Submerged floating tunnels subjected to internal blast loading

(1)

Submerged floating tunnels subjected to internal blast loading

Sondre Rydtun Haug Karoline Osnes

Civil and Environmental Engineering Supervisor: Tore Børvik, KT Submission date: June 2015

(2)

(3)

Department of Structural Engineering Faculty of Engineering Science and Technology

NTNU- Norwegian University of Science and Technology

MASTER THESIS 2015

SUBJECT AREA:

Computational Mechanics

DATE:

10 June 2015

NO. OF PAGES:

16 + 182 + 41

TITLE:

Submerged floating tunnels subjected to internal blast loading

Neddykket rørbru utsatt for eksplosjonslast

BY:

Sondre Rydtun Haug Karoline Osnes

RESPONSIBLE TEACHER: Professor Tore Børvik SUMMARY:

Design of structures against blast loading is of great importance and has received a lot of attention in recent years. The Norwegian Public Road Administration (NPRA) are currently investigating the potential for eliminating all ferries on the highway route E39 between Trondheim and Kristiansand. One of the critical points is the crossing of the Sognefjord, where a submerged floating tunnel (SFT) has been suggested as a possible crossing method. For an SFT, an internal explosion, due to e.g. accidents or terrorist attacks, can be extremely critical. Concrete will most likely be chosen as the main building material, and a comprehensive investigation of the material's response to blast loading is therefore important. As full-scale experiments are not appropriate, numerical tools as the finite element method (FEM) will be applied.

For this thesis, numerical simulations of concrete plates against blast loading were conducted in three different finite element codes: IMPETUS Afea Solver, LS-DYNA and Europlexus. In each of these codes, different material models were investigated: the Holmquist-Johnson-Cook (HJC) model, the K&C Concrete Damage Model (CDM) and the Dynamic Plastic Damage Concrete (DPDC) model. It was mainly focused on the HJC model, while the others were investigated for selected problems. In order to validate the numerical models, it was performed experiments on 50 mm concrete plates subjected to blast loading. The HJC model proved to require an immense tuning of the material parameters to approach the experiments, and in addition, predicted a behaviour that was too ductile. The CDM model showed a much greater potential as the predicted behaviour seemed reasonable with no tuning of the parameters. The DPDC model was applied in a fluid-structure interaction (FSI) analysis, but this type of analyses appeared to be redundant for the problem at hand.

As an alternative to explosive detonations, the new shock tube facility at the Structural Impact Laboratory (SIMLab) was used. The facility proved to be a great alternative to using explosives, in addition to providing a safe and controllable approach when investigating blast loading.

ACCESSIBILITY OPEN

(4)

(5)

Institutt for konstruksjonsteknikk Fakultet for ingeniørvitenskap og teknologi

NTNU- Norges teknisk- naturvitenskapelige universitet

MASTEROPPGAVE 2015

FAGOMRÅDE:

Beregningsmekanikk

DATO:

10 Juni 2015

ANTALL SIDER:

16 + 182 + 41

TITTEL:

Neddykket rørbru utsatt for eksplosjonslast Submerged floating tunnels subjected to internal blast loading

UTFØRT AV:

Sondre Rydtun Haug Karoline Osnes

FAGLÆRER: Professor Tore Børvik SAMMENDRAG:

Prosjektering av konstruksjoner mot eksplosjonslast er av stor betydning, og har fått mye oppmerksomhet de siste årene. Statens vegvesen undersøker nå muligheten for gjøre E39 mellom Trondheim og Kristiansand ferjefri. Et av de kritiske punktene er krysningen av Sognefjorden, og en neddykket rørbru har blitt foreslått som en mulig løsning. En innvendig eksplosjon, på grunn av f.eks. ulykker eller terrorangrep, vil være svært kritisk for en slik bru. Brua vil mest sannsynlig bygges i betong, og det er derfor nødvendig med en

omfattende undersøkelse av materialets respons når det utsettes for eksplosjonslaster. Siden

fullskalaeksperimenter ikke er hensiktsmessig, vil man ta i bruk numeriske verktøy som elementmetoden.

I denne masteroppgaven ble det utført numeriske simuleringer av betongplater utsatt for eksplosjonslast i tre forskjellige elementkoder: IMPETUS Afea Solver, LS-DYNA og Europlexus. I hver av disse kodene ble ulike materialmodeller undersøkt: Holmquist-Johnson-Cook (HJC), K&C Concrete Damage Model (CDM) og Dynamic Plastic Damage Concrete (DPDC). Det ble i hovedsak fokusert på HJC-modellen, mens de andre ble undersøkt for utvalgte problemer. For å validere de numeriske modellene ble det utført eksperimenter på 50 mm betongplater utsatt for eksplosjonslaster. HJC-modellen viste seg å gi en altfor duktil respons, selv etter at materialparameterne var kraftig justert. CDM-modellen viste seg å ha et mye større potensiale, siden responsen virket rimelig selv uten justering av parameterne. DPDC-modellen ble benyttet i en fluid-struktur interaksjons (FSI) analyse, men det viste seg at denne typen analyse var overflødig for denne type problemerr

Det nye shock tube-anlegget ved Structural Impact Laboratory (SIMLab) ble benyttet som et alternativ til eksplosive detonasjoner. Anlegget viste seg å være et bra alternativ til å bruke eksplosiver. I tillegg ga det muligheten for en trygg og kontrollerbar undersøkelse av eksplosjonslaster.

TILGJENGELIGHET ÅPEN

(6)

(7)

Department of Structural Engineering

FACULTY OF ENGINEERING SCIENCE AND TECHNOLOGY

NTNU – Norwegian University of Science and Technology

MASTER’S THESIS 2015

for

Sondre R. Haug and Karoline Osnes

Submerged floating tunnels subjected to internal blast loading

1. INTRODUCTION

Protection of engineering structures against blast loading has received a lot of attention in recent years. The newly proposed coastal highway route E39 seeks to connect Trondheim to Kristiansand along the coast without using any ferry connections. One of the critical points is the crossing of the Sognefjord, where a submerged floating tunnel has been suggested as a means of crossing. Internal blast loading (due to e.g. an accident or a terrorist attack) to a structure like this can be extremely critical, and it is important to verify that the structure is able to withstand a realistic blast load, or at least minimise the damage, as a breach could have disastrous consequences. Concrete will most likely be used for this tunnel due to the scale of the structure. Also, more or less any cross-sectional profile may be cast in concrete and can easily be adjusted to the correct buoyancy.

Computational methods are now available to predict both the loading and structural response in these extreme loading situations, and experimental validation of such methods is necessary in the development of safe and cost-effective structures. In this study blast experiments will be performed, and the data will be used for validation and verification of some frequently used computational methods involving blast loading.

