iii

Shear in Concrete Structural Elements

Subjected to Dynamic Loads

Johan Magnusson

Doctoral Thesis

KTH Royal Institute of Technology

Department of Civil and Architectural Engineering

Division of Concrete Structures

Stockholm, Sweden, 2019

iii

TRITA-ABE-DLT-1916 KTH School of ABE

ISBN: 978-91-7873-229-6 SE-100 44 Stockholm

SWEDEN

Akademisk avhandling som med tillstånd av Kungliga Tekniska Högskolan framlägges till

offentlig granskning för avläggande av teknologie doktorsexamen i Byggvetenskap, med

inriktning mot Betongbyggnad onsdagen den 5 juni 2019 klockan 10:00 i Kollegiesalen,

Kungliga Tekniska Högskolan, Brinellvägen 8, Stockholm.

iii

Abstract

Concrete structural elements subjected to severe dynamic loads such as explosions at

close range may cause shear failures. In the Oklahoma City bombing in 1995 two

concrete columns on the ground level were reported to have failed in shear. Such shear

failures have also been reported to occur in several experimental investigations when

concrete beams and slabs were subjected to blast or impact loads. The dynamic shear

mechanisms are not yet fully understood and it is therefore of research significance to

further investigate these mechanisms. The main objective of the research presented in

this thesis is to experimentally and theoretically analyse shear failures of reinforced

concrete elements subjected to uniformly distributed dynamic loads.

The experimental work consisted of concrete beams of varying concrete grades and

reinforcement configurations subjected to blast loads. One series involved testing of

steel fibre reinforced concrete (SFRC) beams and the other series involved tests with

concrete beams reinforced with steel bars. The former investigation showed that SFRC

beams can resist certain blast loads. In the latter investigation, certain beams subjected

to blast loads were observed to fail in flexural shear while the same beams exhibited

flexural failures in the static tests. Such shear failures specifically occurred in beams

with relatively high reinforcement contents. With these experiments as reference,

numerical simulations with Ansys Autodyn were performed that demonstrated the

ability to predict flexural shear failures.

A direct shear failure mode has also been observed in experiments involving concrete

roofs subjected to intense distributed blast loads. In several cases, the roof slabs were

completely severed from their supporting walls along vertical or near-vertical failure

planes soon after the load had been applied. Theoretical analyses of the initial structural

response of beams subjected to distributed loads were conducted with the use of Euler-

Bernoulli beam theory and numerical simulations in Abaqus/Explicit. These analyses

show that the initial structural response consists of shear stresses and bending moments

developing at the supports. The remaining parts of the beam will be subjected to a rigid

body motion. Further simulations with Abaqus shows that that dynamic direct shear

failure appears to be due to a deep beam response with crushing of the compressive

struts at the supports, and therefore differs from a static direct shear mode. The results

also showed that parameters such as element depth, amount of reinforcement, load level

and load duration played a role in developing a dynamic direct shear failure.

Keywords: Dynamic load, initial response, shear failure, shear capacity, numerical

simulations, bond, shear span, support reactions

iv

v

Sammanfattning

Byggnadselement i betong utsatta för stora dynamiska laster som explosioner på nära

håll kan förorsaka skjuvbrott. I bombådet i Oklahoma City 1995 rapporterades att två

betongpelare i marknivå gick till skjuvbrott. Sådana skjuvbrott har observerats i flera

experimentella undersökningar med betongbalkar eller plattor som utsattes för

explosionslaster eller anslag från fallande föremål. Mekanismerna bakom dynamisk

skjuvning är ännu inte helt klarlagda och det är därför av intresse att utforska dessa

mekanismer. Huvudsyftet med forskningen i föreliggande avhandling är att

experimentellt och teoretiskt analysera skjuvbrott i armerade betongelement utsatta för

jämnt utbredd dynamisk last.

Den experimentella delen av forskningen bestod av betongbalkar med varierande

betonghållfasthet and armeringsutformning utsatta för explosionslaster. En

försöksserie omfattades av stålfiberarmerade balkar och den andra av betongbalkar

med armeringsstänger. Den förra undersökningen visade att de fiberarmerade balkarna

kan bära en viss explosionslast. I den senare undersökningen observerades att de balkar

som utsattes för explosionslast och gick till böjskjuvbrott medans samma balkar gick

till böjbrott i de statiska försöken. Skjuvbrotten uppstod i balkar med relativt höga

armeringsinnehåll. Dessa balkar användes senare som referensbalkar för numeriska

simuleringar med Ansys Autodyn där simuleringarna visade på möjligheten att

förutsäga böjskjuvbrott.

Även direkt skjuvning har observerats i experiment med betongtak utsatta för höga

explosionslaster. I flera fall separerades taken från de stöttande väggarna längs

vertikala eller nära vertikala brottytor kor tid efter pålastningen. Teoretiska analyser av

den tidiga strukturresponsen för balkar utsatta för utbredda laster genomfördes med

Euler-Bernoulli balkteori och numeriska simuleringar med Abaqus/Explicit. Dessa

analyser visar att den initiala strukturresponsen består av skjuvspänningar och böjande

moment som uppstår vid stöden. Områdena på balken från nära stöd mot balkmitt rör

sig i form av en stelkropp. Vid ytterligare simuleringar med Abaqus förefaller ett

dynamiskt direkt skjuvbrott vara resultatet av en respons likt en hög balk med krossning

av de tryckta strävorna vid stöden, och därmed skiljer sig från statisk direkt skjuvning.

Resultaten visar även att balkhöjd, armeringsinnehåll, lastnivå och lastens varaktighet

är parametrar som påverkade utvecklingen av ett dynamiskt direkt skjuvbrott.

Nyckelord: Dynamisk last, initial respons, skjuvbrott, skjuvkapacitet, numeriska

simuleringar, förankring, skjuvspännvidd, upplagsreaktioner

vi

vii

Preface

This thesis presents experimental and theoretical research on concrete beams subjected

to dynamic loads. The experimental research was carried out at the Swedish Defence

Research Agency (FOI) with financial support from the Swedish Armed Forces

Headquarters. The theoretical research was carried out at FOI, KTH Royal Institute of

Technology (Department of Civil and Architectural Engineering, Division of Concrete

Structures), Grontmij and Swedish Fortifications Agency. This research was

financially supported partly by Fortifikationskåren, the Armed Forces, KTH, Grontmij,

Swedish Fortifications Agency and RISE Research Institutes of Sweden. Their support

is greatfully acknowledged.

I wish to express my sincere gratitude to my supervisor Prof Anders Ansell for his

guidance and valuable support. I also wish to express my gratitude to my co-supervisor

Adj. Prof. Mikael Hallgren for invaluable discussions and cooperation. Many thanks

also go to my second co-supervisor Dr. Richard Malm for his advice and support. I

would also like to thank the co-author to one of my research papers Tech. Lic. Håkan

Hansson. I also wish to acknowledge Adj. Prof. Costin Pacoste for his valuable

comments on the thesis. A special thanks to Mr. Göran Svedbjörk at Sweco Structures

for all discussions and suggestions during the theoretical part of the research. Finally,

I wish to thank my fellow doctoral students and the personnel at the Department of

Civil and Architectural Engineering for a friendly and fruitful atmosphere.

Eskilstuna, May 2019

Johan Magnusson

viii

ix

List of Papers

The following journal papers are included in this thesis:

Paper I: Magnusson, J. (2006). Fibre reinforced concrete beams subjected to air

blast loading. Nordic Concrete Research, 35, pp. 18-34.

Paper II: Magnusson, J., Hallgren, M. & Ansell, A. (2010). Air-blast-loaded, high-

strength concrete beams. Part I: Experimental investigation. Magazine of

Concrete Research, 62(2), pp. 127-136.

Paper III: Magnusson, J., Ansell, A. & Hansson, H. (2010). Air-blast-loaded, high-

strength concrete beams. Part II: Numerical non-linear analysis. Magazine

of Concrete Research, 62(4), pp. 235-242.

Paper IV: Magnusson, J., Hallgren, M. & Ansell, A. (2014). Shear in concrete

structures subjected to dynamic loads. Structural Concrete, 15(1), pp. 55-

65.

Paper V: Magnusson, J., Hallgren, M., Malm, R. & Ansell, A. (2019). Numerical

analyses of shear in concrete structures subjected to distributed loads,

submitted to Engineering Structures.

x

xi

List of contents

1 Introduction ............................................................................................................. 1

1.1 Background ......................................................................................................... 1

1.2 Aim ..................................................................................................................... 2

1.3 Scope ................................................................................................................... 3

1.4 Outline of the thesis ............................................................................................ 3

2 Shear of concrete and dynamic loads .................................................................... 7

2.1 Blast loads .......................................................................................................... 7

2.1.1 Incident blast waves ................................................................................. 7

2.1.2 Reflected blast waves ............................................................................... 8

2.1.3 Impulse loads ......................................................................................... 10

2.2 Static shear ....................................................................................................... 10

2.2.1 Flexural shear failure ............................................................................. 13

2.2.2 Shear compression failure ...................................................................... 13

2.2.3 Failure by splitting or crushing of the concrete strut ............................. 14

2.2.4 Direct shear failure ................................................................................. 14

2.2.5 Further notes on static shear .................................................................. 17

2.3 Dynamic shear ................................................................................................. 18

2.3.1 Flexural shear failure ............................................................................. 18

2.3.2 Direct shear failure ................................................................................. 20

2.3.3 Vibration modes ..................................................................................... 21

2.4 Models for calculations of the shear capacity .................................................. 22

2.4.1 Flexural shear capacity .......................................................................... 22

2.4.2 Direct shear capacity ............................................................................. 27

2.4.3 Support reactuions ................................................................................. 27

3 Material models ..................................................................................................... 29

3.1 Material dynamic properties ............................................................................ 29

3.1.1 Concrete tension and compression ........................................................ 29

3.1.2 Concrete elastic modulus ....................................................................... 33

3.1.3 Concrete fracture energy ........................................................................ 34

3.1.4 Steel reinforcement ................................................................................ 36

3.2 Material models for concrete ........................................................................... 37

xii

3.2.1 RHT model ............................................................................................ 37

3.2.2 Concrete damaged plasticity .................................................................. 39

3.2.3 Crack softening for concrete in tension ................................................. 42

3.3 Material models for reinforcement .................................................................. 43

3.3.1 Johnson & Cook ..................................................................................... 43

3.3.2 Isotropic elasto-plastic model ................................................................ 44

3.4 Bond between reinforcing bars and concrete ................................................... 44

4 Dynamic shear of concrete beams ....................................................................... 47

4.1 Experiments ..................................................................................................... 47

4.1.1 Concrete beams ...................................................................................... 47

4.1.2 Concrete roof slabs ................................................................................ 48

4.2 Numerical models ............................................................................................ 49

4.2.1 Flexural shear failures ............................................................................ 49

4.2.2 Direct shear failures .............................................................................. 52

4.2.3 Parametric studies .................................................................................. 54

4.2.4 Material parameters ............................................................................... 57

5 Results from numerical simulations .................................................................... 59

5.1 Initial response and shear ................................................................................. 59

5.2 Flexural shear failures ...................................................................................... 64

5.2.1 Bond between reinforcing bars and concrete ......................................... 64

5.2.2 Flexural shear crack patterns ................................................................ 66

5.2.3 Flexural shear deflections ...................................................................... 68

5.2.4 Flexural shear support reactions ............................................................ 70

5.3 Direct shear failures ......................................................................................... 74

5.4 Parametric studies – failure modes .................................................................. 79

5.5 Parametric studies – support reactions ............................................................. 82

5.5.1 Flexural shear ......................................................................................... 82

5.5.2 Direct shear ........................................................................................... 86

5.6 Parametric studies – shear capacity ................................................................. 91

5.6.1 Evaluation of the shear span .................................................................. 91

5.6.2 Evaluation of reinforcement strains ....................................................... 97

5.6.3 Evaluation of flexural shear capacity ................................................... 101

5.6.4 Evaluation of direct shear capacity ...................................................... 105

xiii

6 Summary of appended papers ........................................................................... 109

6.1 Paper I ............................................................................................................ 109

6.2 Paper II ........................................................................................................... 111

6.3 Paper III ......................................................................................................... 112

6.4 Paper IV ......................................................................................................... 113

6.5 Paper V ........................................................................................................... 114

7 Discussion ............................................................................................................ 115

7.1 General ........................................................................................................... 115

7.2 Failure in dynamic shear ................................................................................ 118

7.3 Support reactions ........................................................................................... 120

7.1.1 Flexural shear ....................................................................................... 120

7.1.2 Direct shear .......................................................................................... 121

7.4 Shear capacity ................................................................................................ 121

7.4.1 Shear span ............................................................................................ 121

7.4.2 Plastic strains in the reinforcement ...................................................... 122

7.4.3 Flexural shear capacity ........................................................................ 122

7.4.4 Direct shear capacity ............................................................................ 123

8 Conclusions and further research ..................................................................... 125

8.1 Conclusions .................................................................................................... 125

8.2 Further research ............................................................................................. 127

Bibliography ............................................................................................................. 129

Appendix A: Parametric study of the strain rate ................................................. 137

Appendix B: Material data .................................................................................... 143

Appendix C: Derivation of the shear span ............................................................ 155

xiv

1

Chapter 1

Introduction

1.1 Background

Dynamic loads such as explosions in air may cause severe damage to surrounding

concrete structures and may lead to varying degrees of damage. Explosions may

originate from different sources such as detonating explosive charges or the rapid

combustion of a fuel-air mixture. The explosion generates blast waves that propagate

through the air in all directions at supersonic velocity. Besides the blast wave, the

effects of explosions may also generate fragments and ground shock resulting from the

energy transmitted to the ground. However, the effects of blast overpressures are

usually the governing load when considering the dynamic response and damage to

structures above ground. There may be cases where the effects of fragments or ground

shock loads are the main cause of structural damage but these loads are not addressed

in this thesis.

As a blast wave strikes an object such as a building wall, the pressure will be reinforced

due to reflections. In a case where the reflected pressures are sufficiently high, local

failures of structural elements such as load-bearing walls or columns may occur. Local

failures of such essential structural elements may be the cause of partial or total collapse

of the entire structure. In the design of structural elements to resist the effects of severe

dynamic loads, a flexural response mode is preferable due to the large energy

absorption capability. The structural elements should therefore be designed for a certain

degree of plastic deformations with concrete cracking and yielding of the

reinforcement. However, blast at close range may subject the structure to loads of

sufficient intensity to cause shear failures. In the Oklahoma City bombing in 1995

(FEMA 1996) two concrete columns on the ground level were reported to have failed

in shear. The third column, even closer to the blast than the other two columns, was

completely shattered by the intensity of the blast. Such shear failures have also been

reported to occur in several investigations involving concrete elements subjected to

blast and impact loads, such as Hughes & Beeby (1982), Niklasson (1994), Kishi et al.

(2002), Morales-Alonso et al. (2011), Slawson (1984), Adhikary et al. (2013) and the

work reported in Paper II. In several cases, these tests also confirm that elements that

2

fail in flexure under a static load may fail in shear under intense dynamic loads. In a

few particular test cases, concrete slabs were observed to fail in a direct shear mode at

the supports with limited deformations in the slab. Shear failures in concrete structures

are brittle in nature and typically occurs prior to the flexural capacity has been reached.

Therefore, shear is regarded as a premature failure that should be avoided.

It is well known that plain concrete is relatively brittle and this is especially the case

for concrete of higher strength (Betonghandbok 2000). However, the introduction of

steel fibres into the concrete matrix results in an enhanced ductility. Steel fibre

reinforced concrete (SFRC) elements are known to have larger energy absorbing

capabilities compared to plain concrete elements. Thus, adding steel fibres in

combination with the use of a proper amount of reinforcement bars results in a ductile

structural element. Tests on SFRC slabs subjected to blast loads and reported by

Luccioni et al. (2017) showed the ability of such slabs to resist intense dynamic loads.

