteristic its front is very abrupt, reaching a

very high value of pressure, then a rapid decrease is visible after which the pressure returns

to initial value of the ambient atmospheric pressure (Figure 2-1).

After a short time, the pressure behind the front may drop below the ambient

pressure (Figure 2-2). During such a negative phase, called also negative-phase or suction

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 [6].

Figure 2-1. Blast wave propagation [6].

Along with propagation of the wave, following observations can be perceived: the

wave decays in strength (the overpressure decreases steadily), lengthens in duration, and

decreases in velocity. This phenomenon is caused by spherical divergence as well as by the

fact that the chemical reaction is completed, except for some afterburning associated with

the hot explosion products mixing with the surrounding atmosphere.

The scheme describing shock wave propagation is very characteristic. It is usually

called blast wave pressure-time profile, also referred to as overpressure curve. The main

components describing the overpressure curve are (followed by [5]):

- peak positive overpressure,- peak negative under pressure,- dynamic pressure,- positive and negative phase durations,- positive and negative phase impulses (integrals with respect to time of the

respective pressures).


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Figure 2-1 shows a typical blast pressure profile of ideal free air blast wave. The

detonation can be considered to take place at time t = 0s. The first important component is

the arrival time that is the time that it takes for the pressure wave to reach the point of

interest. Pressure at that position suddenly increases to a peak value of overpressure; over

the ambient pressure (ambient pressure is marked with dashed line). The pressure then

immediately decays to ambient level (here the point of equalization is at time equal to 1.2

milliseconds), then decays further to an under pressure (creating a partial vacuum) before

eventually returning to ambient conditions at time. The peak pressure is usually referred to

as the peak side-on overpressure, incident peak overpressure or merely peak overpressure.

Throughout the pressure-time profile, two main phases can be observed - portion

above ambient is called positive phase duration, while that below ambient is called

negative phase duration. The negative phase is of a longer duration and a lower intensity

than the positive duration.

The shock wave overpressure curve is important from the standpoint of civil

engineer as it a basis for determination of dynamic pressure. The dynamic pressure

determines the value of loading that is subjecting the structure. Generally blast loading on

a structure caused by a high-explosive detonation is dependent upon several factors:

- the magnitude of the explosion,- the location of the explosion relative to the structure of interest (unconfined or

confined),

- the geometrical configuration of the structure,- the structure orientation with respect to the explosion and the ground surface

(above, flush with, or below the ground).

Experiments prove that for each pressure range there is a particle or wind velocity

associated with the blast wave that causes a dynamic pressure on objects in the path of the

wave. In the free field, these dynamic pressures are essentially functions of the air density

and particle velocity. For typical conditions, standard relationships have been established

between the peak incident pressure, the peak dynamic pressure, the particle velocity, and

the air density behind the shock front. The magnitude of the dynamic pressures, particle

velocity and air density is solely a function of the peak incident pressure, and, therefore,

independent of the explosion size. Figure 2-3 gives the values of the parameters versus the

peak incident pressure. The dynamic pressure is the most important parameter for

determining the loads on structures.


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Figure 2-3. Peak incident pressure versus peak dynamic pressure, density of air behind the

shock front, and particle velocity [7].

Summarizing the size and material of the charge, as well as the stand-off distance

(distance between the detonation point and examined point) and charges shape will all

determine the magnitude and shape of the overpressure curve (the bigger stand-off

distance, the longer duration of the positive-phase, lower amplitude and smaller intensity

of the shock pulse). Additionally, the blast wave and the involved pressure can reflect off of

surfaces (like ground or structures) in various directions, and cause further fluctuations

(and reinforcement) in pressure at a single point. The reinforcement of the blast wave is

described in next paragraph in more details.


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2.3. Blast-loading classificationTwo blast-loading categories have been distinguished (stated after [7]). The division bases

on the confinement of the explosive charge, and so there are unconfined and confinedexplosions. These two categories can be further subdivided into next six basing on the

location of the explosives and the structure. The Figure 2-4 presents full classification:

Figure 2-4. Blast loading categories [7].

Free air burst and surface burst are very important because they result in so-called

ideal blast waves.

Of the six categories, those from air bursts are seldom encountered and the free air

burst is the least likely to occur. The possibility of such blast environments exists where

potentially explosive materials are stored at heights adjacent to or away from protective


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structures such as in manufacturing (process or storage tanks) or missile sites. In the latter,

the rocket propellant would be a source of explosive danger to the ground crew and control

facilities. The other four blast-loading categories can occur in most explosive

manufacturing and storage facilities. In such installations, transportation of explosive

materials between buildings either by rail, vehicle, or in the case of liquid or gases, through

piping, is a possibility. Also, storage and handling of explosives within buildings are

common occurrences.

The Figure 2-4 shows also the five possible pressure loads associated with the blast

load categories, the location of the explosive charge which would produce these pressure

loads, and the protective structures subjected to these loads.

Unconfined explosions

An explosion, which occurs in free air, produces an initial output whose shock wave

propagates away from the centre of the detonation, striking the structure without

intermediate amplification of its wave (Figure 2-5), which takes place during ground

impact (the main part).

Figure 2-5. Free-air burst blast environment [7].


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As the incident wave moves radially away from the centre of the explosion, it will

impact with the structure, and, upon impact, the initial wave (pressure and impulse) is

reinforced and reflected (Figure 2-6). The reflected pressure pulse of Figure 2-6 is typical

for infinite plane reflectors.

Figure 2-6. Pressure-time variation for a free-air burst [7].

The variation of the pressure and impulse patterns on the surface of a structure

between the maximum and minimum values is a function of the angle of incidence. This

angle is formed by the line which defines the normal distance RA between the point of

detonation and the structure, and line R (slant distance) which defines the path of shock

propagation between the centre of the explosion and any other point in question on the

structure surface (Figure 2-5).

The dependency of peak reflected pressure on the angle of incidence can be seen in

Figure 2-7 for a number of different shock strengths.


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Figure 2-7. Reflected pressure coefficient versus angle of incidence [7].

The reflected overpressure can be much larger than the incident overpressure. For

very strong shocks, where the ideal gas approximation is no longer valid, the predicted

upper limit for reflected overpressure is much as 20 times the incident overpressure. It

means that the objects surrounding a structure can create reflected waves that increase the

loading seen by the structure. Additionally, there can be reinforcement in the corner

geometries where multiple reflected waves can interact [19].

Air Burst Explosion

Air burst explosion is an explosion which is located at a distance from and above the

structure, so that the ground reflections of the initial wave occur prior to the arrival of the

blast wave at the structure. When the incident wave is reinforced by the ground effect, two

phenomena can occur: a classical reflection (Figure 2-8) or a reinforcement reflection

(called Mach Front, Figure 2-9).


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Figure 2-8. Air burst blast environment classical reflection [8].