2. OBJECTIVES

The main objective of the research project is to determine how concrete behave under blast loading, and to validate to which extent this can be predicted using computational tools.

3. A SHORT DESCRIPTION OF THE RESEARCH PROJECT The main topics in the research project will be as follows;

1. A comprehensive literature review should be conducted to understand the blast load phenomenon, blast load design, shock tube facilities, constitutive and failure modeling of concrete materials exposed to extreme loadings, and explicit finite element methods.

2. Casting of concrete plates of different thicknesses and of a specified quality (B45) and accompanying material tests.

3. Proper constitutive relations and failure criteria are chosen and calibrated based on the material tests.

4. The SIMLab Shock Tube Facility will be used to expose the plates to blast loading, as an alternative to explosive detonations. The shock tube experiments will be used to investigate typical dynamic responses and failure modes of plated structures exposed to blast loading.

5. Non-linear FE numerical simulations of the shock tube experiments will be performed, and the numerical results shall be compared and discussed based on the experimental findings.

Supervisors: Tore Børvik (NTNU), Martin Kristoffersen (NTNU)

The thesis must be written according to current requirements and submitted to the Department of Structural Engineering, NTNU, no later than June 10^th, 2015.

NTNU, January 14^th, 2015

(8)

(9)

Acknowledgements

This master’s thesis is written for the Structural Impact Laboratory (SIMLab) at the Nor- wegian University of Science and Technology (NTNU) in the spring of 2015. During this thesis we have been supervised by Professor Tore Børvik at SIMLab, and by Ph.D Martin Kristoffersen.

We would like to express our special thanks to our supervisors Professor Tore Børvik and Ph.D Martin Kristoffersen for their weekly guidance and assistance when needed. Their inputs, comments and discussions have been highly valued. We would also like to express our gratitude to Ph.D candidate Vegard Aune for his contributions with the experiments in the shock tube and for all the time he devoted to helping us. Invaluable assistance with IMPE- TUS Afea has been provided by Ph.D Candidate Jens Kristian Holmen, including answering questions whenever we needed. This was highly appreciated.

Further, we wish to thank technical engineer Trond Auestad for the assistance with the experiments at SIMLab. Senior engineer Ove Loraas, senior engineer Gøran Loraas and engineer Steinar Seehuus also deserves credit for their help with material tests and assistance creating the concrete slabs.

Lastly, we would like to thank our fellow students Cecilie Baglo, Therese Dybvik and In- grid Nilsen-Nygaard for inputs and contributions in many discussions.

Trondheim, 10 June 2015

Karoline Osnes Sondre Rydtun Haug

(10)

(11)

Abstract

Design of structures against blast loading is of great importance and has received a lot of attention in recent years. The Norwegian Public Road Administration (NPRA) are currently investigating the potential for eliminating all ferries on the highway route E39 between Trond- heim and Kristiansand. One of the critical points is the crossing of the Sognefjord, where a submerged floating tunnel (SFT) has been suggested as a possible crossing method. For an SFT, an internal explosion, due to e.g. accidents or terrorist attacks, can be extremely critical. Concrete will most likely be chosen as the main building material, and a comprehensive investigation of the material’s response to blast loading is therefore important. As full-scale experiments are not appropriate, numerical tools as the finite element method (FEM) will be applied.

For this thesis, numerical simulations of concrete plates against blast loading were conducted in three different finite element codes: IMPETUS Afea Solver, LS-DYNA and Europlexus.

In each of these codes, different material models were investigated: the Holmquist-Johnson- Cook (HJC) model, the K&C Concrete Damage Model (CDM) and the Dynamic Plastic Damage Concrete (DPDC) model. It was mainly focused on the HJC model, while the others were investigated for selected problems. In order to validate the numerical models, it was performed experiments on 50 mm concrete plates subjected to blast loading. The HJC model proved to require an immense tuning of the material parameters to approach the experiments, and in addition, predicted a behaviour that was too ductile. The CDM model showed a much greater potential as the predicted behaviour seemed reasonable with no tuning of the parameters. The DPDC model was applied in a fluid-structure interaction (FSI) analysis, but this type of analyses appeared to be redundant for the problem at hand.

As an alternative to explosive detonations, the new shock tube facility at Structural Im- pact Laboratory (SIMLab) was used. The facility proved to be a great alternative to using explosives, in addition to being a safe and controllable approach when investigating blast loading.

(12)

(13)

Nomenclature

ALE Arbitrary Lagrangian-Eulerian Formulation

CCFV Cell Centered Finite Volume

CDM K&C Concrete Damage Model

CEA The French Alternative Energies and Atomic Energy Commission

CFD Computational Fluid Dynamics

DPDC Dynamic Plastic Damage Concrete

FE Finite Element

FEA Finite Element Analysis

FEM Finite Element Method

FSI Fluid Structure Interaction

FV Finite Volume

GAZP Perfect gas material in Europlexus

HE High Explosives

HJC Holmquist-Johnson-Cook Concrete Model

JRC Joint Research Center

LE Low Explosives

NCFV Node Centered Finite Volumes

NPRA Norwegian Public Road Administration

(20)

NOMENCLATURE

NS Norwegian Standards

SDOF Singel Degree of Freedom

SFT Submerged Floating Tunnel

SPH Smoothed Particle Hydrodynamics

∆σ_y, ∆σ_m, ∆σ_r Initial, maximum and residual stress surface

Strain

_crush Crushing strain in the HJC model

_vol,lock Locking strain in the HJC model

vol,yield Volumetric strain at yield

_vol Volumetric strain

η Yield scale factor in the CDM model

Df

Dt Material time derivative

γ Heat capacity ratio

λ Effective plastic strain in the CDM model

ν Poisson’s ratio

ω Eigenfrequency

Ω₀, Ω Initial and current configuration

φ Mapping between the initial and the current configuration ρ, ρ₀ Current and original density

σ Cauchy stress

(21)

NOMENCLATURE

σ^∗ Normalized stress

b Body force

F Deformation gradient

P Nominal Stress

q Heat flux

u Material point displacement

v Velocity field

a Speed of sound, Acceleration, Width

A, B, n,sfmax Strength parameters in the HJC model b Decay coefficient in the Friedlander equation b₁, b₂, b₃, f_d, k_d Damage constants in the CDM model

C Strain rate constant

c Wave speed

C_p, C_v Heat capacity at constant pressure and volume

D Damage

d Diameter

D₁, D₂, e^f_min Damage constants in the HJC model

E Young’s modulus

e_int Specific internal energy

F Force

(22)