1.2 Aim

Structural concrete elements subjected to intense dynamic loads have been reported to

fail in shear even if they are designed to fail in a flexural mode under static loads. The

characteristics of a dynamic load can be regarded as significantly different from the

characteristics of a static load and it is of interest to quantify this difference. A concrete

element subjected to dynamic loads accelerates and obtains kinetic energy, which is

never the case in a corresponding static loading case. The structural properties of the

element changes to a certain degree that may influence the behaviour during a dynamic

loading case. A static shear failure is known to follow a certain sequence of events and

it is of interest to analyse the evolution of shear in a dynamic event in order to increase

the understanding of such failures. The force transmitted from the applied static load

to the supports needs to be determined in conventional shear design of concrete

elements. Quantification of this reaction force is naturally also of importance in

dynamic design in order to obtain a concrete element that resists shear.

The shear mechanisms in concrete structures during dynamic loading are not yet fully

understood, especially is this the case for the direct shear mode of failure. It is therefore

of research significance to further analyse the dynamic behaviour of concrete elements

and the nature of dynamic shear. The main objective of the research presented in this

thesis is to experimentally and theoretically analyse the shear failure of reinforced

concrete elements subjected to uniformly distributed dynamic loads. The dynamic load

refers to a shock wave in air with an almost instant increase to its peak value. The brief

discussion in this chapter leads to the following research questions:

3

What characteristics of the dynamic load control failure in shear?

What structural characteristics control failure in shear?

What is the evolution of a shear failure in dynamic events compared to a static

case?

Up to what accuracy can the shear forces at the supports be predicted?

Up to what accuracy can the dynamic shear capacity of concrete elements be

determined?

1.3 Scope

The theoretical work in this thesis is conducted with the use of finite element analyses

in order to investigate details of the different aspects of dynamic shear and a number

of tests were also conducted. The work focused on shear failure modes due to uniformly

distributed dynamic loads with an almost instant increase to peak pressure and with a

linear decay to zero for a certain duration. Any effects on a concrete element due to the

negative phase of a real blast load were neglected and local or arbitrary load

distributions were considered outside the scope of this work. Structures subjected to

blast loads may also be subjected to a variety of fragments from the explosion. The

action of such fragments impacting the concrete element was also left outside the scope

of this work. The concrete elements analysed, with spans and depths that approximately

correspond to 1:2 scale, consisted of conventional concrete and reinforcement.

Elements with a geometry that can be considered as full scale were excluded in the

analyses. Furthermore, the analyses were limited to beams on two supports without the

action of axial loads. The beams may also represent one-way slabs and slab strips that

respond in a similar fashion as that of a beam.

1.4 Outline of the thesis

This thesis presents experimental and theoretical analyses and consists of a

introductory part accompanied with five journal papers. The introductory part discusses

and puts the research presented in the appended papers into a broader context. It also

presents additional background and theoretical analyses.

4

The introductory part includes four main chapters. Chapter 2 presents a review of the

characteristics of blast loads and shear in concrete structural elements. Next, Chapter 3

presents the numerical models for concrete, reinforcement and bond between

reinforcing bars and concrete. In Chapter 4, experiments and numerical models are

presented, followed by Chapter 5 where the results of the numerical analyses are

included. These main chapters are followed by Chapter 6, which presents a summary

of the appended journal papers:

Paper I:

Involves testing of 40 steel fibre reinforced concrete (SFRC) beams subjected

to static and air blast loads. A total of 22 beams were subjected to air blast

loading and the remaining 18 beams were subjected to static loads. The concrete

compressive strength varied between 36 MPa and 189 MPa with a fibre content

of 1.0 percent by volume. Two different fibre lengths having a length-to-

diameter ratio of 80 were used. The author planned and took part in conducting

the experiments, evaluated the results and wrote the paper.

Paper II:

Discusses the structural behaviour of reinforced concrete beams subjected to

blast loads with static tests as reference. The beams were cast in several

compressive strengths and the amount of tensile reinforcement was also varied

accordingly. The investigation showed that beams that failed in flexure in the

static tests could fail in flexural shear in the dynamic tests. The author of this

thesis planned and took part in conducting the experiments, evaluated the

results and wrote this paper. Adj. Prof. Mikael Hallgren supported in the

evaluation and reviewed the paper. Prof. Anders Ansell also reviewed the paper.

Paper III:

This paper presents numerical analyses of a selection of the dynamically tested

beams in Paper II using the software Ansys Autodyn. The analyses

demonstrated the ability to accurately predict the correct failure mode of

reinforced concrete elements subjected to dynamic loads. The author of this

thesis performed the finite element analyses, evaluated the results and wrote

this paper. Tech. Lic. Håkan Hanson supported in the analyses and reviewed

the paper. Prof. Anders Ansell also reviewed the paper.

Paper IV:

Comprises is a literature review of the dynamic shear of reinforced concrete

elements, with a focus on the parameters that control flexural shear and direct

shear. The dynamic loads are referred to as explosions and impacts. The review

show that several structural and material strength parameters need to be

considered when analysing shear. The author of this thesis performed the

5

literature review and wrote this paper. Adj. Prof. Mikael Hallgren and Prof.

Anders Ansell reviewed the paper.

Paper V:

Presents analyses of reinforced concrete beams subjected to extreme dynamic

loads with the use of numerical simulations. The cross-section of the analysed

beams varied using three different depths and two different amounts of

reinforcement for each cross section. The simulations showed that dynamic

direct shear failure appears to be due to a deep beam response and that both

structural and load parameters play a role in developing such a failure mode.

The author of this thesis performed the finite element analyses, evaluated the

results and wrote this paper. Adj. Prof. Mikael Hallgren, Dr. Richard Malm and

Prof. Anders Ansell reviewed the paper.

In Chapters 7, the main findings of the appended papers and the results presented in

Chapter 5 are discussed. Finally, Chapter 8 presents the conclusions of this thesis

accompanied with suggestions for further research.

6

7

Chapter 2

Shear of concrete and dynamic loads

An explosion is a result of a sudden release of energy. There are a number of possible

situations that may cause an explosion, such as that of a sudden relief of compressed

air in a pressure vessel or that from detonation of high explosives. The contents of this

thesis deals only with the severe dynamic loads that arise from the detonation of high

explosives, which generates a shock wave that propagates through the air at supersonic

velocity in all directions.

2.1 Blast loads

2.1.1 Incident blast waves

A shock wave is defined as a discontinuity in pressure, temperature and density

(Meyers 1994). Figure 2.1 shows an idealistic representation of a blast wave profile at

a given distance from the centre of explosion. The blast wave is illustrated with the

time axis at ambient pressure. The arrival of the blast wave creates an almost instant

increase from the ambient pressure to the peak overpressure. The arrival of the shock

front is immediately followed by pressure decay down to the ambient pressure. The

pressure will continue to decrease below the ambient pressure until the minimum

negative pressure is reached, after which the ambient pressure is obtained once more.

The first part of the blast wave, representing overpressure, is termed the positive phase,

and the remaining part is termed the negative phase. The negative phase usually

exhibits a longer duration than the positive phase. In design, only the positive phase is

considered and the negative phase neglected. However, in some cases it is important to

also consider the effects of the negative phase, e.g. when considering damage of

windows or in a case with heavy structures having a response time longer than the

duration of the positive phase. The work in this thesis only involves the load of the

positive phase.

8

Besides the peak overpressure it is also of importance to consider the impulse density

of a blast load. The impulse density of the positive and negative phase, respectively, is

defined by (Baker 1973):

(2.1)

(2.2)

where p denotes the overpressure. The time endpoints of the intervals are chosen as

referred to in Figure 2.1. Due to the expansion of the blast wave, the blast pressure

reduces over an increasing distance while the duration is increased. Thus, blast at close

range produces high pressures with a relatively short duration.

Figure 2.1 Ideal blast wave profile at a given distance from the centre of explosion.

The blast wave is illustrated with the time axis at ambient pressure. Based

on Baker (1973).

2.1.2 Reflected blast waves

As the incident blast wave strikes a solid surface the wave is reflected, which brings

the particle velocity to zero while the pressure, density and temperature are reinforced.

The pressure increase during reflection is due to the conversion of the kinetic energy

of the air immediately behind the shock front into internal energy as the moving air

particles are decelerated at the surface. The actual pressure that develops is determined

by various factors such as the peak overpressure of the incident blast wave and the

p

t

t + t

-

ps

ps-

Negative

phase

Positive

phase

ta

tt

t

s

a

a

tpi )( dt

ttt

tt

s

a

a

tpi )( dt

9

angle between the direction of motion of the wave and the face of the structure, i.e. the

angle of incidence. The largest increase will, for stronger shock waves, arise for normal

reflection where the direction of motion of the wave is perpendicular to the surface at

the point of incidence. The reflected pressure will always be at least twice that of the

incident pressure, but in many cases the reflected pressure will be reinforced by a factor

several times larger in magnitude. The impulse density will in a similar manner be

reinforced by reflection. As a blast wave strikes a solid surface, the resulting load will

exhibit a distribution over the surface as schematically shown in Figure 2.2.

Figure 2.2 Blast wave propagation (a) at different points in time t1–t3 and (b) the

resulting blast load on the surface. Based on Baker (1973).

An explosive charge detonating inside a tunnel will initially give rise to a complicated

event due to the blast wave reflections against the tunnel walls, roof and floor.

However, at a certain distance from the centre of explosion the propagation of the two

shock fronts propagating in opposite longitudinal directions will be mainly one-

dimensional and planar. In a tunnel with a constant cross section this one-dimensional

propagation of the blast waves leads to higher pressures, higher impulse densities and

longer durations of the blast waves compared to the case with a spherical expansion of

the shock front. A shock tube, which can be considered as a small tunnel, was used in

the dynamic tests with concrete beams in order to generate blast loads with relatively

small amounts of explosive charges as presented in Papers I–II and further analysed

using numerical simulations in Paper III.

10

2.1.3 Impulse loads

The intensity of a load generally refers to its rise time to the peak load and the

magnitude of the load. It is well known that the flexural response of structural elements

under dynamic loads depends on the rise time and the peak pressure with respect to the

natural period of vibration and the resistance of the element (Granström 1958; Biggs

1964). Blast waves have a rise time of a fraction of 1 ms (Ross 1983), while loads from

objects impacting on a concrete element typically have rise times of approximately 1

ms (Niklasson 1994; Hughes & Speirs 1982). This applies to impact velocities in the

range of a few metres per second. Another way of describing the load variations in time

is to refer to its frequency content. Accordingly, a short rise time can be regarded as a

load containing a large number of frequencies as opposed to a load with a longer rise

time, which on the contrary contains a lower number of frequencies.

When considering intense dynamic loads such as blast loads, it is convenient to relate

its duration to the natural period of vibration T of the structural element in question.

Impulsive loads typically have high amplitudes and are of short duration in relation to

the natural period of vibration of the loaded element. It is suggested in Biggs (1964)

that a load can be regarded as impulsive if the duration is below 0.1 times the natural

period of the system. Due to this short duration, no significant deflections take place

during this period of time and negligible resistance in the element will be developed.

In such a case, the actual variations of the load can be approximated as of no importance

and the magnitude of the impulse becomes the main load parameter. Explosions at close

range to a wall and impacts can typically be regarded as such impulsive loads. On the

other hand, if the load duration is long with respect to the natural period, the load

variations of the positive phase need to be taken into account in the analysis.

2.2 Static shear

Concrete elements subjected to different types of loading and support conditions need

to resist shear forces. These forces usually act in combination with bending moments

and possibly also with axial forces. This chapter only considers shear in combination

with flexure of a concrete element subjected to static and dynamic loads. Shear in

concrete structures subjected to static loads is briefly reviewed in Paper IV and V. Static

shear is also reviewed here, however, extended to also include more variations of shear

failure modes. In this context, reinforced concrete elements without transverse

reinforcement are only considered. Shear cracks in concrete elements can generally be

related to the tensile strength of concrete. Due to the distribution of principal tensile

stresses in a beam subjected to an external load, shear cracks are inclined (diagonal

11

tension cracks) with respect to the longitudinal axis of the element. A concrete element

resists shear through two main mechanisms, namely through beam action and arch

action (Park & Paulay 1974).

Beam action refers to the mechanism with the assumption of a perfect bond between

the reinforcement and the concrete. The beam action includes the effects of aggregate

friction and interlock in the shear crack, tension and dowel action of the flexural

reinforcement, and the transverse (longitudinal) shear and moment of the concrete

cantilevers (as defined by Park & Paulay) between two cracks. The remaining shear

forces are transferred across the compression zone of the element. The arch action

transfers shear through inclined compression in the element from the load to the

supports. This mechanism requires a horizontal reaction at the supports, which for

simply supported beams, is provided by the flexural reinforcement. The anchorage of

the reinforcement at the support is therefore vital in order for arch action to develop. In

a real concrete element, perfect bond between the reinforcement and the concrete can

not develop due to slip of the reinforcing bars and cracking of the concrete. For these

reasons, the beam and arch mechanisms will provide a combined resistance against

shear in a real element. As the element develops shear cracks, a gradual transition from

beam action to arch action will occur. The element will eventually fail when the

combined beam and arch action no longer are capable of transferring the shear forces.

If, on the other hand, full arch action can be developed the element may be able to

sustain a further load increase.

Kani (1966, 1967) conducted a series of investigations on reinforced concrete beams

without transverse reinforcement. In these investigations where the beams were

subjected to concentrated loads, the beam depth and amount of reinforcement varied.

Based on this work, Kani created a ‘valley of diagonal failure’, see Figure 2.3. This

figure illustrates the relation between the relative beam strength Mu/Mfl, shear span do

depth ratio a/d (or shear slenderness) and geometrical reinforcement ratio . The shear

span a and effective depth d are defined in Figure 2.4. The bending moment at shear

failure is denoted Mu and Mfl is the ultimate bending moment. With reference to Figure

2.3, a transition point exists for each value of at an a/d of approximately 2.5–3.0 that

defines the lowest bending moment at shear failure. For different values the transition

points can be connected and becomes a line that defines the lowest values of Mu/Mfl.

According to the findings of Kani, a beam mechanism governs for a/d values above the

transition point, while the arch mechanism governs for values below the transition

point. Thus, a predominant arch mechanism is present at values of a/d < 2.5–3.0, and

these modes are controlled by failure of the area in the vicinity of the arch or by failure

of the arch itself. Figure 2.3 also illustrates that for low amounts of reinforcement ( <

0.5 %), the failure mode is governed by flexure without a risk of shear failures.

12

Figure 2.3 Relative moment capacity Mu/Mfl versus a/d and (Kani 1966).

While a/d refers to elements subjected to concentrated loads, the shear slenderness for

an element with a distributed load is referred to as the full span to depth ratio (L/d). If

the distributed load is replaced by two resultant forces at the quarter-points closest to

each support, L/d may be translated to a corresponding value of a/d. This way the

parameter L/(4d) can be used, as also mentioned by Ansell et al. (2012). Experimental

investigations on reinforced concrete beams by Leonhardt & Walther (1962) clearly

show that the shear slenderness plays an important role in different shear mechanisms.

Depending on the shear slenderness, the behaviour of reinforced concrete elements in

shear may be divided into four categories with distinct differences in the shear transfer

mechanism as discussed in Park & Paulay (1974), Kotsovos (2014), Leonhardt &

Walther (1962), Ghaffar et al. (2010), Mattock & Hawkins (1972), Ansell et al. (2012).

The literature on shear does not clearly state limits for when these shear mechanisms

are active. This may be due to the fact that the shear mechanism depends on parameters

such as amount of reinforcement, element depth and material properties of the concrete

and reinforcement. Based on the literature, an attempt is made herein to quantify the

four regions for different shear slenderness. Thus, shear failures may generally be

classified into:

flexural shear failure

shear compression failure (initiated by web shear)

failure by splitting or crushing of the compressive strut

direct shear failure.

13

These shear modes will be briefly discussed in the following sections with reference to

Figure 2.4–2.6. Table 2.1 summarises the estimated limits of shear failure modes for

varying shear slenderness.

2.2.1 Flexural shear failure

Flexural shear failures of concrete elements subjected to continuously increasing static

loads are known to originate from a flexural crack that develops into an inclined crack

as discussed in Park & Paulay (1974) and Ansell et al. (2012), see Figure 2.4 (a).