Figure 2-9. Air burst blast environment reinforced reflection [7].

The Mach front is formed by the interaction between initial (incident) and reflected

pressure waves. This reflected wave is the result of the reinforcement of the incident wave

by the ground surface. The occurrence of mentioned interaction depends on the angle of

incidence between ground and incident wave. The critical angle is of around 40 (for < 40

regular reflection takes place). The pressure-time variation of the Mach Front is similar to

that of the incident wave except that the magnitude is somewhat larger [8].


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Surface Explosion

A surface burst explosion occurs when the detonation is located close to or on the ground.

The initial wave of the explosion is reflected and reinforced by the ground surface to

produce a reflected wave. Unlike the air burst, the reflected wave merges with the incident

wave at the point of detonation to form a single wave, similar in nature to the Mach wave

of the air burst but essentially hemispherical in shape (Figure 2-10) [7].

Figure 2-10. Surface burst blast environment [7].

Confined explosions

Confined explosions are a very complex issue. When an explosion occurs inside a building

the presence of the walls and ceiling greatly increases the number of blast wave structure

interactions. Multiple reflections take place, and many waves coalesce to produce

enhancements in corners and other local constrictions. The reflections can extremely

amplify the peak pressures associated with the initial shock front. In addition, and

depending upon the degree of confinement, the effects of the high temperatures and

accumulation of gaseous products produced by the chemical process involved in the

explosion will exert additional pressures and increase the load duration within the

structure. The biggest difference between internal and external explosions, however, is the

presence of the quasi-static gas pressure. Condensed explosives are approximately a

thousand times denser than air. The detonation of an explosive in a building will introduce

a quantity of hot gas into the building, as well as the shock waves mentioned above.


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Depending on the relative magnitudes of the mass of the explosive and the volume of the

building, the gas pressure may be the dominant loading mechanism on the building

elements. Because normal buildings (as opposed to containment buildings) have doors,

windows, heating ducts, etc., which allow the gas to vent into adjacent rooms or the

outside world, the gas pressure does not persist. The decay of the pressure, however, takes

place in a time-scale much longer than the duration of the individual shock reflections, and

the overall duration is typically much longer than the structural response time of elements

loaded in the building. For this reason it is referred to as quasi-static gas pressure.

When an explosion occurs within a confined area, gaseous products will accumulate

and temperature within the structure will rise, thereby forming blast pressures whose

magnitude is generally less than that of the shock pressure but whose duration is

significantly longer [7]. The magnitude of the gas pressures as well as their durations is a

function of the size of the vent openings in the structure.

Figure 2-11. Combined shock and gas pressures for small and/or square chamber [7].

Figure 2-11 illustrates an idealized pressure-time curve considering both the shock

and gas pressures. As the duration of the gas pressures approaches that of the shock

pressures, the effects of the gas pressures on the response of the elements diminishes until

the duration of both the shock and gas pressures are equal and the structure is said to be

fully vented.


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Fully Vented Explosion

A fully vented explosion will be produced within or immediately adjacent to a barrier or

cubicle type structure with one or more surfaces open to the atmosphere. The initial wave,

which is amplified by the nonfrangible portions of the structure, and the products of

detonation are totally vented to the atmosphere (there is no gas pressure build-up) forming

a shock wave (leakage pressures) which propagates away from the structure.

Partially Confined Explosion

A partially confined explosion will be produced within a barrier or cubicle type structure

with limited size openings and/or frangible surfaces. The initial wave, which is amplified by

the frangible and nonfrangible portion of the structure, and the products of detonation are

vented to the atmosphere after a finite period of time. The confinement of the detonation

products, which consist of the accumulation of high temperatures and gaseous products, is

associated with a build-up of quasi-static pressure (gas pressure). This pressure has a long

duration in comparison to that of the shock pressure.

Fully Confined Explosion

Full confinement of an explosion is associated with either total or near total containment

of the explosion by a barrier structure. Internal blast loads will consist of unvented shock

loads and very long duration gas pressures which are a function of the degree of

containment. The magnitude of the leakage pressures will usually be small and will only

affect those facilities immediately outside the containment structure.

2.4. Calculation methods of blast-loadingEmpirical and semi-empirical methods

As it can be seen blast loading is a very complex issue. First attempts to describe it

mathematically were started yet in the 1950s. The crucial notion in this matter is

Hopkinsons law [9]. It is the most widely used approach for blast wave scaling. It

establishes that similar explosive waves are produced at identical scaled distances when

two different charges of the same explosive and with the same geometry are detonated in

the same atmosphere [10]. Thus, any distance R from an explosive charge W can be

transformed into a characteristic scaled distance Z:

3

RZ

W (2-1)


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where Wis the charge mass expressed in kilograms of TNT. The use ofZallows a compact

and efficient representation of blast wave data for a wide range of situations.

One of the first methods estimated peak overpressure Pso due to spherical blast was

introduced by Brode [12] in 1955:

3

2 3

6.71 10

0.975 1.455 5.850.019 0.1 10

so so

so so

P P bar Z

P bar P bar Z Z Z

(2-2)

Next Newmark and Hansen [13] (1961) introduced a relationship to calculate the

maximum blast overpressure, Pso, (in bars) for a high explosive charge detonates at the

ground surface as:

0.5

3 36784 93so W WP

R R

(2-3)

Another expression of the peak overpressure Pso (in kPa) was introduced by Mills [14]

(1987), in which W is expressed as the equivalent charge weight in kilograms of TNT, and Z

is the scaled distance:

3 2

1772 114 108soP

ZZ Z (2-4)

Then the maximum value of dynamic pressure qs can be calculated from the

following formula:

25

2 7so

s

so o

Pq

P P

(2-5)

If the blast wave encounters an obstacle perpendicular to the direction of

propagation, reflection increases the overpressure to a maximum reflected pressure Pr ,

which can be obtained from RankineHugoniot relationships for an ideal gas:

7 42

7o so

r so

o so

P PP P

P P

(2-6)

Used symbols:

Pso - peak overpressure,

Po - ambient pressure.

For more methods check paper [32].

Beside methods that try to approximate the maximum value of peak pressure for

different kinds of explosions (as free air burst or surface burst), the methods describing the

shape of the overpressure curve have been developed. One of the methods represents the


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pressure-time history by linear decay using approximate triangular equivalents. The other

one, probably the most popular represents the pressure-time history by exponential

functions. It is called Friedlanders method [11] and is described by following equation (only

the positive phase):

1

A

o

t t

tA

s so

o

t tP t P e

t

(2-7)

where t is the time, Pso is the peak overpressure, to is the duration of the positive

phase, tA is the arrival time of blast wave, and is a positive constant called the waveform

parameter that depends on the peak overpressure. It is depicted in Figure 2-12.

Figure 2-12. Free-field pressure-time variation [7].

