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Poznań University of Technology

Faculty of Civil and Environmental Engineering

Master’s Thesis

Wojciech Mamrak

Blast wave propagation in the air and

action on rigid obstacles

Supervisors:

Marcin Wierszycki, PhD

Piotr Sielicki, MSc

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Analiza propagacji ciśnienia w powietrzu i obciążenia wybuchem

sztywnej przeszkody

Obciążenia wyjątkowe odgrywają kluczową rolę w projektowaniu wielu obiektów, w

szczególności obiektów użyteczności publicznej. Na przestrzeni ostatnich lat uwzględnienie

wpływu obciążenia wybuchem stało się nierzadko wymogiem w krajach Europy Zachodniej ze

względu na rosnące i coraz częstsze groźby ataków terrorystycznych. W obliczu tak poważnych

wyzwań projektanci wciąż nie zdołali doczekać się norm prawnych opisujących proces

projektowania obiektów zdolnych przeciswstawić się, w możliwym stopniu, wspomnianym

zagrożeniom. Ta sytuacja, wespół z obecną tendencją do wznoszenia konstrukcji lekkich i

delikatnych, w związku z czym także bardziej podantych na obciążenia dynamiczne i lokalne

zniszczenia, dodatkowo potęguje zagrożenia grożące użytkownikom tychże obiektów.

Niniejsza praca stanowi próbę zmierzenia się z wyzwaniami oceny siły fali uderzeniowej

powstałej w skutek wybuchu jak i złożonymi zjawiskami wpływającymi na jej propagację.

W pracy omówiono rodzaje obciążeń wyjątkowych ze szczególnym naciskiem na

obciążenie wybuchem. Przedstawiono definicję i klasyfikację materiałów wybuchowych oraz

precyzyjnie omówiono proces propagacji fali uderzeniowej w powietrzu. Dodatkowo,

zaprezentowano aktualny stan prawny i regulacje, zarówno europejskie jak i amerykańskie,

dotyczące projektowania konstrukcji narażonych na obciążenia wybuchem. Przywołano prace

badawcze z całego świata dotyczące rozpatrywanych zagadnień.

W celu wykonania obliczeń numerycznych posłużono się programem komputerowym

Abaqus. Do tworzenia modeli wykorzystano język programowania Python wraz z Interfejsem

Programowania Aplikacji programu Abaqus dla tegoż języka.

Przeprowadzono zespół analiz dotyczących trzech głównych zagadnień:

- określenia odpowiedniego rozmiaru elementu skończonego do dyskretyzacji modeli,

określenia energii właściwej trotylu do wykorzystania w późniejszych analizach oraz

określenia zgodności wyników dla prostego modelu,

- weryfikacji ciśnień wewnętrznych powstałych w wyniku eksplozji zewnętrznej,

- wpływu zjawiska osłaniania na wartości ciśnień i impulsów wraz z porównaniem dwóch

programów komputerowych wykorzystujących odmienne metody numeryczne i

weryfikacją koncepcji odległości skalowanej.

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Table of contents

1. Introduction .................................................................................. 5

1.1. Preface .................................................................................................. 5

1.2. Thesis.................................................................................................... 7

1.2.1. Motivation ......................................................................................................... 7

1.2.2. Objectives .......................................................................................................... 7

2. Accidental loading ....................................................................... 8

2.1. Introduction ........................................................................................... 8

2.2. Types of accidental loading ................................................................... 8

2.3. Explosives ............................................................................................. 9

2.3.1. Definition and classification ............................................................................... 9

2.3.2. Blast wave propagation .................................................................................... 10

2.4. Explosive loading in Civil Engineering ................................................18

2.5. Previous researches on the topic ...........................................................20

3. Tools and methods ..................................................................... 22

3.1. Abaqus FEA .........................................................................................22

3.2. Explicit dynamics .................................................................................22

3.3. Blast modelling approaches ..................................................................24

3.4. Abaqus Scripting Interface ...................................................................30

4. Python modelling script ............................................................. 32

4.1. Description ...........................................................................................32

4.2. Main script ...........................................................................................34

4.3. Configuration file .................................................................................37

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4.4. Modules ...............................................................................................38

5. Analyses..................................................................................... 43

5.1. Mesh size study ....................................................................................43

5.1.1. Description ...................................................................................................... 43

5.1.2. Analytical solution ........................................................................................... 44

5.1.3. Numerical solution ........................................................................................... 53

5.1.4. Discussion ....................................................................................................... 68

5.2. Interior pressure due to external burst ...................................................71

5.2.1. Description ...................................................................................................... 71

5.2.2. Analytical solution ........................................................................................... 73

5.2.3. Numerical solution ........................................................................................... 75

5.2.4. Discussion ....................................................................................................... 78

5.3. Comparative analysis ...........................................................................79

5.3.1. Description ...................................................................................................... 79

5.3.2. Original study description ................................................................................ 80

5.3.3. Numerical solution ........................................................................................... 83

5.3.4. Discussion ....................................................................................................... 95

6. Conclusions ................................................................................ 99

6.1. Final remarks .......................................................................................99

6.2. Future tasks ........................................................................................ 100

Appendices

Appendix A. Lagrangian and Eulerian descriptions

Appendix B. Model generating script UML class diagram

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1. Introduction

1.1. Preface

Exceptional loads play an important and increasing role while designing engineering structures,

especially in the case of public facilities. These should be designed to resist not only the typical

loads, like permanent (self-weight of the structure and all its permanent elements, fixed equipment,

actions caused by uneven settlements) and variable loads (imposed loads, wind actions, snow

loads etc.), but also exceptional loads, such as fire, earthquake and explosion, since the collapse or

some serious damage of the structure may cause death or severe injuries to hundreds or thousands

of people. To such buildings one can include museums, theatres, cinemas, shopping centres,

stadiums, hospitals, bridges, but also structures of high importance like power plants, dams and

others.

The existing structures are highly vulnerable to explosives. In many cases the extent of the

collapse of a building would be disproportionate to the cause, if the explosive material were

detonated at significant location, for example not well-protected steel column supporting the roof

of a large span. This state is the resultant of both lack of regulations and the assumption, that the

probability of occurrence is negligible, and conviction, that in the case of terrorist attack there is

not much the designer can do. In general, the latter is true, since people with bad intentions will

always find a way to realize their plans, but it shouldn’t be the reason to resign from all

protections. Many disasters and tragic events are a consequence of lack of a very basic security.

Another factor increasing the above mentioned susceptibility is a tendency to design and

construct lighter, more delicate, slender and attractive structures, which are more sensitive to

dynamic loads and local damages. This applies especially to public facilities such as shopping

centres, stadiums and bridges, which should be not only functional, but also impressive and

breathtaking. The latter often affects negatively the safety of the structure and its users.

Except the highly dynamic phenomena caused by a blast (such as already mentioned blast

pressures, flying fragments of a structure and shock loads transmitted through air or ground),

usually other types of loading, inter alia fire, occur as its result. The destructive power of both

shock wave and high temperature complicates already complex issue and protection methods, as

blast wave and objects carried by it can damage the fire protection.

Nowadays, engineers take into account the resistance of the structure to natural phenomena,

for example resistance to dynamic loads caused by earthquake, if there is a risk of occurrence of it.

Similarly, each significant structure is designed to withstand fire for given period and prevent its

further propagation. This is because of existence of design codes and regulations, which instruct

the designers on how to take into consideration and deal with these complex issues.

In contrary, explosive loads are almost completely ignored in the current European

standards. Hence, Unified Facilities Criteria (UFC) [23] document from the Department of

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Defense of the United States of America is used in this thesis as a background for analytical

calculations.

Explosive loads, despite the fact they originate from various sources, are mostly associated

with terrorist attacks. In recent years, due to tragic events of this type, the awareness of the society

about the safety of structures increased considerably. Yet, the observations prove, that the majority

of public facilities is susceptible and vulnerable to explosive loading, and even a small charge

placed near crucial structural elements can cause a serious damage or even total destruction of the

whole building. Except of military facilities, only high-rise buildings, nuclear power plants and

dams are presently designed to resist explosive loads.

The influence of exceptional loading can be analysed in two ways: by means of destructive

and non-destructive tests. The former requires sophisticated permissions, measuring devices and

experience. Both experiments and analyses take a lot of time and are expensive. The latter

approach requires a special numerical software, able to compute these highly dynamic phenomena.

Here, only a verification to experimental data is required. Numerical approach has been chosen

with use of Abaqus Finite Element Analysis (FEA) computer code for modelling and simulation of

explosion and shock wave propagation for the purposes of this thesis.

Since many models were required for each analysis, the decision was made to use Python

programming language together with Abaqus Application Programming Interface (API), which

provides means for interaction with Abaqus and use of its functionalities and capabilities. The

computer script has been written to reduce the modelling time and effort. This way the creation of

model, which varied in such parameters as mesh element size, charge weight, size and location of

obstacles and other parameters, has been automated. What is more, generation of new models to

reflect the individual needs requires little user code and programming knowledge.

To model blasts in Abaqus, use of the explicit solver and Coupled Eulerian-Lagrangian

(CEL) formulation can be used. In this formulation air volume is modelled by Eulerian elements,

and obstacles are modelled by Lagrangian elements. The Lagrangian, Eulerian and CEL

descriptions are broadly presented in this thesis and in Appendix A.

In the thesis three tests are presented. The first one is a mesh size study, which aims in

finding the proper size of the finite element, yet computationally efficient and producing reliable

results. Obtained results are verified for correctness based on analytical solution from the UFC

[24].

In the second analysis the blast wave flow through holes in the front face is simulated.

Pressures inside the structure are compared with each other and with outside, reflected pressure.

Again, results are verified with help of UFC [24].

In the final simulation, the numerical research performed by Remennikov and Rose [17]

regarding the blast wave propagation in complex city geometries is repeated. A comparative

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analysis between Computational Fluid Dynamics (CFD) code used by Remennikov and Rose with

CEL code is performed, and scaled distance concept is verified.

1.2. Thesis

1.2.1. Motivation

A significant factor was an increasing danger of terrorist attack proceeded by increasing awareness

of the society about the issue. Lack of both Polish and European standards regarding the

exceptional loading, especially explosive loading, that could be used by engineers while designing,

was another motive. From personal point of view, a desire to experience and explore topics, that

are not covered during the studies, a desire to improve Abaqus skills, especially in terms of writing

user’s scripts (Abaqus Scripting Interface) and to improve the knowledge about Finite Element

Method in general and Coupled Eulerian-Lagrangian in particular were the most significant

reasons.

1.2.2. Objectives

The major objective is to perform a set of analyses regarding the blast wave propagation arisen

from the explosion. Another goal is to prepare a script allowing for model creation based on

certain parameters. As a result, the pressure-time curve at certain significant points should be

returned. The pressure, that can be later used to specify the loading of the structure or to estimate

the level of its destruction.

Generated models will depend on the following parameters:

- size of examined area of wave propagation,

- size of finite elements used for discretization of the model,

- number of explosives, their weight and location,

- number, geometry and location of obstacles,

- location of the analysed structure.

Models can be summarized in three words: 3D, parameterised, scalable.

Except the air pressure changes, the size of finite elements used for discretization, the

pressure change inside a structure as a result of an external explosion and the influence of

surrounding structures on incident wave propagation will be examined.

Thesis structure

The thesis consists of 6 chapters, bibliography, appendices and a CD containing the Python script.

This introductory chapter contains abstract and description of the thesis. Later in this

document the types of exceptional loadings are presented (Chapter 2). Explosive loading in field

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of civil engineering is described. The current state of knowledge and regulations are discussed.

Numerous researches on the topic are recalled. Chapter 3 describes the tools and methods that

were used while working on the thesis. Abaqus FEA, explicit dynamics and modelling in Abaqus

are presented. Moreover, Python programming language, Abaqus Scripting Interface and basics on

interacting with Abaqus FEA are described. In Chapter 4 the Python script used to create models is

shown. Its structure, possibilities, limitations and usage are discussed. Exemplary script is

provided. In subsequent Chapter 5 the description, results and discussion about three types of

simulations are presented. The ultimate chapter includes final remarks related to explosive loading

and blast modelling in Abaqus FEA. Fields of future development are indicated.

Appendix A contains the theory of Lagrangian and Eulerian descriptions. Appendix B

contains UML class diagram with class’ relationships of the model generating script. On the

attached CD the Python script and short technical notes about its usage are provided.

2. Accidental loading

2.1. Introduction

Two European standards are devoted to the issue of accidental actions. [1] defines the basis of

structural design. It contains primary rules and basic concepts regarding the proposed calculation

method (the limit state concept in conjunction with a partial factor method), design situations,

modelling and others. It defines different types of actions loading the structure, inter alia

accidental actions. [2] is fully devoted to the issue of accidental loading, and contains, among

others, provisions about vehicles’ impact, internal explosions of dusts and gases, risk analysis and

some indications on limits of admissible damage. UFC [24] document describes blast loading and

its effects on structures in details. More about these two can be found in subsection Explosive

loading in Civil Engineering.

2.2. Types of accidental loading

Together with permanent and variable loads, accidental loads can be defined. They are a vast and

diverse group of loading arisen from various reasons and requiring individual design proceeding

and analyses. In many cases they can cause severe consequences. When the structure is subjected

to an accidental loading, one can say that exceptional conditions of the structure occur.

According to Eurocode [1], the following accidental action definition stands ([1] 1.5.3.5):

Accidental action - action, usually of short duration but of significant magnitude,

that is unlikely to occur on a given structure during the design working life

Based on [1] 3.2 and [2], major types of accidental loads are:

- seismic events (earthquakes),

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- fire,

- explosions,

- impacts from vehicles (trucks, cars, lorries, ships, trains, others),

- consequences of localised failure,

- wind,

- snow.

Yet impact, snow, wind and seismic actions (loads) may be variable or accidental actions,

depending on the statistical distributions and probability of occurrence ([1] 1.5.3.5 Note 2).

Accidental actions (loads), based on European standards, should be taken into account in

accidental design situation (see [1] 1.5.2.5) or in seismic design situation (see [1] 1.5.2.7), when

the structure is subjected to a seismic event, and shall be considered to act simultaneously in

combination with other permanent and variable actions ([2] 3.2 (5)).

To clarify, the design situation is a set of physical conditions taking place for certain time

under which the structure is required to fulfil its function, and except of two already mentioned,

the standard specifies another two: persistent and transient design situations.

As the definition of the accidental design situation claims ([1] 1.5.2.5), it is a design

situation involving exceptional conditions of the structure or its exposure, including fire,

explosion, impact or local failure. As can be seen, the resistance to explosion effects is indicated

explicitly.

2.3. Explosives

2.3.1. Definition and classification

From the Encyclopædia Britannica [22], explosive is any substance or device that can be made to

produce a volume of rapidly expanding gas in an extremely brief period. Three fundamental types

can be distinguished: mechanical, nuclear, and chemical. A mechanical explosive is one that

depends on a physical reaction (volcanic eruption, failure of a cylinder of compressed gas, mixing

of two liquids), a nuclear explosive is one in which a sustained nuclear reaction can be made to

take place with almost instant rapidity, releasing large amounts of energy, and a chemical one

depends on a chemical reaction. The rapid oxidation of fuel elements (carbon and hydrogen atoms)

is the main source of energy in this type of explosives [18].

Explosives can be classified by their various properties, such as sensitivity, velocity,

physical form, and others.

Chemical explosives account for virtually all explosive applications in engineering. Two

types of chemical explosives can be distinguished: detonating (high explosives) and deflagrating

(low explosives). Detonating explosives are characterized by extremely rapid decomposition and

development of high pressure. TNT and dynamite belong to this group. Deflagrating explosives

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involve merely fast burning and produce relatively low pressures. Black and smokeless powders

are exemplary deflagrating explosives.

Detonating explosives are usually subdivided into two categories, primary and secondary.

Primary explosives detonate when heat of sufficient magnitude is produced. Flame, spark, impact

and others can cause an ignition. Secondary explosives require a detonator and, in some cases, a

supplementary booster [22].

2.3.2. Blast wave propagation

Detonation is a very rapid chemical reaction. During the detonation, chemical explosives release

rapidly large amount of energy, which previously was stored in strong chemical bonds. Gases,

with temperature up to 4000°C and at a very high pressure are produced and expand instantly, thus

forming a layer of hot, dense, high-pressure gas called a blast wave [18]. This blast wave expands

radially outward from the surface of the explosion (Figure 1) and at a supersonic velocity in the

case of high explosives such as TNT. As the wave expands, it decays in strength, lengthens in

duration, and decreases in velocity (Figure 1 in [18]). In ideal situation (perfectly rigid ground), if

the detonation took place on the ground surface, blast wave propagates spherically and will be

identical to a free-air blast from twice the quantity of explosive.

Figure 1. USS Iowa (BB-61) firing a full of nine 410mm and six 130mm guns during a target

exercise near Vieques Island, Puerto Rico, 1 July 1984 [13][14]. Radial propagation of a shock

wave on water surface clearly seen.

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Figure 2. Explosive shape ([24], Figure 2-17)

Explosive charge can be of a various shape. Most common are specified by UFC [24] and

presented in Figure 2.

Among all, spherical or hemispherical charges produce so-called ideal blast wave.

Pressure-time relation for such a wave is depicted in Figure 3.