NOMENCLATURE

f, g Arbitrary wave functions

f_c Concrete compression strength

f_t Concrete tensile strength

G Shear modulus

H Thickness

I Moment of inertia per unit width

i Impulse

i⁺_s,i⁻_s Positive and negative specific impulse

ir Reflected impulse

J Determinant of Jacobian between spatial and material coordinates J₂⁰ Second deviatoric stress invariant

K Stiffness, Bulk modulus

K₁, K₂, K₃ Pressure constants in the HJC model K_L Load factor for an equivalent SDOF system KM Mass factor for an equivalent SDOF system

L Length

M Mass

m Molecular weight

M_e, K_e, F_e Equivalent mass, stiffness and force

M_s Mach number of shock wave

(23)

NOMENCLATURE

P Pressure

P^∗ Normalized pressure

P₀ Atmospheric pressure

P_r Peak reflected pressure

Pcrush Crushing pressure in the HJC model

Plock Locking pressure in the HJC model Q_x Specific heat of an arbritary explosive

Q_TNT Specific heat of TNT

R Radius, Gas constant, William-Warnke function r_f Strain rate enhancement factor in the CDM model R_spec Specific gas constant

s Specific heat source term

T Temperature, Normalized maximum hydrostatic tensile strength in the HJC model

t time

T^∗ Normalized tensile strength

t_A Arrival time

t_d Positive duration

t⁻_d Negative duration

T_n Natural period of vibration

(24)

NOMENCLATURE

t_rise Rise time

T N T_Equivalent TNT equivalent

U_s Incident shock wave velocity

V_b,V_i,V_r Ballistic limit, initial and residual bullet velocities

W Explosive charge weight, Deflection

w Hyperelastic potential on reference configuration

Z Scaled distance

X, x Material coordinates in initial and current position

(25)

Chapter 1 Introduction

In the summer of 2010, The Norwegian Public Roads Administration (NPRA) was commis- sioned to investigate the potential of eliminating all ferries on the highway route E39 between Trondheim and Kristiansand. The project was named ”Ferry-free coastal route E39”. The crossing of the Sognefjord proves to be a critical point, as the fjord is considered to be the most difficult and challenging fjord to cross. One suggested location for the crossing has a span of 3.7 km and a depth of 1300 m, which challenge existing fjord-crossing alternatives and requires development of new concepts and solutions. The alternatives suggested for the crossing are a submerged floating tunnel (SFT), a suspension bridge, a floating bridge or a combination of the three. For this thesis, only the SFT is considered.

An SFT is a tunnel that floats in water, supported by its buoyancy. A tunnel crossing the Sognefjord will result in a long and slender structure, and will therefore require special measures to provide horizontally and vertical stiffness. Two main concepts solving this problem have been proposed: constructing the tunnel as an arch anchored to the surface with pontoons, or as a straight tunnel anchored to the bottom with tension rods. A combination of the two is also possible. Both solutions make the concept of an SFT feasible in deep waters and does not limit the ship traffic in the fjord. Figure 1.1 shows a possible design for the SFT using pontoons.

An SFT has never been built in the past, which could cause a lot of uncertainties to the design process. One of the design challenges of STFs is to create a structure able to withstand local damage. Possible scenarios causing local damage could be accidents such as sinking ships, internal fire or explosions. Internal explosions could also be caused by terrorist attacks. In fact, terrorist activities have proved to be a growing threat to human civiliza-

(26)

CHAPTER 1. INTRODUCTION

tion and civil infrastructure in the recent decade. For an SFT, an internal explosion can be extremely critical and a comprehensive assessment in terms of blast loads is necessary. The scope of this thesis will be limited to the local structural response due to blast loading.

(a) View from surface (b) View from underwater

Figure 1.1: Possible design for the submerged floating tunnel [1].

Due to the size of the bridge, concrete will most likely be used as the main building material as other alternatives are more expensive. Additionally, concrete makes it possible to cast almost any cross section, which makes it easy to adjust the buoyancy for the structure.

Concrete walls with a thickness of 800 mm have been suggested [2].

In order to secure the tunnel with respect to blast loading, it is important to investigate the dynamic response of the concrete. As full-scale experiments are not an option, the attention must be turned to numerical simulations. The analysis of concrete structures under blast loading is a highly complex problem, and much effort has been made in recent years to develop reliable numerical tools to predict the material response. For this thesis, selected material models will be investigated for the simulation of concrete plates under blast loading.

Experiments on concrete plates exposed to blast loading will also be carried out in order to validate the numerical models. By comparing the experimental results to the simulations,

(27)

CHAPTER 1. INTRODUCTION the numerical models’ ability to recreate the physical behaviour under blast loading can be evaluated.

The NPRA has earlier suggested that the inner tunnel ring must withstand a pressure of 700 kPa (7 bar) [3]. The documentation for this guideline pressure was however difficult to obtain. Despite this fact, the provided pressure is worth looking into and will be used as a starting point for investigating the response of the concrete plates. The experiments will be conducted on concrete plates reinforced with shear stirrups, although the guideline applies to SFTs with full reinforcement.

As an alternative to explosive detonations, the new SIMLab Shock Tube Facility will be used. The shock tube is a safe alternative to using explosives and allows for great repeatabil- ity and the use of advanced instrumentation monitoring the experiments. As the shock tube facility at SIMLab is new, i.e. build in October 2014, a calibration of the shock tube will be a part of the work for this thesis.

A short overview of each chapter is presented below.

Chapter 2 - Theory

In this chapter, relevant theory for the thesis will be presented. This includes blast theory, featured continuum mechanics, an introduction to Fluid Structure Interaction (FSI), Lagrangian Finite Element Analysis (FEA), Eulerian FEA, Arbitrary Lagrangian-Eulerian (ALE) FEA and one dimensional stress wave theory.

Chapter 3 - Materials

This chapter presents a qualitative description of concrete’s material properties. In addition, failure mechanisms and material modelling of concrete are included.

(28)

CHAPTER 1. INTRODUCTION

Chapter 4 - Introductory Experimental Work

This chapter describes the necessary preparations conducted in the laboratory prior to the shock tube experiments. This includes casting of the concrete plates and test specimens, and the material testing.

Chapter 5 - Shock Tube

In this chapter, a description of the shock tube and shock tube theory will be presented. The calibration of the shock tube is also included.

Chapter 6 - Preliminary Studies

In this chapter, a 2D Europlexus model is validated by the calibration tests. A mesh study and a preliminary load study are performed in IMPETUS Afea Solver.

Chapter 7 - Experimental Work

This chapter describes the procedure and results of the experiments performed in the shock tube.