Also many charts and tables predicting the blast pressures and blast durations have

been developed. The method presented in UFC [7] is an example of semi-empirical

method. It will be used in Verification Analysis to obtain peak incident overpressure and

peak reflected pressure.

Nowadays to simulate the blast one can use developed by US Army ConWep [16, 18],

a blast loading predictive tool. It contains a big database of experimental records of blast

loading parameters from a wide range of explosive charge masses and stand-offs produced

by Kingery and Bulmash [15]. The main advantage of this model is that the loading is

applied directly to the structure subject to the blast. There is no need to include the fluid

medium in the computational domain. In the ConWep model, empirical data for two types


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of waves are available: spherical waves for explosions in midair and hemispherical waves for

explosions at ground level in which ground effects are included [21]. It is also available in

Abaqus.

Also AtBlast program [17] funded by US General Services Administration is available

for free. Both of these applications can calculate free-field and reflected blast pressure

histories (from free-air and surface burst explosions), average peak pressure and impulse

from a hemispherical surface burst on a specified reflected wall area.

Numerical methods

A blast analysis using pressure time history predicted by the ConWep or AtBlast

applications produces reasonable results. However it cannot consider the effects, such as

confinement due to geometry of structure, reflection of multiple blast waves, and

shadowing occurring when an object is blocking a surface of structure from direct blast

wave. These factors can change the blast pressure dramatically. If more accurate results

have to be obtained, also more accurate methods than the semi-empirical ones used in

mentioned applications need to be applied.

One of the possibilities is to use software which supports finite element analysis.

One can mention here Autodyn, Dyna3D, LS-Dyna, and Abaqus. The mentioned software

supports a coupled analysis, for which the blast simulation module is linked with the

structural response module. In this type of analysis the CFD (computational fluid

mechanics) model for blast-load prediction is solved simultaneously with the CSM

(computational solid mechanics) model for structural response. By accounting 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.

The similar simulations can be conducted in Abaqus using the Coupled-Eulerian-

Lagrangian (CEL) method.

3.Finite Element Analysis techniquesChoice of an appropriate kinematical description of the continuum is crucial while dealing

with mechanics of fluids and solids because it determines the relationship between the

deforming continuum and the finite grid or mesh of computing zones. The algorithms of

continuum mechanics usually make use of two classical descriptions of motion: the

Lagrangian description and the Eulerian description [20, 21]. Additionally, for use in more

sophisticated problems, when functionality of these two methods is insufficient, two other


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techniques have been developed: Coupled Eulerian-Lagrangian and Arbitrary Eulerian-

Lagrangian. The more detailed description is given in Appendix A.

4.Tools and methods4.1. SoftwareThe shock wave propagation analyses have been performed with use of three basic tools.

First one was Abaqus/CAE which provided a complete interface for creating simulations,

which then were solved by Abaqus/Explicit. Second one was Python language, one of few

supported languages for the purpose of use in Abaqus. Also Abaqus Scripting Interface was

used. It is, as referenced in [1], an application programming interface (API) to the models

and data used by Abaqus.

Abaqus

Abaqus is commercial software for performing numerical analyses of different physical

phenomena. The most common ones in the case of civil engineering can be simulations of

statics and dynamics of 2D and 3D structures. However Abaqus, originally released in 1978

has a much wider range of application, as it is commonly and successfully used in

automotive, aerospace and industrial products industries. Its important feature is that it

provides numerous multiphysics simulations (that is such, which involve multiple physical

models or multiple simultaneous physical phenomena). By the description of Abaqus often

two shortcuts appear: FEA and CAE. First one is the name of numerical technique (Finite

Element Analysis) for finding approximate solutions of partial differential equations (PDE)

as well as of integral equations which this software uses. The second is abbreviation of

Computer Aided Engineering and refers to a group of software applications that have been

produced to help in engineering tasks.

Abaqus/CAE is an environment that generates an input file (that is a file that

contains all necessary data) that is next submitted to the Abaqus solver.

Abaqus/Explicit

Abaqus/Explicit as its name points uses explicit dynamics analysis procedure. It is based

upon the implementation of an explicit integration rule together with the use of diagonal


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(lumped) element mass matrices. The equations of motion for the body are integrated

using the explicit central-difference integration rule:

11 1

2 22

i iN N Ni

i i

t tu u u

(4-1)

1 1 12

N N N

i i ii

u u t u

(4-2)

In the formulas (4-1) and (4-2) Nu is degree of freedom (a displacement or rotation

component) and the subscript i refers to the increment number in an explicit dynamics

step. The central-difference integration operator is explicit in the sense that the kinematic

state is advanced using known values of1

2

N

uu

and Niu from the previous increment.

Explicit procedure uses diagonal element mass matrices. Therefore the accelerations

at the beginning of the increment are computed by:

1N NJ J J

i i iu M P I

(4-3)

where:

NJM - Mass matrix,

JP - Applied load vector,

JI - Internal force vector.

The benefits of lumped mass matrix are first of all significant computational

advantages of calculations. Lumped mass matrix is a sparse matrix thus it gives much

better performance of computer calculations than a full one. A diagonal mass matrix

negates also the need to integrate mass across the deformed element and to build tangent

stiffness matrix. The internal force vector JI is assembled from contributions from the

individual elements such that a global stiffness matrix need not be formed.

To assure the stability of the procedure, which integrates through time by using

many small time increments, special conditions need to be fulfilled. The central difference

operator is conditionally stable, and the stability limit for the operator (with no damping)

is given in terms of the highest frequency of the system as:

max

2t

(4-4)

And with damping:

2max maxmax

21t

(4-5)


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where max is the fraction of critical damping in the mode with the highest frequency.

An approximation to the stability limit is often written as the smallest transit time of

a dilatational wave across any of the elements in the mesh:

min

d

Lt

c (4-6)

where minL is the smallest element dimension in the mesh and dc is the dilatational wave

speed in terms of 0 and 0 . The current dilatational wave speed dc is described by the

formula:

2dc

(4-7)

where is the density of the material and , are effective Lames constants. For an

isotropic, elastic material Lames constants can be defined in terms of Youngs modulus E

and Poissons ratio by:

0

0

1 1 2

2 1

E

E

(4-8)

This estimate for t is only approximate and in most cases is not a conservative

(safe) estimate. In general, the actual stable time increment chosen by Abaqus/Explicit is

less than this estimate by a factor between1

22

and 1 in a two-dimensional model and

between1

23

and 1 in a three-dimensional model. Description is based on [25, 26].

The equations presented above are based on CourantFriedrichsLewy condition

(CFL condition), which is a necessary condition for convergence while solving certain

partial differential equations (usually hyperbolic PDEs) numerically by the method of finite

differences. It arises when explicit time-marching schemes are used for the numerical

solution. As a consequence, the time step must be less than a certain time in many explicit

time-marching computer simulations; otherwise the simulation will produce wildly

incorrect results [31].