Key blast wave parameters associated with ideal blast waves are peak positive

overpressure, peak negative underpressure, dynamic pressure, positive and negative phase

durations, and positive and negative phase impulses (integrals with respect to time of the

respective pressures) [25].

soP , a peak positive overpressure, is a pressure over the value of ambient pressure op .

soP ,

a negative underpressure, is a pressure below the ambient pressure. Dynamic pressure oq is the

pressure formed by the movement of gas particles behind the moving shock front. The magnitude

of the dynamic pressures, particle velocity and air density is solely a function of the peak incident

pressure [24]. The relation between these parameters is shown in Figure 4.

The positive phase of a blast wave (Figure 3) is described by Friedlander formula [21]:

1 expA A

s so

o o

t t t tP t P

t t

(1)

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where:

At - arrival time,

ot - positive phase duration,

- wave form constant depending on the shape of the wave front (see Table 1)

Table 1. Wave form constant in relation to scaled distance Z [28]

1/3

mZ

kg

0.4 50

8.50 0.5

Figure 3. Typical ideal free-air blast wave pressure-time graph ([25], Fig. 1)

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Figure 4. Peak incident pressure versus peak dynamic pressure, density of air behind the shock

front, and particle velocity ([24], Figure 2-3)

Three principle effects caused by a burst load the structure: blast overpressures, fragments

generated by the explosion and the shock loads produced by the shock wave and transmitted

through the air or ground. The latter are loads which generate transient vibrations of the ground

and the structure. Usually they do not cause severe damages.

Various methods of estimating the blast peak (incident) overpressure were collected in [18].

All they base on a scaled distance, which is denoted as:

1/3 1/3 1/3

,R m ft

ZW kg lb

(2)

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where:

R - distance to the charge,

W - charge’s mass

Another important concept is a so-called TNT equivalence. Based on it, particularly any

explosive material can be represented by scaling its weight according to the formula ([24],

Equation 2-1):

d

EXP

E EXPd

TNT

HW W

H (3)

where:

EW - effective charge weight,

EXPW - weight of the explosive,

d

EXPH - heat of detonation of explosive,

d

TNTH - heat of detonation of TNT

This relation is valid for unconfined detonations of explosive material of similar shape.

Depending on the location of the explosive, internal and external explosions can be

identified. Later, only external explosions will be discussed, yet there is no big difference between

these two types except that blast wave-structure interactions are even more complex and multiple

reflections take place in the case of internal explosions.

External air blasts can be separated into free air burst, air bursts and surface air burst.

Free air burst occurs when the wave reaches the structure before being strengthened. When

the wave reaches the ground before reaching the structure, it might be necessary to take into

account its reflection from the ground. Two types of reflections can occur: classical (Figure 5) or

reinforcement reflection (Mach Front) [21] (Figure 6). This phenomenon depends on the angle of

incidence between ground and incident wave. 40° is assumed as a critical angle of occurrence of

the Mach Front. Magnitude of the resultant – incident and reflected – wave is higher and the

pressure-time relation is modified. Surface air bursts take place when the charge detonation occurs

on the ground or close to it. In such a case incident and reflected waves are merged near the

detonation point.

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Figure 5. Classical ground reflection ([21], Figure 3)

Figure 6. Reinforced ground reflection ([24], Figure 2-11)

When the shock wave reaches an obstacle, the reflection occurs. The reflected, or face-on

pressure, can be higher from 2 up to 20 times the incident pressure in the case of zero angle of

incidence [25], while up to 8 times for typical explosions [28]. The increase of the pressure

depends mainly on magnitude of incident pressure, angle of incidence, distance to the explosion

source (stand-off distance) GR and dimensions of the loaded obstacle’s face. The latter splits

obstacles’ faces into finite and infinite faces. The difference is defined by the clearing effect.

When the face is finite, after certain time called clearing time, the face-on pressure is relieved

completely due to expansion of the wave outside from the face. The clearing time can be less than

the positive phase duration of the wave. Then, the total load encountered by the face is less than

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the face-on load and is greater than the side-on load. The difference in pressure history for finite

and infinite faces is presented in Figure 7.

Figure 7. Pressure histories on front face of large and small buildings ([25], Fig. 5)

The wave pressure for unconfined environment (no obstacles affecting the incident wave) is

straightforward and can be achieved with use of analytical/semi-empirical formulae. When the

geometry of streets and buildings is complex, two major effects arise, i.e. channelling and

shielding effects. They are graphically outlined in Figure 8.

Impact of shielding and channelling on the wave pressure is variable and depends,

including factors mentioned earlier, also on streets width, height of the buildings, and what is

obvious, on explosive location and stand-off distance. In certain cases, positive phase blast wave

impulse can be reinforced even more than 5 times (compare Figure 9), and the importance of the

negative phase impulse increases significantly [25]. Numerical methods, i.e. CFD or CEL must be

used in order to receive accurate results in such cases.

Refer to the subchapter Previous researches on the topic for more data on the influence of

confinement on the blast wave.

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Figure 8. Schematic representation of shielding and channelling ([16], Fig. 19)

Figure 9. Peak reflected overpressure on target building with and without buildings along the street

([17], Fig. 1) [31]

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2.4. Explosive loading in Civil Engineering

Firstly, the current provisions, both European ([1, 2]) and American ([24]) are recalled, then some

general remarks are presented.

As it states in Eurocode - Basis of structural design [1] 2.1 (1):

A structure shall be designed and executed in such a way that it will, during its

intended life, with appropriate degrees of reliability and in an economical way

- sustain all actions and influences likely to occur during execution and use,

- remain fit for the use for which it is required.

Later ([1] 2.1 (4)) in the same standard, the risk arising from explosion and requirement of

taking it into account is explicitly expressed:

A structure shall be designed and executed in such a way that it will not be damaged

by events such as:

- explosion,

- impact, and

- the consequences of human errors,

to an extent disproportionate to the original cause.

This implies, that the designer must take into account the risk coming from the events (inter

alia explosion) and design the structure in a way to minimize the effects caused by them, if such

exceptional conditions can be reasonably foreseen to occur during the execution and use of the

structure ([1] 3.2 (3)). Events to be taken into account are those agreed for an individual project

with the client and the relevant authority ([1] 2.1.4 Note 1).

Eurocode suggests methods of reducing the potential damage resulting from accidental and

explosive loading [1] 2.1 (5). This can be achieved by:

- avoiding, eliminating or reducing the hazards to which the structure can be subjected,

- selecting a structural form which has low sensitivity to the hazards considered,

- selecting a structural form and design that can survive adequately the accidental removal

of an individual member or a limited part of the structure, or the occurrence of acceptable

localised damage,

- avoiding as far as possible structural systems that can collapse without warning,

- tying the structural members together.

Eurocode [2], General Actions – Accidental actions, is the only standard devoted to

accidental and explosive loading applied to structures. It describes principles and application rules

for the assessment of accidental actions on buildings and bridges, including the following aspects:

- impact forces from vehicles, rail traffic, ships and helicopters,

- internal explosions,

- consequences of local failure.

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The internal explosions (dust explosions, gas and vapour explosions) regard all parts of the

building where gas is burned or regulated or where otherwise explosive material such as explosive

gases or liquids forming explosive vapour or gas are being stored. [2] is not applicable to the so-

called solid high explosives [2] 5.1 (1) and does not specifically deal with accidental actions

caused by external explosions, warfare and terrorist activities[...]([2] 1.1), thus it is not a real aid

for the designers. Yet it mentions, that prevention or reduction of the action and design the

structure to sustain the action are strategies for dealing with accidental actions ([2] Figure 3.1).

Unified Facilities Criteria document [24] presents methods of design for protective

construction used in facilities for development, testing, production, storage, maintenance,

modification, inspection, demilitarization, and disposal of explosive materials and establishes

design procedures and construction techniques.

According to UFC, two main groups of blast loads on structures can be distinguished:

confined and unconfined explosions. Unconfined explosions can be later subdivided, based on the

blast loading, into free air burst, air burst and surface burst explosions, whereas confined

explosions can be subdivide into fully vented, partially confined and fully confined explosions.

First three one can associate with the external explosions, and last three with internal once. More

detailed specification of blast loading categories can be found in [24] Section 2-10 and subsequent

sections.

UFC provides methods for finding the reflected pressure on different faces of the structure,

for both internal and external explosions. It describes the trajectories and influence of fragments

carried by the blast wave and influence of air and ground shocks on the structure.

Reinforced concrete and steel design rules as well as design rules for masonry structures

and principles of dynamic analysis are discussed.

Explosive loadings originate from various sources. Terrorist attacks are the one the society

is most aware and scared of, hence these are the major reasons of the development of protective

materials and more resistant structures. The war industry has a big account in the researches, since

it tries to protect soldiers exposed to blasts and explosives at every turn. Storage of explosive

materials and protection against accidental detonation and a desire to protect people working in

companies, that deal with dangerous chemicals are another important factors that contribute to the

development of standards.

Major types and origins of explosives, after [15], are:

- Natural Explosions

- Lightning

- Volcanoes

- Meteors

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- Intentional Explosions

- Nuclear weapon explosions

- Condensed phase high-explosives

- Blasting

- Military

- Pyrotechnic separators

- Vapor phase high explosives (FAE)

- Gun powder/propellants

- Exploding spark

- Exploding wires

- Laser sparks

- Contained explosions

- Accidental Explosions

- Condensed phase explosions

- Combustion explosions in enclosures

- Gasses and Vapors

- Dusts

- Pressure vessels (gaseous content)

- BLEVE’s (Boiling Liquid Expanding Vapor Explosion)

- Unconfined vapour cloud explosions

- Physical vapour explosions

2.5. Previous researches on the topic

The availability of other researches is limited. The results of experiments are usually confidential

because of the high value and importance of them, and because of the risk, that their use by

terrorists may be dangerous. Some data is widely available, but usually it does not represent the

level of detail required to perform a valuable comparison.

Due to lesser complexity, most common are destructive analyses performed to examine the

behaviour of some particular, simple objects, such as window glasses, which are extremely

dangerous since they shatter after the shock wave reaches them. Laminated glasses and anti-shatter

films are tested to resist the shock wave. Other tests investigate the protection of different types of

materials to the structural elements such as columns or beams.

Extensive works on the subject have been conducted by Remennikov, Smith, Rose and

others. They investigated inter alia propagation of blast wave in space, influence of shadowing,

reflection and channelling, phenomena related to wave propagation, decreasing and increasing the

peak pressure. They discussed pressure-time curves for various obstacles’ configurations,

21

conducted both numerical and experimental researches on the topic. In the final analysis, the

numerical experiment described in [17] is repeated and obtained results are compared.

In Blast loading and blast effects on structures - an overview [18] Ngo, Mendis, Gupta and

Ramsay present an explanation of the nature of explosion and the mechanism of blast wave

propagation in free air, as well as introduce different methods to estimate blast load and structural

response. Moreover, they provide a response of structural elements of a building.

Modelling blast loads on buildings in complex city geometries by A.M. Remennikov and

T. A. Rose [17] is devoted to an accurate prediction of the effects of adjacent structures on the

blast loads on a building in urban terrain.

As the authors claim, historical records indicate that the majority of terrorist incidents

have occurred in an urban environment in the presence of nearby buildings forming the street

geometries. Streets create complex geometries, in which the actual blast loads can either be

reduced due to shadowing by other buildings or can be enhanced due to the presence of other

buildings in the vicinity. In such cases, typically used empirical relationships are not satisfactory

since they assume, that there are no obstacles between the charge and the target (so does UFC and

ConWep).

The study proves, that adjacent structures play a very important role when determining the

blast loads on buildings, as they can enhance the peak pressure (effect of congestion/effect of

confinement) significantly (up to 4 times in certain cases of channelling, see Figure 9).

Blast wave propagation in city streets - an overview by P. D. Smith and T. A. Rose [16],

experimentally and analytically investigates the blast wave interactions with buildings and

structures in an urban landscape. Street width, height of buildings and influence of the location of

the explosive within the street layout on the blast load experienced by buildings is examined. Also

the phenomena of channelling and shielding are presented. Works of different researchers are

recalled. The requirement of use of advanced numerical software is confirmed, as it might be

difficult to develop the rules predicting the blast resultants on buildings. The results of numerical

analyses are referred to as reasonably accurate prediction [16].

Another work by the same authors [25] deals with the nature of shock wave in both external

and internal explosions, and provides advices for increasing protection and robustness of a

structure. It describes the concept of blast-resistant design and principles of strengthening existing

buildings.

Peak pressures causing failure of various structural elements and whole buildings have been

assembled in doctoral dissertation by P. W. Sielicki Masonry Failure under Unusual Impulse

Loading [28]. According to it, pressure increase of 1kPa is a typical pressure causing glass failure,

whereas windows shatter when subjected to pressure of 3.5-7kPa. Pressure of 5.2kPa causes minor

house damages, while of 7kPa destructs them partially and makes them uninhabitable. Pressure of

22

about 50-70kPa results in collapse of both steel frame buildings and brick walls. Pressure above

70kPa entirely destroys any un-reinforced buildings.

3. Tools and methods

3.1. Abaqus FEA

Abaqus Finite Element Analysis (FEA) is a computer application for finite element analysis (FEA)

and computer-aided engineering (CAE). Abaqus delivers solutions for challenging nonlinear

problems, large-scale linear dynamics applications, and routine design simulations. It contains user

programmable features, scripting interface and Graphical User Interface (GUI) customization

features.

It is commonly used by automotive and aerospace industry for analysing structural

components of vehicles, it is used in analyses of behaviour of materials, by engineers for analyses

of structural responses, distribution of stresses, analyses of displacement field, and during the

design of various products. Abaqus FEA suite consists of four products, i.e. Abaqus/CAE,

Abaqus/CFD, Abaqus/Standard and Abaqus/Explicit. The first one is a pre- and post-processor,

which function is mainly to visualize the model and or results and allow the user to interact with

the application by providing input data, while the three others are components of the processor

(solver). Abaqus’s Computational Fluid Dynamics (CFD) solver is devoted to nonlinear coupled

fluid-thermal and fluid-structural problems. Abaqus/Standard is intended to be applied to the

analyses that are well-suited to an implicit solution technique, such as static, low-speed dynamic,

or steady-state transport analyses [4], while Abaqus/Explicit may be applied to those analyses,

where high-speed, nonlinear, transient response dominates the solution [5].

In general, FEA is a numerical technique for finding approximate solutions of partial

differential equations (PDE) and integral equations, which describe the real behaviour of

object/material, by means of numerical integration. The object is being divided info finite number

of small elements and solution is found for certain points called nodes. Alternative approaches are

boundary element method (BEM) and finite difference method (FDM) and others, however they

are not as powerful and robust as FEA.

3.2. Explicit dynamics

In this thesis, Abaqus FEA has been used for all numerical simulations and Abaqus/Explicit was a

solver of choice, since it is devoted to high-speed phenomena.

Explicit dynamic analyses are computationally efficient for large models with relatively

short dynamic response times and for analyses of extremely discontinuous events or processes, as

23

they allow for the definition of very general contact conditions and as they use a consistent, large-

deformation theory, hence models can undergo large rotations and large deformations [7].

The explicit methods of numerical integration are, as opposed to implicit methods,

conditionally stable, which means, that some requirements regarding time step need to be met in

order to obtain proper results. The advantage over implicit methods is, that they do not require the

computation of global set of equations after each time increment. Central difference method is one

of explicit methods.

In Abaqus the explicit dynamics analysis procedure is based upon the implementation of an

explicit integration rule together with the use of diagonal (lumped) element mass matrices. The

equations of motion for the body are integrated using the explicit central-difference integration

rule [6, 7]:

11/2 1/2

2

i ii i it t

u u u

(4)

1 1 1/2i i i iu u t u

(5)

where u is velocity, u is acceleration, superscript i is the increment number and 1/ 2i and

1/ 2i are midincrement values. The central difference integration operator is explicit in that the

kinematic state can be advanced using known values of 1/2iu

and i

u from the previous

increment (in explicit approach unknown values are obtained from information already known -

obtained in previous time steps).

The explicit procedure requires neither iteration nor convergence checking (as in implicit

approach), but the time increment must be less than the so-called stable time increment.

With no damping, the stable time increment is described by the highest eigenvalue in the

system:

max

2t

(6)

With damping, the stable time increment is given by:

2

max max

max

21t

(7)

where max is the highest frequency of the system and 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 (8)

24

where minL is the smallest element dimension in the mesh and

dc is the dilatational wave speed.

This condition is often referred to as Courant-Friedrichs-Lewy (CFL) condition, which describes

the necessary condition for convergence while solving certain partial differential equations

numerically by the method of finite differences, which are commonly used in explicit

algorithms. [27]

The dilatational wave speed dc can be expressed for a linear elastic material (with

Poisson’s ratio equal to zero) as:

d

Ec

(9)

where E is the Young’s modulus and is the density of the material, or more generally as:

ˆ ˆ2

dc

(10)

where ̂ and ̂ are the effective Lamé’s constants. In the case of isotropic, elastic material they

can be defined in terms of Young's modulus E and Poisson’s ratio by:

ˆ1 1 2

E

(11)

ˆ2 1

E

(12)

The Abaqus software finds the proper time increment automatically using above mentioned

techniques. In Eulerian analyses, the stable time increment size is adjusted this way to prevent

material from flowing across more than one finite element in each time increment. When the

material velocity approaches the speed of sound (for example in simulations involving blasts and

shocks), further restrictions on the time increment size may be needed to maintain accuracy and

stability [10].