Chapter 8 - Numerical Studies

This chapter presents the FSI analyses conducted in Europlexus, in addition to the Lagrange analyses in IMPETUS. A parameter study and simulations of the experiments in IMPETUS are included. Simulations of the experiments are also performed in LS-DYNA.

(29)

Chapter 2 Theory

2.1 The blast phenomena

In this section, relevant theory for the blast phenomena will be presented.

2.1.1 General definition of an explosion

The definition of an explosion is a rapid increase in volume and release of energy [4]. A common way to classify explosions is the way energy is transferred from reacted to un-reacted material, i.e. as a detonation or a deflagration [4]. A deflagration describes a subsonic combustion propagating through heat transfer [5], while a detonation is defined as a supersonic front that travels through a medium that drives a shock front. This will lead to the formation of a shock wave [6]. It is also differed between deflagration and detonation by the speed of which they are burning material. If a material is burning with a rate slower than the speed of sound, it is called a deflagration. If the rate of burning is greater than the speed of sound, it is classified as a detonation [7]. Figure 2.1 shows a typical pressure profile for a deflagration and a detonation.

Figure 2.1: Pressure profile for a detonation and deflagration [4].

(30)

CHAPTER 2. THEORY

As seen in Figure 2.1, deflagrations are characterized by a gradual increase to the peak pressure with a long duration followed by a gradual decrease. Detonations are characterized by a rapid rise to peak pressure followed by a steady decrease [4].

Combustion of an explosive material releases energy under a short period of time that have several effects on its surroundings. They are listed by Krauthammer in [7] as shock waves, ground shocks, fragments, fire, radiation and electromagnetic pulse. The scope of this thesis is limited to the response from shock waves.

2.1.2 Classification

Explosions can be divided into three groups by their nature, namely physical, nuclear and chemical. In a physical explosion, energy is released from a compressed gas, while in a nuclear explosion, energy is released from a nuclear reaction. In a chemical explosion, rapid oxidation of materials is the main source of energy [8]. Explosive materials can also be divided into groups of high explosives (HE) and low explosives (LE). LE burns through deflagration and will not explode without a confinement, while HE have a supersonic reaction and will explode without a confinement. The latter results in a detonation [9]. There exist a large variety of HE, and in order to compare them, it is common to refer to the ratio of specific heats of the HE (Q_x) and the TNT (Q_TNT). This is called TNT equivalence. In this way, different explosives can be modelled to a known experiment with TNT.

TNT_Equivalent = Qx

Q_TNT (2.1)

Explosive materials may also be classified by the amount of energy that is needed to ignite the material. Primary explosives can easily be detonated from a single spark or flame. Sec- ondary explosives are relatively insensitive and need a greater amount of energy to detonate.

However, secondary explosives are much more powerful when they detonate than primary explosive materials.

(31)

CHAPTER 2. THEORY

2.1.3 Process of an explosion

As an explosive charge is heated, its temperature will rise. If the rate of supplied heat is greater than the release of heat, a runaway reaction can develop, which in turn can ignite the explosive charge. Typically a temperature between 150 and 350 ^◦C is enough to ignite explosives [10]. The ignition of an explosive can generate hot gases at pressures up to 300 kilobar and temperatures of about 3000-4000 ^◦C. As a result, the surrounding air is forced out of the volume it occupies. The layer of surrounding air is compressed and will contain most of the energy released from the initial explosion. The air formed in front of the explosive is called a blast wave [10].

Because of a pressure difference between the blast wave and the air, the blast wave will propagate away from the initial source. As the explosive gases expand, their pressure will fall to the atmospheric pressure. The same happens for the blast wave as the distance from the initial source increases [10]. The explosive gases will cool and their pressure will fall below the atmospheric pressure. This is referred to as the negative phase. This pressure fall occurs because the gases over-expand because of their momentum. Afterwards, a reversal flow is driven due to the difference in pressure before equilibrium is regained [10].

2.1.4 Blast wave interaction

When a blast wave hits an object, which is more dense than the material transmitting the wave, the wave will be reflected from it, or depending on its geometry, diffract around it.

The reflected wave has the same shape as the original (incident) wave, but the peak pressure is higher than in the original wave. The air molecules that are moving forward, are stopped at the object. The air then compresses, which will give a rise in pressure.

At a 90 degree attack angel of the blast wave, there is no reflection and the surface is loaded by the incident pressure, which is often called ”side-on” pressure (see Figure 2.2a).

At an attack angel of zero degrees, the pressure can reach a value up to 8 times the incident

(32)

CHAPTER 2. THEORY

overpressure for a strong shock and up to 2 times for a weak shock [11]. This is reffered to the reflected pressure (see Figure 2.2b).

(a) Side-on pressure (b) Reflected pressure

Figure 2.2: Side-on and reflected pressure [12].

When the attack angle exceeds 40 degrees, Mach reflection occurs. The Mach reflection is a complex process which is of great importance to a structural response. For an explosion in air, it can be assume a spherical shock wave formation where the incident shock will reflect from the ground. This reflected wave will interact with the incident shock and form a third shock front, i.e. the Mach front (see Figure 2.3). It is assumed that the Mach front is a plane wave and has the same characteristics as the incident wave. However, the pressure magnitude can be larger. The point where the three waves meet is referred to as the triple point and defines the height of the Mach front. The same reflections are obtained from explosions on the ground surface [7].

Figure 2.3: Mach front formation [13].

For design purposes, the triple point is of great importance to structures exposed to blast

(33)

CHAPTER 2. THEORY loading. If the height of the triple point is greater than the structure, it can be assumed that the structure will be loaded with a uniform pressure. However, if the structure is taller than the triple point, the pressure distribution will not be a plane wave, and numerical simulations are needed to describe the blast loading. This thesis is limited to shock waves with an attack angle of zero degrees and Mach reflections are therefore disregarded.

2.1.5 The ideal blast wave

Figure 2.4: Pressure profile for an ideal blast wave [7].

Figure 2.4 shows a typical pressure-time history for a shock wave. Before the shock front impacts an object, the pressure is the atmospheric pressure P0. The pressure at the current position suddenly rises to P_r at the arrival time t_A with a rise time t_rise ≈ 0. Then, the pressure decays to the original pressure at time t_A+t_d, and further to a value P_s less than the original pressure P₀, before it returns to equilibrium. The quantity P_r is referred to as the peak reflected pressure [14].