Abaqus Scripting Interface

Abaqus Scripting Interface is an application programming interface which is (after [2]) a

particular set of rules and specifications that software programs can follow to communicate

with each other. Entering the data is different - instead of clicking on specific icons and


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fulfilling numerous boxes, user has to set every parameter using functions provided by ASI.

This way traditional procedure of entering data can be omitted.

As it is written in [1], the Abaqus Scripting Interface is a customized extension of

standard Python. This means that in order to write scripts or programs one has to do it in

Python language. Python is dynamic, interpreted and interactive high-level programming

language. It enables object-oriented, structural and functional programming. It is

developed as an open source project, what means it is free and its source code is publicly

available. The feature distinguishing Python from other languages is its syntax, in which

the scope of the loop or conditional statement is designated through indentation.

4.2.Creation of CEL model in Abaqus

Creating a model in Abaqus requires specifying numerous data. In this part, to show the

concept of Coupled Eulerian-Lagrangian analysis technique, individual steps of creating a

fully-working model are described.

As it is written in Chapter 3 and Appendix A, there are significant differences in

Lagrange and Euler descriptions. These differences influence significantly on the creation

of model. The main differences are presented in following paragraphs.

In Lagrange analysis technique nodes are fixed within the material, which fulfils it

entirely, so the element boundary coincides with the material boundary. In Eulerian

analysis nodes are fixed in space, so the material flows through non-deformable elements.

In contrast to Lagrange elements Eulerian ones may not always be 100% full of material -

they can be partially or completely void. The Eulerian material boundary must, therefore,

be computed during each time increment and generally does not correspond to an element

boundary. Eulerian models typically consist of a single Eulerian part, what is the notable

difference comparing to Lagrange models, where several parts are created and then

assembled into one model [21]. This part can be arbitrary in shape but typically it is a

simple rectangular grid of elements. It represents the domain within which Eulerian

materials can flow (area of simulation). The necessity of Eulerian model to consist of a

single part induces the problem of defining different objects. It is solved by creating

multiple regions within the Eulerian-type part instance. Then a particular material is

assigned to a particular region. This way complex body geometry can be defined. Details

can be found in paper [21].


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Generally, the entire procedure involves accomplishing following steps (from [21]):

1. Create Eulerian-type part/parts that defines/define the geometric region withinwhich Eulerian materials can flow. Create partitions that will represent the

initial boundaries between different materials in the part (needed if materials

are assigned uniformly across a region).

2. Define materials.3. Define and assign an Eulerian section for the model.4. Create an instance of each Eulerian part. Merge instances to obtain single part

instance.

5. Create a field output request (in order to view the deformation of materials in anEulerian model, output variable EVF is necessary).

6. Create a predefined field that defines the topology of materials in the initialconfiguration of the Eulerian part instance.

7. Define any loads and/or boundary conditions acting on the model.8. Create a hexagonal mesh for the Eulerian part.

Parameters

During the preparation of the model for blast simulations, following material parameters

have to be set (the given values are for typical air conditions and TNT as an explosive

charge):

1. Air described by following parameters:a. Density - 31.293 kg m ,b. EOS parameters for ideal gas:

i. gas constant - 287 J kg K ,ii. ambient pressure - 2101325 N m ,

c. Specific heat - 717.6 J kg K ,d. Data describing the viscosity kg s m - temperature dependency K

(in tabular form),

e. Physical constants:i. Stress - 2101325 N m ,

ii. Specific energy between 193300 J kg and 219780 J kg ,


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2. Explosive charge (TNT):a. Density - 31630 kg m ,b. Jones-Wilkins-Lee EOS parameters:

i. detonation wave speed - 6930 m s ,ii. coefficients:

1. A - 11 23.7377 10 N m ,2. B - 11 20.037471 10 N m ,3. R1 - 4.15 ,4. R2 - 0.9 ,5. - 0.35 ,

iii. detonation energy density J kg ,iv. pre-detonation bulk modulus 2N m ,

c. Physical constants:i. Stress - 2101325 N m ,

ii. Specific energy between 3680 kJ kg (for classical TNT from1970s) and 5000 kJ kg (in the case of TNT-C4 compounds).

Ideal gas law is used to describe the compressible flow phenomena that include

microscopic properties such as density, pressure and internal energy. The formula of ideal

gas:

ZAp p R (4-9)

where p is current pressure, pA is ambient pressure, is density, R is gas constant, is

current temperature, Z is the absolute zero on the temperature scale being used.

Jones-Wilkins-Lee equation of state is widely used to model both the detonation and

the expansion phase of the explosion (unlike the ideal gas EOS which can be only used to

model expansions phase) [22]:

1 2

int

1 2

1 1

R R

p A e B e eR R

(4-10)


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where p is current pressure, coefficients A, B, R1, R2 and are experimentally determined

material constants, 0 is the ratio of densities: - of the detonation products, 0 - of

the explosive. e

int

is internal energy per unit mass.All above values are specified in SI units, that is: length - metres [m], mass -

kilograms [kg], time - seconds [s], thermodynamic temperature - kelvins [K]. Force is given

in newtons (N = kgm/s2) while pressure in pascals (Pa = N/m2).

4.3.Assumptions and restrictions in the scriptBasic elements building the structure are straight sections and connections. In the script

following assumptions have been made:

- Straight sections are defined in their axes, by specifying start and finish points(hereinafter referred to as important points),

- Sections may intersect freely,- If two or more sections have to be connected, this connection has to take place

in sections important points only,

- To avoid unpleasant gaps, connection should be in horizontal or vertical planeonly,

- Depending on the type of connection (vertical or horizontal) sections that arebeing joined should have respectively same width or height,

- From the fact that it is impossible to obtain horizontal angle from sectionsdirected vertically (which would sometimes result in inadequate orientation of

this sections faces) an additional, non-vertical section, from which the vertical

one inherits the horizontal angle, has to be specified,

- When sections of different width or height (later called important dimension)connect at angle 90, the precedent of them is one which has smallest important

dimension it results in smooth connection.

In process of modelling the structure, the user has to have in mind following

limitations:

- Although the sections can intersect at any angle, angles close to 180will resultin very long connection (because of fact that tan 180 ),


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- If the connecting sections differ in value of important dimension, depending onthe mentioned difference for angles bigger than 90 some strange artefacts may

appear,

- In case of failure to comply with assumption number 5, unpleasant connectionwill be created.

Meeting above requirements one can be sure about correctness of the model and its

compatibility with his own assumptions and predictions.