Diagonal tension failure by flexural shear occurs at locations where both shear and

flexural stresses exist. Flexural shear typically occurs for point-loaded beams with

approximately a/d of 3–7 (Park & Paulay 1974; Leonhardt & Walther 1962). This

failure mode can also occur for distributed loads as shown by Leonhardt & Walther

(1962). These tests indicate that flexural shear can occur for L/d of approximately

11–20. Larger shear slenderness will result in a flexural failure. Figure 2.5 (c) and 2.6

(b) show test results of beams failing in flexural shear.

2.2.2 Shear compression failure

Diagonal tension cracks may occur where relatively large shear forces and

comparatively small bending moments exist. Such a case is typically where a point load

is located relatively close to the support, i.e. at values of a/d < 3 for rectangular cross

sections. The diagonal tension crack is initiated in regions near the neutral axis of the

element, see Figure 2.4 (b), and does not progress from a flexural crack. Web shear

cracks can also appear in the webs of flanged beams or near inflection points for

continuous beams. As the shear crack propagates under an increasing load, the

compression zone of the element is reduced and eventually becomes too small to resist

the compression stresses in that region. This subsequently leads to crushing of the

concrete and failure of the element. Such a failure is denoted a shear compression

failure and typically occurs for a/d = 1.5–3 (Park & Paulay 1974; Leonhardt & Walther

1962; Ghaffar et al. 2010; Kotsovos 2014), see Figure 2.4 (c). For a distributed load

the corresponding limits are estimated to approximately L/d = 5–11. Tested beams that

failed in shear compression are shown in Figure 2.5 (b) and 2.6 (a). The formation of

the diagonal crack does not lead to immediate failure, and the applied load must be

increased further to cause failure. Figure 2.7 illustrates the shear force at failure for low

values of a/d, i.e. with a significant increase in shear capacity compared to a/d > 3.

14

2.2.3 Failure by splitting or crushing of the concrete strut

Failure by crushing or splitting of the concrete compressive strut may occur for a point

load close to the support, i.e. at a/d < 1.5 (Park & Paulay 1974; Leonhardt & Walther

1962; Ghaffar et al. 2010), see Figure 2.4 (d). In this case, the load is carried by the

inclined compression strut that develops between the load and the support. Since a

considerable portion of the load is carried by inclined compression struts at each

support, the final failure is due to splitting or crushing of the concrete in these struts.

One example of a tested beam failing in this mode is shown in Figure 2.5 (a). Such

failures have also been observed in other tests with deep beams for a/d ≤ 1, see for

instance Rogowsky & MacGregor (1983) and Gedik (2011). It is uncertain if a

distributed load can cause crushing of the compressive strut, but it is estimated that L/d

needs to be less than 5. Figure 2.7 illustrates the shear force at failure for low values of

a/d, i.e. with a significant increase in shear capacity compared to the case where

a/d > 3.

2.2.4 Direct shear failure

Direct shear failure is a sliding type of failure along a well-defined plane through the

depth of the element. This type of failure is only critical in a case where the

concentrated load is very close to the support, i.e. at a/d < 0.5 (Park & Paulay 1974).

The failure mechanism of direct shear is different in initially uncracked and cracked

concrete. In the latter case, the shear transfer mechanism will be due to aggregate

interlock and dowel action of the longitudinal reinforcement. Shear transfer for an

initially uncracked concrete cross section will cause several short diagonal tension

cracks to develop along the shear plane (Mattock & Hawkins 1972), as shown in

Figure 2.4 (e). Direct shear failure will occur when the small concrete struts fail under

the combined action of compression and shear, or when the local shear stresses at the

ends of the struts reach the ultimate capacity. The final stage is characterised by a

sliding type of failure along the shear plane through the depth of the element. It is not

probable that a distributed load can cause a direct shear failure due to the necessary

concentration of a load close to the support.

15

Figure 2.4 Schematic view of (a) diagonal tension by flexural shear, (b) diagonal

tension by web shear, (c) shear compression, (d) crushing/splitting of the

concrete strut and (e) direct shear.

(a)

(b)

(c)

Figure 2.5 Tested beams with two concentrated loads for a/d = (a) 1.0, (b) 1.5 and

(c) 4.0. Tests by Leonhardt & Walther (1962).

(a) (b) (c)

(d) (e)

a

d

F

F

a

a

F

F

a

F

a

16

(a)

(b)

Figure 2.6 Tested beams with a distributed load for L/d = (a) 5.2 and (b) 14.7. Tests

by Leonhardt & Walther (1962).

(a) (b)

Figure 2.7 The shear capacity at varying shear slenderness for (a) point loaded

beams and (b) beams subjected to uniformly distributed loads according

to the results reported by Leonhardt & Walther (1962). From Ansell et al.

(2012).

Table 2.1 Failure modes for concrete elements with estimated variations in shear

slenderness.

Failure mode Shear slenderness

Concentrated load a/d Distributed load L/d

Flexural shear 3–7 11–20

Shear compression 1.5–3 5–11

Splitting / crushing of

concrete strut < 1.5 < 5

Direct shear < 0.5 -

17

2.2.5 Further notes on static shear

Beams with an applied load at a distance less or equal to 2d are in the literature referred

to as deep beams, e.g., Gedik (2011) and Adhikary et al. (2013). In this case, a

considerable portion of the load can be carried by inclined compression struts at each

support. The shear failures of deep beams have been reported to involve crushing and

splitting of the concrete struts or diagonal tension along the struts, e.g. Rogowsky &

MacGregor (1983), Gedik (2011), Birgisson (2011), Londhe (2011) and Malm &

Holmgren (2008a). Furthermore, tests performed by Birgisson (2011) indicated that a

diagonal tension cracks were able to form at the supports and that caused a shear failure

without crushing of the concrete in the compression zone. The loads were positioned

relatively close to the supports with a shear slenderness a/d of 1.86. Two of the beams

failed as soon as the diagonal tension crack appeared, and for the third beam the load

could be further increased by approximately 17 %. Thus, this was a brittle shear failure

and deviates from the description of a shear compression failure in Section 2.2.2. The

fact that the shear cracks formed at the supports may be due to a web shear crack but

this was not confirmed in the investigation.

It is well known that the shear strength of a concrete beam is reduced with an increased

depth commonly referred to as a size effect. Based on experiments on concrete beams,

Kani (1967) reported that the shear stresses decrease as the beam depth increase.

Figure 2.8 presents the results of four tests on beams with different depth. The

parameter Mu denotes the bending moment at shear failure and Mfl is the ultimate

bending moment of the cross section. It is clear that the relative load capacity of the

beams are reduced at an increasing depth. Bažant & Yu (2005a, 2005b) discuss this

issue using fracture mechanics concepts. Size effects in deep concrete beams were also

reported by Gedik (2011). The same investigation also showed that the beam width

influenced the post-peak behaviour after the maximum load had been obtained. Gedik

reported that wider beams exhibited an increased ductility after the maximum load had

been attained. However, the peak load was not affected by beam width.

18

Figure 2.8 Normalized bending moment for different shear slenderness (a/d) and

beam depth. From Kani (1967).

2.3 Dynamic shear

A structure subjected to dynamic loads may exhibit a significantly different behaviour

compared to the same structure subjected to static loads, especially if the applied load

is impulsive in nature. The abrupt changes in the applied load give rise to accelerations

of the structural elements and, consequently, the effects of inertia and kinetic energy

must be considered in the dynamic analysis. Several investigations have shown that

structural concrete elements are more susceptible to failing in shear when subjected to

dynamic loads (Magnusson 2000; Niklasson 1994). Two types of shear failures have

been reported due to dynamic loads, namely flexural shear and direct shear. These two

modes of dynamic shear failures are discussed in the following sections. Soon after a

dynamic load has been applied, a concrete element will exhibit deflections and

distributions of bending moments and shear forces which significantly deviate from

that of the same element subjected to a static load. This behaviour affects the direct

shear failure. Therefore, in the subsequent section, the response of concrete elements

soon after the load has been applied is discussed.

2.3.1 Flexural shear failure

As thoroughly discussed in Paper IV, several investigations have shown that concrete

elements that failed in flexure under a static load could fail in shear under a dynamic

load, such as in Hughes & Beeby (1982), Niklasson (1994), Morales-Alonso et al.

19

(2011) and Kishi et al. (2002). This is also shown in Paper II. These investigations

involved concrete beams subjected to blast and impact loads. The literature review by

Ansell (2005) also highlights the change in failure mode in several investigations.

Flexural shear failures of concrete elements subjected to continuously increasing static

loads are known to originate from a flexural crack that develops into an inclined crack

(Ansell et al. 2012). In the investigations involving blast and impact loads previously

listed in this section, shear failures exhibited similar types of inclined cracks.

Numerical simulations of concrete beams subjected to blast loading as reported in

Paper III showed that the flexural shear crack developed from a flexural crack in the

same manner as in the static case, which is also reported in Paper IV. This indicates

that flexural shear has the same characteristics for both static and dynamic loading

conditions.

The tests by Niklasson (1994) indicate that the rise time of the load and the subsequent

applied load level control the initiation of shear cracks. Hard impact conditions

generate a relatively short rise time and an increased amplitude compared to the case

with softer impacts. Hard impacts thereby increase the beams’ susceptibility of shear

failures. Such intense loads have a high frequency content and are therefore able to

excite higher vibration modes in the element. This fact results in a larger portion of the

strain energy being due to shear rather than flexure in the element. It should also be

noted that a blast wave having an extremely short rise time exhibits a high-frequency

content that induces higher vibration modes in the element. Furthermore, the beam

stiffness plays an important role such that beams of a relatively high reinforcement

content and under dynamic loading exhibited shear failures, while beams with lower

reinforcement contents failed in bending. Strain rate effects of the concrete and

reinforcement also contribute to an even further increase in stiffness. The influence of

the beam stiffness is more thoroughly discussed in Paper IV.

It is well known that adding steel fibres to the concrete matrix enhances its ductility

and may also increase the flexural strength of a beam compared to a corresponding

beam without any reinforcement. This is thoroughly discussed in Paper I. Other

investigations, involving reinforced concrete beam subjected to blast loads, showed

that beams containing steel fibres failed in bending while similar beams without fibres

failed in shear. Thus, the presence of steel fibres prevented shear cracks from

developing, which increased the shear strength of the beams. This discussion is

included in Paper II

20

2.3.2 Direct shear failure

The direct shear mechanisms during dynamic events are thoroughly discussed in Papers

IV and V. As previously mentioned in Section 2.2.4, it is not probable that a distributed

static load can cause a direct shear failure. However, tests have shown that uncracked

concrete elements can fail in direct shear under the action of a distributed blast load

with high intensity (Slawson 1984; Ross 1983), see also Paper IV and V. In several

cases, the test results reported by Slawson show that the slabs were completely severed

from the walls along vertical and near-vertical failure planes at the supporting walls. It

was further reported that the central portion of the roof slab remained relatively flat as

though no flexural deformations had taken place, see Figure 2.9. In a theoretical

analysis based on Timoshenko beam theory (Ross 1983; Ross & Krawinkler 1985), it

was stated that the actual failure process of the dynamic direct shear failure is unknown.

However, Ross (1983) provides a discussion of a possible dynamic failure mechanism,

which involves the propagation of a near-vertical crack close to the supports due to

wave propagation effects. Further theoretical investigations of dynamic shear are

presented in Krauthammer et al. (1986), Krauthammer et al. (1993a), Krauthammer et

al. (1993b), Chee (2008) and Krauthammer & Astarlioglu (2017) that involve a

Timoshenko beam and a single-degree-of-freedom approach. A direct shear-slip

relationship with an elastic slope, post-cracking slope and a softening slope beyond the

peak value was used in these investigations. The shear-slip relationship refers to the

separation of the structural element along a vertical failure plane. Direct shear stress-

slip relationships for normal and high performance concrete with and without fibre

reinforcement are also proposed for both static and impact loads in French et al. (2017).

The analyses by Ross (1983) also concludes that shorter rise times initiates direct shear

failures at lower pressure levels compared to the case with longer rise times, see also

Section 2.3.1 and Paper IV.

Figure 2.9 Illustration of a post-test view of a slab based on test DS2-3 by Slawson

(1984).

21

2.3.3 Vibration modes

A structural element subjected to transient dynamic loading will experience modes of

vibration that are higher than the fundamental mode. Figure 2.10 shows the first free

modes of vibration for a simply supported (pin-ended) beam. Hughes & Speirs (1982)

made a comparison between the response of the static mode versus first and third free

vibration modes for pin-ended reinforced concrete beams subjected to mid-span impact

of a falling mass, see also Hughes & Beeby (1982). The deformed mode shapes for the

calculated equal potential energies are schematically illustrated in Figure 2.11. This

figure shows reductions in displacements and bending moments for the first and third

vibration modes compared to the corresponding properties for the static mode. The

shear forces, on the other hand, are greater for both vibration modes in relation to the

static mode and this is especially the case for the third mode response. Thus, larger

shear forces can therefore be associated with higher vibration modes excited in a beam

under dynamic loading compared to that of a static loading case. These results show

similarities to the discussion in the previous section and in Paper IV, where it was

shown that higher modes are the major contributors to shear. See also the discussion

on load rise times and soft and hard impacts in Section 2.3.1.

Figure 2.10 The first three free vibration modes of a simply supported beam subjected

to a uniform symmetric load.

22

Figure 2.11 Comparisons between different modes with equal potential energies.

Modified from Hughes & Speirs (1982).

2.4 Models for calculations of the shear capacity

There exists different models for calculation of the shear capacity of concrete structural

elements. In this section, models for calculations of the flexural and direct shear

capacities are described. A description of the model for calculations of the support

reactions is also included.

2.4.1 Flexural shear capacity

Swedish design manual for protective construction

In this section, a brief overview of the model in the Swedish design manual for

protective design FKR (Swedish Fortifications Agency 2011) for calculating the shear

capacity of reinforced concrete elements is given. The shear capacity is calculated

according to:

(2.3)

Static mode for mid-span

loading

Free vibration modes for pin-ended beams

1st mode 3rd mode

Displacement

Shear

1

1

1

1 1.28

3.84

0.82 0.82

0.99 0.11

𝑉𝑐 = 𝑘𝑐𝑏𝑑

23

where b and d denote the element width and the effective depth, respectively. The

strength parameter kc is determined by:

(2.4)

where s is a safety factor and k and k are calculated according to Eq. (2.5–2.6) and

Eq. (2.7–2.8), respectively.

(2.5)

The geometrical reinforcement ratio is determined by:

(2.6)

for (2.7)

for (2.8)

In these expressions, As denotes the area of flexural reinforcement and fck is the

characteristic compressive strength of concrete. The shear span is denoted a.

Determining the shear span plays an important role in the calculations of the shear

capacity in FKR. The method employed in these design rules uses the initial response

where the beam is regarded as temporarily responding with an apparently low shear

slenderness L’/d soon after the load has been applied. This structural behaviour in

dynamic events is further explained in Paper IV. Eq. (2.9) presents the expression for

calculations of the shear span for simply supported beams and Eq. (2.10) is presented

without the 0.025 term.

(2.9)

(2.10)

Parameters p, q and L refer to the applied peak pressure, the static load capacity of the

structural element and its span, respectively. In the derivation of Eq. (2.10), the

𝑘𝑐 = 𝑘𝜏𝑘𝜇

𝑠

𝑘𝜇 = 0.7 +𝜇 − 0.1

3

𝜇 =𝐴𝑠𝑏𝑑

𝑘𝜏 = 0.450.25𝑓𝑐𝑘𝑎𝜏𝑑

𝑘𝜏 = 0.25𝑓𝑐𝑘 𝑎𝜏𝑑

< 0.45

𝑎𝜏𝑑≥ 0.45

𝑎𝜏𝐿

= 0.025 + 0.25 ∙ 𝑞

𝑝

𝑎𝜏𝐿

= 0.25 ∙ 𝑞

𝑝

24

influence of inertial forces are neglected. When inertial forces instead are included the

expression changes into Eq. (2.11).

(2.11)

This expression Appendix C presents the background to Eq. (2.9–2.11). It should be

noted that Eq. (2.11) is not included in FKR.