3.3. Blast modelling approaches

In civil engineering, modelling is a technique of representing the real life objects in such a way

(usually strongly simplified), that it is possible to perform certain analyses and computations on

them, which result in data about their real behaviour. The model simplification cannot influence

the obtained results dramatically. Typically, few models of given object, varying in its complexity

(e.g. 2D, 3D) and software capabilities can be created. Most common analyses concern statics and

dynamics.

25

In blast modelling approaches, a distinction can be made on methods that produce the value

of pressure only, and those, which additionally allow for air pressure propagation analysis. Both

groups are presented below.

The simplest approach to include the influence of a blast wave is to use UFC [24] and

calculate time variable, plane pressure and other crucial values as a function of distance, explosive

weight, type and location of explosive (ground, air) and other parameters. UFC provisions and

rules originate from empirical tests and experiments. This method does not include the influence

of reflection and shadowing due to surrounding obstacles. Based on this, a software called

ConWep (Conventional Weapon) was developed.

In ConWep, loading is applied directly to the structure subject to the blast and any fluid

medium (air) is not needed to transport the wave, thus making it impossible to analyse wave

propagation. The ConWep model provides data for two types of waves: spherical waves for

explosions in mid-air (air blasts) and hemispherical waves for explosions at ground level in which

ground effects are included (surface blasts) [19]. Yet, the influence of other structures, that may

reflect the wave, enhance it (channelling) or reduce (shadowing), can’t be taken into account (so

called free-field predictions). Thus, ConWep is valuable for analyses with direct shock wave

loading, such as analysis of a reinforced concrete column loaded with a blast. A typical pressure

history of a blast wave used by ConWep is depicted in Figure 3.

In ConWep, a scaled distance concept (see Chapter 2), describing the distance of the

loading surface from the source of the explosion and weight of the explosive charge, is used. For a

given scaled distance, following data, based on real-life experiments can be obtained: the peak

overpressure, the arrival time, the duration of the positive phase and the exponential decay

coefficient (for both the incident pressure and the reflected pressure). The time history of incident

and reflected pressure is presented in [21], Figure 30.4.5-3.

The total pressure P is described as [20]:

For cos 0 :

2 21 cos 2cos cosincident reflectP t P t P t (13)

For cos 0 :

incidentP t P t (14)

where:

incidentP t is a function of the incident pressure,

reflectP t is a function of the reflected pressure,

is the angle of incidence, i.e. the angle between the normal of the loading surface and the

vector connecting the loading surface and the explosion point.

26

Except that, several numerical approaches can be used to describe and analyse propagation

of the blast wave and air pressure changes. The article [3] is partially devoted to this issue. Some

of the modelling techniques used in the past and nowadays are given:

- reduction of the structural system to a single degree of freedom (SDOF) model,

application of the SDOF to the modern structural finite element analysis code, together

with full 3D geometry and realistic boundary conditions. For more details refer to [18],

Paragraph 4.

- use of one-way coupling of a computational fluid dynamics (CFD) code with a structural

analysis code,

- use of a (Fully) Coupled Eulerian-Lagrangian (CEL) code.

In the latter model, the Lagrangian structure is surrounded by a volume of air modelled

with use of Eulerian elements. Refer to the Appendix A for more information about Lagrangian

and Eulerian descriptions of motion. One of the main arguments in favour of this method is a

chance to analyse complex geometries, that would be impossible to handle with other approaches.

Authors of [3] claim, that there is a belief, that growth of complexity won’t cause the significant

increase of errors. Yet, in some cases, a comparison to empirical data should be performed to

verify the correctness of CEL code.

While writing this thesis, the latter approach has been used.

Model creation in Abaqus FEA

Blast wave propagation with influence of obstacles is an example of fluid-structure interaction.

Fluid (air) is a medium in which the pressure change propagates. To enable such highly dynamic

analysis, an Eulerian description must be used. In the Eulerian analysis nodes are fixed in space,

and material flows through elements that do not deform. These elements may not be 100% full of

material. This implicates the need of material boundary recomputation after each time increment.

If the Eulerian material flows out of the Eulerian mesh, it is lost from the simulation. In contrary,

the obstacles must be defined as Lagrangian elements, where nodes are fixed within the material,

and elements deform as the material deforms [10]. If the interaction between two types of elements

can be assured (by means of Eulerian-Lagrangian contact), then the analysis is denoted as coupled

Eulerian-Lagrangian (CEL) analysis. Solid elements are advised over shell elements as Lagrangian

components.

In Abaqus, Eulerian elements may simultaneously contain more than one material. Within

each element, the Eulerian volume fraction (EVF) of each material is computed. In each time

increment the boundaries of each Eulerian material are rebuilded. The interface reconstruction

algorithm approximates the material boundaries within an element as simple planar facets. This

27

produces a simple, approximate material surface that may be discontinuous between neighbouring

elements.

To introduce materials into Eulerian element, an Eulerian section, with list of materials that

may appear, must be defined. Additionally, initial condition, i.e. material assignment predefined

field in an initial step must be used to fill the Eulerian mesh with the material, because by default,

all Eulerian elements are initially void (having neither mass nor strength). To fill an Eulerian

element, initial volume fraction for each available material instance must be defined.

The Eulerian time incrementation consists of two steps: traditional Lagrangian phase

followed by an Eulerian, or transport, phase. Such formulation is known as Lagrange-plus-remap.

During the Lagrangian phase of the time increment, nodes are assumed to be temporarily fixed

within the material, and elements deform with the material. During the Eulerian phase of the time

increment deformation is suspended, elements with significant deformation are automatically

remeshed, and the corresponding material flow between neighbouring elements is computed [10].

This concept is depicted in Figure 10.

Figure 10. The Eulerian time incrementation algorithm scheme [26]

To introduce the contact between Eulerian and Lagrangian components, any type of

contact, e.g. general or automatic contact must be defined. Then, the interface between the

Lagrangian structure and the Eulerian material is calculated automatically. In CEL there is no need

to generate a conforming mesh for the Eulerian domain. Eulerian-Lagrangian contact also supports

failure and erosion in the Lagrangian body, e.g. Lagrangian element failure can open holes in a

surface through which Eulerian material may flow. Moreover, it is possible to define independent

inflow and outflow boundary conditions at an Eulerian boundary. In all simulations for the

purpose of this thesis, inflow free and outflow non-reflecting Eulerian boundary conditions have

been used. The former enables the material to flow into the Eulerian domain freely, while the latter

is used to simulate an infinite domain of air.

28

Air and TNT materials, and in general all liquids and gases, are modelled with use of

equations of state (EOS), i.e. air with use of ideal gas EOS, and TNT with use of Jones-Wilkins-

Lee (JWL) EOS. EOS are described later in this Chapter.

In blast wave propagation, following field output values are needed:

- EVF – gives the Eulerian volume fraction for each material in the Eulerian section

definition. It is important to request output for EVF in all Eulerian analyses because

visualization of Eulerian material boundaries is based on the material volume fractions.

- SVAVG – gives a single value of stress for each element computed as a volume fraction

average of stress over all materials present in the element.

Equations of State

An equation of state is a constitutive equation that defines the pressure as a function of the density

and the internal energy [11].

In the thesis, to model the air, an ideal gas equation of state has been used, which is an

idealization to real gas behaviour, and can be used to model any gases. TNT has been defined by

means of Jones-Wilkins-Lee equation of state, which models the pressure generated by the release

of chemical energy in an explosive.

The JWL equation of state can be written in terms of the internal energy per unit mass mE

as [12]:

0 0

1 2

1 0 2 0

1 exp 1 exp mp A R B R ER R

(15)

where:

A , B , 1R ,

2R , (adiabatic constant) are material constants,

0 is the density of the explosive,

is the density of the detonation products.

The ideal gas equation of state can be written in the form of [12]:

Z

Ap p R (16)

where:

Ap is the ambient pressure,

R is the gas constant,

is the current temperature,

Z is the absolute zero on the temperature scale being used. This is different than 0 only

when a non-absolute temperature scale is used.

29

Properties and their respective values used to define air and TNT materials are presented in

Table 2.

Table 2. Definition of materials

Material Type Property Value Unit

Air

Density Mass density (1) 1.293 kg/m3

Eos

Specific gas constant

(2) 287 J/kgK

Ambient pressure (3) 101325 N/m2

Specific heat Specific heat 717.6 J/kgK

Viscosity Viscosity (4) 6.924e-06 at 100.0K kg/s*m

Initial state Specific energy 193300 J/kg

Ambient pressure 101325 N/m2

TNT

Density Mass density (1) 1630 kg/m3

JWL Eos

Detonation wave speed 6930 m/s

A 373770000000 N/m2

B 3747100000 N/m2

0.35 -

1R 4.15 -

2R 0.9 -

Detonation energy

density 0.0 J/kg

Pre-detonation bulk

modulus 0.0 N/m2

Initial state Specific energy (5) 3680000 J/kg

Ambient pressure 101325 N/m2

(1) at 0°C

(2) for dry air

(3) approximately at sea level

(4) only one row from the whole set presented

(5) see below

The proper identification of the TNT specific energy is crucial for the reliability of

analyses. Different values have been proposed in the literature, with 3.68MJ/kg as defined in

30

computer code LS-DYNA [32], 4.52MJ/kg as in [33] and 5MJ/kg (for C4 like components). The

first set of simulations (Subchapter 5.1) deals with this topic.

3.4. Abaqus Scripting Interface

Python is a programming language, which is multi-platform, object oriented, interpreted,

dynamically typed and with automatic memory management (garbage collector). It is popular due

to straightforward semantics and small learning curve, yet it is powerful as it allows to program

practically any functionality and application. Large number of libraries devoted to certain issues

and an active community simplify both learning and solving the problems. Due to abovementioned

advantages, it is a scripting language of choice of many application vendors, which try to enhance

their software by implementing an Application Programming Interface (API) to allow the users to

interact and customize their products. This is also the case of Abaqus FEA, where Python API,

called Abaqus Scripting Interface, so a set of routines and functions, has been implemented. The

Abaqus manual, together with Abaqus Scripting Interface (ASI) description, contains introduction

to Python programming language.

Abaqus Scripting Interface is an application programming interface (API) to the models

and data used by Abaqus, which allows to create and modify components of an Abaqus model,

create and submit analyses, read and write to Abaqus output database file and view the results [8].

Abaqus Scripting Interface is an extension of Python programming language and provides a set of

classes (The Abaqus Object model) reflecting Abaqus internal data structure. Programming with

use of Abaqus Scripting Interface requires the knowledge about such concepts of object-oriented

programming (OOP) as classes, objects, constructors, object- and class- methods and members,

inheritance, encapsulation, polymorphism, references and others.

One of the components of Abaqus suite is Abaqus Python development environment

(PDE), which provides an interface that can be used to develop - create, edit, test, and debug -

Python scripts. It has been used extensively by the author during his work on the model generating

script.

The following flowchart (Figure 11) illustrates how Abaqus Scripting Interface commands

interact with the Abaqus/CAE kernel. Arbitrary Python command can be triggered by any of

graphical user interface (GUI), command line interface (CLI), or a script. These commands are

parsed and executed by Python interpreter before they reach the Abaqus/CAE kernel. After a job is

submitted, an input file, containing all user-defined data, including the geometry, is created and

passed to the solver. The result is saved in an output database file.

31

Figure 11. Abaqus Scripting Interface commands and Abaqus/CAE [9]

32

4. Python modelling script

4.1. Description

Described below is a set of Python scripts, that generate a model in Abaqus FEA for the shock

wave analysis.

The package of scripts is composed of the following components: the main script, the

configuration file and the set of modules using Abaqus Scripting Interface. All of them are

attached to the thesis and briefly described in succeeding paragraphs and subsections.

The script allows the user to generate the model based on parameters defined in

configuration file and routines defined in the main script. The Main script subsection contains

indications on how to use the modules and configuration file in order to produce a model.

No restrictions are made on the number of obstacles and charges that can be defined.

Obstacles of rectangular, circular and polygonal cross-sections can be modelled. Moreover it is

possible to form more complex shapes by rotating the obstacles or placing them one on another.

The script automatically merges all obstacles into one part, hence this does not imply any issues.

Any charge, based on its weight, is converted into a cube of appropriate size, and its explosion can

be triggered at any, user-specified time. Field output does not have to be saved for the whole

model, but for user-defined points or cuboid spaces. Details about fundamental steps conducted

during model generation can be found in the previous section.

Some functionalities can be altered from the inside of the module files – please refer to the

file of interest and comments and suggestions within it.

Most of the limitations related to obstacles’ geometry can be solved by appropriate usage of

attainable functions. More advanced methods have not been implemented mainly due to Abaqus

CAE limitations.

The most severe limitation is related to the meshing step. After the parts have been

instantiated and merged into one instance in the assembly step, the mesh can be applied. For the

instance of Eulerian type, the only elements that can form a mesh are elements of hexahedral

shape. Since the geometry of instance and its cells (cells are 3D objects from which the instance is

composed of) can be complex, mainly due to complicated geometry of obstacles, and since each

cell is treated as a separate meshing region, different meshing techniques have been developed in

order to handle the mesh generation. The advised technique is a structured technique, which

generates a structured mesh, mainly because of its efficiency. The other one is a sweep technique,

which generates a swept mesh. These two can coexist together in one instance. The partition

algorithm tries to partition the instance in such a way, that any structured or sweep meshing

technique can be applied to regions. In some cases manual intervention may be required.

33

Even when the same meshing technique has been used for adjacent regions, issues with

meshing can appear. Again, this is an individual case that emerges from internal Abaqus

limitations.

Few general remarks, valid for all models, are presented below.

1. All units are standard SI units, i.e. m, s, kg, J, N.

2. After instantiating Analysis class, all methods generating and operating on a model must

be called in exactly specified order (see Figure 12).

3. The air volume is always a cuboid. Any obstacles located outside of it will not be

analysed, since the air is a medium in which the pressure propagates.

4. The air volume in any model has a displacement boundary condition set to zero on the

bottom face, which simulates the non-deformable and non-penetrable ground. Each face

is loaded with the air pressure. All faces except the bottom one have inflow free and

outflow nonreflecting Eulerian boundary conditions.

5. It is possible to either merge the obstacles with the air instance or cut them from it. The

latter produces holes in Eulerian domain, into which Eulerian materials can’t flow, hence

forming a rigid barriers influencing the material propagation. The former requires

additional displacement boundary conditions on each face of the obstacle to achieve this.

Additionally, in the first approach, ambient pressure stress is applied on each face to

reflect the reality.

6. The field output can be recorded at each incrementation. This obviously has a huge

impact on results available after the simulation, length of the calculation process and size

of the output file. This can be changed in the Config class. Refer to it for more detailed

description.

7. In the case of large simulations it is recommended to record the values of interest from

limited area or even from single mesh elements. This can be achieved with use of

fieldOutputHandler class.

34

Figure 12. Block scheme presenting execution steps of the main script

4.2. Main script

The main script in general is composed of two parts. The first one is constant and is responsible

for informing Python interpreter about the location of the BlastAnalysis package. This is achieved

35

by setting the library_path variable, which has to contain the absolute path to BlastAnaylis folder.

When this is done, all modules from this package are imported.

The second part requires user involvement. It starts with the initiation of Config class and

optional change of its properties. Then, the obstacles, as well as charges, are being defined. They

are added to the respective handler classes, which take care of them. These handlers, together with

the Config object, are passed to the constructor of the Analysis class. By calling methods of

Analysis object, it is possible to create a model, apply a mesh to it, and trigger a simulation.

Subsequently provided exemplary main script and information on all classes and modules are

intended to give more details on usage of the package.

Exemplary main script

Note: some comments which are available in the main file were removed for clarity. Only

significant statements are presented. The script can be found on the attached CD under the name

simple_examlpe.py. This file can be also used as a template for other scripts.

The following script creates the air volume of a size of10 10 20m , sets the duration of

explosion step to 0.05s and sets the size of the mesh to 10cm. It defines one charge, located at

point x=5, y=3, z=5, with the weight of 10kg and detonation delay time of 91 10 s . It defines a

cuboid obstacle at point x=2.5, y= 0, z=15.0 and a size of 5 5 0.1m . It declares a field output

point with the name Point1 at the front wall of the obstacle (position x=5, y=3, z=15), for which

value of the pressure will be collected during the analysis. Later, the analysis object is created and

all previously declared data is passed to it. Next, the model is created. The main instance will be

partitioned by planes defined by obstacles' and charges' faces. Only faces with normal parallel to z

axis will be used for partitioning in the case of charges. Method createMesh of Analysis object

generates the mesh (forceSweep is False, hence structured meshing technique will be used when

applicable). Method prepareAnalysis makes the final changes to the Abaqus keyword input file

and should be called just before runAnalysis. Its argument, analysisName, specifies the name of

Abaqus input and output files that will be created in the last step. The job is submitted by method

runAnalysis. OnlyInputFile is set to False, thus entire analysis will be performed. Otherwise,

Simple Example.inp input file would be created.