4.4. Simplified block scheme of the scriptThis script automates operations required to be done in Abaqus/CAE modules, requiring

the user to specify just the key values, like points describing geometry of the structure,

basic material properties and duration of analysis. Although some of the actions can be

done at any time, in this application a certain order of execution of some modules has been

introduced. This treatment results from the fact that for the correct execution of the script

(as well as the Abaqus solver) it is necessary to enter all required data. Some of the

functions are, however, non-obligatory, that is one can choose if any (if yes then which

one) to execute. Figure 4-2 presents applications functioning block scheme.


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Figure 4-2. Block scheme presenting the functioning algorithm.


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The first main step is the initialization of applications main class called MyModel

(A). Next one is to define data concerning the arrangement of corridors (B). The more

detailed graph showing this procedure is presented in Figure 4-3.

Figure 4-3. Block scheme presenting major steps of geometry definition.

At the beginning one has to define number of geometry data sets. These sets should

be filled with information on particular corridors and connections. To assure flexibility of

the script, no check of correctness of the input data is performed. While setting the

geometry, data is stored in complex list structures.


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Figure 4-4. Block scheme presenting draw method execution.

Subsequent step is call to draw method (step C). After some computations each

record from data structures becomes the basis for the creation of corridors and connections

(in both cases firstly sketches, then parts and in the end instances are created). It was one

of the assumed conventions to firstly define all the necessary geometry and then, finally

draw it.

In this place it is important to mention that creation of simple connection of two

corridors is not always a process involving just creation of two cuboids. In order to avoid

gaps or discontinuities in corridors other elements need to be placed. Then, in fact such a

connection can be made even of several elements, which in the end are merged into new

element. The complexity of the connection is dependent on the relative angle between the

corridors. If angle is equal to zero, no additional elements are required. If the angle is

smaller than or equal to 90, two elements are sufficient. The most complex case takes

place when the angle is wider than 90, then four elements have to be used and the

procedure itself involves operations such as merging and cutting, performed in a certain


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way. When all the declared elements of each set have been drawn, they are merged into

one element.

Next step, which is partitioning of the elements (D), is non-obligatory (denoted as

No partitioning in Figure 4-2). However, usually when the geometry is complex, it is

necessary to execute it. We can choose from two different partitioning algorithms (Auto

partitioning and Raw partitioning), it is recommended to check each one and compare

which one produces better results for particular structure. Unfortunately author cannot

assure that these algorithms will always produce elements appropriate for meshing, as the

complexity of the different connections is unlimited, and list of options available in Abaqus

is very short. Fortunately in many cases just a slight manual alteration may solve the

problem. This issue is described in part Problems and difficulties. Except cutting the model

into regions, this part of script merges all geometry data sets and creates calculation step

(Explicit dynamic), loads acting on the walls (air pressure, including possibility of an inflow

and outflow of air) and boundary conditions (fixed displacements).

In step E user has to specify parameters describing the medium (air) and explosive

charge, like number of explosives, their location and weight. Also moments in time when

the detonations take place have to be specified.

Subsequent step (F) covers declaring the output data one would like to obtain. Two

methods differing in way of data recording are possible to execute. The data recording

frequency can be chosen via appropriate attribute.

The next step is to call mesh method (G). Though the suggestive name, the function

does a bit more than just create a mesh of a specified size, additionally it modifies the

special keywords, what is an important issue.

The penultimate step (H) is optional and related to point (F). It concerns defining

individual finite elements as output data.

The last operation is a call to job method (I), which in this case does nothing else

and nothing more beyond what its name implies. The method allows declaring number of

CPUs and threads involved in process of calculations.

4.5. Structure of the scriptOne of the properties of Python is that it is an object-oriented programming language. This

important feature has been taken into account while creating the script. Thanks to object-

oriented approach it was possible to divide the code into separate parts classes

consisting of appropriate properties and methods. This way greater clarity of code


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(comparing to procedural approach) has been achieved. Moreover, it allows easier and

error-free further development. Also the usage is simpler, as the functions being used are

precisely specified.

The application consists of 23 classes, which one can divide into 4 groups: creation of

geometry (marked as B, C in Figure 4-2), partitioning into regions (D), setting material

properties (E) and output parameters (F and H), and job execution (G and I). The classes

and methods, on which user can operate are briefly described below.

The script has been designed in such a way, that all necessary commands (in order to

create a functional and complete Abaqus model) are localized within the main class

MyModel. The object of this class instance is responsible for managing of all data and

subordinate classes. The user can call following methods:

Model object initialization (A)

__init__(name : str)

Constructor method called when the MyModelobject is being initialized. The objects of all

subordinate classes are created. The only attribute is the name of the model, under which it

will appear in Abaqus. Exemplary call:

mod = MyModel('Corridor')

Methods for creating geometry (B, C)

addCMs(GDSList : list)

The method generates geometry data sets specified in GDSList list. GDSList should contain

their names. Exemplary execution:

GDSList = ['GeometrySet_1', 'GeometrySet_2']

mod.addCMs(GDSList)

The first line creates list of data sets. Then it becomes an input parameter to addCMs()

method.

addCorridor(name : str, startPoint : vec3f, endPoint : vec3f,

width : float, height : float, oocc : str, addCorr : CorridorData,

newName : str = '') : CorridorData

Method is used to create straight corridor section assigned to set specified by its name

name, with geometry defined by start and end points (startPoint and endPoint) and cross-


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section dimensions width and height. By parameter ooccuser can specify corridors closure.

Possible options are following:

- oo corridor open at both points,- oc corridor open at start point and closed at end point,- c0 corridor closed at start point and open at end point,- cc corridor closed at both points.

addCorridorIndexspecifies the corridors parent corridor, from which it will inherit vertical

angle. This parameter shall take value other than None only if the corridor is pointing

vertically. The last parameter is optional and is a new name of the corridor. This option is

useful if geometry data set contains a single corridor section. Example:

Corr1 = mod.addCorridor(GDSList[0], vec3f(0,0,0),

vec3f(10,0,0), 2, 3.5, 'oo', None)

The corridor is created from point [0,0,0] to [10,0,0], with width of 2m and height of 3.5m.

The corridor has both ends open and no parent section. It will be assigned to geometry

data specified by the name contained in GDSList[0].

addConnection(name : str, corr1 : CorridorData, corr2 :

CorridorData, verORhor : str, connType : str)

This method adds connections between corridors. The input parameters are: name of the

geometry data set, to which the connection will be associated, followed by the

CorridorData objects of two corridor sections that the connection will concern. Next

attribute is string specifying the plane in which the connections take places. It can take two

values:

- hor if the connection is in horizontal plane,- ver if the connection is in vertical plane.

The last parameter concerns which parts of corridors to merge. Possible options:

- sf the part will connect starting point of first specified corridor with endingpoint of the second one,

- fs the part will connect ending point of first specified corridor with startingpoint of the second one,

- ss the part will connect starting points of both corridors,- ff the part will connect ending points of both corridors.