FKR states that Eq. (2.3–2.8) are valid for the case that a/d ≤ 1.5, and that shear design

according to a conventional design code shall be employed for a/d > 1.5. However,

Eq. (2.7) states that k is valid for a/d ≥ 0.45 without an upper limit. In this thesis, the

interpretation was made that these equations are to be used to calculate the shear

capacity during the initial response of the element and before deflections in the

fundamental vibration mode take place. Thus, it is assumed that the element responds

in an apparently low shear slenderness with an enhanced shear strength. The Swedish

Betonghandbok (1990) provides a model for enhancing the shear strength for elements

subjected to a uniformly distributed load under static conditions. This model allows for

an increased shear strength at reduced values of L/d down to the lowest slenderness of

6.0, see Figure 2.12. The model in FKR is adjusted to fit the function in Betonghandbok

(1990) starting from an L/d of 6.0 in a continuous curve down to a value of 1.8. This

part of the curve is denoted ‘FKR’ in the figure and can be interpreted as an

extrapolation of the original curve in Betonghandbok. In the same figure, a curve

denoted ‘FKR+’ is included, which is based on Eq. (2.7) without using an upper limit

of a/d. Thus, it appears as though Eq. (2.7) is allowed between the limits 0.45 ≤ a/d ≤

1.5, and that the function in Betonghandbok is to be employed for a/d > 1.5, which

corresponds to an L/d of 6.0 for distributed loads. It should be noted, however, that the

function in Betonghandbok is based on the former Swedish design code that is no

longer valid for design of conventional concrete structures. Thus, Eq. (2.3–2.8) may be

valid for determining the direct shear capacity of a concrete section.

𝑎𝜏𝐿

= 0.43 ∙ 𝑞

𝑝

25

Figure 2.12 The increase in shear strength for a varying shear slenderness L/d.

Eurocode 2

The expression for calculations of the shear capacity without shear reinforcement or

normal forces is in Eurocode 2, EN 1992-1-1:2005 (Swedish Standards Institute 2005),

as follows:

(fck in MPa) (2.12)

where

(2.13)

and

(2.14)

and CRd,c = 0.18 (for c = 1.0)

The parameter fck is the characteristic concrete compressive strength, and bw and d is

the element width and effective depth, respectively. The area of reinforcement is

denoted As1. The maximum shear at the support shall not exceed VRd,max, which is a

0

1

2

3

4

5

6

7

0 5 10 15 20 25 30

Ince

rase

in s

he

ar c

apac

ity

L/d

FKR

FKR+

Betonghandbok

𝑉𝑅𝑑 ,𝑐 = 𝐶𝑅𝑑 ,𝑐𝑘 100𝜌1𝑓𝑐𝑘 1 3 𝑏𝑤𝑑

𝑘 = 1 + 200

𝑑≤ 2.0

𝜌1 =𝐴𝑠1

𝑏𝑤𝑑≤ 0.02

26

measure of the compression strength of the concrete compressive strut at the supports

and is calculates as:

(2.15)

where

(2.16)

Here, fcd is the design concrete compressive strength. The shear capacity is compared

to the reactions at a distance d from the face of the supports. Any reductions of the

reactions due to any applied forces placed close to the supports were not accounted for

in the calculations herein.

Draft revision of Eurocode 2

A simplified and closed form expression proposed for a revised Eurocode 2 (prEN

1992-1-1:2018, CEN 2018) for calculations of the shear capacity without shear

reinforcement or normal forces derived from the critical shear crack theory by Muttoni

& Ruiz (2008) is as follows:

(fck in MPa) (2.17)

where

mm (for fck ≤ 60 MPa) (2.18)

and

(2.19)

(2.20)

In the calculations herein, a Dlower was set to 16. In the calculation of acs, the design

values of the applied bending moment MEd and shear force VEd in the critical cross

section are used. The analyses in this thesis only consider uniformly distributed loads

and, therefore, the value of acs was approximated to d according to Eq. (2.20). The

parameter c was set to 1.0.

𝜈 = 0.6 1−𝑓𝑐𝑘250

𝜏𝑅𝑑𝑐 =0.6

𝛾𝑐 100𝜌1𝑓𝑐𝑘

𝑑𝑑𝑔

𝑎𝑣

1 3

𝑑𝑑𝑔 = 16 +𝐷𝑙𝑜𝑤𝑒𝑟 ≤ 40

𝑉𝐸𝑑 ≤ 𝑉𝑅𝑑 ,𝑚𝑎𝑥 = 0.5𝑏𝑤𝑑𝜈𝑓𝑐𝑑

𝑎𝑣 = 𝑎𝑐𝑠4∙ 𝑑

𝑎𝑐𝑠 = 𝑀𝐸𝑑

𝑉𝐸𝑑 ≥ 𝑑

27

2.4.2 Direct shear capacity

The direct shear capacity of reinforced concrete beams may be calculated according to

Department of Defense (2008):

(2.21)

where fdc is the dynamic compressive strength of concrete, b is the element width and

d denotes the effective depth.

Another model for calculating the direct shear capacity was originally proposed by

Hawkins (1982), which utilizes a piecewise linear approach to relate direct shear

strength and the corresponding shear slip values. This model was later modified for

applications to dynamic loads by applying an enhancement factor of 1.4 to account for

the effects of strain rate and in-plane compression Krauthammer et al. (1986). For the

purpose of calculating the direct shear capacity of the maximum shear strength m in

the modified model was used and multiplied by the beam width and the effective depth

as follows:

(2.22)

In Eq. (2.22), fc is in psi, and b and d are in inches. Converting this equation for

calculations in SI units becomes:

(2.23)

Thus, fc is in MPa, and b and d are in metres. The geometrical reinforcement ratio is

denoted by v. These equations account for direct shear with shear reinforcement across

the shear plane while the concrete beams modelled herein do not contain such shear

reinforcement other than the tensile and compression bars. The tensile reinforcement

was included in the calculations herein even though it is probable that this might

overestimate the influence of the reinforcement on the direct shear capacity.

2.4.3 Support reactions

The support reactions of a structural element subjected to a dynamic load is usually

regarded as a function of the flexural resistance and the applied load as discussed by

Biggs 1964 and in the Swedish design manual for protective design FKR (Swedish

Fortifications Agency 2011). Both refer to similar expressions and also provide similar

𝑉𝑑 = 0.18𝑓𝑑𝑐𝑏𝑑

𝑉𝑑 = 1.4 8 145𝑓𝑐 + 0.8𝜌𝑣 ∙ 145𝑓𝑦 6.895𝑏𝑑

𝑉𝑑 = 1.4 8 𝑓𝑐 + 0.8𝜌𝑣𝑓𝑦 𝑏𝑑

28

results. The basis for these expressions is the dynamic equilibrium of the beam

subjected to a uniformly distributed load, and the assumption that the beam deflects in

its fundamental mode. The expression for calculating the reactions at each support for

simply supported beams in FKR is as follows:

(2.24)

where for beams

p = 0.64 and m = 0.50 for 𝑝𝑞𝑑⁄ ≤ 2

p = 0.50 and m = 0.33 for 𝑝𝑞𝑑⁄ > 2

The parameters p and qd denote the applied peak pressure and the static load bearing

capacity of the beam, respectively. The beam span and width are denoted with L and b,

respectively.

𝑅𝑑 = 0.5 ∙ 𝑝 ∙ 1−𝜒𝑝

2

𝜒𝑚 + 𝑞𝑑 ∙

𝜒𝑝2

𝜒𝑚 ∙ 𝐿 ∙ 𝑏

29

Chapter 3

Material models

Finite element modelling and analyses enables detailed investigations of the strain and

stress states in a concrete element subjected to complex loads. Certain material models

also permits analyses of local damage initiation and evolution of failure zones of the

element, which would not be possible in experiments. This enables an increased

understanding of different types of failures that may occur in concrete elements

subjected to intense blast loads.

3.1 Material dynamic properties

The dynamic load will introduce a certain degree of deformation rate in the different

parts of the structural elements as these deform. This is commonly referred to as strain

rate effects in the material, which results in increases of the material strength and this

affects the structural response of the element. Reinforced concrete structures subjected

to blast loads will respond by deforming over a relatively short period of time, and the

strain rates in the concrete and reinforcement reach magnitudes considerably higher

than that of a statically loaded structure. For concrete, a static compressive and tensile

load is defined at a strain rate of 3·10-5 s-1 and 1·10-6 s-1, respectively, according to CEB

(1993). As a comparison, strain rates of around 1 s-1 can be expected for concrete

elements subjected to blast loading (Palm 1989; Magnusson & Hallgren 2000). The

material strength increase is normally referred to as a dynamic increase factor (DIF),

which is the ratio of the dynamic to the static value. Several aspects on the effects on

the material properties of concrete and steel reinforcement are given in the following

sections.

3.1.1 Concrete tension and compression

Several researchers have experimentally studied the compressive and tensile strength

of concrete at different strain rates. A selection of their commonly published results for

concrete in compression and tension are presented in Figure 3.1–3.2. For strain rates

exceeding static loads there is an increasingly larger scatter in test results for increasing

strain rates. Bischoff & Perry (1991) suggest that this relatively large scatter for the

compressive behaviour of concrete may depend on factors such as experimental

techniques used and methods of analysis employed. Other factors that can influence

30

the results are specimen size, geometry and aspect ratio as well as moisture content in

the specimens. Therefore, care should be taken when comparing the results of concrete

strength properties from different research programs.

Figure 3.1 Strain rate effects on the concrete compressive strength. From Bischoff &

Perry (1991).

Figure 3.2 Strain rate effects on the concrete tensile strength. From Johansson

(2000) and based on Malvar & Crawford (1998a).

31

For both compressive and tensile loading, two intervals with different strain rate

dependencies exist and with a relatively sharp transition zone between these. The first

more moderate strength increase has been explained by different authors and is

summarised by Johansson (2000) and later in Magnusson (2007). The increase in both

compressive and tensile strength for this interval may be explained by viscous effects

with free water in the micropores. At strain rates exceeding the transition zone there

will be a sharper strength increase. Explanations for this behaviour can mainly be

ascribed to inertia effects and lateral confinement. Weerheijm (1992) studied strain rate

effects on the tensile strength of concrete by using linear elastic fracture mechanics.

His studies showed that changed stress and energy distributions due to inertia effects

around the crack tips were the cause of the rapid strength increase. At an increasing

static compressive loading the behaviour will be affected by the propagation of

microcracks (Zielinski 1984). When the specimen instead is exposed to a rapid load,

the time available for initiation and propagation of microcracks will be reduced

(Bishoff & Perry 1991). This could be an explanation of the strain-rate dependent

behaviour at higher strain rates. Johansson (2000) reasons that the effects of the inertia

effects around the crack tips, which explains tensile strain rate sensitivity, also is a

reasonable explanation since the compressive failure is also governed by cracking.

Bishoff & Perry (1991) suggest that the sudden increase in compressive strength also

can be ascribed to lateral inertia confinement.

A model for the strain rate dependence of concrete in compression and tension is

presented in CEB (1993) and fib (2012). However, the former version of the Model

Code provides models that also include the reduced strain rate sensitivity of higher

concrete strength, which was excluded in Model Code 2010. Thus, the strain rate model

for concrete compression in Model Code 1990 was used in the work presented herein.

The ratio of dynamic to static compressive strength for strain rates up to 30 s-1 is given

as:

(3.1)

where fcd and fc denote the dynamic and static compressive strength, and 휀�̇� and 휀�̇�0

(= 310-5 s-1) are the actual and static strain rates, respectively.

The strain rate factor s is given as

(3.2)

𝑓𝑐𝑑𝑓𝑐

= 휀�̇�휀�̇�0

1.026𝛼𝑠

𝛼𝑠 = 5 + 9𝑓𝑐𝑓𝑐0 −1

32

where fc0 = 10 MPa. This model is presented graphically in Figure 3.3 and appears to

properly fit the test data as shown in Figure 3.1.

In Model Codes 1990 and 2010, the change in a moderate strength increase into the

more dramatic strength enhancement is set to a strain rate of 30 s-1 and 10 s-1 for tensile

loading. This does not agree well for tensile stresses. Malvar & Crawford (1998a) argue

that available test data reveal that the change in slope should instead occur around

1 s-1. Therefore, they proposed a formulation similar to the Model Code 1990, which

was fit against test data. According to Malvar & Crawford, the ratio of dynamic to static

compressive strength is given as:

for 1.0 s-1 (3.3)

for 1.0 s-1 (3.4)

where fctd and fct denote the dynamic and static tensile strength, and 휀�̇�𝑡 and 휀�̇�𝑡0

(= 110-6 s-1) are the actual and static strain rates, respectively. The strain rate factor

is given as

(3.5)

where fc0 = 10 MPa. The strain rate factor is given as:

(3.6)

This model is presented graphically in Figure 3.3 and appears to properly fit the test

data as shown in Figure 3.2. The higher strain rate sensitivity of concrete in tension is

also noted in these figures.

𝑓𝑐𝑡𝑑𝑓𝑐𝑡

= 휀�̇�𝑡휀�̇�𝑡0

𝛿

휀�̇�𝑡 ≤

𝑓𝑐𝑡𝑑𝑓𝑐𝑡

= 𝛽 휀�̇�𝑡휀�̇�𝑡0

1/3

휀�̇�𝑡 >

𝛿 = 1 + 8𝑓𝑐𝑓𝑐0 −1

𝑙𝑜𝑔𝛽 = 6𝛿 − 2

33

Figure 3.3 Strain rate dependence of concrete in compression (fc = 45 MPa)

according to the CEB (1993) and in tension with the modified model

according to Malvar and Crawford (1998a).

3.1.2 Concrete elastic modulus

The elastic modulus of concrete is also affected by changes in strain rate. According to

Bishoff and Perry (1991), the enhancement of the elastic modulus at dynamic loading

can be ascribed to the decrease in internal microcracking at a given stress level with an

increasing strain rate. This behaviour results in a stress-strain curve that remains linear

up to higher stress values. The strain-rate dependence of the elastic modulus is also

included in fib (2012) as presented in Figure 3.4. The ratio of dynamic to static

compressive strength for different strain rates is given as:

(3.7)

where Ecd and Eci denote the dynamic and static elastic modulus, and 휀�̇� and 휀�̇�0 are the

actual and static strain rates, respectively. The strain rate 휀�̇�0 is given the value

310-5 s-1 for compression and 110-6 s-1 for tension. This model is presented graphically

in Figure 3.4. It is noted that the elastic modulus in tension exhibits a higher sensitivity

compared to the modulus in compression. Also, the model does not account for a sharp

transition zone with two different strain rate dependencies. Furthermore, the model

does not account for possible strain rate dependencies due to varying concrete

strengths.

0

0.5

1

1.5

2

2.5

3

3.5

1.0E-06 1.0E-05 1.0E-04 1.0E-03 1.0E-02 1.0E-01 1.0E+00 1.0E+01

Dyn

amic

in

cre

ase

fac

tor

Strain rate (s-1)

Compressive strength

Tensile strength

10-6 10-5 10-4 10-3 10-2 10-1 100 101

𝐸𝑐𝑑𝐸𝑐𝑖

= 휀�̇�휀�̇�0

0.026

34

Figure 3.4 Strain rate dependence of concrete elastic modulus according to fib

(2012).

3.1.3 Concrete fracture energy

Energy is consumed in the process of crack initiation and propagation in concrete. The

fracture energy is the total energy consumed per unit area of the fracture surface during

failure. Previous research has shown that also the fracture energy of concrete in tension

is sensitive to strain rate. Two investigations used the Split-Hopkinson-Bar test set-up

as a basis for their spall tests on concrete specimens. Weerheijm & Van Doormaal

(2007) used a detonating charge at a short distance from the end of the steel bar to

create the pressure pulse. Schuler et al. (2006), on the other hand, used a projectile that

impacted on the end of the bar. Both investigations concluded that the fracture energy

is enhanced at an increasing strain rate. The test results in Schuler et al. (2006) show a

relationship between the relative increase of the fracture energy with the crack opening

velocity and the strain rate. Herein, these results were interpreted as presented in

Figure 3.5. The tests indicate that the fracture energy is constant up to a strain rate of

about 0.3 s-1, where after the energy increases at a constant rate.