# imports all the modules

from BlastAnalysis import *

from config import Config

config = Config()

config.size = Vector(10.0, 10.0, 20.0)

36

config.explosion_step_duration = 0.05

config.mesh['size'] = 0.10

# define charges

charge0 = Charge(config, Point(5.0, 3.0, 5.0), 10.0, 1.0e-09)

chargeHandler = ChargeHandler()

chargeHandler.add(charge0)

# define obstacles

obstacle0 = Cuboid(Point(2.5, 0, 15.0), Vector(5.0, 5.0, 0.1))

obstacleHandler = ObstacleHandler()

obstacleHandler.add(obstacle0)

# define field output points

fieldOutputHandler = FieldOutputHandler()

fieldOutputHandler.add(FieldOutputHandler.FO_TYPE.ONE,

Point(5.0, 3.0, 15.0), None, 'Point1')

a = Analysis(config=config,

chargeHandler=chargeHandler,

obstacleHandler=obstacleHandler,

fieldOutputHandler=fieldOutputHandler)

# createmodel

a.createModel(requestEntireFieldOutput=True,

partitionObstacles=True,

partitionCharges=True,

charges_partitionPlanes=[ AXIS.Z])

a.createMesh(forceSweep=False) # create a mesh

a.prepareAnalysis(analysisName='Simple Example') # create a

job and merge keywords

a.runAnalysis(onlyInputFile=False) # run the job

Example 1. Exemplary main script

37

4.3. Configuration file

Configuration file config.py contains a class which defines customizable parameters used by all

other modules. Configuration file is imported in the main script file, and the Config class is

instantiated. Parameters can be declared directly in the constructor of the class, or later, when

Config object is available, by direct setting its properties.

It is possible to copy the config.py file to redeclare object properties, and then import it and

use alternatively to the original configuration file, as shown below:

from config import Config #original configuration file

from new_config import Config as New_Config #new configuration

file

config1 = Config() #old Confg instance

config2 = New_Config() #new Config instance

Example 2. Usage of few (two) configuration files in the main script file.

List of customizable parameters in Config object (see Figure 13 for more details):

- size – size of the air space,

- airDef – material definition – air (contains inter alia mass density, gas constant, specific

heat and others),

- TNTDef – material definition – TNT (contains inter alia mass density, detonation wave

speed and others),

- initialCond_stress_air – stress initial condition for air set,

- initialCond_stress_TNT – stress initial condition for TNT set,

- initialCond_specificEnergy_air – specific energy initial condition for air set,

- initialCond_specificEnergy_TNT – specific energy initial condition for TNT set,

- explosion_step_duration – time period of Explosion step,

- air_pressure – magnitude of the Pressure load,

- numIntervals – number of intervals at which field output is recorded. If set to 0, field

output will be recorded at each increment. This applies only to the global field output -

any output set by FieldOutputHandler is always saved at each increment.

- mesh – mesh properties,

- airflow – defines whether air flow should be modelled,

- merge_or_cut – specifies whether obstacles should be merged with the air instance and

displacement boundary conditions should be defined on obstacles’ faces, or obstacles

should be cut out from the air instance.

38

Figure 13. Config class (UML)

4.4. Modules

The following are the modules contained in BlastAnalysis folder. Public properties and methods’

prototypes of each class have been represented in form of UML’s (Unified Modelling Language)

class diagram. See Appendix B for the full class diagram with class relationships. All methods

called inside the main script, that the user is most interested in, are described.

Analysis

The core of all modules. Contains the definition of Analysis class which is used to create a model,

create a mesh and run the simulation. It imports Abaqus Scripting Interface modules, classes and

constants. Analysis object requires objects of Config, ChargeHandler, ObstacleHandler and

FieldOutputHandler classes as its arguments. Methods of this object must be called in the correct

order:

1. createModel(requestEntireFieldOutput,

partitionObstacles,

partitionCharges,

charges_partitionPlanes)

2. createMesh(forceSweep)

3. prepareAnalysis(analysisName=’’)

4. runAnalysis(onlyInputFile=False)

39

Meaning of the parameters:

- requestEntireFieldOutput – if set to true, the whole model will be queried for field

output,

- partitionObstacles – if true, then the model will be partitioned by Datum planes parallel

to obstacles’ surfaces,

- partitionCharges – when true, model will be partitioned by Datum planes specified by

charges’ faces, which in turn are specified by charges_partitionPlanes,

- charges_partitionPlanes – list of normal to planes. Model instance will be partitioned by

planes defined by charges' faces. Only faces with normal parallel to these axes will be

used for partitioning in the case of charges. Note, that the least axes specified, the better.

Usually one axis is enough,

- forceSweep – boolean indicating which meshing technique to use – sweep or structured,

- analysisName – specifies name of the analysis. If not defined or set to empty string (by

default), the current timestamp (i.e. the number of seconds elapsed since 00:00 hours, Jan

1, 1970 UTC) will be used after the name Analysis to uniquely distinguish each

simulation.

- onlyInputFile – boolean describing whether to submit the analysis or just save the input

file. False by default.

__init__

File responsible for import of all modules to the local namespace when following import statement

from BlastAnalysis import * is executed.

Charge

Class representing a charge. Parameters taken: configuration object, location point, mass of a

charge in kilograms, detonation delay time from the start of Explosion step in seconds.

ChargeHandler

Class responsible for managing the charges. It partitions the air part/instance to define the charges.

Charges are added to the ChargeHandler object with use of add method.

FieldOutputHandler

Class creating the field output requests for given mesh elements. Two calls to add method, which

is used to specify the field output area, are possible:

40

fieldOutputHandler.add(FieldOutputHandler.FO_TYPE.ONE,

Point(5,3,8), None, 'Point1')

fieldOutputHandler.add(FieldOutputHandler.FO_TYPE.MANY,

Point(0,0,0), Vector(3,5,7), 'Set1')

First call creates a field output request for a single finite element located at point (5, 3, 8)

and named Point1. In the second call a cuboid space, named Set1, is defined by specifying its

origin in point (0, 0, 0) and setting its size to 3 5 7m . Results for all finite elements contained

in this cuboid will be saved.

ObstacleHandler

Class grouping the obstacles and performing some common operations on them. Obstacles are

added to the ObstacleHandler object with use of add method.

AbstractObstacle

An abstract base class for all obstacle classes. Defines methods that each obstacle class should

define. These methods are called internally by the script to create a part or an instance of obstacle

in Abaqus. Since its abstract, this class mustn’t be instantiated at all.

obstacles

Defines various types of obstacles, that can be later added to the obstacleHandler. Each class is a

subclass of AbstractObstacle class and requires parent’s instantiation in its constructor. It

overrides createPart and createInstance methods. Available obstacle types are: Polygon, Cuboid

and Cylinder. Their prototypes and brief descriptions are shown below.

Polygon(Point position, float height, list xz_points) – creates a polygon with vertices specified by a

list of tuples of x and z coordinates xz_points, with given height and at given position. xz_points

are relative to the position. Shape defined by xz_points will be automatically closed, i.e. first and

last points from the list will create the last side of the polygon.

Cuboid(Point position, Vector dimension, AXIS axisOfRotation=AXIS.Y, float angle=0.0) – creates

a cuboid at specified position and with given dimensions, which can be rotated by given angle

about axis axisOfRotation.

41

Cylinder(Point position, float radius, float height, AXIS axisOfRotation=AXIS.X, float

angle=270.0) – creates a cylinder at given position, defined by radius and height, which can be

optionally rotated by given angle about axis axisOfRotation.

basicMath

Defines two basic mathematical types – Point and Vector. These are commonly used across all

modules to specify the location of the point in a 3D space and dimensions of an object. Arguments

passed to each of them are x, y, z coordinates or x, y, z dimensions, respectively.

constants

Defines enumerated type and two constant values, from which AXIS is commonly used as a

parameter passed to objects and methods.

Figure 14. Analysis class

Figure 15. Vector, Point and Charge classes

42

Figure 16. ChargeHandler, ObstacleHandler and FieldOutputHander classes

Figure 17. AbstractObstacle abstract base class and three classes: Polygon, Cylinder and Cuboid

inheriting after it.

Figure 18. Enumeration types: AXIS and AXIS_DIRECTION

43

5. Analyses

5.1. Mesh size study

5.1.1. Description

The aim of the study is to obtain the pressure-time blast loading curves on faces of a rectangular

structure from an external, unconfined, surface, TNT burst. The example comes from the UFC

[24] Problem 2A-10. Results from UFC and Abaqus, for different density of the mesh are

compared in order to find the size of finite element at which the results can be considered as

similar. Positive and negative loading for the front wall, roof and rear half of side walls of the

structure are calculated. Additionally, different values of TNT specific energy are considered to

show its influence on results and to find the one used across subsequent analyses.

The geometry of the structure and location of a hemisphere charge of 2267.96kg (5000lbs)

are shown in Figure 19. Symbols used in subsequent Figures, Tables as well as in the Python script

are consistent and presented below.

Figure 19. Scheme of the simulation model: a) view from the side, b) view from the top

Table 3. Symbols used

Symbol Value

[ft] [m]

R 155 47.244

B 30 9.144

H 12 3.6567

44

5.1.2. Analytical solution

According to [24], the shock front is assumed plane. For more comprehensive data and description

of symbols refer to [24] and example 2A-10.

Determination of blast wave parameters

Determination of following free-field blast wave parameters at Point A:

- peak positive incident pressure soP

- time of arrival of blast wave At

- wave length of positive pressure phase WL

- duration of positive phase of blast pressure ot

Scaled ground distance:

1/3 1/3 1/3

1558.53

6000G

R ftZ

W lb

From Figure 2-15 [24] for 1/3

8.53G

ftZ

lb :

1/3 1/3

1/3

1/3 1/3

1/3

1/3 1/3

1/3

12.8 88253

3.35

3.35 6000 60.9

2.10

2.10 6000 38.2

2.35

2.35 6000 42.7

so

A

W

W

o

o

P psi Pa

t msA

W lb

t ms

L ft

W lb

L ft

t ms

W lb

t ms

Determination of unit positive incident impulse si from Figure 2-15 [24]:

1/3 1/3 1/3

1/3

8.53 9.0

9.0 6000 163.5 1127293

s

G

s

ift psi msZ

lb W lb

i psi ms Pa ms

45

Table 4. Free-field blast wave parameters for Points A, B, C, calculated as for Point A – part 1

Point

R GZ soP

ft

m

1/3/ft lb 1/3/m kg

psi

Pa

A 155 47.244 8.53 3.3838 12.8 88253

B 170 51.816 9.35 3.7091 10.8 74463

C 185 56.388 10.18 4.0384 9.0 62053

Table 4. Free-field blast wave parameters for Points A, B, C – part 2

Point

1/3/At W

At

ms

1/3/WL W WL

1/3/ms lb 1/3/ms kg

1/3/ft lb 1/3/m kg

ft

m

A 3.35 4.360 60.9 2.10 0.833 38.2 11.64

B 3.90 5.076 70.9 2.24 0.889 40.7 12.41

C 4.60 5.987 83.6 2.35 0.932 42.7 13.02

Table 4. Free-field blast wave parameters for Points A, B, C – part 3

Point

1/3/ot W

ot

ms

1/3/si W si

1/3/ms lb 1/3/ms kg 1/3

psi ms

lb

1/3

Pa ms

kg

psi ms

Pa ms

A 2.35 3.059 42.7 9.0 80762 163.5 1127293

B 2.48 3.228 45.1 - - - -

C 2.62 3.410 47.6 - - - -

Front wall peak positive reflected pressure (Point A):

From Figure 2-193 [24]:

12.8

2.70 2.70 12.8 34.6 2385590

so

r r r so

P psiC P C P psi Pa

where is angle of incidence of the pressure front

Unit positive reflected impulse, from Figure 2-194 [24]:

1/3

1/3

12.817.0 17.0 6000 308.9 2129791

0

so r

r

P psi ii psi ms Pa ms

W

46

Front wall loading, positive phase

Calculation of sound velocity in reflected overpressure region rC from Figure 2-192 [24]:

12.8 1.325so r

ftP psi C

ms

Clearing time for reflected pressures ct calculation (from Equation 2-3 [24]):

4

1c

r

St

R C

where:

3012.0

2

3015 12.0

2

12.00.80

15.0

S ft

G ft ft

SR

G

S - height of front wall or one-half its width, whichever is smaller

G - maximum of wall height or one-half its width

then:

4 1220.1

1 0.80 1.325ct ms

Calculation of fictitious positive phase duration oft from Equation 2-11 [24]:

2 2 163.5

25.512.8

s

of

so

it ms

P

From Figure 2-3 [24], peak dynamic pressure:

12.8 3.5 24132so oP psi q psi Pa

Drag coefficient, based on from Section 2-15.3.2 [24]:

1.0 12.8 1.0 3.5 16.3 112385D so D oC P C q psi Pa

Calculation of fictitious duration of the reflected pressure according to Equation 2-11 [24]:

2 2 308.917.9

34.6

r

r

r

it ms

P

The pressure time curve is plotted in Figure 20.

47

Front wall loading, negative phase

Peak reflected pressure at angle of incidence from Figure 2-15:

1/3 1/3

34.6 8.5

17.0 10.4

r r

r r

P psi Z P

i ipsi ms Z

W W

Determination of peak negative reflected pressure rP

and peak negative reflected impulse ri

:

1.3 1/3 1/3

8.5 3.25 22408

10.4 14.6

r r

r r

Z P P psi Pa

i i psi msZ

W W lb

1/314.6 6000 265.3 1829179ri psi ms Pa ms

Fictitious negative reflected pressure duration rft :

2 2 265.3163.3

3.25

r

rf

r

it ms

P

Negative phase rise time:

0.27 0.27 163.3 44.1rft ms

The negative pressure-time parameters:

42.7

0.27 42.7 44.1 86.8

42.7 163.3 206.0

o

o rf

o rf

t ms

t t ms

t t ms

The negative pressure-time curve is plotted in Figure 20.

Side wall loading, positive phase

Calculation of loading on the rear half of the side wall (Point B to C)

Distance between Points B and C:

15.0L ft

Wave length to span length ratio wfL

L:

40.72.71

15

wfL

L

Based on Figures 2-196, 2-197 and 2-198 [24], for Point B:

2.71wfL

L

48

1/3

1/3

0.76

10.8 0.66

2.47

E

d

sof

of

C

tP psi

W

t

W

where EC - equivalent load factor,

dt - rise time

Hence:

1/3

1/3

0.76 10.8 8.2 56537

0.66 6000 12.0

2.47 6000 44.9

E sof

r

of

C P psi Pa

t ms

t ms

where rt - fictitious reflected pressure duration,

oft - fictitious positive phase pressure duration

Peak dynamic pressure from Figure 2-3 [24]:

8.2 1.55 10687E sof oC P psi q psi Pa

Based on Section 2-15.3.2 [24], the drag coefficient equals:

0.40DC

Calculation of peak positive pressure from Equation 2-12 [24]:

8.2 0.40 1.55 7.6 52400E sof D oC P C q psi Pa

The pressure-time curve is plotted in Figure 21.

Side wall loading, negative phase

Calculation of loading on the rear half of the side wall (Point B to C)

According to Figures 2-196 [24] and 2-198 [24]:

1/3 1/3

0.28

2.7110.7

Ewf

of

CL

t msLW lb

Calculation of peak negative normal reflected pressure rP and fictitious negative phase pressure

duration oft :

1/3

0.28 10.8 3.0 20684

10.7 6000 194.4

r E sof

of

P C P psi Pa

t ms

Negative phase rise time:

0.27 0.27 194.4 52.5oft ms

The negative pressure-time parameters (Point B):

45.1ot ms

49

0.27 45.1 52.5 97.6

45.1 194.4 239.5

o of

o of

t t ms

t t ms

The negative pressure-time curve is plotted in Figure 21.

Roof loading, positive phase

Calculation of roof loading (Point A to C)

Distance between Points A and C:

30.0L ft

wfL

L ratio:

38.21.27

30.0

wfL

L

Based on Figures 2-196, 2-197 and 2-198 [24], for Point A:

1.27wfL

L

1/3

1/3

0.52

12.8 1.25

3.10

E

d

sof

of

C

tP psi

W

t

W

Hence:

1/3

1/3

0.52 12.8 6.66 45919

1.25 6000 22.7

3.10 6000 56.3

E sof

r

of

C P psi Pa

t ms

t ms

Peak dynamic pressure from Figure 2-3 [24]:

6.66 1.05 7239.5E sof oC P psi q psi Pa

Based on Section 2-15.3.2 [24], the drag coefficient equals:

0.40DC

Calculation of peak positive pressure from Equation 2-12 [24]:

6.66 0.40 1.05 6.24 43023E sof D oC P C q psi Pa

The pressure-time curve is plotted in Figure 22.

50

Roof loading, negative phase

Calculation of roof loading (Point A to C)

According to Figures 2-196 [24] and 2-198 [24]:

1/3 1/3

0.26

1.2711.7

Ewf

of

CL

t msLW lb

Calculation of peak negative normal reflected pressure rP and fictitious negative phase pressure

duration oft :

1/3

0.26 12.8 3.33 22960

11.7 6000 212.6

r E sof

of

P C P psi Pa

t ms

Negative phase rise time:

0.27 0.27 212.6 57.4oft ms

The negative pressure-time parameters (Point B):

42.7ot ms

0.27 42.7 57.4 100.1

42.7 212.6 255.3

o of

o of

t t ms

t t ms

The negative pressure-time curve is plotted in Figure 22.