Exemplary use:

mod.addConnection('GS1', Corridor2, Corridor1, 'hor',

'sf')


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The connection is created between starting point of corridor identified by Corridor2 and

ending point of corridor Corridor1. The connection is in horizontal plane. It is saved to

geometry data set specified by name GS1.

draw()

AddCorridor() and addConnection() methods append new elements to appropriate lists,

respectively for sections of corridors and connections between them. Method draw()

executes classes, which relying on the mentioned lists, generate the defined geometry

elements. Finally, after all elements have been drawn, they are merged and the unnecessary

instances are suppressed. The only possible call:

mod.draw()

Methods for partitioning (D)

merge(name : str, timePeriod : float)

First of three alternative methods, it merges all defined geometry sets into a new one. No

partitioning takes place. First argument is name of resulting data set. The second one is

duration of the simulation, needed for dataPrePartition() method, which is executed within

the merge() method. It defines the analysis step (Explicit Dynamics), creates boundary

conditions and loads. Example:

timePeriod = 0.007

mod.merge("Merged", timePeriod)

The method creates data set Merged. The duration of the analysis is 0.007 second.

autoPartitionAndMerge(name : str, PartHeights : list,

timePeriod : float)

Method performs same operations as the above one; additionally it creates regions based

on Abaqus algorithm regarding auto partitioning of faces. This method enhances its usage

to cells. First argument is name of resulting data set. The second is a list of floats, in whichone can define additional horizontal planes for partitioning. The last parameter defines

duration of the simulation. Exemplary use:

timePeriod = 0.07

horPartitions = [-2.0, 2.0, 6.5]

mod.autoPartitionAndMerge("Merged", horPartitions,

timePeriod)


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In the example data set named Merged is created. The geometry will be cut by three

planes, at heights defined in horPartitions list. The analysis will be performed for 0.07s time

period.

rawPartitionAndMerge(name : str, timePeriod : float, angPart :

bool)

This method performs simple partitioning algorithm, based on planes defined by corridors

faces. The arguments are: name of the new data set, length of the simulation and Boolean

parameter specifying whether the angular partitions should be taken into account.

Example:

timePeriod = 0.07

mod.rawPartitionAndMerge("Merged", timePeriod, False)

In above example the method creates data set Merged. The analysis will be performed for

time equal to 0.07s, the partitioning along bisectors of angles between intersecting

corridors is disabled.

Methods for setting material properties (E)

dataPrePartition(timePeriod : float)

The method shall be considered as private method. It is called by methods belonging to the

group Methods for partitioning. It is responsible for creating Explicit Dynamics step, load

and boundary conditions. The only argument it takes is the duration of analysis.

dataPostPartition(tntLoc : list, tntWeight : list,

detTime : list)

This complex method creates fist of all air and TNT properties and Eulerian section

consisting of these materials. Next it assigns them to the part. Subsequently the space for

explosive charge is cut from the model, thus creating two sets - air and TNT. At the end, in

Abaqus Predefined Field Manager, appropriate section materials are assigned to particularsteps. The method takes following arguments: list of vectors specifying centre points of

explosives, lists of floats describing weight of charges and time of the explosion. Exemplary

call:

tntLoc = [vec3f(3,2.7,1.2), vec3f(15,14.8,0)]

tntWeight = [270, 300]

detTime = [0.01, 0.4]

mod.dataPostPartition(tntLoc, tntWeight, detTime)


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In this example two explosive charges are created. Their location within the model is set in

tntLoc list, weight (270 and 300kg) in tntWeight list. The explosions take place in moment

of time contained in detTime list.

Methods concerning output data (F, H)

HO(HOPoint : vec3f)

Method for creating history output for specified point (HOPoint). Example:

mod.HO(vec3f(20.3, 14, 27))

FO(numIntervas : int)

This method creates field output for the whole model. The parameter it takes is number of

data recordings the application will take during the simulation. If the value is 0, then all

increments are stored to output file. Example:

mod.FO(700)

Here the field output containing 700 records is created.

FOSet(centrePoints : list, FOSSizes: list,

numIntervals : int)

Method for setting a region of the model as a field output domain. The localizations of

points being centres of these regind and sizes of the sets are defined by the first two

parameters the function takes. The last argument works in the same why as in the function

above. Exemplary use:

centrePoints = [vec3f(3, 0.4, -0.3), vec3f(5, 0.1, 0.3)]

FOSSizes = [vec3f(0.15, 0.15, 0.15), vec3f(0.1, 0.1,

0.1)]

mod.FOSet(centrePoints, FOSSizes, 0)

This call will effect in creation of two field output sets located in points specified by

components of centrePoints list, and with sizes contained in FOSSizes list. ThenumIntervals value equal to 0 indicates that every simulations increment will be saved.

elementFOSet(pointList:list, numIntervals : int)

Unlike three methods described above, this one has to be executed after creating mesh. It

generates field output for finite elements defined inpointList list. The second parameter it

takes is number of data recordings the application will take during the simulation. If the

value is 0, then all increments are stored to output file. Example:


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pList = [vec3f(10,1,0), vec3f(1,1,0.5), vec3f(20,1,0)]

mod.elementFOSet(pList, 123)

In this example three element field output sets are created (defined in pList). The

numIntervals specifies how many increments will be saved (here 123 increments).

Methods related to job execution (G, I)

mesh(meshSize : float)

This method creates mesh of specified size. It edits keywords associated with Explicit

Dynamics step in order to make calculations possible. Example:

mod.mesh(0.15)

The mesh of 15cm size will be created.

job(jobName : str, writeInput : bool, numCpus : int, numDomains :

int, activateLoadBalancing : bool)

The method creates job with name specified byjobName. The second argument is a

Boolean and determines whether to save .inp file. The three recent parameters describe the

parallelization options available in Abaqus. The first of them defines number of CPUs

which will be engaged in calculations, second denotes number of threads for each CPU, the

latter one should be set to True if the dynamic load balancing should be activated. For

more details check Abaqus documentation [21, 26].

save(pathName : str)

This method enables to save .cae file under the named specified bypathName parameter:

mod.save("C:/myAnalysis")

All classes are presented schematically in Appendix B.

4.6. Problems and difficultiesDuring the preparation of the script several obstacles connected with Abaqus limitations

have been encountered. Certain of them have been just partly resolved and in order to

submit a job some modifications of the model can be required. In many cases they forced

author to revise initial assumptions and modify the code to cooperate with Abaqus more

properly. In this part these issues are extensively described.


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The script allows to define arbitrary geometry. However because of the necessity to

use single Eulerian part instance, in order to distinguish different objects in its interior, it is

required to cut it with datum planes (partitioning). In the script this method is used to

separate explosives' cubes from the air domain. The problems with this approach arise

during meshing. Eulerian parts use unique meshing rules, which are more restrictive

comparing to those used in Lagrangian ones, that is the finite elements need to form right

prisms (so the faces joining the two bases have to be perpendicular to the bases) with bases

being tetrahedral. In certain cases the algorithm fails to mesh the object. First of it is

impossible to mesh complex 3D shapes (e.g. if corridors are directing at different horizontal

and vertical angles simultaneously). The author has taken efforts to solve these problems.