In terms of fracture toughness, the fracture energy is not an adequate measure. Instead,

the characteristic length is a more adequate parameter to describe the toughness of a

material Betonghandbok (2000). The characteristic length depends on the fracture

energy GF, the elastic modulus E and the tensile strength ft and is calculated according

to:

0.8

1

1.2

1.4

1.6

1.8

2

1.0E-06 1.0E-05 1.0E-04 1.0E-03 1.0E-02 1.0E-01 1.0E+00 1.0E+01

Dyn

amic

in

cre

ase

fac

tor

Strain rate (s-1)

Compression

Tension

10-6 10-5 10-4 10-3 10-2 10-1 100 101

35

Figure 3.5 Strain rate dependence of the fracture energy based on the research

conducted by Schuler et al. (2006) for normal strength concrete.

(3.8)

This equation may be regarded in such a way that lch is proportional to the ratio of

energy consumed per unit area and the elastic energy per volume unit that is released.

Increasing values of lch results in an increased toughness and decreasing values results

in a reduced toughness. Considering the strain rate effects of GF, E and ft in dynamic

events results in different values of lch. Using the strain rate dependence in

Section 3.1.1–3.1.2 and Figure 3.5 as a basis, Figure 3.6 presents the results of how lch

is reduced at strain rates 0.001 s-1, 0.1 s-1 and 10 s-1. It is clear that the toughness of

concrete in tension is reduced considerably at an increasing strain rate.

0.8

1

1.2

1.4

1.6

1.8

2

0 1 2 3 4 5 6 7 8 9 10

Dyn

amic

in

cre

ase

fac

tor

Strain rate (s-1)

𝑙𝑐ℎ =𝐸𝐺𝐹

𝑓𝑡2

36

Figure 3.6 The characteristic length for three different strain rates.

3.1.4 Steel reinforcement

Steel reinforcing bars are also affected by dynamic loads and will exhibit different

degrees of strength increases depending on steel grade. The elastic modulus, however,

is usually found to remain constant as stated by Malvar & Crawford (1998b). The

reasons for the strength increase of steel reinforcement are explained to be due to

dislocation effects originating from the crystalline structure of the steel under shear

stresses, Palm (1989) and Meyers (1994). Malvar & Crawford proposed a formulation

of the dynamic increase factor (DIF) for both the yield strength and ultimate strength

for reinforcing bars at different strain rates as follows:

(3.9)

where for the yield stress:

(3.10)

and for the ultimate stress:

(3.11)

where 휀̇ is the strain rate (s-1), and fy is the yield strength (MPa) of the reinforcing bar.

This formulation is valid for bars with yield strengths between 290 MPa and 710 MPa

0.00

0.05

0.10

0.15

0.20

0.25

0.001 0.01 0.1 1 10

Ch

arac

teri

stic

le

ngt

h (

m)

Strain rate (s-1)

𝐷𝐼𝐹 = 휀̇

10−4 𝛼

𝛼 = 0.074− 0.040𝑓𝑦

414

𝛼 = 0.019− 0.009𝑓𝑦

414

37

and for strain rates of 0.0001–225 s-1. According to this formulation the magnitude of

the DIF will decrease with an increasing steel yield strength, which is the case for real

bars according to Malvar and Crawford (1998b). Figure 3.7 illustrates the strain rate

dependence of the yield and ultimate strength of the reinforcing steel. The figure shows

that the yield strength increase is more significant compared to that of the ultimate

strength. Thus, the ductitliy of the reinforcement is reduced at higher strain rates.

Figure 3.7 Strain rate dependence on the yield and ultimate strength of reinforcing

steel according to Malvar and Crawford (1998b) for a static yield and

ultimate strength of 500 and 600 MPa, respectively.

3.2 Material models for concrete

3.2.1 RHT model

The RHT material model is a general constitutive model for brittle materials (Riedel

2000). This model has been used in several investigations in the analyses of projectile

penetration in concrete and structural response of concrete beams subjected to blast

loads such as in Hansson (2011) and the work in Paper III. The stress states in the model

is described by three pressure dependent yield surfaces that define the elastic limit

surface, failure surface and the residual strength surface for crushed material, see

Figure 3.8. All surfaces are scaled with reference to the uniaxial compressive strength

for the concrete. In addition, the elastic limit and failure surfaces are strain rate

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

0.0001 0.001 0.01 0.1 1 10

Dyn

amic

in

cre

ase

fac

tor

Strain rate (s-1)

Yield strength

Ultimate strength

10-4 10-3 10-2 10-1 100 101

38

dependent. During a continuously increased loading of the material, the elastic surface

is eventually reached after which the shear modulus is reduced and the stresses will

increase during strain hardening up to the failure surface. After the failure surface has

been reached increasing plastic strains will lead to damage evolution and strength

reduction. The damage model describes the strength degradation due to increasing

plastic strains pl until the residual strength surface is finally reached. The damage

growth in the material is calculated according to:

(3.12)

(3.13)

where DRHT1 and DRHT2 are specific material parameters, and p* is the pressure

normalised to the uniaxial compressive strength (p/fc). The parameter is defined

in Figure 3.9, and is the lower limit of the failure strain. The damage variable

D can take values 0–1, where the former expresses undamaged material and the latter

fully damaged material. The relationship for the compressive meridian in Riedel (2000)

is not used for pressures below fc/3. A piecewise linear approximation is instead used

as shown in Figure 3.9. Furthermore, an associated flow rule was employed, which

considers the plastic volume increase of the concrete that occurs near failure.

Figure 3.8 The three surfaces of the RHT concrete model as 3D projections.

Modified from Hansson (2011).

Failure surface Residual strength

surface

Elastic limit

surface

𝐷 = ∆휀𝑝𝑙

휀𝑝𝑙𝑓𝑎𝑖𝑙𝑢𝑟𝑒

휀𝑝𝑙𝑓𝑎𝑖𝑙𝑢𝑟𝑒

= 𝐷𝑅𝐻𝑇1 𝑝∗ − 𝑝𝑠𝑝𝑎𝑙𝑙

∗ 𝐷𝑅𝐻𝑇 2

≥ 휀𝑝𝑙 ,𝑚𝑖𝑛𝑓𝑎𝑖𝑙𝑢𝑟𝑒

𝑝𝑠𝑝𝑎𝑙𝑙∗

휀𝑝𝑙 ,𝑚𝑖𝑛𝑓𝑎𝑖𝑙𝑢𝑟𝑒

39

Figure 3.9 The linear approximation of the compressive meridian at pressures below

fc/3. From Hansson (2011) and Riedel (2000).

3.2.2 Concrete damaged plasticity

The Concrete Damage Plasticity (CDP) model was used for calculations of the stress-

strain states of the concrete (Abaqus 2011). This model is intended for analyses of

concrete structures under cyclic or dynamic loading and has been used in previous

investigations for both static and dynamic loads such as in Malm & Holmgren (2008b)

and Kamali (2012). The CDP model is based on the models proposed by Lubliner et al.

(1989) and by Lee and Fenves (1998).

As a concrete specimen is unloaded from any point on the strain softening branch of

the stress-strain curves, the unloading response is weakened, see Figure 3.10. Thus, the

elastic stiffness of the material appears to be weakened and this effect becomes more

pronounced at an increasing plastic strain. Therefore, the degraded response is

characterised by the two independent uniaxial damage variables dt (in tension) and dc

(in compression). The degree of damage is associated with the failure due to cracking

and crushing of the concrete, respectively. The damage variables can take values 0–1,

where the former expresses undamaged material and the latter fully damaged material.

The stress-strain relations under uniaxial tension and compression, respectively,

become (Abaqus 2011):

(3.14)

(3.15)

𝜎𝑡 = 1− 𝑑𝑡 𝐸0 휀𝑡 − 휀 𝑡𝑝𝑙

𝜎𝑐 = 1− 𝑑𝑐 𝐸0 휀𝑐 − 휀 𝑐𝑝𝑙

40

where the parameters are explained in Figure 3.10. The values of the maximum damage

parameter were limited to represent 1 % of the tensile strength and a maximum of 0.97

for the damage in compression, see Appendix B.

The strain rate dependent concrete strength can be tabulated for both compression and

tension. However, simulations on prisms subjected to uniaxial compressive and tensile

loads, respectively, revealed that the rate dependence of tensile stresses was not

functional. Therefore, the static concrete compressive and tensile strength was

increased with a DIF based on the strain rate of the tensile reinforcement in all

simulations. It is recognised that the strain rates vary throughout the concrete element

and at different points in time during a dynamic response. Thus, using a constant DIF

is a simplified approach. Therefore, simulations were performed with varying DIF

factors to specifically study the effects on the support reactions and the failure modes

in Appendix A. Strain rate effects for concrete compressive and tensile strength were

accounted for by using the relationships in fib (2012) and Malvar & Crawford (1998a),

respectively. The relationship in fib (2012) was used for the corresponding strain rate

effects of the elastic modulus.

The CDP model employs a non-associative flow rule and, thus, the plastic potential

function and the yield surface do not coincide. The dilation is a measure of the plastic

volume increase that occurs near failure of the concrete in compression. In Abaqus, the

dilation angle is used as a material parameter where a low value will produce a brittle

behaviour and higher values will produce a more ductile behaviour (Malm, 2009). This

author conducted a literature survey on the value of the dilation angle, which was used

as a basis for the simulations performed herein and in Paper V, see Appendix B.

According to Chen (1982) the yield surface of concrete has a triangular cross-sectional

shape in the deviatoric plane and at low hydrostatic pressures, but will show a more

circular shape at higher pressure levels. At low pressures, the failure in concrete is

typically brittle in nature, which is the case for tensile stresses and compressive stresses

at low pressures. On the other hand, if concrete is subjected to higher hydrostatic

pressures the material can deform plastically on the failure surface like a ductile

material before failure strains are obtained (Chen, 1982). Furthermore, the CDP model

uses a factor Kc to control the shape of the yield surface in the deviatoric plane, see

Figure 3.11. A value of 0.67 was used in the simulations performed herein.

41

Figure 3.10 Response of concrete subjected to uniaxial loading in (a) tension and (b)

compression. From Abaqus (2011).

Figure 3.11 Yield surface in the deviatoric plane with Kc values of 0.67 and 1.0,

respectively.

42

3.2.3 Crack softening for concrete in tension

Even though concrete is a relatively brittle material in tension, the tensile failure is not

characterised by an instantaneous strength loss. Hillerborg (1978) introduced the

concept of a fictitious crack, which is not a real crack but instead a simplified portrayal

of the fracture zone in the crack tip region. According to Hillerborg, after the ultimate

tensile stress of the concrete is reached the fictitious crack is portrayed as a

continuously descending stress as a function of the crack width (w) of the fracture zone.

Gylltoft (1983) proposed a bi-linear crack softening law for concrete in tension as

shown in Figure 3.12. Leppänen (2004) implemented this softening law in Autodyn to

be used together with the RHT material model. The slopes of the bi-linear crack

softening are:

(3.16)

(3.17)

where fct and GF are the tensile strength and the fracture energy, respectively. The

ultimate crack width wu is calculated as:

(3.18)

This model for crack softening of concrete in tension was used in the simulations in

Papers III and V, and also in the simulations in this thesis. The crack softening model

was implemented in Ansys Autodyn as part of the work later presented in Paper III and

reported by Westerling (2005).

𝑘1 =𝑓𝑐𝑡

2

𝐺𝐹

𝑘2 =𝑓𝑐𝑡

2

10𝐺𝐹

𝑤𝑢 = 4𝐺𝐹𝑓𝑐𝑡

43

Figure 3.12 The bi-linear crack softening model for concrete in tension.

3.3 Material models for reinforcement

3.3.1 Johnson & Cook

The constitutive model by Johnson & Cook (1983) for metals is an empirical expression

where the flow stress () is a function of plastic strain (), plastic strain rate (휀̇) and

temperature (T) according to:

(3.19)

The parameter A describes the yield strength, and B and n describe the strain hardening.

The parameter C considers the strain rate dependence of the material and, finally, m

takes the effects of temperature into account. The reference temperature (Tr) is

normally set to 293 K and Tm is the melting temperature. The reference strain rate 휀0̇ is

in the original model set to 1.0 s-1. Therefore, the Johnson & Cook model was therefore

modified as a part of the work later presented in Paper III and reported by Westerling

(2005). This modified model was implemented in Ansys Autodyn and gives the user

the opportunity to choose the value of the reference strain rate. A reference strain rate

of 10-4 s-1 was employed in the work of paper III, which is also used by Malvar &

Crawford (1998b)

w

fct/3

fct

wu wu/6

k1

k2

𝜎 = (𝐴+ 𝐵휀𝑛) 1 + 𝐶𝑙𝑛휀̇

휀0̇ 1−

𝑇 − 𝑇𝑟𝑇𝑚 − 𝑇𝑟

𝑚

44

3.3.2 Isotropic elasto-plastic model

An isotropic elasto-plastic material model for the calculation of stresses in the

reinforcing steel was used in the work presented in Paper V and also in the simulations

presented in this thesis. The static and dynamic stress-strain relations were specified as

piecewise linear approximations with isotropic strain hardening. The dynamic increase

of the yield and ultimate strength for steel reinforcement was employed as previously

discussed in Section 3.1.4. Figure 3.13 presents stress-plastic strain approximations for

a static yield stress of 500 MPa, see also Appendix B. The broken lines in the figure

represents the stresses using a DIF for the yield stress only. However, the DIF for the

ultimate stress at different strain rates are lower than the factors for the yield stress, see

Figure 3.7. Therefore, the stress-strain curves were adjusted to account for the larger

increase in yield stress compared to the ultimate stress according to Figure 3.13.

Figure 3.13 Piecewise linear approximations of the stress and plastic strain for

reinforcement with a static yield stress of 500 MPa (adjusted yield stress).

3.4 Bond between reinforcing bars and concrete

The bond between concrete and reinforcement is fundamental in reinforced concrete

structures. In a case where a diagonal shear crack is initiated in a reinforced concrete

beam, there is a sudden increase of the tensile force in the reinforcing bars across the

crack (Magnusson 2000, Chalmers University of Technology). For a diagonal crack

that develops close to one of the supports, the anchorage of the bars becomes more

400

450

500

550

600

650

700

750

800

850

900

0.00 0.02 0.04 0.06 0.08 0.10 0.12

Stre

ss (M

Pa)

Plastic strain (-)

10

1

0.1

0.01

0.001

Static

s-1

s-1

s-1

s-1

s-1

Yield stressAdjusted yield stress

45

critical. It may therefore be important to consider this interaction between concrete and

reinforcement when performing numerical analyses. Some of the parameters affecting

the bond strength are concrete strength, confinement, shrinkage of the concrete, rib

geometry and concrete cover. The bond resistance mechanisms are mainly due to

mechanical interlocking and friction (Magnusson 2000, Chalmers University of

Technology). The stresses transferred from the ribs to the concrete can be divided into

a longitudinal and a radial component, where the former is usually referred to as the

bond stress, see Figure 3.14. The tensile stresses that are initiated near the rib tips cause

transverse microcracks in the concrete at the tips, which allow the bar to slip. As the

applied tensile load on the rebar is further increased the shear resistance of the concrete

between two adjacent ribs determines the maximum bond stress. The concrete between

the two ribs will be completely sheared off at a certain slip and the rebar will start to

slide inside the concrete. At this point the transferred stresses are merely due to friction

and the rebar will be pulled out of the surrounding concrete at gradually reducing bond

stress.

The ribs give rise to a wedging action of the surrounding concrete and circumferential

tensile stresses are developed, which may cause longitudinal splitting cracks to

develop, see Figure 3.14. The local confinement of the concrete influences the initiation

and propagation of these splitting cracks. Such confinement enhances the bond strength

and occur e.g. at the supports of a loaded beam. In a case where the concrete cover is

relatively small, the splitting action may cause longitudinal cracks to propagate to the

surface and there may be a more severe abrupt drop in bond resistance. Such a case

may typically develop along the tensile reinforcement across a diagonal shear crack of

a beam near a support. This was observed in tests with concrete beams subjected to

blast loads, see Paper II.

(a) (b)

Figure 3.14 Stresses and cracks in the surrounding concrete as a reinforcing bar is

pulled out. From (a) Ansell et al. (2012) and (b) Betonghandbok (1990).

46

47

Chapter 4

Dynamic shear of concrete beams

This chapter presents experiments on reinforced concrete beams and roof slabs

subjected to various blast loads. These experiments served as a basis for the numerical

models that are also presented in this chapter. Further numerical modelling is also

included in order to conduct parametric studies.