Results

In Table 5 peak over- and under- pressures for all faces are presented. Table 6 presents same data

after including ambient pressure (101325Pa). In Figures 20, 21 and 22, the pressure-time relations

are depicted. As can be seen, empirically obtained graphs are estimated with straight lines.

Peak reflected overpressure (on the front wall) in comparison to peak side on pressure (on

the side wall) is about 4.55 times greater, and in comparison to roof pressure, about 5.55 times

greater. Side wall peak pressure is about 20% greater than its roof counterpart. Negative peak

underpressures do not vary considerably across walls (difference of 2kPa – 10%).

51

Table 5. Positive and negative peak over- and under- pressures for various faces

Face

Peak over/under- pressure

positive negative

[psi] [Pa] [psi] [Pa]

Front wall 34.60 238559 -3.25 -22408

Side wall 7.60 52400 -3.00 -20684

Roof 6.24 43023 -3.33 -22960

Table 6. Positive and negative peak pressures for various faces

Face

Peak pressure

positive negative

[psi] [Pa] [psi] [Pa]

Front wall 49,30 339884 11,45 78917

Side wall 22,30 153725 11,70 80641

Roof 20,94 144348 11,37 78365

Figure 20. Pressure-time curve for the front wall. As can be observed, negative phase does not

follow the positive phase immediately.

70000,00

120000,00

170000,00

220000,00

270000,00

320000,00

370000,00

0 50 100 150 200 250

Pre

ssu

re [

Pa]

Time [ms]

Positive pressure Positive pressure Negative pressure

339884rP Pa

78917rP Pa

1829.2ri s Pa

2129.8ri s Pa

52

Figure 21. Pressure-time curve for the side wall

Figure 22. Pressure-time curve for the roof

70000,00

80000,00

90000,00

100000,00

110000,00

120000,00

130000,00

140000,00

150000,00

160000,00

0 50 100 150 200 250

Pre

ssu

re [

Pa]

Time [ms]

Positive pressure Negative pressure

70000,00

80000,00

90000,00

100000,00

110000,00

120000,00

130000,00

140000,00

150000,00

0 50 100 150 200 250

Pre

ssu

re [

Pa]

Time [ms]

Positive pressure Negative pressure

153725soP Pa

1176.4si s Pa

2010.48si s Pa

80641soP Pa

144348soP Pa

78365soP Pa

1211.1si s Pa

2441si s Pa

53

5.1.3. Numerical solution

Crucial assumptions:

- the air medium exceeds any obstacle or charge by the distance equal to 4m,

- a cube TNT charge instead of hemisphere one has been used,

- obstacles and ground have been assumed as rigid, non-deformable bodies,

- the values of the pressure have been measured at each face in its centre in three points

equally spaced across height of the face. Additional points have been added on side wall

at one fourth from the back wall,

- three values of TNT specific energy has been used: 3.68MJ/kg, 4.52MJ/kg and 5.0MJ/kg,

- TNT and air parameters that have been used in corresponding equations of state are

presented in Table 2,

- sizes of finite elements equal to 0.8m, 0.4m, 0.2m, 0.15m and 0.1m have been

investigated,

- additional analysis has been carried out for the model that has been scaled according to

the scaled distance concept. Thus, the use of finite elements of the size of 0.05m was

possible.

The script, that has been used to generate Abaqus model, can be found on the attached CD

under the name analysis_1.py.

As the wave propagates and recedes from its origin, it decays in strength, lengthens in

duration, decreases in velocity, and the hemispherical front becomes more flat. Hence, in the case

of large stand-off distance, the assumption taken by UFC about the plane wave front is acceptable

(see Figure 2-14 from [24]).

Analysis of results starts with verification and comparison of peak positive overpressures,

which are the most significant parameter describing the blast wave, and later proceeds to the

analysis of pressure-time curves. Influence of TNT specific energy and size of finite elements on

the results is investigated. Later, negative underpressures are analysed.

Pressure variation across face analysis

As it was mentioned in the introduction to this subchapter, values of the pressure were measured at

three points. In following Tables and Figures, the point located in the centre of each face has been

taken into account. Yet, the values between points from the same face differ from each other, and

for front and side walls the highest pressure value occurs at the lowermost point. In the case of the

roof, the point closest to the explosion origin owns the greatest value. Comparison between values

at each face and finite element size shows Table 7.

54

Table 7. Peak positive overpressures at each face for TNT specific energy equal to 3.68MJ/kg and

various finite element size

Face, FE size

Peak positive overpressure [Pa]

Top/Front

point

Middle/Centre

point

Difference

[%]

Bottom/Rear

point

Difference

[%]

Front wall, 0.2cm 151326 153702 1,57 163130 7,80

Front wall, 0.15cm 192183 194670 1,29 203729 6,01

Front wall, 0.10cm 275460 281076 2,04 288430 4,71

Front wall, 0.05cm 291434 299587 2,80 309940 6,35

Side wall, 0.20cm 52859 52238 -1,17 53061 0,38

Side wall, 0.15cm 62811 60830 -3,15 60077 -4,35

Side wall, 0.10cm 84303 79437 -5,77 77946 -7,54

Side wall, 0.05cm 85733 80182 -6,47 78572 -8,35

Roof, 0.20cm 63994 57678 -9,87 50816 -20,59

Roof, 0.15cm 78321 70851 -9,54 63328 -19,14

Roof, 0.10cm 99727 88810 -10,95 79778 -20,00

Roof, 0.05cm 102869 90015 -12,49 80220 -22,02

It can be seen, that variation of overpressure across front wall or side wall, for the given

finite element size, is less than 10%, which can be considered as insignificant. Moreover, the mesh

densification does not influence the difference in the results between points greatly, yet it

influences the results itself. Other performed analyses have proven, that the differences are not

dependant on the TNT specific energy, hence the conclusion, that the stand-off distance and wave

propagation itself influence the results at most.

The variation of pressure is more significant in the case of roof, but here the meaningful

factor was the change in the stand-off distance.

Since in the UFC solution the pressure is assumed as plane (equal on all the face), thus

points from the middle/centre of each face (the average ones) were used in incoming Tables and

Figures.

Positive pressure analysis

As can be seen in Figures 23, 24 and 25, the peak positive pressure grows while the mesh size

decreases. This is valid for any face. The gradient of pressure growth decays and stabilizes for

finite element size of 0.1m - 0.05m. From this it can be assumed, that further mesh refinement

won’t have an important effect on obtained results. The less dense meshes than 0.1m produce

incorrect results.

55

Influence of TNT specific energy on peak overpressures is negligible for the front wall reflected

pressure (~2%), and is better observable for the pressures on two remaining faces, where the

difference reaches 15%. The values described above present Tables 8, 9 and 10.

Figure 23. Peak positive pressure for the front wall for various finite element sizes and various

TNT specific energy

Figure 24. Peak positive pressure for the side wall for various finite element sizes and various

TNT specific energy

150000,00

200000,00

250000,00

300000,00

350000,00

400000,00

450000,00

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9

Pre

ssu

re [P

a]

FE size [m]

5.0MJ/kg 4.52MJ/kg 3.68MJ/kg

100000,00

110000,00

120000,00

130000,00

140000,00

150000,00

160000,00

170000,00

180000,00

190000,00

200000,00

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9

Pre

ssu

re [P

a]

FE size [m]

5.0MJ/kg 4.52MJ/kg 3.68MJ/kg

56

Figure 25. Peak positive pressure for the roof for various finite element sizes and various TNT

specific energy

Table 8. Peak positive overpressures [Pa] for the front wall for various finite element sizes and

various TNT specific energy

TNT specific

energy [MJ/kg]

Finite element size [m]

0.80 0.40 0.20 0.15 0.10 0.05

5.00 107381,8 218094,3 174033,1 206456,1 280533,1 -

4.52 99982,0 205260,8 165533,1 203102,6 286668,7 -

3.68 81042,0 181185,5 153701,5 194669,7 281076,1 299586,6

Table 9. Peak positive overpressures [Pa] for the side wall for various finite element sizes and

various TNT specific energy

TNT specific energy

[MJ/kg]

Finite element size [m]

0.80 0.40 0.20 0.15 0.10 0.05

5.00 20530,4 48438,6 60643,1 67612,5 91383,1 -

4.52 19236,1 45849,5 57334,8 65827,8 89215,7 -

3.68 15830,0 40753,0 52238,4 60829,8 79437,1 80182,3

100000,00

120000,00

140000,00

160000,00

180000,00

200000,00

220000,00

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9

Pre

ssu

re [P

a]

FE size [m]

5.0MJ/kg 4.52MJ/kg 3.68MJ/kg

57

Table 10. Peak positive overpressures [Pa] for the roof for various finite element sizes and various

TNT specific energy

TNT specific

energy

[MJ/kg]

Finite element size [m]

0.80 0.40 0.20 0.15 0.10 0.05

5.00 29127,6 62649,8 67268,8 77504,0 102435,3 -

4.52 27339,0 59240,2 63156,1 75789,1 99979,1 -

3.68 22458,5 52608,8 57678,2 70851,4 88809,8 90015,3

Figures 26, 27 and 28 depict the pressure-time dependency for each face for three

investigated values of TNT specific energy. As it was already stated, the relations differ in

magnitude. Moreover, by similar factor they differ in configuration. Positive impulses, presented

in Table 11, seem to prove the previous statement, as the variations do not exceed 10% in the

worst case (side wall), and are only about 2% in the case of front wall. From this it implies, that

TNT specific energy has comparable impact on both pressure magnitude and pressure

configuration. Impulses for FE size of 0.05m have been presented to satisfy ones curiosity, but

have been excluded from the analysis of results. Influence of charge weight with the preserved

scaled distance is investigated in Subchapter 5.3 (Table 32).

Table 11. Positive impulses for each considered face and TNT specific energy, for finite element

size of 0.1 and 0.05m

Finite element size [m]

Face TNT specific energy

[MJ/kg] Positive impulse

[s*Pa]

0.1

Front wall

3.68 2520,9

4.52 2464,8

5.0 2536,6

Side wall

3.68 621,0

4.52 683,4

5.0 672,4

Roof

3.68 881,8

4.52 912,0

5.0 925,5

0.05

Front wall

3.68

1657,8

Side wall 312,2

Roof 404,4

58

Figure 26. Pressure-time blast loading curves for the front wall for various TNT specific energy

and finite element size equal to 0.1m

Figure 27. Pressure-time blast loading curves for the side wall for various TNT specific energy and

finite element size equal to 0.1m

90000

140000

190000

240000

290000

340000

390000

440000

0 0,02 0,04 0,06 0,08 0,1 0,12 0,14 0,16

Pre

ssu

re [P

a]

Time [s]

5.0MJ/kg 4.52MJ/kg 3.68MJ/kg

90000

110000

130000

150000

170000

190000

210000

0 0,02 0,04 0,06 0,08 0,1 0,12 0,14 0,16

Pre

ssu

re [P

a]

Time [s]

5.0MJ/kg 4.52MJ/kg 3.68MJ/kg

387994

2464

382401

2520

381858

253

.8

.9

6.6

r

s

s

r

s

r

P Pa

P P

i s Pa

a

i s P

P Pa

i s Pa

a

190541

683.4

192708

672

180762

62

.4

1.0

so

so

s

so

s

s

P Pa

i

P Pa

i

s

s P

P

a

a

Pa

Pa

i s P

59

Figure 28. Pressure-time blast loading curves for the roof for various TNT specific energy and

finite element size equal to 0.1m

What is reasonable, increase of TNT specific energy causes the wave reaches the target

faster. Yet, much higher effect on these times has the mesh density (see Figure 29). The above is

valid for any face of the analysed structure. The analysis for the finite element size of 0.05m has

been excluded from the arrival time analysis from obvious reasons. Arrival times are presented in

Figure 29 and Table 12.

Figure 29. Blast wave arrival times in relation to finite element size for various TNT specific

energy

90000

110000

130000

150000

170000

190000

210000

0 0,02 0,04 0,06 0,08 0,1 0,12 0,14 0,16

Pre

ssu

re [P

a]

Time [s]

5.0MJ/kg 4.52MJ/kg 3.68MJ/kg

0,03

0,035

0,04

0,045

0,05

0,055

0,06

0,065

0,07

0 0,2 0,4 0,6 0,8 1

Tim

e [s

]

FE size [m]

5.0MJ/kg 4.52MJ/kg 3.68MJ/kg

201

190135

881.8

203760

925.5

304

912.0

r

s

r

s

r

s

P Pa

i

P Pa

i

P Pa

i s

P

a

P

s P

s a

a

60

Table 12. Blast wave arrival times [s] for various finite element sizes and various TNT specific

energy

TNT specific

energy

[MJ/kg]

Finite element size [m]

0.8 0.4 0.2 0.15 0.1

5.00 0.062 0.0540 0.0490 0.0455 0,0405

4.52 0.063 0.0551 0.0495 0.0460 0,0408

3.68 0.066 0.0570 0.0525 0.0466 0,0406

In Figures 30, 31 and 32, peak positive pressures for each face, certain TNT specific energy

and various element sizes are shown. Figure 33 assembles all the data. Table 13 gathers all peak

positive overpressures at each face for various TNT specific energy for the most dense meshes.

Both side wall and roof overpressures account for about 30% of front wall peak reflected

overpressure. This relation is more or less preserved for any TNT specific energy and finite

element size (see the last mentioned Figures).

Figure 30. Peak positive pressure for the front wall, side wall and roof for various finite element

sizes and TNT specific energy equal to 5MJ/kg

100000,00

150000,00

200000,00

250000,00

300000,00

350000,00

400000,00

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9

Pre

ssu

re [P

a]

FE size [m]

Front wall Side wall Roof

61

Figure 31. Peak positive pressure for the front wall, side wall and roof for various finite element

sizes and TNT specific energy equal to 4.52MJ/kg

Figure 32. Peak positive pressure for the front wall, side wall and roof for various finite element

sizes and TNT specific energy equal to 3.68MJ/kg

100000,00

150000,00

200000,00

250000,00

300000,00

350000,00

400000,00

450000,00

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9

Pre

ssu

re [P

a]

FE size [m]

Front wall Side wall Roof

100000,00

150000,00

200000,00

250000,00

300000,00

350000,00

400000,00

450000,00

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9

Pre

ssu

re [P

a]

FE size [m]

Front wall Side wall Roof

62

Table 13. Peak positive overpressures [Pa] at each face for various TNT specific energy, and

differences between them

TNT specific

energy [MJ/kg]

FE size

[cm]

Front wall

[Pa]

Side wall

[Pa]

Difference

[%]

Roof

[Pa]

Difference

[%]

3.68 0.1 281076,1 79437,1 -71,74 88809,8 -68,40

0.05 299586,6 80182,3 -73,24 90015,3 -69,95

4.52 0.1 286668,7 89215,7 -68,88 99979,1 -65,12

5.00 0.1 280533,1 91383,1 -67,43 102435,3 -63,49

Figure 33. Peak positive pressure for the front wall, side wall and roof for various finite element

sizes and various TNT specific energy

In Figures 34, 35 and 36, the influence of element size on the pressure configuration and

the arrival time is investigated. It can be clearly seen, that both positive and negative peak

pressures rise and the wave arrival time decrease while the mesh density grows. The difference in

peak pressure is substantial, which stresses the importance of using correct mesh. This situation is

100000,00

150000,00

200000,00

250000,00

300000,00

350000,00

400000,00

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9

Pre

ssu

re [P

a]

FE size [m]

Front wall, 5.00MJ/kg Side wall, 5.00MJ/kg Roof, 5.00MJ/kg

Front wall, 4.52MJ/kg Side wall, 4.52MJ/kg Roof, 4.52MJ/kg

Front wall, 3.68MJ/kg Side wall, 3.68MJ/kg Roof, 3.68MJ/kg

63

partially caused by the fact, that pressure is calculated for each finite element, and element of a big

size contains more or less average value of pressures from several finite elements in the case of

more dense mesh. Together with the increase of peak pressures gradients of pressure changes are

more rapid. The graphs become less smooth.

Figure 34. Pressure-time blast loading curves for the front wall for various finite element sizes and

TNT specific energy equal to 3.68MJ/kg

Figure 35. Pressure-time blast loading curves for the side wall for various finite element sizes and

TNT specific energy equal to 3.68MJ/kg

50000

100000

150000

200000

250000

300000

350000

400000

450000

0 0,05 0,1 0,15 0,2

Pre

ssu

re [P

a]

Time [s]

0.4m 0.2m 0.15m 0.1m 0.05m

90000

100000

110000

120000

130000

140000

150000

160000

170000

180000

190000

0 0,05 0,1 0,15 0,2

Pre

ssu

re [P

a]

Time [s]

0.4m 0.2m 0.15m 0.1m 0.05m

295994.

282

40

3

0

8

911.

2

510.5

2401.1

55026.

6

5

7r

r

r

r

r

P P

P P

P Pa

P

a

a

P a

a

P P

162154.

142

18

1

1

8

507.

1

078.0

0762.1

53563.

3

4

8r

r

r

r

r

P P

P P

P Pa

P

a

a

P a

a

P P

64

Figure 36. Pressure-time blast loading curves for the roof for various finite element sizes and TNT

specific energy equal to 3.68MJ/kg

Figures 37 and 38 combine the pressure loading curves for each face. As can be seen, front

wall loading is much more rapid and complex, while side wall and roof loadings are relatively

more smooth. Based on Figure 38, clearing time on side wall and roof is about two times shorter

than in the case of front wall.