Despite using three different partitioning procedures, finally not all of them have been

eliminated. Starting from the assumption that the fundamental objective of every computer

script is that it has to work, to make partitioning methods work properly, some usage

limitation have been imposed. Therefore the following structure of the geometric data has

been adapted: elements have to be grouped into so called sets, where each set should

consist of corridors lying in one level (in horizontal plane). Also the vertical distance

between sets should be greater than height of the storey, so that individual sets would not

overlap each other. Then one of two available partitioning methods will result in model

correctly partitioned. As it was mentioned, to assure flexibility, no special restrictions are

set, so one can arrange the geometry arbitrarily. The partitioning methods are non-

obligatory, but before creating mesh and running job, it is possible that one would have to

manually adjust the model for calculations.

4.7. Further developmentNow the application supports only cuboids as elements to define geometry. A good

solution would be to add some other methods of the structure generation, for example to

give the possibility to model rooms or halls of more complex shape. As far as creation ofthe script, which would handle it, seems not be a huge problem, it appears to be harder to

figure out how to solve the problem of connection with corridors in order to avoid

discontinuities.

Also the script would be more functional if point 5 from part Assumptions and

restrictions would lose its validity. However, this involves solving problem of this matter:

how to model connection of corridors, when each has different height, merging at

horizontal angle different than 0 or 180?


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Another issue is that now the model consists of only rigid elements, which are fixed

and unable to displace. Adding deformable materials inside the structure (e.g. building)

would be a nice option, which would extend applicability of the script.

The most problematic issue is one related to partitioning of the model.AutoPartition

method is sufficient for structures which consist of horizontal layers of sets of corridors and

additional corridors directed vertically. However for cases described in part Problems and

difficulties this method is insufficient. Implementation of fully working code seems to be

impossible; however this is author's private opinion.

For now all faces of corridors (except the ones which normal vector coincides with

the corridors longitudinal axis) have fixed outflow and inflow of the air. An interesting

option would be to add windows and doors, so that the outflow would be enabled.

5.Analyses5.1. Benchmarking analysis of blast wave propagation5.1.1. DescriptionThe analysis is aimed to compare peak incident overpressure obtained using Abaqus

numerical approach and UFC analytical method. The subject of the analysis is a simple air

cuboid with dimensions:

9.2

0.4

0.4

L m

l m

h m

The TNT explosive weights 1kg and is located 1m from one end in centre of the cross

section. The model as shown in Figure 5-1 has been prepared using script. The code can be

found on the CD in folderModels under the name SimpleCuboid.py.

Figure 5-1. Perspective projection of an air cuboid.


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5.1.2. Analytical solutionMethod described in UFC is used. For convenience calculations are performed for points

based on scaled distance Z (here presented for1 3

0.25 ftZlb

). According to UFC guidelines

the charge is increased by 20%.

1 3 1 3

1.2 2.65

0.25 0.0992

0.105

W kg lbs

ft mZ

lb kg

R m

Peak incident overpressure can be calculated from Table 2-7 [7]:

0.25 4600 31.82soP Z psi MPa

Some of the results are shown in Table 5-1.

5.1.3. Numerical solutionThe numerical simulations have been performed in Abaqus. Dimensions of the model have

been taken as in the analytical solution. The time of the analysis has been set to 0.1s, while

air and TNT parameters have been set according to chapter 4. Internal energy has been set

to 3680kJ/kg. The pressure has been measured uniformly across the model in elements

located on the longitudinal axis (in red points shown in Figure 5-2). Five different mesh

refinements have been studied: 10, 8, 6, 4and 2cm. The results are given in Table 5-1.

Figure 5-2 shows distribution of points across the model and location of the explosive

charge for model meshed with 2cm elements.

Figure 5-2. Plan view of the air cuboid meshed with 2cm elements. Measuring points are in

red, while blast charge in blue.


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5.1.4. Discussion of the resultsFigure 5-3 shows the comparison of maximum pressures obtained using UFC algorithm

(blue line) and Abaqus.

Figure 5-3. Comparison of peak pressure obtained for each distance from the charge for

UFC solution and Abaqus.

It can be seen that mesh refinement has significant influence on the results.

Generally the finer the mesh the higher values of pressure one obtains. While for stand-off

distance smaller than 2.0m the differences between Abaqus results for all kinds of meshes

are visible, for the remaining part those for 10, 8, 6 and 4cm meshes are minimal, except

0,1

1,0

0,80 1,80 2,80 3,80 4,80 5,80 6,80 7,80

Peakpressure[MPa]

Distance from the charge R [m]

UFC

10cm, 1473 FE

8cm, 2875 FE

6cm, 6874 FE

4cm, 23000 FE

2cm, 184000 FE


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the 2cm mesh, which still provides higher results (minimum of 65%). The pressure

differences for 2cm mesh are presented in Table 5.1.

Table 5-1. Comparison of peak incident pressure (Pso) obtained by UFC method and in

Abaqus for selected points.

R [m]Pso [MPa] Pressure difference [%]

with respect to UFCUFC Abaqus 2cm mesh

0.8 1.77097 1.24762 30

0.9 1.36714 1.03842 24

1 1.15650 0.88089 24

1.1 0.96297 0.74037 23

1.2 0.75270 0.61475 18

1.6 0.37646 0.26066 31

2 0.22305 0.11699 48

2.4 0.15001 0.05419 64

2.8 0.10886 0.03204 71

3.2 0.08225 0.02208 73

3.6 0.06420 0.01511 76

4 0.05218 0.01039 80

4.4 0.04495 0.00749 83

4.8 0.04077 0.00562 865.2 0.03658 0.00432 88

5.6 0.03239 0.00342 89

6 0.02820 0.00275 90

6.4 0.02444 0.00227 91

6.8 0.02248 0.00198 91

7.2 0.02051 0.00192 91

7.6 0.01856 0.00202 89

8 0.01655 0.00200 88

Basing on Table 5-1 it can be concluded that in the case of 1kg TNT charge, the

convergence of results obtained using UFC method and Abaqus CEL simulations is similar

for stand-off distance smaller than 1.6m. However for R outside this range the differences

grow very significantly. For R = 3.5m pressure calculated using Abaqus with 2cm mesh is

76% smaller than UFC value. For R = 8m pressure calculated using UFC is 14.5kPa, while

using Abaqus is 2.1kPa. The discrepancy is enormous, for example in order to assess the

damage that the pressure can induce. A pressure of 7kPa would already partially demolish


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houses, while 2.1kPa would only damage windows and ceilings (for more details

check [32]).