(34)

CHAPTER 2. THEORY

Two phases can be observed in Figure 2.4. The part where the pressure is greater than the ambient pressure is called the positive part, and the part where the pressure is less, is called the negative part [14]. The negative part is often of longer duration and a lesser intensity than the positive part [8]. The specific impulses are defined as:

Positive impulse =i⁺_s =

Z tA+td

tA

[P(t)−P₀]dt (2.2)

Negative impulse =i⁻_s =

Z tA+t_d+t⁻_d tA+td

[P₀−P(t)]dt (2.3)

wheret_Ais the arrival time of the shock wave,t_dis the positive duration andt⁻_d the is negative duration. The pressure of the positive phase is reduced as the distance to the original source increases, but the duration grows. This results in a lower amplitude and longer shock wave.

Representation of an ideal blast wave

A number of authors have recommended formulas by empirical curve-fitting to describe an ideal blast wave. The simplest way to describe a blast-wave shape is by linear decay of the pressure. An example is given by Equation 2.4.

P(t) =







P₀+P_r 1− _t^t

d

if t≤t_d

0 if t≥t_d

(2.4) A somewhat more complex and common formulation is the modified Friedlander equation.

The Friedlander equation involves three parameters and is able to match the following blast characteristics: reflected peak pressure Pr, impulse i, positive phase time td and the decay- coefficient b.

P(t) = P₀+P_r 1− t

t_d

e

−bt

td (2.5)

The Friedlander equation allows for a good compromise of accuracy and complexity when representing a blast load [12].

(35)

CHAPTER 2. THEORY

2.1.6 Internal blast loading

When an explosion occurs inside a structure, two blast loading phases occur. The first phase is the shock pressure, which consists of the initial blast load followed by reflected pulses from the walls (see Figure 2.5). These reflected pulses will decay in amplitude and cause a complex loading situation because each reflected wave will propagate and interact with other surfaces.

The duration of this period is short [7, 11].

Figure 2.5: Shock reflections on walls for an internal detonation [7].

The shock phase is followed by the gaseous phase. This phase is a result of the high pressure and high temperature gaseous products of the detonation expanding inside the structure.

(36)

CHAPTER 2. THEORY

It is characterized by a complex blast environment, a high pressure and has a much longer duration than the shock phase. Because of the longer duration, it is also known as the quasi-static phase [7].

2.1.7 Design of blast loads

There exist three approaches to calculate blast loads on structures. The empiric, semi- empirical and computational fluid dynamics (CFD) [3]. CFD is a numerical method that solves and describe fluid flows. Only the empirical approach will be discussed in this section.

The most basic empirical approach is based on scaling laws. The following scaling law is based on the principle that the energy released from a point explosion will propagate as a spherical shock wave. Since the volume of a sphere is proportional to its radius cubed, the scaling law is called cube-root scaling. If R₁ is the distance from a reference explosion with charge weight W₁, the blast wave properties can be related to a distance R₂ from another explosion with charge weight W₂ by [7]:

R₁ R2

= ³ rW₁

W2

(2.6) The scaled distance is further defined as:

Z = R

√3

W (2.7)

By using these relations, parameters such as overpressure, dynamic pressure, particle velocity etc. can be calculated from a reference explosion. The concepts of cube-root scaling implies that all physical quantities with dimensions of pressure and velocity must remain unchanged in the scaling [7]. The scaling relationship only applies when identical conditions exist, the compared charges have the same geometric shape and the surface geometry is the same.

Under other conditions, the scaling laws should only be used as an approximation and used with caution [7].

With a known peak pressure P_r and standoff distance R, it is possible to estimate the

(37)

CHAPTER 2. THEORY corresponding mass of TNT for explosions in air and at surface level by using Figures 2.6 and 2.7. By further using the TNT equivalence, the weight of other high explosives can be found.

Figure 2.6: Positive phase shock wave parameters for a spherical TNT explosion in free air [15].

(38)

CHAPTER 2. THEORY

Figure 2.7: Positive phase shock wave parameters for a hemispherical TNT explosion on the surface [15].

(39)

CHAPTER 2. THEORY

2.2 Continuum mechanics

The following section presents topics related to nonlinear continuum mechanics, and are essential for the understanding of the nonlinear finite element method, a tool that is widely used for this thesis.

Continuum mechanics is concerned with the deformation and motion of materials modeled as a continuous mass (continuum), and ignores inhomogeneities like molecular, grain and crystal structures. The objective is to establish models for the macroscopic behaviour of both fluids, solids and structures [16].

To develop a model for material behaviour, it is necessary to describe the state of the body, in other words, its configuration. The domain of the initial configuration is denoted Ω0, and for the current configuration, Ω (see Figure 2.8). Equations will be referred to a reference configuration and can be the same as both the initial and the current state of the body. The initial configuration will often be the undeformed state.

Figure 2.8: Initial and current configuration of a body.

(40)

CHAPTER 2. THEORY

The position of a material point in the initial configuration is given by X and the current is given by x. u is the deformation and φ is the mapping between the initial and the current configurations, given by x=φ(X,t).

2.2.1 Eulerian and Lagrangian description

When analysing the deformation and motion of a continuum, two different descriptions are used: a Lagrangian (material) and an Eulerian (spatial). The Lagrangian description uses the initial configuration as a reference, while the Eulerian uses the current configuration.

Consequently, the Eulerian description focuses on a fixed point in space describing particles passing through the point, while the Lagrangian focuses on the movement of individual particles [16]. The behaviour of solids is usually dependent on the deformation history and the undeformed configuration must therefore be known. A Lagrangian description is therefore dominant in solid mechanics. An Eulerian description is typically used for fluid mechanics, since it is often impossible to describe a fluid on the basis of the initial configuration. In addition, fluids are generally independent of deformation history, making it unnecessary to use a Lagrangian description.

Figure 2.9: Deforming block with Lagrangian (L) and Euler (E) elements.

(41)

CHAPTER 2. THEORY Figure 2.9 illustrates the difference between an Eulerian and a Lagrangian mesh. For the Eulerian mesh, the coordinates of the nodes are fixed, whereas for the Lagrangian mesh the, node coordinates are independent of time. The white dots in Figure 2.9 illustrates the element material points. In a Lagrangian mesh, the nodes and the material points will stay coincident with the element integration points. Because of this, the boundary nodes will remain on the boundary when the body deforms. This makes it easier to apply boundary conditions in Lagrangian meshes. Adding boundary conditions is much more complicated for an Eulerian mesh, since the boundary nodes do not remain coincident with the boundary.

On the other hand, since the elements deforms with the material, elements in a Lagrangian mesh can become severely distorted for large deformations. In an Eulerian mesh, the element length remains constant and large deformations will not be a problem.

2.2.2 Arbitrary Lagrangian-Eulerian Formulation (ALE)

The Arbitrary Lagrangian-Eulerian (ALE) formulation is a hybrid technique combining the advantages of Eulerian and Lagrangian formulations while seeking to minimize the disadvan- tages of the different methods. ALE will not be discussed in detail here, since the method has not been applied in any numerical studies for this thesis. It is however important to know about the method, since ALE is an alternative approach for solving problems that will be presented later.