Figure 37. Pressure-time blast loading curves for the front wall, side wall and roof for finite

element size equal to 0.1m and TNT specific energy equal to 3.68MJ/kg

90000

110000

130000

150000

170000

190000

210000

0 0,05 0,1 0,15 0,2

Pre

ssu

re [P

a]

Time [s]

0.4m 0.2m 0.15m 0.1m 0.05m

80000

130000

180000

230000

280000

330000

380000

430000

0 0,02 0,04 0,06 0,08 0,1 0,12 0,14 0,16

Pre

ssu

re [P

a]

Time [s]

Front wall Side wall Roof

172176.

153

19

1

1

9

340.

1

933.8

0134.8

59003.

3

2

4r

r

r

r

r

P P

P P

P Pa

P

a

a

P a

a

P P

180762.

382

1901

1

401.1

34.8

r

r

rP P

P

a

a

a

P P

P

65

Figure 38. Pressure-time blast loading curves for the front wall, side wall and roof for finite

element size equal to 0.05m and TNT specific energy equal to 3.68MJ/kg

Negative pressure analysis

According to Figures 39, 40 and 41, the peak negative front wall pressure decreases with the

increase of TNT specific energy, and apart from this, it grows while the size element decreases. In

contrary, roof pressure decreases as the mesh refinement occurs. As it was with peak positive

pressures, slight stabilization of results at each face occurs for finite element size of 0.1m - 0.05m.

Yet, further investigation and mesh refinement should be performed to prove the correctness of

stabilization assumption. From obtained results it is impossible to claim the possible impact of

mesh refinement on obtained results. Definitely, the less dense meshes than 0.1m produce

incorrect results. Described data in tabularized form contain succeeding Tables (14, 15, 16). The

pressure-time relations can be found in Figures 30, 31 and 32.

80000

130000

180000

230000

280000

330000

380000

430000

0 0,02 0,04 0,06 0,08 0,1 0,12 0,14

Pre

ssu

re [P

a]

Time [s]

Front wall Side wall Roof

181507.

400

1913

3

911.6

40.3

r

r

rP P

P

a

a

a

P P

P

66

Figure 39. Peak negative pressure for the front wall for various finite element sizes and various

TNT specific energy

Figure 40. Peak negative pressure for the side wall for various finite element sizes and various

TNT specific energy

50000,00

60000,00

70000,00

80000,00

90000,00

100000,00

110000,00

120000,00

130000,00

140000,00

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9

Pre

ssu

re [P

a]

FE size [m]

5.0MJ/kg 4.52MJ/kg 3.68MJ/kg

100000,00

102000,00

104000,00

106000,00

108000,00

110000,00

112000,00

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9

Pre

ssu

re [P

a]

FE size [m]

5.0MJ/kg 4.52MJ/kg 3.68MJ/kg

67

Figure 41. Peak negative pressure for the roof for various finite element sizes and various TNT

specific energy

Table 14. Peak negative underpressures [Pa] for the front wall for various finite element sizes and

various TNT specific energy

TNT specific

energy

[MJ/kg]

Finite element size [m]

0.80 0.40 0.20 0.15 0.10 0.05

5.00 -36074,9 -31880,0 -36769 -24725,7 16085,6 -

4.52 -34011,5 -30557,9 -34708 -21235,6 18043,6 -

3.68 -28927,2 -28138,7 -26140 -10616,3 29475,3 32144,6

Table 15. Peak negative underpressures [Pa] for the side wall for various finite element sizes and

various TNT specific energy

TNT specific energy

[MJ/kg]

Finite element size [m]

0.80 0.40 0.20 0.15 0.10 0.05

5.00 -180,2 3277,9 4002,6 5426,7 8434,1 -

4.52 -40,5 3262,6 4013,9 5836,9 9467,8 -

3.68 26,8 1316,1 3684,8 6829,0 9475,5 8770,0

98000,00

99000,00

100000,00

101000,00

102000,00

103000,00

104000,00

105000,00

106000,00

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9

Pre

ssu

re [P

a]

FE size [m]

5.0MJ/kg 4.52MJ/kg 3.68MJ/kg

68

Table 16. Peak negative underpressures [Pa] for the roof for various finite element sizes and

various TNT specific energy

TNT specific

energy

[MJ/kg]

Finite element size [m]

0.80 0.40 0.20 0.15 0.10 0.05

5.00 -453,5 619,7 -1477,6 1563,6 -2864,0 -

4.52 -162,8 955,9 -792,6 2404,6 -2880,2 -

3.68 437,7 1638,4 803,2 4104,2 50,1 -436,0

Table 17 gathers all peak negative underpressures at each face for various TNT specific

energy for the most dense meshes. Side wall and roof pressures are much smaller than front wall

peak negative underpressure.

Table 17. Peak negative underpressures [Pa] at each face for various TNT specific energy and

differences between them

TNT specific

energy [MJ/kg]

FE size

[cm]

Front wall

[Pa]

Side wall

[Pa]

Difference

[%]

Roof

[Pa]

Difference

[%]

3.68 0.1 29475,3 9475,5 -67,85 50,1 -99,83

0.05 32144,6 8770,0 -72,72 -436,0 -101,36

4.52 0.1 18043,6 9467,8 -47,53 -2880,2 -115,96

5.00 0.1 16085,6 8434,1 -47,57 -2864,0 -117,80

5.1.4. Discussion

Table 18 contains the comparison of peak overpressures from UFC and from Abaqus. In the case

of Abaqus solution, results for 0.05m and 0.01m mesh element size has been taken into account as

the most accurate.

In general, peak positive overpressure differences are much smaller than their negative

counterparts. The difference in front wall peak positive overpressure equals to 18% for both TNT

specific energy of 3.68MJ/kg and 5.00MJ/kg, and to 20% for specific energy of 4.52MJ/kg. For

other faces these distinctions are much higher. In the case of side wall, overpressures are

overestimated by 51 to 74%, and in the case of roof by 106 to 138%. The minimum pressures are

underestimated even more significantly.

The most accurate are results obtained for the TNT specific energy value of 3.68MJ/kg.

Since all the results tend to the exact solution with the mesh refinement, pressures for specific

energy of 3.68MJ/kg and finite element size of 0.05m should be treated as most reliable.

The smallest distinctions in peak overpressures between Abaqus and UFC can be observed

for the front wall. Comparison of positive phase impulses (Table 18) affirms additionally, that

69

configurations of these pressures vary significantly less from UFC solution than configurations for

side wall or roof.

What is interesting, the front wall underpressure rises as the mesh density grows (Figure

39), which causes the decrease of accuracy compared to UFC. Another important difference is,

that UFC produces the negative pressure (with value below the ambient pressure) for each face,

while Abaqus, in contrary, produces such negative pressure only for the roof. Since UFC results

base on empirical experiments, their quality should not be questioned. Thus, some defects of the

numerical approach, especially for the tensile elements, can explain these differences.

The differences in both positive and negative phase durations are meaningful, since in

Abaqus the negative phase merely does not exist, and the positive phase is extended. This can lead

to the statement, that Abaqus solution is valid only for the positive phase of pressure-time relation.

Based on the performed analyses, results for the TNT specific energy of 3.68MJ/kg are the

nearest to the UFC solution, hence this value will be used in the following studies. Moreover, the

general convergence of results has been proven for the front wall, and further investigations should

be carried out to find the source of inaccuracies, especially in the case of negative pressures. The

study confirms the results’ dependence on the mesh density. The determined size of 0.05m, for

smaller scaled distance (i.e. smaller stand-off distance, greater explosive weight, or both), due to

increasing rapidness of the blast phenomenon, can be insufficient - more dense mesh might be

required, and separate set of analyses should be carried out to investigate that.

Table 18. Comparison of positive impulses between UFC and Abaqus

Face Positive phase impulse [s*Pa]

Difference [%] UFC Abaqus, FE size 0.05m

Front wall 2129,8 1657,8 28,5

Side wall 1176,4 312,2 276,8

Roof 1211,1 404,4 199,5

70

Table 19. Comparison of peak over- and under- pressures

Face Parameter

TNT

specific

energy

[MJ/kg]

UFC

solution

[Pa]

Abaqus solution,

element size 0.1m

[Pa]

Difference

[Pa]

Difference

[%]

Front wall

Peak positive overpressure

[Pa]

3.68*

238558,6

299586,6 61027,96 25,58

3.68 281076,1 42517,50 17,82

4.52 286668,7 48110,10 20,17

5.00 280533,1 41974,50 17,60

Peak negative underpressure

[Pa]

3.68*

-22408

32144,64 54552,60 -243,45

3.68 29475,3 51883,26 -231,54

4.52 18043,6 40451,56 -180,52

5.00 16085,6 38493,56 -171,79

Side wall

Peak positive overpressure

[Pa]

3.68*

52400,16

80182,25 27782,09 53,02

3.68 79437,1 27036,94 51,60

4.52 89215,7 36815,54 70,26

5.00 91383,1 38982,94 74,39

Peak negative underpressure

[Pa]

3.68*

-20684,3

8770,047 29454,32 -142,40

3.68 9475,5 30159,77 -145,81

4.52 9467,8 30152,07 -145,77

5.00 8434,1 29118,37 -140,78

Roof

Peak positive overpressure

[Pa]

3.68*

43023,29

90015,28 46991,99 109,22

3.68 88809,8 45786,51 106,42

4.52 99979,1 56955,81 132,38

5.00 102435,3 59412,01 138,09

Peak negative underpressure

[Pa]

3.68*

-22959,5

-436,016 22523,52 -98,10

3.68 50,1 23009,64 -100,22

4.52 -2880,2 20079,34 -87,46

5.00 -2864 20095,54 -87,53

* values for finite element size of 0.05m

71

5.2. Interior pressure due to external burst

5.2.1. Description

The aim of the study is to determine the pressure-time loads acting on the exterior front wall and

all interior surfaces of a rectangular structure with front wall openings due to an external shock

load. The example comes from the UFC [24] Problem 2A-11. Results from UFC and Abaqus are

compared.

This type of simulation might be especially interesting while dealing with the air pressure

impact on objects inside of the structure, inter alia human bodies, and damages dealt by it.

The geometry of the structure and location of a hemisphere TNT charge of 2267.96kg

(5000lbs) are shown in Figure 43. The geometry of the front wall is depicted in Figure 44.

Symbols used in subsequent Figures, Tables as well as in the Python script are compatible and

presented below.

Figure 42. View on the structure subjected to the blast load (from Abaqus CAE)

7eL H

4eH V

eB

72

Figure 43. Simulation model: a) view from the side, b) view from the top

Figure 44. Front wall of the structure and the central point P1

73

Table 20. Symbols used

Symbol Value

[ft, in] [m]

R 155 47.244

Be 32 9.7536

Bi 30 9.144

Le 22 6.7056

Li 20 6.096

He 16 4.8768

Hi 15 4.572

w 1 0.3048

Table 21. Front wall symbols

Symbol Value

[ft, in] [m]

V1 4 1.2192

V2 3 0.9144

V3 9 2.7432

V4 16 4.8768

H1 5 1.524

H2 5-6 1.6764

H3 3-6 1.0668

H4 3 0.9144

H5 5 1.524

H6 13 3.9624

H7 22 6.7056

5.2.2. Analytical solution

The below mentioned results (Table 22) were taken from Example 2A-11 from [24]. Note, that

only the positive loading phase is investigated. For the detailed description and procedure of

calculation refer to the respective document. For the graphical representation of results check

Figure 45.

The peak positive over pressure on the internal front wall is almost 1.7 times lower than the

exterior wall peak reflected overpressure. Overpressures on other internal walls are more than 2.5

times lower than the internal front wall overpressure. Internal pressures rise immediately after the

74

shock wave reaches the object and decay few times longer than the external pressure. The clearing

time is 71ms from the occurrence of peak reflected pressure. The pressure-time relation for the

external wall is approximated by two straight lines which approximate the real-life configuration

of the pressure (see Figure 3). Interior front wall overpressure is assumed to decay more than 3

times faster than side wall and roof pressures. Back wall overpressure appears about 21ms after

reaching the object and is assumed as a very short peak.

Table 22. Peak positive pressures for the exterior and interior walls of the structure and positive

pressure phases

Location

Peak positive

overpressure

Peak positive

pressure T1

[ms]

T2

[ms]

T3

[ms]

T4

[ms] [psi] [Pa] [psi] [Pa]

Exterior

front wall 31.9 219942.76 46.6 321267.76 5.3 26 - -

Interior

front wall 18.9 130310.91 33.6 231635.91 1.6 7.4 17.9 -

Interior

side wall 7.3 50331.73 22.0 151656.73 4.4 5.9 19 71

Interior back wall

7.5 51710.68 22.2 153035.68 21.1 23.5 - -

Interior roof

6.2 42747.50 20.9 144072.50 1.82 3.2 35 65

Figure 45. Pressure-time curves according to UFC

100000,00

150000,00

200000,00

250000,00

300000,00

350000,00

0 10 20 30 40 50 60 70 80

Pre

ssu

re [

Pa]

Time [ms]

Exterior front wall

Interior front wall

Interior side wall

Interior back wall

Interior roof

75

5.2.3. Numerical solution

Crucial assumptions:

- the air medium exceeds any obstacle or charge by the distance equal to 4m,

- a cube TNT charge instead of hemisphere one has been used,

- obstacles and ground have been assumed as rigid, non-deformable bodies,

- the values of the pressure has been measured at each face in three points equally spaced

across height of the face,

- based on previous study (Mesh size study in this Chapter), finite elements of a size of

0.1m have been used,

- TNT specific energy equal to 3.68MJ/kg has been used,

- TNT and air parameters that have been used in corresponding equations of state are

presented in Table 2.

The script, that has been used to generate Abaqus model, can be found on the attached CD

under the name analysis_2.py.

Peak positive overpressure for the interior front wall accounts for about 24% of the external

front wall pressure. Pressures on other internal faces are comparable and vary by 30% of the peak

interior overpressure at most. It can be observed, that the external pressure decays rapidly, while

the internal pressure, due to limited escape routes, is significantly greater and persists longer than

the external one.

An interesting observation can be made when comparing front and rear pressure

configurations. First of all wave front reaches the front face and propagates inside the object. After

about 17ms it touches the rear wall. At that time, the peak rear wall pressure is approximately

two times higher than its front wall counterpart. After another 20ms, the pressure wave reflected

from the rear wall reaches the front wall and causes yet another rapid growth of pressure. This

pressure grows a bit higher than the peak rear wall pressure. At that time value of the rear face

pressure equals to about half of front wall pressure. From that point on, the repeating reflections

from both front and rear faces occur. They can be observed on the pressure configuration graph as

local pressure maxima, which are shifted from each other in time by about 20ms, i.e. the travel

time of the wave. This phenomenon is combined with the pressure compensation between internal

and external environments.

Side wall pressure configurations, despite unsymmetrical holes in the front face, preserve

significant convergence. Side 2 pressure is slightly higher than side 1 pressure which is related to

the doorway existence. Discussed data has been assembled in Table 23 and graphically presented

in Figures 46, 47, 48 and 49.

76

Table 23. Peak positive pressures for exterior and interior walls of the structure

Location Peak positive pressure

[Pa]

Peak positive overpressure

[Pa]

Exterior front wall 374569,2 273244,2

Interior front wall 165884,4 64559,4

Interior side wall 1 151168,5 49843,5

Interior side wall 2 157873,3 56548,3

Interior back wall 169601,0 68276,0

Interior roof 160554,6 59229,6

Figure 46. Pressure-time curves for external front face and all internal faces according to Abaqus

80000

130000

180000

230000

280000

330000

380000

430000

0 0,05 0,1 0,15 0,2

Pre

ssu

re [P

a]

Time [s]

Front Rear Roof Side 1 Side 2 External front

77

Figure 47. Pressure-time curves for front and rear internal faces according to Abaqus

Figure 48. Pressure-time curves for internal side faces according to Abaqus

80000

90000

100000

110000

120000

130000

140000

150000

160000

170000

180000

0 0,05 0,1 0,15 0,2

Pre

ssu

re [P

a]

Time [s]

Front Rear

80000

90000

100000

110000

120000

130000

140000

150000

160000

0 0,05 0,1 0,15 0,2

Pre

ssu

re [P

a]

Time [s]

Side 1 Side 2

78

Figure 49. Pressure-time curves for all internal faces according to Abaqus

5.2.4. Discussion

As presented in Table 24, external front wall peak reflected overpressure varies between UFC and

Abaqus solution by about 24%. The differences for other walls, except the interior front wall,

where the difference is significant and equals to more than 50%, are small (side walls) or slight

(back wall and roof). This implies, that internal pressure estimation can be performed, with well

correctness, by means of Abaqus.

Substantial distinctions can be observed in the case of positive phase durations and clearing

times. According to Abaqus, pressure decays slower and clearing time occurs later. Yet, both

methods yield very similar value of the difference between external front wall arrival time and

back wall arrival time (which equals to ~20ms).

Obtained results, in general, in the case of peak positive pressures are convergent, and in

the case of phase durations, are not convergent. The huge difference in back wall pressure duration

is here the most important doubt, as well as 50% difference in the interior front wall peak

overpressure. Either UFC simplifications or too large element size can be the reason of it.