4.1 Experiments

4.1.1 Concrete beams

The experiments on reinforced concrete beams were conducted inside a shock tube

where a detonating high explosive charge generated the blast load, see Figure 4.1.

These tests are thoroughly described by Magnusson & Hallgren (2000) and in Paper II.

The beam width and depth measured 300 mm and 160 mm, respectively. The beams

were assembled in a test rig and placed vertically with a span of 1500 mm as shown in

Figure 4.1. Bolts were used to secure the beams to the supports during the tests. The

end restrictions of these bolts appeared to have negligible influence on the test results

as discussed by Magnusson & Hansson (2005). Even though the beams were secured

with bolts, At the used distance between the charge and the beam of 10 m the beams

can be regarded as subjected to a uniformly distributed blast load. The instrumentation

of the tests consisted of two pressure gauges, load cells at the supports, and a deflection

gauge and an accelerometer at mid-span of the beam. A strain gauge on the concrete

surface at the compressive zone of the beam and on one reinforcing bar were also used

in a few tests. These gauges were placed at the beam mid-span.

48

Figure 4.1 Configuration of the blast tests inside a shock tube. From Magnusson &

Hallgren (2000).

4.1.2 Concrete roof slabs

In these tests, the roofs of reinforced concrete box structures were subjected to loads

from detonating explosive charges. These tests are thoroughly described by Slawson

(1984) and in Paper IV. The box structures were cast monolithically with two open

ends with a wall, roof and floor thickness of 184 mm. In this configuration, the free

span of the roof slab was 1140 mm. Strips of high explosive charges were evenly placed

across the sand backfill covering the entire roof and walls of the box structure as

illsustrated in Figure 4.2. Such configuration ensured that the roof load was due to the

propagation of a planar wave. Pressure gauges at different locations measured the load

on the box structure, and strain gauges were used on certain principal reinforcement.

Accelerometers on the roof slab at at mid-span were also used in a few tests.

Figure 4.2 Configuration of the blast tests of concrete roof slabs. From Slawson

(1984).

49

4.2 Numerical models

Analysing dynamic shear in concrete structures with large non-linearities makes a finite

element software with an explicit dynamics analysis procedure a useful tool. For this

purpose, the software Abaqus/Explicit 6.11 (Abaqus 2011) was used for modelling of

concrete beams and roof slabs subjected to varying blast loads.

4.2.1 Flexural shear failures

The tested beams B40-D3 and B40-D4 reported in Magnusson & Hallgren (2000) were

used as verifications of a flexural shear failure. The results of the latter beam were also

included in Paper II and further analysed in Paper III. The beam has a cross-section of

300 160 mm (width depth) and a span of 1500 mm. The concrete, the tensile

reinforcing bars and the reinforcement bar-concrete interface were modelled using

solid elements with linear interpolation and reduced integration. A mesh size of

4 4 4 mm3 was used for the concrete while the mesh of the interface varied in size

depending on the bar diameter, see Figure 4.3. A mesh size of 2 2 2 mm3 for the

concrete was also used in a limited number of simulations as a comparison. A mesh of

5 5 5 mm3 was used for the supports. The transverse and compression

reinforcement, with bar diameters of 8 mm and 10 mm, was modelled using 4 mm

linear truss elements. Figure 4.4 (a) illustrates the configuration of the reinforcement.

Due to symmetry, half the beam span was modelled, with a symmetry plane at mid-

span. An interaction with friction between the support and beam surfaces was employed

to enable sliding as the beam deflects. One line of nodes at the middle of the bottom

surface of the support was used as a boundary condition, which enabled the support to

rotate during the response. This beam was subjected to a uniformly distributed blast

load over the top surface as shown in Figure 4.4 (b). A piecewise linear approximation

of the registered pressure-time curve was used as the applied load, see Figure 4.5.

The material properties of the concrete and reinforcing steel followed those reported

from the tests as reported by Magnusson & Hallgren (2000), see Appendix B. The

Concrete Damaged Plasticity model was used for calculating the stress-strain states of

the concrete in combination with the bi-linear crack softening model for concrete in

tension. In these simulations, the concrete strength properties were calculated for a

strain rate of 1.0 s-1, see Appendix B. The isotropic elasto-plastic material model was

used for the reinforcing steel with strain rate dependent data, as listed in Appendix B.

50

Figure 4.3 Modelled beam cross section with reinforcing bars and interface material

(in white).

(a)

(b)

Figure 4.4 Modelled beam with the (a) configuration of the reinforcement and the

(b) blast load applied over the top surface.

51

Figure 4.5 Piecewise linear approximation of the average pressure from two

measurements in the test of B40-D4 (Magnusson & Hallgren 2000).

In Paper III, the concrete beams were modelled using an interface material between the

reinforcing bars and the concrete to allow for slip of the bars. The results showed that

the crack pattern and possibly the failure mode could change when allowing for a

certain slip. Therefore, an interface material was used between the tensile reinforcing

bars and the concrete. Herein, the description of the different bond stress-slip

relationships in Model Code 1990 (CEB 1993) and Model Code 2010 (fib 2012) were

used as a basis to establish the shear strength and the slip. Abaqus/Explicit (Abaqus

2011) was used for modelling and simulations of a real pull-out test reported in

Magnusson (2000, Chalmers University of Technology). A reinforcement bar with a

diameter of 16 mm and a yield strength of 569 MPa was embedded with a length of

0.22 m in a concrete prism measuring 0.4 0.4 0.4 m3, see Figure 4.6. Due to

symmetry, half the pull-out test was modelled. The prism was modelled with two mesh

sizes where the solid elements, with linear interpolation and reduced integration, in the

region closest to the bar measured 4 4 4 mm3. The interface elements were modelled

with approximately the same size. A constant velocity of 0.002 m/s was used at the end

of the bar for pulling the bar out of the concrete. A rectangular area at the left edge of

the front surface of the concrete prism in Figure 4.6 served as boundary condition, with

restraints in all directions. The condition of an unconfined concrete (failure by splitting

of the concrete) ‘all other bond conditions’ according to CEB (1993) was employed.

For the maximum shear strength max, the concrete strength of the B40 beam in Paper

II was used. The strength parameters of the interface are presented in Appendix B. An

isotropic elasto-plastic material model was used for the calculation of stresses and

strains in the interface material.

0

200

400

600

800

1000

1200

1400

0 5 10 15 20 25 30

Pre

ssu

re (

kPa)

Time (ms)

Registration

Simulation

52

The bond strength between concrete and reinforcement increases at an increasing strain

rate (Gebbeken & Greulich 2005) but this strain rate effect was not considered in the

bond model.

Figure 4.6 The model for pull-out of a reinforcing bar out of concrete.

4.2.2 Direct shear failures

Test DS1 and DS4 in Slawson (1984) were used as reference for verification of a direct

shear mode. In these tests, the entire roof surface of a concrete box structure was

subjected to a uniformly distributed blast load. The box structure was cast

monolithically with two open ends as shown in Figure 4.7. The roof slab thickness

measured 140 mm with a clear span of 1220 mm and the reinforcement consisted of

bars in both directions at the top and bottom surfaces and vertical stirrups (Slawson

1984). Half the roof width and the upper region of the side walls were modelled, see

Figure 4.8. The boundary conditions of the bottom surfaces of the walls were modelled

as fixed in all directions. The concrete was modelled using solid 4 4 4 mm3 elements

with linear interpolation and reduced integration, and the reinforcing bars and stirrups,

with bar diameters of 12.7 mm (No. 4 bar) and 9.5 mm (No. 3 bar), respectively, were

modelled with 4 mm linear beam elements. A mesh size of 2 2 2 mm3 for the

concrete was also used in a limited number of simulations as a comparison. The applied

blast load was modelled across the entire top surface using a piecewise linear

approximation of the average values of the registered pressure-time curves from gauges

53

IF-2 and IF-3 in Slawson (1984), see Appendix B. Figure 4.9 presents the piecewise

linear approximations of the registered pressure-time curves in tests DS1 and DS4. The

material properties of the concrete and reinforcing steel followed those reported from

the tests and presented in Appendix B. The Concrete Damaged Plasticity model was

used for calculating the stress-strains states of the concrete in combination with the bi-

linear crack softening model for concrete in tension. The concrete strength properties

were calculated for a strain rate of 1.0 s-1, see Appendix B. The isotropic elasto-plastic

material model was used for the reinforcing steel with strain rate dependent data as

presented in Appendix B.

Figure 4.7 Concrete box structure used in the direct shear tests. From Slawson

(1984).

(a)

(b)

Figure 4.8 Modelled (a) roof slab with walls and (b) reinforcement configuration.

54

Figure 4.9 Piecewise linear approximations of the average pressure from two

measurements in each test (DS1 and DS4).

4.2.3 Parametric studies

Beam models with a width and span of 300 mm and 1500 mm with three different

depths were considered in the simulations, i.e. depths of 260 mm, 160 mm and 84 mm

as shown in Figure 4.10. Also, the amount of tensile reinforcement was varied such

that each beam section had a reinforcement content of approximately 0.6 % and 1.5 %,

respectively. No transverse reinforcement was included in the models. Table 4.1

presents the geometric properties and reinforcement employed for each beam type.

Each beam type (B7, B12 and B27) is labelled with reference to the corresponding

shear slenderness L/d. The number in parenthesis refers to the number of reinforcement

bars. The beam depth and, thereby, the value of L/d was particularly chosen to resemble

beams subjected to uniformly loads and typically responding in a beam and an arch

mechanism as discussed in Section 2.2. Beam types B7 and B27 can typically be related

to responding in an arch mechanism and beam mechanism, respectively. With L/d of

approximately 12, beam type B12 relates to the transition point between these two

mechanisms. Another reason for modelling beams of varying depths is the known size

effects on shear for concrete elements. Furthermore, the amount of reinforcement is

also an important parameter that affects the shear of concrete elements. Thus, it is of

interest to analyse the behaviour of the same cross section but with different amounts

of reinforcement.

0

5

10

15

20

25

30

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Pre

ssu

re (

MPa

)

Time (ms)

DS1 test

DS4 test

55

(a) B7(2) (b) B7(5)

(a) B12(2) (b) B12(5)

(a) B27(2) (b) B27(5)

Figure 4.10 Modelled cross sections with three beam types with two amounts of

reinforcement.

Table 4.1 Geometry and reinforcement of the modelled beams.

Beam type h (m) d (m) L/d Reinforcement (%)

B7(2) 0.260 0.228 6.6

216 0.59

B7(5) 516 1.47

B12(2) 0.160 0.128 11.7

212 0.59

B12(5) 512 1.47

B27(2) 0.084 0.056 26.8

28 0.60

B27(5) 58 1.50

56

The same mesh for the beam models as for the beam in Section 4.5.1 was employed

using half symmetry at mid-span and the same boundary conditions. For beam types

B7 and B12, two reinforcing bars with a diameter of 10 mm as compression

reinforcement were also modelled using truss elements. For beams B27, two bars with

a diameter of 6 mm were modelled. Truss elements do not consider bending and dowel

action but this effect was considered as negligible for the compression reinforcement

in the analyses. The blast load was idealised as a triangular pressure pulse with an

almost instant rise to peak pressure, immediately followed by a linear decay to zero,

see Figure 4.11. The load was uniformly applied over the top surface between the

interior faces of the supports. The peak pressure and duration was varied between the

different simulations.

The material properties of the concrete and reinforcing steel are listed in Appendix B.

A compressive concrete strength and yield strength of the reinforcement of 45 MPa and

500 MPa was employed, respectively. The Concrete Damaged Plasticity model was

used for calculating the stress-strains states of the concrete in combination with the bi-

linear crack softening model for concrete in tension. The isotropic elasto-plastic

material model was used for the reinforcing steel with strain rate dependent data. The

properties for the interface material is presented in Appendix B. Strain rate effects for

the interface material around each reinforcing bar were not considered.

Figure 4.11 Idealised blast load employed in the simulations.

Pre

ssu

re

Time (ms)

57

4.2.4 Material parameters

The strain rate dependence of concrete in tension can not be considered in the Concrete

Damaged Plasticity (CDP) model. Therefore, the concrete strength properties in tension

and compression were specified for a typical strain rate that occurred in the concrete

beam. For the simulations of flexural and direct shear failures, this strain rate was

determined in the concrete elements adjacent to the tensile reinforcement. In

simulations where flexural shear was analysed, the strain rate in the vicinity of the shear

crack was used, while the strain rate at the inner face of the support was used when

direct shear failures were analysed. It is recognised that this approach is a simplification

that may influence the simulation results. Therefore, separate simulations were

performed with beam types B7(5), B12(5) and B27(5) in order to verify the dependence

on used material strength data in compression and tension. These simulations are

presented in Appendix A. In the models described in Sections 4.2.1–4.2.2, the concrete

strength properties were calculated for a strain rate of 1.0 s-1, see Appendix B. In the

parametric studies of the B7 and B12 beams, concrete properties for a strain rate of

5.0 s-1 were used at pressure levels of 15 MPa and above. Concrete properties for a

strain rate of 1.0 s-1 were used for pressure levels below 15 MPa. For the B27 beams,

concrete properties for a strain rate of 1.0 s-1 were used.

Furthermore, the dilation angle of concrete may typically be selected between

approximately 25° and 40° (Malm 2009). Using the upper values within this range is

more suitable for a low degree of hydrostatic pressures, while lower values provide a

better description of the material behaviour in biaxial stress states. Chen (1982), on the

other hand, states that an associated flow rule should be employed in structural

analyses, i.e. at a dilation angle of approximately 56° in CDP. Such a flow rule was

employed in the simulations in Paper III. In the simulations performed herein, a dilation

angle of 30° and 45° was used for description of the direct shear and flexural shear

failure, respectively. The relatively high value of the hydrostatic pressure at the

supports where direct shear failure occurs and the relatively low pressure values at the

location of flexural shear cracks motivated this choice.

The strain-rate dependence of the reinforcement was accounted for by specifying piece-

wise linear approximations of the stresses and plastic strains at a variation of strain

rates. These piecewise values were tabulated for the isotropic elaso-plastic model in

Abaqus, see Appendix B.

58

59

Chapter 5

Results from numerical simulations

This chapter includes the results from different analyses of dynamic shear in concrete

beams and roof slabs. The analyses were conducted with the use of Abaqus/Explicit

6.11 (Abaqus 2011) in order to investigate the initiation and propagation of different

shear failures. A part of the work focused on verifying the numerical models against

tests and the remaining part consisted of parametric studies. Prior to these simulations,

the initial response of a beam subjected to intense dynamic loads is reviewed.

5.1 Initial response and shear

Under static loads, material fractures such as cracks in concrete are initiated and

propagate according to the stress and strain fields existing throughout the concrete

element. The weakest regions will thereby control locations and levels of cracking.

However, under dynamic conditions, local high stresses and strains can develop whose

location may change before an initiated crack has time to propagate. Due to such

conditions, wave propagation effects become increasingly important in the analyses.

Shear failures typically may occur at an early stage and it is therefore of interest to

analyse the initial structural response soon after the load has been applied. The initial

response on aluminium beams have been experimentally reported by Menkes & Opat

(1973) and on reinforced concrete slabs by Slawson (1984). Theoretical analyses have

also been conducted on reinforced concrete beams with the Euler-Bernoulli and

Timoshenko beam theories (Svedbjörk 1975; Hughes & Beeby 1982; Ross 1983), and

with the use of finite element analyses (Ardila-Giraldo 2010; Andersson & Karlsson

2012). The theoretical analyses in Papers IV and V also agree with the findings in these

references.