In an ALE method, the nodes can either move with the material in a Lagrangian sense, be kept in place as in an Eulerian description or the nodes can move arbitrarily to accommo- date rezoning needs and avoid mesh entanglement (see Figure 2.10)[17]. In a fluid-structure analysis involving ALE, the fluid mesh follows the deformation of the (Lagrangian) structure.

This is in contrast to an embedded analysis, which will be discussed further in Section 2.3.

(42)

CHAPTER 2. THEORY

Figure 2.10: 1D example of ALE mesh and particle motion [17].

2.2.3 Conservation laws

The conservation laws (balance laws) must be satisfied in all physical systems, and are consequently an essential tool in continuum mechanics. In short, they state that certain properties of an isolated physical system do not change as the system evolves. For a thermomechanical system, the following conservation laws are relevant [16]:

• Conservation of mass

• Conservation of linear momentum

• Conservation of energy

• Conservation of angular momentum

The conservation laws can be used to derive the conservation equations, which may be of either Eulerian or Lagrangian description. Derivation of the equations can be found in the literature [16].

(43)

CHAPTER 2. THEORY

Conservation of mass Eulerian description:

Dρ

Dt +ρdiv(v) = 0 (2.8)

Lagrangian description:

ρ(X, t)J(X, t) = ρ₀(X) (2.9) Conservation of linear momentum

Eulerian description:

ρDv

Dt = div(σ) +ρb (2.10)

ρ₀δv(X, t)

δt =5₀P+ρ₀b (2.11)

Conservation of energy Eulerian description:

ρDw^int

Dt =D:σ− 5q+ρs (2.12)

ρ0w˙^int=ρ0

δw^int(X, t)

δt = ˙F^T :P− 50q˜+ρ0s (2.13) Conservation of angular momentum

Eulerian description:

σ=σ^T (2.14)

FP=P^TF^T (2.15)

(44)

CHAPTER 2. THEORY

The following describes the different variables used for the conservation equations.

ρ, ρ₀: Current and original density v: Velocity field

J: Determinant of Jacobian between spatial and material coordinates, J=det[δx_i/δX_j] σ: Cauchy stress

b: Body force P: Nominal stress q: Heat flux

w: Hyperelastic potential on reference configuration s: Specific heat source term

F: Deformation gradient, F_ij =δx_i/δX_j

Conservation equations in an ALE description is similar to those of an Eulerian description, with the only difference being a material time derivative ^Df_Dt. Please refer to the literature for more details [16].

2.2.4 Constitutive equations

To mathematically describe the behaviour of a material, a constitutive equation must be used.

A response of the material is then approximated by relating physical quantities specific to that material. For example, in solid mechanics, the constitutive equation is often referred to the stress-strain relation of a material. These relations are often purely phenomenological.

There exist numerous constitutive equations, some more comprehensive than others. Only a short description of the constitutive relation adopted for a fluid (air) will be presented here.

The constitutive relations for the solid (concrete) will be discussed in Section 3.3, where selected material models are presented.

(45)

CHAPTER 2. THEORY

The ideal gas law

The pressure P of an ideal gas is given by the equation:

P =ρ(γ−1)e_int (2.16)

where ρ is the density,γ = ^C_C^p

v is the heat capacity ratio and e_int the specific internal energy.

The gas may also be described by the equation:

P =ρR_specT (2.17)

where R_spec is the specific gas constant and T is the temperature.

The conservation equation listed in Section 2.2.3 makes up an indeterminate set of equations. In order to close the system, one extra equation is required, namely a constitutive equation such as the ideal gas law. The ideal gas law, together with the conservation laws, are able to describe a compressible flow, e.g. a flow that experiences significant changes in fluid density [18].

2.3 Fluid-structure interaction (FSI) in Europlexus

Fluid-structure interaction (FSI) is the interaction between a fluid and a movable or de- formable structure. The phenomena plays an important role in many disciplines, including civil, offshore, aeronautical engineering and more [19]. The following section addresses the treatment of FSI in the finite element code Europlexus. Europlexus is a computer code for simulations of fluid-structure systems under transient dynamic loading developed by the Joint Research Centre (JRC) in Ispra and the French Alternative Energies and Atomic En- ergy Commission (CEA) [20].

The fluid may be discretized by different techniques: finite elements (FE), finite volumes (FV) and Smoothed Particle Hydrodynamics (SPH). If we disregard SPH, three different approaches for discretization are available for the fluid: Finite Elements (FE), Node-Centered Finite Volumes (NCFV) and Cell-Centered Finite Volumes (CCFV) (see Figure 2.11).

(46)

CHAPTER 2. THEORY

Figure 2.11: Ways of discretizing the fluid domain [19].

For FE, kinematic variables (e.g. velocity) are discretized at the element nodes, while state variables (e.g. fluid pressure) are discretized at the Gauss points. For NCFV, a dual mesh is built with the fluid node at its centre and all variables are discretized at the fluid nodes. For CCFV, all variables are discretized at the volume centres [19]. Figure 2.12 shows an overview of the FSI algorithms available in Europlexus when discretizing the fluid by either FE or FV.

Figure 2.12: FSI Algorithms available in Europlexus [20].

(47)

CHAPTER 2. THEORY As seen in Figure 2.12, there are a few ways of classifying the different algorithms. At first they may be divided into strong or weak approach. The strong approach is based on constraints imposed on the fluid and structure at the interface between the fluid and the structure. In the weak approach, fluid pressure is applied as external forces on the structure [19]. Traditionally, strong algorithms are used in FE and weak in FV. Further, the algorithms can be classified as basic or embedded. A basic algorithm can either be classified as conforming or non-conforming, and Figure 2.13 illustrates the difference. A conforming FE mesh has a corresponding fluid node for each Lagrangian (structure) node and vice versa.

For a non-conforming mesh, there is a corresponding fluid node for every structure node, but not necessarily the other way around.

Figure 2.13: FSI problem with a conforming and non-conforming discretization [19].

Basic algorithms are suitable for problems without structural failure or too large rotations and deformations. In a basic algorithm, an ALE formulation is applied and this will lead to some limitations. As already mentioned, the fluid mesh will follow the structure mesh at the interface. The mesh will then quickly become irregular, and in the event of structure failure, erosion or flying debris are difficult to achieve. An embedded algorithm can be used for problems with damage and even fragmentation. The structure mesh is in that case immersed within the fluid mesh because the structure and fluid are discretized independently in the interface. An ALE formulation for the fluid is therefore no longer necessary and the fluid can

(48)

CHAPTER 2. THEORY

be purely Eulerian [19]. Figure 2.14 illustrates the embedded structure in the fluid mesh.