Numerical analysis with use of more refined element mesh and verification of results by means of

some other numerical software seem to be reasonable way of finding the cause of inaccuracies.

80000

90000

100000

110000

120000

130000

140000

150000

160000

170000

180000

0 0,05 0,1 0,15 0,2

Pre

ssu

re [P

a]

Time [s]

Front Rear Roof Side 1 Side 2

79

Table 24. Comparison of peak positive overpressures according to UFC and Abaqus

Location Peak positive overpressure [Pa]

Difference [%] UFC solution Abaqus solution

Exterior front wall 219942,76 273244,2 24,23

Interior front wall 130310,91 64559,4 -50,46

Interior side wall 1 50331,73 49843,5 -0,97

Interior side wall 2 50331,73 56548,3 12,35

Interior back wall 51710,68 68276,0 32,03

Interior roof 42747,50 59229,6 38,56

5.3. Comparative analysis

5.3.1. Description

The comparative analysis to the study entitled Modelling blast loads on buildings in complex city

geometries by Remennikov and Rose [17] has been performed.

The purpose of the original study was to demonstrate the influence of adjacent structures on

blast loads acting on a given target building and to underline the need of use of advanced

numerical software when dealing with shock wave propagation in complex urban geometries.

The original results, presented later in this subsection, were obtained with use of CFD

application called Air3D. Repeating the analyses makes possible the verification of two different

computer codes: Air3D and Abaqus.

Apart from this, the correctness of the scaled distance concept is investigated. According to

this concept, blast parameters, such as peak incident or reflected pressure, are constant for the

given scaled distance Z. The correctness of this assumption is verified.

The model consists of a hemispherical TNT charge located on the ground surface, and of

two buildings situated one behind the other.

Two simulations are performed: single building, baseline simulation, where only the second

building is present, and two-building simulation with both buildings present. In the second

analysis, a smaller building (Building 1) provides partial shielding to an adjacent larger building

(Building 2).

The aim of this is to stress the influence of the foremost structure on the loading

experienced by the second one and vice-versa.

The geometry of the model shows Figure 50. Symbols used in subsequent Figures, Tables

as well as in the Python script are consistent.

80

Figure 50. Scheme of the simulation model: a) view from the side, b) view from the top

5.3.2. Original study description

Air3D CFD computer code was used for analyses. It uses an explicit, finite volume formulation to

solve Euler equations.

The grid consisted of cubic cells in a regular Cartesian mesh. The size of each cell was

10mm, and there was total number of about 5 million cubic cells.

The model was extended in each direction to neglect the effect of boundaries. Blast

pressure and impulses were measured, and it was done in points distributed over the front and rear

walls of both buildings, on the ground level. 10msec period of wave propagation was simulated.

Hemispherical charge was used.

81

The concept of scaled distances has been used by the authors to describe the relation

between TNT equivalent weight, dimensions of buildings and distances between them. Table 25

presents all required scaled distances and their equivalents in meters, for the examined charge

weights.

Three-dimensional analysis has been carried out to capture such effects as multiple

reflection, diffraction, blast focusing and shielding. The buildings were modelled as rigid

reflective surfaces.

Results

As it can be seen in Figure 51, the rear wall of Building 1 experienced the second shock, which is

about two and a half times as high as the pressure shock and impulse that initially loaded the

building. If the second building were neglected, this would lead to a significant underestimation of

the blast loads on the rear wall of Building 1. As authors conclude, these results clearly indicate

the importance of considering adjacent structures in simulations.

Moreover, the relative negative phase impulse (1.3) is three times greater than the positive

phase impulse delivered by the first shock (0.42) on the rear of Building 1, and twice as much as

the positive phase impulse delivered by the second shock (0.65). Hence the conclusion, that the

negative phase of the blast pulse may have an important influence on lightweight facade panel

behaviour by causing the facade material to fail outward.

Figure 52 shows the pressure and impulse histories at the ground level on the front wall of

Building 2. The results at this location for the two-building simulation are compared to the

pressure and impulse at the same point for the single building simulation, where Building 1 was

directly exposed to the blast effects from the explosion. What follows from the Figure, the peak

reflected pressure shock on the front wall of Building 2 would be overestimated by a factor of 3.5

if the building in front of it were not present. The same is true for the peak positive phase impulse,

which would be over-predicted by a factor of 2.6 [17]. For more comprehensive analysis refer to

the original document.

Note, that the pressure and impulse histories in Figure 51 are normalized by the peak

pressure and peak positive impulse, respectively, associated with the first pressure pulse. Pressures

and impulses in Figure 52 are scaled by the peak pressure and peak positive phase impulse,

accordingly, associated with the two-building model.

82

Figure 51. Blast pressure and impulse histories on rear wall of Building 1 ([17], Figure 7)

Figure 52. Comparison of pressure/impulse histories on front wall of Building 2 for single- and

two-building simulations ([17], Figure 8)

83

5.3.3. Numerical solution

Crucial assumptions:

- the air medium exceeds any obstacle or charge by the scaled distance equal to 0.4,

- a cube TNT charge instead of hemisphere one has been used,

- single and two-building simulations have been performed,

- the results has been measured at the ground level,

- obstacles and ground have been assumed as rigid, non-deformable bodies,

- TNT specific energy equal to 3.68MJ/kg has been used,

- charge weights of 10, 50, 100, 200, 500 and 1000kg has been examined,

- size of finite elements and their number, related to given charge weight, have been placed

in Table 26,

- TNT and air parameters that have been used in corresponding equations of state are

presented in Table 2.

Size of finite elements for each explosive weight has been chosen in such a way, so that

their number was similar throughout all analyses. Refer to Table 26 for more details.

The script that has been used to generate Abaqus model can be found on the attached CD

under the name analysis_3.py. It allows the user to create a model scaled by any TNT weight.

Table 25. Scaled and real dimensions used in analyses

Symbol

Scaled

dimensions

1/3

m

kg

Real dimensions for various W

[m], based on Formula (2)

10kg 50kg 100kg 200kg 500kg 1000kg

Z1 0.5 1.08 1.84 2.32 2.92 3.97 5.00

Z2 2.0 4.31 7.37 9.28 11.70 15.87 20.00

B1 0.8 1.72 2.95 3.71 4.68 6.35 8.00

Z3 0.7 1.51 2.58 3.25 4.09 5.56 7.00

B2 1.3 2.80 4.79 6.03 7.60 10.32 13.00

H1 1.0 2.15 3.68 4.64 5.85 7.94 10.00

H2 1.7 3.66 6.26 7.89 9.94 13.49 17.00

84

Table 26. Number of finite elements in each analysis

Charge weight

[kg]

Finite element

size [m]

Number of finite

elements

10 0.032 5585428

50 0.055 5526072

100 0.069 5565264

200 0.087 5565264

500 0.118 5565264

1000 0.15 5506050

Results

The scaled distance concept is verified for the pressure on various walls of both Building 1 and

Building 2, including the shielding effect in two-building simulation.

Tables 27 and 29 contain the peak pressures for various charge weights for the front face of

Building 2, while Tables 28 and 30 assemble peak side-on pressures. Figures 54, 55, 56 and 57

depict the respective pressure-time curves.

The continuation of scaled distance concept verification is provided in Figures 58, 59, 60,

which present pressure changes on walls of Building 1, and in Table 31.

As it follows from the above mentioned Tables and Figures, the peak positive overpressures

vary insignificantly across all the analyses (compare also Figure 61), as the greatest difference

equals to less than 1.8%, and in vast majority of analyses is lesser than 0.5%. The pressure

configurations show very close convergence as well, yet together with various arrival times a

distinction in phase durations is observable. Table 32, which contains positive phase impulses,

proves this statement. Impulses grow considerably with charge weight increase. This growth has

been presented in Figure 53. Presented results confirm the validity of scaled distance concept in

the case of peak pressures.

85

Table 27. Peak reflected pressures for the front wall of Building 2 (single building simulation) for

various charge weights

Charge weight [kg] Peak pressure [Pa] Peak overpressure [Pa] Difference [%]

10 1724332,9 1623007,9 -

50 1718022,6 1616697,6 -0,389

100 1752213,0 1650888,0 1,718

200 1733256,5 1631931,5 0,550

500 1739282,5 1637957,5 0,921

1000 1712706,3 1611381,3 -0,716

Table 28. Peak side-on pressures for the side wall of Building 2 (single building simulation) for

various charge weights

Charge weight [kg] Peak pressure [Pa] Peak overpressure [Pa] Difference [%]

10 210759,1 109434,1 -

50 211636,9 110311,9 0,802

100 210896,1 109571,1 0,125

200 211023,3 109698,3 0,241

500 211068,6 109743,6 0,283

1000 211590,0 110265,0 0,759

Table 29. Peak reflected pressures for the front wall of Building 2 (two-building simulation) for

various charge weights

Charge weight [kg] Peak pressure [Pa] Peak overpressure [Pa] Difference [%]

10 317075,7 215750,7 -

50 317256,5 215931,5 0,084

100 317150,0 215825,0 0,034

200 317234,9 215909,9 0,074

500 317156,5 215831,5 0,037

1000 317405,9 216080,9 0,153

86

Table 30. Peak side-on pressures for the side wall of Building 2 (two-building simulation) for

various charge weights

Charge weight [kg] Peak pressure [Pa] Peak overpressure [Pa] Difference [%]

10 178525,5 77200,5 -

50 178839,9 77514,9 0,407

100 178737,4 77412,4 0,274

200 178803,4 77478,4 0,360

500 178771,9 77446,9 0,319

1000 178692,4 77367,4 0,216

Table 31. Peak overpressures on walls of Building 1 for various charge weights

Charge

weight

[kg]

Peak overpressure [Pa]

B1, front B1, side B1, rear

Peak

overpressure

[Pa]

Difference

[%]

Peak

overpressu

re [Pa]

Difference

[%]

Peak

overpressure

[Pa]

Difference

[%]

10 29186327,0 - 316482,4 - 215323,3 -

50 28787351,0 -1,367 316199,7 -0,089 215764,9 0,205

100 29160197,0 -0,090 316378,7 -0,033 215427,4 0,048

200 29158059,0 -0,097 316413,4 -0,022 215434,6 0,052

500 29185843,0 -0,002 316428,0 -0,017 215456,0 0,062

1000 28758245,0 -1,467 316815,7 0,105 216081,5 0,352

Table 32. Positive impulses on walls of Building 1 and Building 2

Location Analysis Positive impulses [s*Pa]

10kg 50kg 100kg 200kg 500kg 1000kg

Front wall, B2 Single building 1390,2 2387,0 3060,2 3802,0 5211,8 6441,1

Two-building 233,7 400,1 503,6 634,8 861,3 1085,4

Side wall, B2 Single building 141,9 243,2 307,5 387,7 525,9 660,8

Two-building 56,6 97,2 122,7 154,7 210,0 263,8

Front wall, B1

Two-building

5844,2 10041,8 12632,2 15965,1 21497,0 27013,2

Side wall, B1 111,2 190,7 239,5 302,0 409,5 516,4

Roof, B1 133,4 228,9 287,7 362,7 492,0 621,5

87

Figure 53. Charge weight influence on positive impulses on walls of Building 1 and Building 2

Figure 54. Pressure-time blast loading curves for the front wall of Building 2 (single building

simulation) for various charge weights

32,0

128,0

512,0

2048,0

8192,0

32768,0

0 200 400 600 800 1000

Imp

uls

e [s

*Pa]

Charge weight [kg]

Front wall, B2, Single building Front wall, B2, Two-building

Side wall, B2, Single building Side wall, B2, Two-building

Front wall, B1 Side wall, B1

Roof, B1

-200000

300000

800000

1300000

1800000

2300000

0 0,01 0,02 0,03 0,04 0,05 0,06

Pre

ssu

re [P

a]

Time [s]

4.31m (10kg) 7.37m (50kg) 9.28m (100kg)

11.70m (200kg) 15.87m (500kg) 20.00m (1000kg)

2387.0

5211

1390.2

6441

3802.0

3060.

8

.

2

1

.s

s

s

s

s

s

i

i s

s Pa

i

i s Pa

i s P

P

s Pa

a

a

i s Pa

88

Figure 55. Pressure-time blast loading curves for the side wall of Building 2 (single building

simulation) for various charge weights

Figure 56. Pressure-time blast loading curves for the front wall of Building 2 (two-building

simulation) for various charge weights

70000

90000

110000

130000

150000

170000

190000

210000

230000

0 0,02 0,04 0,06 0,08 0,1

Pre

ssu

re [P

a]

Time [s]

5.71m (10kg) 9.77m (50kg) 12.30m (100kg)

15.50m (200kg) 21.03m (500kg) 26.5m (1000kg)

20000

70000

120000

170000

220000

270000

320000

370000

0 0,02 0,04 0,06 0,08 0,1

Pre

ssu

re [P

a]

Time [s]

4.31m (10kg) 7.37m (50kg) 9.28m (100kg)

11.70m (200kg) 15.87m (500kg) 20.00m (1000kg)

243

660.8

52

141.

387.7

307.5

.

9

2

9

.

5

s

s

s

s

s

s

i

i s

i

P

i s Pa

i s P

i s P

s P

a

Pa

a

a

a

s

400.

1085.

86

233.

634.8

4

1.3

7

03.6

1

5

s

s

s

s

s

s

i

i s Pa

s Pa

i s Pa

i s

i s Pa

i s

P

Pa

a

89

Figure 57. Pressure-time blast loading curves for the side wall of Building 2 (two-building

simulation) for various charge weights

Figure 58. Pressure-time blast loading curves for the front wall of Building 1 for various charge

weights

80000

100000

120000

140000

160000

180000

200000

0 0,02 0,04 0,06 0,08 0,1

Pre

ssu

re [P

a]

Time [s]

5.71m (10kg) 9.77m (50kg) 12.30m (100kg)

15.50m (200kg) 21.03m (500kg) 26.5m (1000kg)

-1000000

4000000

9000000

14000000

19000000

24000000

29000000

34000000

0 0,001 0,002 0,003 0,004 0,005 0,006

Pre

ssu

re [P

a]

Time [s]

1.08m (10kg) 1.84m (50kg) 2.32m (100kg)

2.92m (200kg) 3.97m (500kg) 5.00m (1000kg)

2

1

154.7

97.2

10.0

56

263

.

.

6

8

22.7

s

s

s

s

s

s

i s

i

i s Pa

i s Pa

s

i s Pa

a

P

s P

P

a

i

a

10041.

21497

5844.

270

1

15965

13.

26

.0

3

8

2

.

2

2.2

1

s

s

s

s

s

s

i

i s

i s

i s Pa

i s P

Pa

i s Pa

s a

Pa

a

P

90

Figure 59. Pressure-time blast loading curves for the side wall of Building 1 for various charge

weights

Figure 60. Pressure-time blast loading curves for the rear wall of Building 1 for various charge

weights

0

50000

100000

150000

200000

250000

300000

350000

400000

450000

0 0,01 0,02 0,03 0,04 0,05 0,06

Pre

ssu

re [P

a]

Time [s]

1.94m (10kg) 3.32m (50kg) 4.18m (100kg)

5.26m (200kg) 7.15m (500kg) 9.00m (1000kg)

0

50000

100000

150000

200000

250000

300000

350000

0 0,02 0,04 0,06 0,08 0,1

Pre

ssu

re [P

a]

Time [s]

2.80m (10kg) 3.79m (50kg) 6.03m (100kg)

7.60m (200kg) 10.32m (500kg) 13.00m (1000kg)

190

516.4

40

111.

302.0

239.5

.

2

7

5

.

9

s

s

s

s

s

s

i

i s

i

P

i s Pa

i s P

i s P

s P

a

Pa

a

a

a

s

228

621.5

49

133.

362.7

287.7

.

4

9

0

.

2

s

s

s

s

s

s

i

i s

i

P

i s Pa

i s P

i s P

s P

a

Pa

a

a

a

s

91

Figure 61. Peak positive pressures for various charge weights on various walls of Building 1 and

Building 2

As it can be seen in Table 33 and in Figures 62 and 63, the front wall peak overpressure is

about 7.6 times lower when the shielding caused by Building 1 occurs. Simultaneously, in the case

of side wall, the decrease is by the magnitude of about 1.4.

The rear wall of Building 1 experienced the second shock, which is about 2.3 times as high

as the initial shock that loaded the building (see Figure 64 and Figure 65). If the second building

were neglected, this would lead to a significant underestimation of the blast loads on the rear wall

of Building 1. Again, the importance of considering adjacent structures in simulations is clearly

indicated. After that, a considerable negative phase takes place, which magnitude is of about 0.5 of

initial pressure shock.

Figure 62 shows the pressure histories on the front wall of Building 2. The results at this

location for the two-building simulation are compared to the pressure at the same point for the

single building simulation, where Building 1 was directly exposed to the blast effects from the

explosion. The shielding effect is easily visible. Figure 66 shows the same pressures as well as

impulse histories, both normalized. The peak reflected pressure shock is about 7.6 times higher in

the case of unshielded simulation. Peak positive phase impulse is greater by a factor of about 6.

50000,0

500000,0

5000000,0

50000000,0

0 200 400 600 800 1000

Pre

ssu

re [P

a]

Charge weight [kg]

B1, front B1, side

B1, rear B2, front, single building

B2, side, single building B2, front, two-building

B2, side, two-building

92

The pressure and impulse curves in Figure 65 are normalized by the peak pressure and peak

positive impulse, respectively, associated with the first pressure pulse. Pressures and impulses in

Figure 66 are scaled by the peak pressure and peak positive phase impulse, accordingly, associated

with the two-building model.