For the case of a beam subjected to a uniformly distributed dynamic load, such as

shown in Figure 5.1, a pressure wave will develop and propagate through the thickness

of the beam. As soon as the wave strikes the rear face and reflects, there will be a

difference in particle velocity and part B-C begins to move downward. This movement

will occur with twice the velocity of the particles in the supporting walls. The portion

60

of the slab between the supporting walls will move downwards, which results in shear

stresses and bending moments developing in the cross section at the face of the

supports. At this early point in time, the remaining parts of the beam will be subjected

to a rigid body motion without any deformations. The distributions of deformations,

bending moments and shear forces at the early stages of a structural response are

presented in Paper IV and V and also in Andersson & Karlsson (2012). As concluded

in Paper IV, the concrete beam will initially exhibit deformations, shear forces and

bending moments with significantly different distributions compared to those under

static loads. Figure 5.2 presents the calculated deflections, bending moments and shear

forces at two points in time using Euler-Bernoulli beam theory, and with the same

values of the different parameters as presented in Paper IV. The values of the vertical

axes are normalized to the corresponding static quantities, and the same diagrams for

bending moment and shear force are used as in Paper IV. The bending moment and

shear distributions close to each support in Figure 5.2 (b) and (c) show similarities to

the distributions of smaller beams. Thus, at this stage, the entire beam may initially be

regarded as divided into two smaller beams, each responding with an apparently low

shear slenderness L/d. Structural wave motions originating from each support will over

time change the moment and shear distributions to eventually become similar to that of

the entire beam being statically loaded. The apparent shear slenderness of the smaller

beams will thereby increase and eventually be transformed into a response where the

entire beam deflects according to its fundamental mode.

The maximum values of the different properties in Figure 5.2 (a)–(c) reveal that the

relative magnitudes of the shear forces are larger than the corresponding magnitudes of

the bending moments and deflections. This is illustrated in Figure 5.3 at two points in

time. It is clear that shear is the dominant response mode over bending moments and

deflections and, thus, shear dominates the early response. The deflections are

comparatively small with a distinct approximately straight central portion of the beam.

The results presented in Figure 5.2–5.3 appears to, at a qualitative level, explain the

behaviour of the tested roof slabs that failed in a direct shear mode, as previously

discussed in Section 4.1.2.

Figure 5.1 Wave propagation through the roof slab. Based on Ross (1983).

A D C B

Open space

Pressure wave

v0 2v0 v0

61

(a)

(b)

(c)

Figure 5.2 Normalized (a) deflected shapes, (b) bending moments and (c), shear

forces for a simply supported beam subjected to a uniformly distributed

load according to Euler-Bernoulli theory. Based on the work in Paper IV.

-0.007

-0.006

-0.005

-0.004

-0.003

-0.002

-0.001

0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

y/y s

tat

x/L

0.025

0.125

1t :

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

M/M

stat

x/L

0.025 0.125

1t :

Small beam

Small beam

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

V/V

stat

x/L

0.025

0.125

1t :1t :1t :

Small beam

Small beam

62

Figure 5.3 Calculated peak deflection y, moment M and shear force V using Euler-

Bernoulli theory and normalized to the corresponding static quantities.

The results in Figure 5.2–5.3 are related to the value of a similar beam subjected to a

static load of the same magnitude. However, it is also of interest to compare the bending

moments and shear forces to the ultimate capacities of the concrete beam. For the

purpose of performing such comparison, simulations using Timoshenko beam elements

were performed for beam types B7(5) and B27(5). The Timoshenko beam theory also

includes the effects of shear deformations and rotary inertia that become increasingly

significant for deep beams and dynamic loads. These calculations were analysed in

Abaqus using linear beam elements of 38 mm length and with the same dimensions as

the B7 and B27 beams. The same elastic modulus and load were used as in the

corresponding simulations. The beams were subjected to a uniformly distributed load

with a peak pressure of 15 MPa and a triangular duration of 4.0 ms.

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.025 0.125

y/y s

tat ;

M/M

stat

; V

/Vst

at

1t

y

M

V

63

(a) B7 beam. The moment shown at 133 mm ( 0.6d) distance from the support.

(b) B27 beam. The moment shown at 57 mm ( 1d) distance from the support.

Figure 5.4 The bending moment (M) and shear forces (V) at the support over time

normalized to their ultimate capacities for a beam of type (a) B7 beam

and (b) B27. Calculations using Timoshenko beams subjected to a

uniformly distributed load of 15 MPa and a duration of 4.0 ms.

These simulations were part of the work in Paper V and the results in Figure 5.4 are

extracted from that work. The figure illustrates the increase in bending moments and

0

0.25

0.5

0.75

1

1.25

1.5

1.75

2

0 0.025 0.05 0.075 0.1 0.125 0.15 0.175 0.2

M/M

u ;

V/V

u

Time (ms)

M

V

B7 beam

0

0.25

0.5

0.75

1

1.25

1.5

1.75

2

0 0.025 0.05 0.075 0.1 0.125 0.15 0.175 0.2

M/M

u ;

V/V

u

Time (ms)

M

V

B27 beam

64

shear forces over time. The bending moment is taken at the locations where the

maximum bending moment occurs at an early time, i.e. at 133 mm (0.6d) and 57 mm

(1d) distance to the supports for the B7 and B27 beam, respectively. The shear is taken

at the supports. These two quantities are normalized to their ultimate calculated

capacities. The bending moment capacity was calculated according to FKR (Swedish

Fortifications Agency 2011):

(5.1)

where fd and As is the steel yield strength and area of the reinforcement, respectively,

and d denotes the effective depth. The same reinforcement area was used as for the

B7(5) and B27(5) beams, see further Section 4.2.3 and Paper V. A dynamic increase

factor at a strain rate of 1 s-1 was used for the reinforcement as presented in Figure 3.7.

The ultimate shear capacity refers to the direct shear strength, which was calculated as

the maximum shear capacity at a/d < 0.45, see further Section 2.4.1.

For beam B7, the simulations show that the shear forces reach the ultimate value at

approximately half the time (i.e., at 0.060 ms) after the load was applied compared to

that of the bending moments (i.e., at 0.13 ms), see Figure 5.4 (a). However, for the B27

beam, the shear forces and bending moments reach their ultimate values rather close to

each other (i.e., at 0.038 ms and 0.051 ms, respectively). Thus, there is a tendency that

a shallow beam may reach either its shear or its moment capacity first, while shear

clearly reaches its ultimate capacity much earlier that the bending moments in the case

of a deeper beam. These simulations indicate that a deeper beam is more susceptible to

a shear failure while a shallower beam may be more susceptible to failing in shear or

possibly flexure close to the support. Furthermore, it is recognised that this elastic

response is a simplified approach compared to non-linear finite element analyses using

solid elements. However, the calculations indicate which failure mode to expect.

5.2 Flexural shear failures

5.2.1 Bond between reinforcing bars and concrete

Figure 5.5–5.6 present the resulting damage evolution due to concrete cracking and the

force-slip obtained in the simulation compared to the test result. The damage shows a

similar local conic failure of the concrete as observed by Magnusson (2000). Figure 5.6

shows that the entire force-slip response obtained in the simulation is completely below

the curve from the test. This is expected since the test can be regarded as a pull-out

failure with failure by shearing of the concrete between the ribs, while the simulation

𝑀 = 0.95𝑓𝑑𝐴𝑠𝑑

65

is based on a model that describes failure by splitting of the concrete. The result in

Figure 5.5 shows a shearing failure since the cracks were not extended away from the

bar itself. It is also noted that the slip at maximum force and the slope of the descending

part of the force-slip curve were approximately the same as in the test.

Figure 5.5 The concrete damage in tension from the simulation.

Figure 5.6 The result from the pull-out simulation compared to the test result

(Magnusson 2000, Chalmers University of Technology).

0

10

20

30

40

50

60

70

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Forc

e (

kN)

Slip (mm)

Test

Simulation

66

5.2.2 Flexural shear crack patterns

The resulting concrete damage due to cracking of the concrete in the simulations of the

B40-D3 and B40-D4 beams reported in Magnusson and Hallgren (2000) are presented

in Figure 5.7–5.8. The simulations show flexural and shear cracks in a similar fashion

as obtained in the tests, even though the location of the shear cracks appear further

away from the supports in the simulations. The simulations using Ansys Autodyn and

the RHT concrete model in Paper III also resulted in a shear crack further away from

the supports than in the tests. One reason may be the employment of the modelled

interface between the concrete and reinforcement. The simulations reported in Paper

III show that such bond-slip model results in a shear crack further away from the

supports compared to the case without this model. Therefore, simulations were

performed herein without the interface material model such that the reinforcing bars

were connected directly to the concrete elements, see Figure 5.8. Without the interface

material, the number of flexural cracks increased and the shear cracks developed

slightly closer to the supports. Such increase of flexural cracks were also noted in Paper

III. The concrete cracks along the reinforcement from the shear crack and towards the

supports are also noted in Figure 5.8 (b).

The results in Figure 5.9, using a concrete element size of 2 2 2 mm3 also with and

without the interface material, were obtained. For both cases, shear cracks appeared

although with different shapes and further away from the supports compared to the

simulations using a 4 mm mesh. The number of flexural cracks decreased compared to

the corresponding simulation with the 4 mm mesh, which indicates that the interface

material responds in a softer manner as the element size with constant interface

properties. Furthermore, similar to the coarser mesh, the number of flexural cracks

increased without an interface material. The simulations show flexural cracks pointing

away from midspan at a slight inclination, except for Figure 5.9 (a), or a slight tendency

for a few cracks with such inclination. Such cracks were not observed in the tests and

the reason for these crack inclinations is unclear to the author. Figure 5.10 presents a

sequence of the simulation in Figure 5.8 (a) illustrating the propagation of flexural and

shear cracks. This figure shows that flexural shear failures in dynamic events follow

the same series of events as in the case of static loads. A similar result was obtained

with another software as presented in Paper IV.

67

Figure 5.7 Simulation with concrete damage in tension using a 4 mm concrete mesh

compared to tested beam B40-D3. The simulation image is mirrored at

mid-span.

(a) With interface between the reinforcing bars and the concrete.

(b) Without interface between the reinforcing bars and the concrete.

Figure 5.8 Simulations (a) with and (b) without an interface. The results show the

concrete damage in tension using a 4 mm concrete mesh compared to

tested beam B40-D4. The simulation images are mirrored at mid-span.

68

(a) With interface

(b) Without interface

Figure 5.9 Simulation results with concrete damage in tension using a 2 mm concrete

mesh compared to tested beam B40-D4. The simulation images are

mirrored at mid-span.

(a)

(b)

(c)

Figure 5.10 Simulation results of the propagation of concrete cracks (damage in

tension). The simulation sequence is shown at (a) 2.0 ms, (b) 3.0 ms and

(c) 4.0 ms after the load was applied.

5.2.3 Flexural shear deflections

Using the 4 mm mesh, the deflections at mid-span could be predicted relatively well

for the B40-D4 simulation, while the B40-D3 simulations exhibits a stiffer response

compared to the test result, see Figure 5.11. A similarly stiff response was obtained for

the B40-D3 beam in a corresponding simulation with Ansys Autodyn as reported by

Magnusson & Hansson (2005).

69

(a)

(b)

Figure 5.11 Simulations of midspan deflections compared to test results for beam (a)

B40-D3 and (b) B40-D4.

The deflections decreased when the reinforcement was tied directly to the concrete

without the interface material, which is valid also for the simulation using a 2 mm

mesh. This is in agreement with the results in Paper III. The simulation herein using

the same mesh but with the interface resulted in softer response with increased

0

2

4

6

8

10

12

14

0 2 4 6 8 10 12 14

Mid

span

de

fle

ctio

n (

mm

)

Time (ms)

Test

Sim. 4 mm mesh

Beam B40-D3

0

5

10

15

20

25

0 2 4 6 8 10 12 14

Mid

span

de

fle

ctio

n (

mm

)

Time (ms)

Test

Sim. 4 mm mesh

Sim. 4 mm mesh; no interface

Sim. 2 mm mesh

Sim. 2 mm mesh; no interface

Beam B40-D4

70

deflections compared to the test result. In this case, the deflections follow the test result

quite well up to approximately 6 ms. At approximately this point in time the shear

failure occurred, and the tensile reinforcement across the shear crack will be subjected

to an increased tensile force. This force is trying to pull the reinforcing bars towards

the supports out of the concrete, which puts a higher demand on the anchorage length

of the bars. As previously mentioned in Section 5.2.2, the interface appears to respond

in a weaker manner as the mesh size is reduced. The increased deflections after 6 ms is

interpreted to be the cause of reduced reinforcement anchorage with the risk of bars

being pulled out of the concrete when the shear failure occurs.

5.2.4 Flexural shear support reactions

The support reactions in the simulations were calculated using element stresses in the

support top surface. Figure 5.12–5.14 shows that the reactions could be estimated

relatively well in comparison to the tests on beams B40-D3 and B40-D4. However, the

simulation could not capture the reaction peak at 2.7 ms from the B40-D3 test in

Figure 5.12. It is also noted that the reaction forces in the simulations include high-

frequency oscillations that were not present in the test results. The presence of the

oscillations will be further discussed below in this section. In spite of the oscillations,

the main trend of the calculated reactions is that they appear to follow the registered

reactions up to the peak value. During the descending part of the reactions, there are

some discrepancies between the simulations and the test. The reactions in Figure 5.13

follow the test result relatively well. These results indicate that assuming full bond

between the reinforcing bars and concrete elements results in slightly greater reactions

compared to the case with a modelled interface material. It is probable that this is due

to the increased stiffness of the beam when the bars are tied directly to the concrete.

The simulation results in Figure 5.14, using a 2 mm mesh, show the same results. In

this figure, it is further noted that the simulation with an interface material was able to

capture the relatively large dip in reactions that was observed in the test at

approximately 5 ms. This dip can be referred to the point in time when the shear failure

occurred in the test, which is also shown in the corresponding simulation.

71

Figure 5.12 Simulations of total support reactions (using a 4 mm mesh) compared to

test results for beam B40-D3.

Figure 5.13 Simulations of total support reactions compared to test results for beam

B40-D4 using a 4 mm mesh

0

50

100

150

200

250

300

350

400

450

0 2 4 6 8 10 12 14

Sup

po

rt r

eac

tio

ns

(kN

)

Time (ms)

Test

Simulation

Beam B40-D3

0

50

100

150

200

250

300

350

400

450

500

0 2 4 6 8 10 12 14

Tota

l su

pp

ort

re

acti

on

s (k

N)

Time (ms)

Test

Sim. interface

Sim. no interface

Beam B40-D44 mm mesh

72

Figure 5.14 Simulations of total support reactions compared to test results for beam

B40-D4 using a 2 mm mesh.

The average frequency of the oscillations in both simulations were determined to

2128 Hz. The air blast tests used a sampling frequency of 50 kHz and analogue filters

with a cut-off frequency of 15 kHz for the support reactions (Magnusson & Hallgren

2000). Thus, high-frequency oscillations similar to the simulations did not appear in

the tests. However, similar high-frequency oscillations were observed in a separate

series of analyses using Ansys Autodyn and the RHT concrete model reported in

Magnusson & Hansson (2005). If these frequencies are filtered, the reactions would

become smoother and obtain values approximately between the peak and the minimum

of each oscillation. However, such approach appears to underestimate the reactions

measured in the tests at certain points in time.

As discussed in Section 2.3.3, it is known that dynamic loads with a small rise time

such as blast loads, excites several vibration modes in the loaded element. Such

vibrations appear to influence the support reactions. The expression for calculations of

the natural frequencies of the beam is given in Eq. (5.2). In order to determine the

natural frequencies, the magnitudes of the elastic modulus of concrete Ec and the

second moment of area I, need to be established. For the concrete beam, I depends on

the crack state of the beam. The expression for the calculation of I in the cracked state

is according to Mårtensson (1996) given by Eq. (5.4–5.5), and the expression for

calculating the average value of I of the uncracked and cracked sections as suggested

by Biggs (1964) is given in Eq. (5.6). The results from calculations of the natural

frequencies of the beam with the use of Eq. (5.2–5.6) with varying values of Ec and I

is presented in Table 5.2. In this context, b, d and L denote the beam width, depth and

0

50

100

150

200

250

300

350

400

450

500

0 2 4 6 8 10 12 14

Tota

l su

pp

ort

re

acti

on

s (k

N)

Time (ms)

Test

Sim. interface

Sim. no interface

Beam B40-D42 mm mesh

73

span, respectively. Furthermore, x denotes the depth of the compressive block, while

and A are the mass density and the cross section area, respectively. Finally, As and Es

are the area of reinforcement and elastic modulus of steel, respectively.