It can be assumed that for 1cm mesh the numerical results would be even closer to

the UFC, especially for small stand-off distances, where the biggest disturbances are

observed.

5.2.Analysis of the cubicle5.2.1. DescriptionThe analysis concerns comparing empirical and numerical computational methods of

calculating blast wave properties induced by an explosion for a simple structure (shown

schematically in Figures 5-4 and 5-5). The calculations presented in subsequent part

Analytical solution are based on methods described in UFC [7]. The numerical simulations

were conducted in Abaqus/Explicit. The main point of this example is to compare values

obtained by these two methods. Also verification of influence of the finite element size on

the accuracy of the results is performed.

Figure 5-4. Cubicle in perspective projection. Explosive charge in red, analysed wall in blue.

16H ft

32L ft

6h ft

12l ft

5.33AR ft


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Figure 5-5. Plan and section view of the cubicle [7].

5.2.2. Analytical solutionThe analytical solution is performed using method described in UFC [7]. This approximate

method has been developed using theoretical procedures based on semi-empirical blast

data and on the results of response tests on slabs [7].

The subject of the task is to calculate average peak reflected pressure and average

scaled reflected impulse on the side wall of a structure, which accordingly to the

nomenclature given by UFC is described as a fully vented, three-wall cubicle. A TNT

explosive charge weights 245lbs. Additionally peak incident pressure in line between the

charge and the wall, and peak reflected overpressure on the side wall in 10 points are

calculated.

Data:

16 4.87

32 9.75

6 1.83

12 3.66

5.33 1.62A

H ft m

L ft m

h ft m

l ft m

R ft m

Charge weight:

W 245 111.13lbs kg


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Calculation of peak incident overpressure

For a freely chosen point located 1m from the charge, the maximum pressure is calculated:

1 3 1 3 1 3

1 3.283.28

0.5243245

R m ftftR

ZW lb

Peak incident overpressure can be calculated from Table 2-7 [7]:

0.5243 2150 14.82soP Z psi MPa

All the results are shown in Table 5-2.

Table 5-2. Peak incident overpressure for different distances from the charge.

Distance fromthe charge R [m]

Scaled distance Z1 3

ftlb

Pso [psi] Pso [MPa]

0.29 0.15 6850 47.23

0.39 0.20 5500 37.92

048 0.25 4650 32.06

0.58 0.30 3950 27.23

0.66 0.35 3370 23.24

0.76 0.40 2950 20.34

0.86 0.45 2630 18.130.95 0.50 2280 15.72

1.05 0.55 2030 14.00

1.14 0.60 1800 12.41

1.24 0.65 1695 11.69

1.33 0.70 1590 10.96

1.43 0.75 1449 9.99

1.53 0.80 1270 8.76

1.61 0.84 1150 7.93

Calculation of peak reflected overpressure

Peak reflected overpressure has been calculated on the side wall in 10 points (9 of them are

distributed evenly) as in the Figure 5-6. The distance from each point to the edge of the

wall has been set to 0.5m.


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Figure 5-6. The view from the centre of the cubicle towards the wall on the left side

(according to Figure 5-5) - scheme showing analysed points on the wall.

Calculation scheme for point 1 is presented below. The results for all points are given in

Table 5-5.

Coordinates of point 1 relative to the charge:

5.60

1.62

2.55

x m

y m

z m

Distance to the charge:

2 2 2 6.36R x y z m

Scaled distance Z:

1 3 1 3 1 3 1 3

6.361.323 3.335

111.13

ftR mZ

W kg lb

For Z=3.335 one can read from Figure 2-7 [7] peak incident overpressure:

77soP psi

Angle of incidence :

1

2 2tan 75.21y

x z

Basing on peak incident overpressure and angle of incidence a peak incident overpressure

multiplier can be obtained (Figure 2-193 [7]):

1.2rC

Peak reflected overpressure:

1.22 77 92.4 0.637r r soP C P psi MPa


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Table 5-3. Summary of peak reflected overpressure in chosen points on the side wall.

Pr [psi] Pr [MPa]

Point

1 92.4 0.6372 924 6.371

3 283.9 1.957

4 561 3.868

5 4408 30.392

6 119 0.820

7 102.5 0.707

8 3900 26.890

9 462 3.18510 10005 68.982

Calculation of average peak reflected pressure and scaled average unitreflected impulse

Average peak reflected pressure and scaled average unit reflected impulse; unlike their

peak reflected equivalents are assumed to act across the whole face.

Calculation of chart parameters h/H, l/L, L/RA, and the scaled distance ZA:

1 3

1 3 1 3

0.3750.375

6.00

2.00

5.330.85

245

A

AA

h Hl L

L R

L H

RZ ft lb

W

Interpolation is required for ZA, L/H, l/L, and h/H.

Number of adjacent reflecting surfaces N = 2. Then:

6.00

0.8523

A

A

L R

Z

The values of rP and1 3

ri W can be determined and tabulated from Table 2-3 [7] for

calculated above AL R and AZ and the following variables:

0.625,1.25,2.50,and5.00

0.10,0.25,0.50,and0.75

0.10,0.25,0.50,and0.75

L H

l L

h H

The results are given in Tables 5-4 and 5-5.


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Table 5-4. Average pressure rP , part 1.

h H 0.10 0.25

l L 0.10 0.25 0.50 0.75 0.10 0.25 0.50 0.75

0.625L H 462 569 598 569 533 665 701 665

1.25L H 749 932 980 932 943 1178 1238 1178

2.50L H 1200 1488 1562 1488 1432 1796 1881 1796

5.00L H 2032 2519 2635 2519 1870 2334 2437 2334

Figure [7] 2-64 2-65 2-66 2-67 2-68 2-69 2-70 2-71

Table 5-4. Average pressure rP , part 2.

h H 0.50 0.75

l L 0.10 0.25 0.50 0.75 0.10 0.25 0.50 0.75

0.625L H 546 681 718 681 533 665 701 665

1.25L H 1017 1267 1333 1267 943 1178 1238 1178

2.50L H 1609 2028 2120 2028 1432 1796 1881 1796

5.00L H 1987 2456 2563 2456 2623 3119 3210 3119

Figure [7] 2-72 2-73 2-74 2-75 2-76 2-77 2-78 2-79

Table 5-5. Average unit impulses 1 3ri W , part 1.

h H 0.10 0.25

l L 0.10 0.25 0.50 0.75 0.10 0.25 0.50 0.75

0.625L H 73 71 70 66 65 61 59 55

1.25L H 96 92 90 84 96 92 90 83

2.50L H 126 121 121 111 139 131 129 120

5.00L H 172 164 164 153 167 153 154 143

Figure [7] 2-113 2-114 2-115 2-116 2-117 2-118 2-119 2-120

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