Table 33. Peak overpressures on walls of Building 2 for single- and two-building simulation

Charge weight

[kg]

Peak overpressure [Pa]

Front wall Side wall

Single building

Two-building

Decrease factor

Single building

Two-building

Decrease factor

10 1623007,9 215750,7 7,523 109434,1 77200,5 1,418

50 1616697,6 215931,5 7,487 110311,9 77514,9 1,423

100 1650888,0 215825,0 7,649 109571,1 77412,4 1,415

200 1631931,5 215909,9 7,558 109698,3 77478,4 1,416

500 1637957,5 215831,5 7,589 109743,6 77446,9 1,417

1000 1611381,3 216080,9 7,457 110265,0 77367,4 1,425

Figure 62. Pressure-time blast loading curves for the front wall of Building 2 for single- and two-

building simulation

0

200000

400000

600000

800000

1000000

1200000

1400000

1600000

1800000

2000000

0 0,01 0,02 0,03 0,04 0,05 0,06

Pre

ssu

re [P

a]

Time [s]

100kg, shielding 100kg, no shielding

1752213rP Pa

317150rP Pa

503.6

3060.2

s

si

i s

s

Pa

Pa

93

Figure 63. Pressure-time blast loading curves for the side wall of Building 2 for single- and two-

building simulation

Figure 64. Pressure-time blast loading curve for the rear wall of Building 1

50000

70000

90000

110000

130000

150000

170000

190000

210000

230000

0 0,01 0,02 0,03 0,04 0,05 0,06 0,07 0,08

Pre

ssu

re [P

a]

Time [s]

100kg, shielding 100kg, no shielding

0

50000

100000

150000

200000

250000

300000

350000

0 0,02 0,04 0,06 0,08

Pre

ssu

re [P

a]

Time [s]

Pressure, B1, rear wall, 100kg

210896soP Pa

178737soP Pa

122.

30 .5

7

7

s

si

i s

s a

P

P

a

94

Figure 65. Blast pressure and impulse histories on rear wall of Building 1

Figure 66. Comparison of pressure/impulse histories on front wall of Building 2 for single- and

two-building simulations

-1

-0,5

0

0,5

1

1,5

2

2,5

0 0,01 0,02 0,03 0,04 0,05 0,06

Scal

ed

pre

ssu

re/i

mp

uls

e

Time [s]

Pressure, B1, rear wall, 100kg Impulse, B1, rear wall, 100kg

-1

0

1

2

3

4

5

6

7

8

0 0,01 0,02 0,03 0,04 0,05 0,06

Scal

ed p

ress

ure

/im

pu

lse

Time [s]

Pressure, B2, front wall, single building, 100kg

Pressure, B2, front wall, two-building, 100kg

Impulse, B2, front wall, single building, 100kg

Impulse, B2, front wall, two-building, 100kg

95

5.3.4. Discussion

For the results’ comparison, the model with charge of 100kg and respective dimensions from

Table 25 has been chosen. Comparison of both numerical methods and some crucial parameters

presents Table 34.

Table 34. Comparison of models in Air3D and Abaqus FEA

Air3D Abaqus FEA

Solver CFD, explicit CEL, explicit

Formulation finite volume formulation Lagrange-plus-remap

formulation

Size of finite element 10x10x10mm 69x69x69mm

Number of finite elements ~5,000,000 ~5,565,000

Charge hemispherical cubic

Comparison of blast pressures on rear wall of Building 1, according to both solutions, is

presented in Figure 67 and Table 35. In both cases two positive pressure shocks appear, and are

succeeded by the negative pressure phase. The first positive shock is related to the initial pressure

wave that wrapped around the rear edges of the building while moving forward from the

detonation source. The second pressure shock is a resultant of the wave reflection off the front

wall of second building. The second shock is about 17% lower in the case of Abaqus solution than

in Air3D, which can be recognized as satisfactory. The negative pressure varies more significantly,

since the difference is equal to 33%. The pressure configuration is nicely convergent till the end of

the negative phase of Abaqus solution. Then, in Abaqus solution, the growth of pressure occurs,

while according to Air3D, the pressure decreases once again. In general, the results for the positive

phase can be considered as close, having especially in mind, that two different computer codes and

models have been compared to each other. The negative phases differ considerably, but not

extremely. The comparison of impulses of each of three phases gives similar conclusions – the

relation between two positive impulses is retained, while for the negative phase the distinction is

greater.

Figure 68 presents the pressure and impulse histories on the front wall of Building 2.

According to Abaqus, the peak reflected overpressure on the front wall of Building 2 would be

overestimated more than 7.6 times if the building 1 were not taken into account, whereas Air3D

estimated this factor to 3.5 (Table 36). Distinction of similar severity can also be observed for the

impulses (Abaqus – 6.1, Air3D – 2.6). What is interesting, in both cases the peak underpressure

for single building simulation equals to about one third of the peak over pressure from the two-

building simulation. The negative phases for both pressure histories retain similar configuration.

96

Table 35. Scaled pressures and scaled impulses on rear wall of Building 1 according to Air3D and

Abaqus

Description Air3D Abaqus

Scaled pressure Relative impulse Scaled pressure Relative impulse

First shock,

positive phase 1.00 0.42 1.00 5.50

Second shock,

positive phase 2.75 0.65 2.30 6.14

Negative phase -0.75 -1.30 -0.50 -4.98

Table 36. Decrease of peak reflected pressure and peak positive phase impulse due to shielding

phenomenon according to Air3D and Abaqus

Parameter Decrease factor

Air3D Abaqus

Peak reflected pressure 3.5 7.65

Peak positive phase impulse 2.6 6.08

In general, results obtained with use of Abaqus numerical code are not comparable to those

from Air3D, since not all analyses have produced similar results. Pressures on rear wall of first

building are related, whereas shielding phenomenon and its influence on pressures on front wall of

second building is very divergent. From this it follows, that the reduction of pressure is

overestimated in Abaqus, which excludes its use in analyses of shock wave propagation in

complex geometries.

97

Figure 67. Blast pressure and impulse histories on rear wall of Building 1 according to Air3D (top)

and Abaqus (bottom)

-1

-0,5

0

0,5

1

1,5

2

2,5

3

0 1 2 3 4 5

Scal

ed

pre

ssu

re/i

mp

uls

e

Time [ms]

Pressure, B1, rear wall Impulse, B1, rear wall

-1

-0,5

0

0,5

1

1,5

2

2,5

3

0 0,01 0,02 0,03 0,04 0,05 0,06

Scal

ed

pre

ssu

re/i

mp

uls

e

Time [s]

Pressure, B1, rear wall, 100kg Impulse, B1, rear wall, 100kg

2 0.65i

1 1.30i

1 0.42i

2 2.75rP

0.75rP

2 6.14i 1 4.98i

1 5.50i

2 2.30rP

0.50rP

98

Figure 68. Comparison of pressure/impulse histories on front wall of Building 2 for single and

two-building simulations, according to Air3D (top) and Abaqus (bottom)

-1

0

1

2

3

4

5

6

7

8

0 1 2 3 4 5

Scal

ed

pre

ssu

re/i

mp

uls

e

Time [ms]

Pressure, B2, front wall, single building

Pressure, B2, front wall, two-building

Impulse, B2, front wall, single building

Impulse, B2, front wall, two-building

-1

0

1

2

3

4

5

6

7

8

0 0,01 0,02 0,03 0,04 0,05 0,06

Scal

ed p

ress

ure

/im

pu

lse

Time [s]

Pressure, B2, front wall, single building, 100kg

Pressure, B2, front wall, two-building, 100kg

Impulse, B2, front wall, single building, 100kg

Impulse, B2, front wall, two-building, 100kg

99

6. Conclusions

6.1. Final remarks

In this thesis, an approach has been made to compare the analytical and numerical methods

of solving the shock wave propagation problems arisen from the high explosive detonation.

Analytical method based on the Unified Facilities Criteria [24] from the Department of Defense of

the United States of America. Results obtained this way have been compared to numerical ones

from the Abaqus FEA. Also the comparison between results received in two different numerical

codes – Computational Fluid Dynamics used in Air3D and Coupled Eulerian Lagrangian used in

Abaqus have been conducted.

Moreover, accidental loads have been shown. Theoretical basis of blast phenomena, shock

wave propagation and how it is affected by objects and structures in vicinity of the explosion have

been presented. The scientific researches on the topic have been recalled. European standards and

American regulations regarding the accidental loading have been cited and commented.

Principle steps required to produce a blast simulation in Abaqus have been shown. To ease

the process of model creation, the Python script has been written. This set of modules and classes

enables the user to generate Abaqus models representing the air medium, obstacles and charges

inside it based on crucial parameters.

In total, 29 simulations, from which 17 had more than 5 millions of finite elements, and of

three distinct issues have been carried out.

The aim of the first problem (Subchapter 5.1) was to determine TNT specific energy and

the size of finite element, that would later be used across succeeding issues. The specific energy of

3.68MJ/kg and size of about 10-5mm have been determined. Obtained results have proven, that in

the case of analyses using explicit approach, the grid size plays a very important role. Another

conclusion is, that various finite element sizes are proper for problems of different classes, hence

establishment of the computationally efficient size of finite element, that would produce reliable

results, should be performed individually. The convergence of results for front wall pressures has

been proven, yet significant underestimations of underpressures, overestimations of overpressures

and inaccuracies in phase durations, across various faces, have been indicated as most problematic

issues.

The second problem (Subchapter 5.2) regarded the blast wave flow through the holes in the

front face of a structure. The internal overpressures have been compared between each other and to

external reflected overpressure. It has been concluded, that the internal peak overpressure accounts

for about 25% of external reflected overpressure, and that the internal pressures can be estimated

by means of Abaqus with acceptable convergence. Inaccuracies in phase durations have been

encountered again.

100

The third issue (Subchapter 5.3) concerned the shielding phenomenon and its effect on

pressure approaching building’s faces. Here, the study [17] has been reproduced in order to

compare two computer codes. Additionally, verification of scaled distance has been performed

and, in the case of peak pressures, with no doubt confirmed. The obtained results have shown nice

convergence for pressure waves reflected off the surrounding objects, and meaningful divergence

in influence of shielding phenomenon on pressures on shielded building. The overestimation of

pressure reduction in Abaqus excludes its use in analyses of shock wave propagation in complex

geometries.

Results from Abaqus tool proved to be accurate for simple geometries, yet for complex

ones CFD codes seem to be more reliable. The type of finite element might be probable cause of

such state.

In all analyses, obstacles were treated as rigid objects. This assumption is valid since

deformations are relatively small comparing to dimensions of the obstacle, however the

phenomenon of absorption of wave and its energy by obstacles and ground, and by their

deformation, was not considered. Also the damage criteria were not taken into account.

6.2. Future tasks

One of two major phenomena related to wave propagation in urban districts, i.e. shielding,

has been described. The influence of the second one – channelling – could be a possible and

reasonable extension of this thesis. However, some reference researches would be required to

estimate the correctness of results. Another reasonable improvement could be to estimate the

inaccuracies caused by use of cubic charges instead of hemispherical ones. To model the ground as

non-rigid domain could be further field of enhancement.

101

Bibliography

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2. EN 1991-1-7 General Actions – Accidental actions

3. C. Mougeotte, P. Carlucci, S. Recchia, J. Huidi, Novel Approach to Conducting Blast

Load Analyses Using Abaqus/Explicit-CEL, SIMULIA Customer Conference 2010

4. http://www.simulia.com/products/abaqus_standard.html

5. http://www.simulia.com/products/abaqus_explicit.html

6. Abaqus Theory Manual (6.10), Section 2.4.5: Explicit dynamic analysis, Dassault

Systems SIMULIA Corp. Providence, RI, USA (2010)

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9. Abaqus Scripting User's Manual V6.10, Part I: An introduction to the Abaqus Scripting

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Systems SIMULIA Corp. Providence, RI, USA (2010)

11. Abaqus Analysis User's Manual V6.10, Section 22.1.1: Hydrodynamic Behavior:

Overwiev, Dassault Systems SIMULIA Corp. Providence, RI, USA (2010)

12. Abaqus Analysis User's Manual V6.10, Section 22.2: Equations of state, Dassault

Systems SIMULIA Corp. Providence, RI, USA (2010)

13. http://www.history.navy.mil/photos/images/dod/8505379c.htm

14. http://www.history.navy.mil/photos/images/dod/8505379.jpg

15. W. E. Baker, Explosion hazards and evaluation, Elsevier Scientific Pub. Co., Amsterdam

and New York 1983

16. P. D. Smith, T. A. Rose, Blast wave propagation in city streets - an overview, Cranfield

University, John Wiley & Sons 2005

17. A.M. Remennikov, T. A. Rose, Modelling blast loads on buildings in complex city

geometries, 2005

18. T. Ngo, P. Mendis, A. Gupta, J. Ramsay, Blast loading and blast effects on structures -

an overview, EJSE 2007

19. Abaqus Analysis User's manual V6.10, Section 6.10.1: Acoustic, shock, and coupled

acoustic-structural analysis, Dassault Systems SIMULIA Corp. Providence, RI, USA

(2010)

102

20. Abaqus Analysis User's manual V6.10, Section 30.4.5: Acoustic and shock loads,

Dassault Systems SIMULIA Corp. Providence, RI, USA (2010)

21. G. Le Blanc, M. Adoum, V. Lapoujade, External blast load in structures – Empirical

approach, 5th European LS-DYNA Users Conference

22. "explosive" Encyclopædia Britannica. Encyclopædia Britannica Online. Encyclopædia

Britannica, 2011. Web. 30 May. 2011.

http://www.britannica.com/EBchecked/topic/198577/explosive

23. http://www.wbdg.org/references/pa_dod.php

24. UFC 3-340-02, 2008, Structures to resist the effects of accidental explosions, Department

of Defense, USA, 2008

25. P. D. Smith, T. A. Rose, Blast loading and building robustness, John Wiley & Sons 2002

26. M. Spinelli, R. Vitali et al, Virtual Treaded Tire Simulation as a Design Predictive Tool:

Application to Tire Hydroplaning, SIMULIA Customer Conference 2009

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Technology, 2011

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Gdańsk University of Technology, 2003

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International Symposium on Impact Engineering, Warsaw, Poland, 2010

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103

Appendix A

Lagrangian and Eulerian descriptions

Two methods of description of motion can be used – so called Lagrangian and Eulerian

descriptions [29][30].

Consider a three dimensional Euclidean space as a reference space:

3

1 2 3 ,x x x x x Ε (1)

i ixx e (2)

where ie are unit vectors or versors, and a continuous body B inside of it.

Figure A-1. Deformed and undeformed configurations of a body [29]

It is assumed, that there exists a continuous and unambiguous projection for this body

which depends on time:

:B B (3)

104

Two configurations are taken into account. Current configuration is a set of projections of

particles of body B at time t , and is denoted as t C . Second configuration is an initial

configuration 0C (at 0t ).

The relation between them is as follows:

0t C C (4)

Location of an arbitrary particle c in configuration 0C can be described by so-called

material coordinates:

1 2 3c c cc (5)

Figure A-2. Lagrangian description concept visualization [29]

The motion of the chosen particle c we denote as:

,t x c x c (6)

Coordinates of point P in the initial configuration can be denoted as:

3

1 2 3 ,X X X X x Ε (7)

which results in following description of motion:

,tx x X (8)

The independent variables are in this case the coordinates of particle c in the initial

configuration X and time t . The motion of given particle of matter is being tracked. This

105

description of motion is referred to as a material or Lagrangian description. It is useful in all cases

where the initial conditions of a body influence significantly the succeeding configurations. This is

especially related to solid bodies.

In Finite Element computer codes the material particles are associated with nodes of

computational mesh, which in turn limit the deformations the particles can undergo. From this one

of the biggest flaws of Lagrangian description arises – its inability to follow large deformations.

Due to above mentioned constraints, structural mechanics is the main field of application of this

method.

The displacement of the material point:

, , ,0t t u X x X x X x X (9)

Based on the above equation, velocity and acceleration formulae can be obtained:

, ,

,t t

tt t t

u X x X xv X u (10)

2

2

, ,,

t tt

t t t

v X x X xa X v u (11)

The alternative description of motion presents the equation:

,tX X x (12)

Here, the independent variables are time t and coordinates in current configuration x . The

changes of parameters of a medium at given location and time are being tracked. This description

of motion is referred to as a spatial or Eulerian description. It is useful in all cases where the initial

configuration influences insignificantly the succeeding configurations, or it has no influence at all.

This is especially important for fluids.

In computer codes the computational mesh is fixed to certain points, whilst material flows

through it. The limitations regarding the mesh distortions are not present, yet large amount of

finite elements is required, since not only the object, but also surrounding space needs to be

modeled. This method allows for existence of multiple materials in single finite element. Fluid

materials behavior, large deformations and damage simulations are main fields of application of

this method.

106

The velocity and acceleration in spatial coordinates can be denoted as:

, , ,t t tv x v x X (13)

, , ,

,ji i i i

j

j j

xD t v t v t v vt v

Dt t x t t x

v x x xa x (14)

Figure A-3. Eulerian description concept visualization [29]