Figure 2.14: FSI problem with an embedded discretization [19].

To run an FSI analysis with an embedded discretization, the structure’s ”influence domain”

is needed. The influence domain is defined by the fluid nodes close enough to influence the structure. This information is needed for each time step as it will change with the deformation of the structure. Circles (or spheres in 3D) is specified based on mesh size and quadrangles (cones, prisms and hexahedral in 3D) connects the circles (spheres) forming an area (volume) that defines the influence domain. Fluid nodes within the domain will become coupled [17].

2.4 Numerical investigation of blast load on structures

The finite element method is a helpful tool when it comes to investigation of blast loads and the response of structures. The method is often employed if empirical methods are insufficient or suitable experimental tests are difficult to perform. Many techniques are available for describing the blast load response and the choice depends on both accuracy, complexity and computational power. Some techniques are listed below [21].

• A pure Lagrangian simulation

• An uncoupled Eulerian and Lagrangian simulation

• A coupled Eulerian and Lagrangian simulation

(49)

CHAPTER 2. THEORY

2.4.1 Pure Lagrangian

For a pure Lagrangian simulation, the blast load is approximated as an idealised pressure- time curve, such as the Friedlander curve. The curves can then be used in a simulation which uses a load-time function, where the load is applied to a given surface of a structure. During a simulation, the load will stay in the initial direction independent of a structural deformation.

This approach will give reasonable results for small deformations [21], and should otherwise only be used as an estimation of the structural behaviour [22].

2.4.2 Uncoupled Eulerian and Lagrangian

This approach uses both an Eulerian and a Lagrangian approach, but do not account for any fluid-structure interactions. A pure Eulerian simulation, involving only the fluid, is first conducted to find the load on the structure at a number of points. In the Eulerian simulation, the structure is assumed rigid such that there are no coupling effects between the load and the structure. The load from the Eulerian simulation is then applied to the structure to find the response. This method will only give reasonable results if there are minimal or no deformations of the structure during the blast loading. If the approach is applied to flexible structures, the response will be overestimated [21].

2.4.3 Coupled Eulerian and Lagrangian

To account for the interaction between the fluid and the structure, a coupled simulation must be applied. During a blast loading, the deformation of the structure and the fluid motion are coupled and a coupled approach should ideally be carried out for all blast load simulations. However, this type of simulations leads to a significant increase in computational time compared with the two other approaches [21]. A careful assessment must be made to find a good compromise between accuracy and computational time.

(50)

CHAPTER 2. THEORY

2.5 One dimension stress wave

A possible failure mechanism for concrete subjected to blast loading is spalling. Spalling is defined as material getting broken off from the side opposite to the one where the impact occurs. This is made possible in concrete due to the difference in compression and tensile strength. A valuable tool for understanding the phenomenon is the one dimensional stress wave theory. When a body is not in equilibrium, a stress wave will propagate through the material. There are two basic elastic stress waves; longitudinal waves and shear waves, but longitudinal waves will be discussed here. The following descriptions are given according to the compendium ”Støt og Energiopptak” by Langseth and Clausen [23].

When longitudinal stress waves move through a long stationary bar, they are divided into compression and tensional waves. For a compression stress wave, the individual particles move in the same direction as the stress wave, while for a tensional stress wave they move in the opposite direction. By assuming that the Poisson’s ratio effect is negligible and that the Hooks law is valid, the one dimensional wave equation can be derived as:

δ²u

δt² =c²·δ²u

δx² (2.18)

where u is the displacement in x-direction, and c is the wave speed given by:

c= s

E

ρ (2.19)

(a) Long stationary bar (b) Infinitesimal element Figure 2.15: Dynamic equilibrium of a bar.

(51)

CHAPTER 2. THEORY Equation (2.18) is valid for an infinitesimal bar element with thickness dx (see Figure 2.15).

N is the normal force, M is mass and a is the acceleration. A general solution to the one dimensional wave equation is:

u(x, t) = f(x−ct) +g(x+ct) (2.20) where f represents a wave moving in the positive direction andg in the negative. When the wave hits the end of the bar, it will be reflected. For a free end, the net stress must be zero and a compression wave is reflected as a tension wave and vice versa. Figure 2.16 illustrates the reflection of a triangular stress wave at a free end. The left side of the illustration shows the behaviour of the wave, while the right side shows the total stress. Please refer to the compendium [23] for the complete derivations of the listed equations.

Figure 2.16: Compression stress wave reflection at a free end [24].

(52)

CHAPTER 2. THEORY

(53)

Chapter 3 Materials

This chapter presents a qualitative description of concrete together with selected material models. A short description of possible failure mechanisms for concrete plates subjected to dynamical loads are also presented.

3.1 Qualitative description of concrete

Concrete behaves differently from other common construction materials like steel or aluminium. Steel and aluminium are often assumed to be homogeneous isotropic and have the same mechanical properties in compression and tension. For concrete, the situation is more complex. The complexity derives mostly from the concrete’s non-homogeneous material behaviour. Concrete is a composite material created by two main building blocks, aggregate and mortar. Both the interaction between them and the material properties to each of them influence the material behaviour. It is possible to create a concrete with a specific strength and properties suitable for specific situations. Adding harder aggregate will provide a higher concrete strength and a distribution of aggregate can create strength gradients over the material [25]. The tensile strength is dependent on the bond between the aggregate and the mortar [26]. The aggregate and the mortar also decide the level of micro-cracks present in the material before any load is applied, and the interaction between them causes new crack to appear and existing cracks to propagate when a load is applied. To fully understanding all aspects of concrete’s behaviour, knowledge on a micro-scale is necessary. This is however out of the scope of this thesis and will therefore not be presented. In the following sections, the general characteristics of concrete behaviour will be discussed.

(54)

CHAPTER 3. MATERIALS

3.1.1 Compression response

The behaviour of concrete in compression can be illustrated with a uniaxial compression test as shown in Figure 3.1, whereσ is the stress,is the strain andfcis the uniaxial compressive strength. When concrete is loaded in compression, its strain-stress relationship is assumed linear until the micro-cracks between the aggregate and the mortar begin to increase in size.

This happens at a load of approximately 30% of the ultimate strength [27]. From this point, the stress-strain relationship behaves non-linearly and the stiffness will decrease. When the concrete approaches its ultimate strength, the micro-cracks start to connect with each other forms continuous cracks throughout the concrete. The stress-strain relationship after the post-peak is generally assumed to be a softening phase. This is however disputed in the literature. Some clearly state that the softening phase exists and is influenced mainly by the principal tension strain [28]. Others state that the reason for the softening phase is not due to material behaviour, but the result of the interaction between the loading plates and the test specimen [29]. It is believed that the friction between the loading plates and the specimen limit the specimen from expanding near the loading plate [30]. This creates a complex tri-axial stress-state from which the softening phase develops. When the friction is reduced, the softening phase is reduced [31].

Figure 3.1: Compression response of concrete [27].

(55)

3.1.2 Tensile response

A major characteristic of concrete is that the compression and tension strength are not equal.

Due to its non-homogeneous behaviour, the tensile strength of concrete is approximately 5- 10% of the compressive strength [27]. The tensile response of concrete can be observed by conducting both a load controlled test and a displacement controlled test. In the load controlled test, the concrete behaves almost linearly up to about approximately 70% of its ultimate strength (see Figure 3.2a). By unloading from this state, no irreversible deformations other than some creep-related displacement can be observed [26]. Above this elastic limit, the concrete behaves non-linearly until it reaches its ultimate strength and fails. However, concrete has a softening phase in the tensile response. To observe this effect, a displacement controlled test must be performed (see Figure 3.2b). Such a test is difficult to perform due to the brittle behaviour of concrete. There are three different phases of the tensile response where the two first are the same as in the load controlled test, and the last is a softening phase. This softening is a result of a continuous growth of micro-cracks which will finally form a discrete crack. The softening phase can influence structural components, but for ordinary design of concrete structures, this softening phase has no or little importance. This is because concrete’s tensile strength is assumed to be zero and the reinforcement is assumed to carry all tensile forces.

(a) Load controlled (b) Displacement controlled Figure 3.2: Tensile response of concrete [26].

(56)

3.1.3 Pressure dependency

Concrete is a pressure dependent material and both concrete strength and stiffness are known to be sensitive to multiaxial stress conditions [32]. For example, in biaxial stress conditions, the compressive strength increases with an average of 30% with regards to the uniaxial compression strength f_c. Research shows that the cylinders subjected to lateral confinement in combination with axial load can reach a compression strength of 3.5 times the uniaxial strength (see Figure 3.3) [33]. The strength and stiffness increase is due to microscopic cracks being inhibited from propagating because of lateral confining stresses [27]. This also affects the ductility of the concrete, as shown in Figure 3.3, where concrete is able to withstand larger strains with an increased lateral confining pressure. For lateral tension stresses, the strength of the concrete will decrease.

Figure 3.3: Pressure dependency [27]. Figure 3.4: Volumetric response [27].

3.1.4 Volumetric expansion

Concrete is a porous material and it will compact under compression, until it reaches a value of approximately 80% of the ultimate strength. At this point, the minimal volume is reached and the volume will start to increase because micro-cracks will propagate through the

(57)

CHAPTER 3. MATERIALS material and form crack patterns (see Figure 3.4). Further loading will eventually cause the concrete to expand past its initial volume. As areas with high compressive stresses expand, tensile stresses can induce to neighbouring areas, which will result in crack formation [27].

3.1.5 Rate dependency

When dealing with fast transient dynamic loads, it is important to have knowledge of the material’s rate dependency. Studies have shown that concrete has a non-linear rate dependency, which can be separated into two regions. The location of the transition region is somewhat disputed due to a difference in experimental methods [34], but a general view is that the transition happens between a strain rate of 1s⁻¹ to 20s⁻¹ [32]. In the first region, the strength of the concrete increases almost linearly with the strain rate up to a dynamic increase factor of approximately 1.3. After the transition zone, the concretes strength increases almost exponential with increasing strain rate.

Figure 3.5: Increase of concrete strength under high strain rates [32].

(58)

As shown in Figure 3.5, the dynamic increase factor is greater in tension than in compression.

Experiments show that the maximum dynamic strength can be 3.5 times the quasi-static strength in compression and the dynamic strength can be up to 13 times the quasi-static strength in tension [34].

3.2 Failure mechanisms of a concrete plate

In this section, a short introduction to failure modes of concrete plates subjected to dynamic loads will be given. Possible failure modes are listed by Krauthammer [35] as bending, shear and punching shear (see Figure 3.6). A special type of shear behaviour is also listed, namely the direct shear response, which is caused by high shear inertia forces. If this type of failure occurs, it will be early in the response and before the occurrence of any bending deformation [36]. For the given failure modes, in-plane forces are not considered.

(a) Bending (b) Shear (c) Punching shear

Figure 3.6: Possible failure mechanisms for plates.

The probability of which failure mode will occurs, depends on both the rate of the load and the rigidity of the plate. There is also a possibility of failure modes occurring simultaneously.

For quasi-static loads, bending failure is most common, and if the same load is applied dynamically, the failure mode will most likely be shear. Shear failure is also favoured by an increase of rigidity. If a harder shock impacts the plate, the probability of a more local failure increases [35]. This is also the case if the distance between the plate and a possible charge is decreased [36]. It should also be noted that phenomena such as spalling and scabbing generally takes place during bending failure [35]. A ductile failure is preferred as

Submerged floating tunnels subjected to internal blast loading

Submerged floating tunnels subjected to internal blast loading

Sondre Rydtun Haug Karoline Osnes

MASTER THESIS 2015

MASTEROPPGAVE 2015

MASTER’S THESIS 2015

Submerged floating tunnels subjected to internal blast loading

Acknowledgements

Abstract

Contents

Nomenclature

Chapter 1 Introduction

Chapter 2 Theory

2.1 The blast phenomena

2.1.1 General definition of an explosion

2.1.2 Classification

2.1.3 Process of an explosion

2.1.4 Blast wave interaction

2.1.5 The ideal blast wave

2.1.6 Internal blast loading

2.1.7 Design of blast loads

2.2 Continuum mechanics

2.2.1 Eulerian and Lagrangian description

2.2.2 Arbitrary Lagrangian-Eulerian Formulation (ALE)

2.2.3 Conservation laws

2.2.4 Constitutive equations

2.3 Fluid-structure interaction (FSI) in Europlexus

2.4 Numerical investigation of blast load on structures

2.4.1 Pure Lagrangian

2.4.2 Uncoupled Eulerian and Lagrangian

2.4.3 Coupled Eulerian and Lagrangian

2.5 One dimension stress wave

Chapter 3 Materials

3.1 Qualitative description of concrete

3.1.1 Compression response

3.1.2 Tensile response

3.1.3 Pressure dependency

3.1.4 Volumetric expansion

3.1.5 Rate dependency

3.2 Failure mechanisms of a concrete plate