The natural frequencies for different vibration modes were calculated to find any

frequencies that correspond to that of the support reactions from the simulations. Thus,

the frequencies for an uncracked (state 1) and a cracked (state 2) cross section, as well

as for an average value of the uncracked and cracked sections according to Biggs

(1964) were calculated. In addition, the same calculations were conducted using the

dynamic increase factor for Ec, denoted Ec,d. Assuming a strain rate of 1.0 s-1 for

concrete, the dynamic increase factor for Ec is approximately 1.3, see Figure 3.4. The

results of these calculations are presented in Table 5.1. As previously mentioned, the

average frequency of the oscillations in both simulations were determined to 2128 Hz.

This frequency is close to the calculated EcI2 = 2126 Hz, which indicates that the

oscillations may be the result of a beam response in mode 5. The measured reactions

appear to correspond to the third vibration mode.

(5.2)

(5.3)

(5.4)

(5.5)

(5.6)

𝑓 =𝜔

2𝜋

𝑥

𝑑=𝐴𝑠𝑏𝑑∙𝐸𝑠𝐸𝑐 1 + 2

𝑏𝑑

𝐴𝑠∙𝐸𝑐𝐸𝑠− 1

𝜔 = 𝑛𝜋

𝐿

2

∙ 𝐸𝑐𝐼1𝜌𝐴

𝐸𝐼2 = 0.5𝑏𝑑3𝐸𝑐 𝑥

𝑑

2

1−𝑥

3𝑑

𝐸𝐼1.5 = 𝐸𝑐𝑏𝑑3

2 5.5

𝐴𝑠𝑏𝑑

+ 0.083

74

Table 5.1 Calculated natural frequencies for the first five modes of vibration of

concrete beam B40 subjected to a uniformly distributed blast load.

Vibration

mode

f (Hz)

Ec I1 Ec I1.5 Ec I2 Ec,d I1 Ec,d I1.5 Ec,d I2

1 119 101 85.0 136 115 97.0

3 1070 910 765 1220 1038 873

5 2973 2528 2126 3390 2883 2424

I1 = second moment of area (I) for an uncracked section; I1.5 = average value of I for a cracked

and uncracked section; I2 = I for a cracked section; Ec,d = dynamic elastic modulus of concrete

5.3 Direct shear failures

The DS1 test showed that the roof slab was completely separated from the walls along

vertical failure planes, while the failure planes in the DS4 test were observed to be

inclined Slawson (1984), see Figure 5.15. In the DS1 test, the centre part of the slab

was observed to remain relatively flat, which indicates that the main deformations

occurred in a narrow region around the supports. In the corresponding simulations, the

slabs exhibit damage zones at both supports with severe concrete crushing throughout

the entire depth, see Figure 5.16. The crushed compressive struts at each support were

fully developed at approximately 0.5 ms after the load was applied. Both simulations

exhibit inclined failure planes, which deviates from that observed in the DS1 test but,

is in agreement with the observations from test DS4. Corresponding simulations using

a 2 mm mesh for the concrete results in similar damage zones in compression and

cracking compared to the results in Figure 5.16 (a) and (b), see Figure 5.17.

Furthermore, the simulations show that a centre portion of the slab is relatively flat at

0.5 ms, which was also reported in the DS1 test. In this test, the walls were pushed in

approximately 100 and 130 mm from vertical at the top, respectively, and the

corresponding measurements for test DS4 was 65 and 75 mm, see Figure 5.15. Such

deformations in the supporting walls could not be observed in the simulations due to

the chosen boundary conditions of the model. In the DS1 simulation, the deflection of

the central portion of the slab was approximately 7 mm at 0.5 ms and the walls would

not be forced to bend at such a small deflection. The deformations of the walls are

estimated to occur at a much later time compared to the evolution of the failure planes

at the supports. Therefore, the displacements of the walls are believed not to have

influenced the response of the slab in the simulations. Furthermore, the first flexural

cracks appeared at a distance of approximately one beam depth from the supports in

75

both simulations at 0.3 ms, see Figure 5.16–5.17. This indicates that the slab is mainly

responding in shear during the initial response. Finally, Figure 5.18 presents a

simulation sequence that illustrates the propagation of compressive damage. This

figure shows that the damage in the compressive struts originate in the interior corner

of the slab and supporting wall, and propagates in the direction upwards over time.

In both tests, several reinforcing bars across the supports were pulled to failure and the

remaining bars were pulled out of the concrete. The simulations show that the top

reinforcing bars reach plastic strains prior to 0.5 ms. This is presented in Figure 5.38

and further discussed in Section 5.6.2. It is probable that the strains in these bars will

continue to increase as the slab moves downwards and eventually pulled to failure or

pulled out of the concrete. Furthermore, the analyses show that the region in the vicinity

of the supports were subjected to a triaxial stress state in compression, see

Figure 5.19–5.20. Von Mises stresses in Figure 5.20 are merely plotted to illustrate the

stress distribution and to point out which elements are used for the plots of the

compressive stresses in Figure 5.19. It is recognised that the compressive and tensile

stresses are the ones that control a failure initiation and propagation in a concrete

structural element. It is clear that the compressive stresses in Figure 5.19 change rapidly

over time and at different locations that are relatively close to one another.

(a) DS1 (b) DS4

Figure 5.15 View of the failed slabs after the DS1 and DS4 tests. Modified from

Slawson (1984).

Failed slabs

76

(a) DS1 compressive damage

(b) DS1 tensile damage

(c) DS4 compressive damage

(d) DS4 tensile damage

Figure 5.16 Simulations of the DS1 and DS4 slabs using a 4 mm mesh, and with

concrete compressive and tensile damage 0.5 ms after the load was

applied. The deformations are exaggerated with a factor of 10.

(a) DS1 compressive damage

(b) DS1 tensile damage

Figure 5.17 Simulation of the DS1 slab using a 2 mm mesh, and with concrete

compressive and tensile damage 0.5 ms after the load was applied. The

deformations are exaggerated with a factor of 10.

77

(a) 0.1 ms

(a) 0.2 ms

(a) 0.3 ms

(a) 0.4 ms

(a) 0.5 ms

Figure 5.18 A simulation sequence of the compressive damage of DS1. Each sequence

is shown at 0.1, 0.2, 0.3, 0.4 and 0.5 ms after the load was applied. The

deformations are exaggerated with a factor of 10.

78

(a) Element 1

(b) Element 2

Figure 5.19 Compressive stress distribution at the locations of the two elements as

specified in Figure 5.20. The directions 11 and 22 refer to the longitudinal

and vertical directions, and 33 refers to the direction of the slab width.

-90

-80

-70

-60

-50

-40

-30

-20

-10

0

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

Stre

ss (

MPa

)

Time (ms)

S11

S22

S33

11

22

33

Element 1

-90

-80

-70

-60

-50

-40

-30

-20

-10

0

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

Stre

ss (

MPa

)

Time (ms)

S11

S22

S33

11

22

33

Element 2

79

Figure 5.20 Mises stresses at the right inner corner of the slab-wall junction for DS1.

5.4 Parametric studies – failure modes

The simulations with beam types B7, B12 and B27 that were subjected to varying peak

pressures and load durations showed that the resulting failure mode depended on

several parameters. The results from these simulations are thoroughly discussed in

Paper V but will be briefly portrayed herein. One of the conclusions made in the paper

is that both the applied pressure level and the load duration are important parameters

that control a direct shear failure mode. The applied pressure level needs to be of

sufficient magnitude to drive the beam to a direct shear failure. Also, the applied load

needs to possess a sufficiently long duration in order to cause such a failure. It is evident

that these two load parameters interact and may cause a change in failure mode

depending on the load intensity. Another conclusion of the analyses is that deeper

elements, such as the B7 and B12 beams, are more susceptible to respond in a shear

failure mode compared to slimmer elements, such as the B27 beam. This is a known

fact in static loading cases (Kani, 1967) and this also seems to be the fact for

dynamically loaded structural elements. Figure 5.21 presents simulations with a B7(5)

Element 1

Element 2

80

and a B27(5) beam subjected to the same load. The B7(5) beam appears to fail in a

direct shear mode while the B27(5) beam appears to fail in flexural shear.

Also, the reinforcement appears to influence the behaviour such that the beam becomes

more susceptible to shear as the reinforcement is increased. This was observed in the

analyses of B12(5)-1 and B12(2)-1 in Paper V where the former indicates a more severe

damage zone due to concrete crushing. The results of these simulations are also

included here as shown in Figure 5.22. The behaviour that a beam having more

reinforcement is more susceptible to shear failure is in agreement with concrete

elements subjected to static loads such as the findings of Kani (1966). Tests on concrete

beams subjected to blast loads also show the same tendency for flexural shear as

reported in Paper II.

Figure 5.21 Simulations of B7(5) and B27(5) beams subjected to a pressure of 15 MPa

and with a duration of 0.5 ms. The plots are at 0.7 ms after the load was

applied.

(a) B7(5) damage in compression

(b) B7(5) damage in tension

(c) B27(5) damage in compression

(d) B27(5) damage in compression

81

Figure 5.22 Simulations of B12(2) and B12(5) beams subjected to a pressure of

15 MPa and with a duration of 0.5 ms. The plots are at 0.7 ms after the

load was applied.

Finally, the analyses presented in Paper V show that the dynamic direct shear mode is

a combination of bending moment and shear in the vicinity of the supports, which

results in a deep beam response in a static case. Therefore, a dynamic direct shear mode

will not be governed by the same mechanisms as static direct shear. Once the failure

zone has evolved throughout the depth of the element, a sliding motion commences

that separates the element along a near-vertical plane. The direct shear mode may also

involve the formation of a web shear crack near the supports. In a case where the failure

is due to web shear, the failure planes appear to have a larger inclination to the vertical.

(a) B12(5) damage in compression

(b) B12(5) damage in tension

(c) B12(2) damage in compression

(d) B12(2) damage in tension

82

5.5 Parametric studies – support reactions

5.5.1 Flexural shear

A series of simulations were performed using the B12(2) and B12(5) beam types

subjected to triangular blast loads with varying peak pressures and a duration of 2.0

and 10 ms. The results from these simulations and the corresponding calculated support

reactions, using Eq. (2.24) in Section 2.4.3, are given in Figure 5.23–5.24. These

figures show that the calculated reactions underpredict those from the simulations in

all cases except for B12(5) subjected to a pressure of 2000 kPa and a 2.0 ms duration

in Figure 5.24 (a). In this case the calculated value of reactions is very close to the peak

reactions from the simulation. For the cases with a load duration of 2.0 ms, the

calculated reactions underpredict the first reaction pulse but predicts the second pulse

better. The reaction calculations for the case with a 10 ms load duration shows

underprediction of several reaction pulses, see Figure 5.23 (b) and Figure 5.24 (b).

Similarly to the discussion in Section 5.1.3, the simulations in Figure 5.23–5.24 exhibit

high-frequency oscillations with a frequency of approximately 1550–1600 Hz. If these

frequencies are filtered, the reactions would become smoother and obtain values

approximately between the peak and the minimum of each oscillation. This would

result in calculated reactions above those from the simulations in Figure 5.23 and

Figure 5.24 (a), and a slight underprediction in Figure 5.24 (b).

Comparing the amplitudes of the reactions in diagrams (a) and (b) in Figure 5.23, shows

that a reaction of approximately 270 kN causes a shear failure for a 2.0 ms duration

load, while 120–130 kN is sufficient for a duration of 10 ms. Figure 5.24 shows a

similar result where the corresponding reactions are approximately 310 kN and

165–205 kN, respectively. Thus, comparing the reactions of beams that are subjected

to different load durations, it is apparent that the duration is also a driving parameter

that influences the reactions. Also, the duration of each reaction pulse appears to be a

key parameter that controls shear failure. This becomes clearer when comparing the

reactions that caused shear failure in Figure 5.25 (a). The impulse of these reactions

was evaluated by numerical integration of the reaction-time curves in Figure 5.24 (a)

and (b) and presented in Figure 5.25 (b). Two curves represent the impulse that caused

shear failure, while the two other curves represent the impulse without a failure. These

latter curves are below the values of the former curves, which indicates that a certain

reaction impulse is necessary do cause a shear failure. This evaluation would still be

valid in a case where the reaction oscillations are filtered since the impulse of the

reactions needs to be maintained,

83

(a)

(b)

Figure 5.23 Calculated support reactions for beam type B12(2) subjected to blast

loads of (a) 2.0 and (b) 10 ms duration.

0

50

100

150

200

250

300

0 2 4 6 8 10

Re

acti

on

s (k

N)

Time (ms)

p = 3000 kPa

p = 2000 kPa

FKR

FKR

B12(2)td = 2.0 msShear failure

FKRFKR

0

20

40

60

80

100

120

140

160

0 2 4 6 8 10

Re

acti

on

s (k

N)

Time (ms)

p = 1200 kPa

p = 800 kPa

FKR

FKR

B12(2)td = 10 ms

Shear failure

84

(a)

(b)

Figure 5.24 Calculated support reactions for beam type B12(5) subjected to blast

loads of (a) 2.0 and (b) 10 ms duration.

0

50

100

150

200

250

300

350

0 2 4 6 8 10

Re

acti

on

s (k

N)

Time (ms)

p = 3500 kPa p = 2500 kPaFKRFKR

B12(5)td = 2.0 ms

Shear failure

0

50

100

150

200

250

0 2 4 6 8 10

Re

acti

on

s (k

N)

Time (ms)

p = 1500 kPa

p = 1000 kPa

FKR

FKR

B12(5)td = 10 msShear failure

85

(a)

(b)

Figure 5.25 Comparison between (a) calculated support reactions and (b)

corresponding reaction impulse for beam type B12(5) subjected to blast

loads of 2.0 and 10 ms duration.

0

50

100

150

200

250

300

350

0 2 4 6 8 10 12

Re

acti

on

s (k

N)

Time (ms)

p = 1500 kPa, = 10 ms

p = 3500 kPa, = 2.0 ms

td

td

B12(5)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 2 4 6 8 10 12

Re

acti

on

imp

uls

e (

kNs)

Time (ms)

p = 1000 kPa, = 10 ms

p = 1500 kPa, = 10 ms

p = 2500 kPa, = 2.0 ms

p = 3500 kPa, = 2.0 ms

td

td

td

Shear failure

td

B12(5)

86

5.5.2 Direct shear

A series of simulations were performed using the B7(5), B12(5) and B27(5) beam types

subjected to triangular blast loads with varying peak pressures and duration. The results

from these simulations show that the support reactions depend on both the amplitude

and the load duration of the applied load, see Figure 5.26–5.27. The latter figure

summarises the reactions from the simulations of all the beam types. These results

indicate that the reactions level off at a load duration of approximately 4 ms for the

used beam geometries, span and applied load. The results also show a slightly larger

relative increase between the reactions at 8 ms duration compared to the reactions at

0.5 ms duration for a deeper beam. Furthermore, Figure 5.26 also illustrates that the

reactions depend on the type of beam section, where the deepest beam exhibits the

greatest reactions compared to slimmer beams. Thus, apart from the applied pressure,

also the beam stiffness appears to influence the reactions.

The support reactions were calculated with the use of Eq. (2.24) in Section 2.4.3. One

may question this approach because the deflected shape of the beam at this relatively

early point in time where the maximum reactions occur is very different from the

fundamental flexural mode. Figure 5.26 shows that the maximum support reactions

occur at approximately 0.3 ms for the B12(2) beam type, while the maximum reactions

occurred at approximately 0.35 ms and 0.2 ms for beam types B7(2) and B27(2). At

these early times, the beams have not yet deflected in its fundamental mode, see

Paper V. This is especially the case for beam type B27(2).

The results from the simulations and the corresponding calculated support reactions,

using Eq. (2.24), are given in Figure 5.28–5.30. Figure 5.28 shows that the calculated

reactions exhibit a steeper slope with respect to the applied pressure compared to the

reactions in the simulations. Thus, for the B7 beams, there is a tendency to underpredict

the reactions at lower pressures and overpredict the reactions at higher pressures. A

similar result was obtained for the B27 beams, although the calculated reactions do not

underpredict the reactions at lower pressures. The calculated reactions for the B12

beams show relatively similar calculated reactions compared to the reactions in the

simulations. The general tendency is that the calculations appear to underpredict the

actual reactions at lower pressure levels and overpredict the reactions at higher

pressures. In Figure 5.28–5.30, the simulated reactions using concrete strength

parameters of varying strain rates are also included to illustrate the variation in

reactions due to strain rate effects. This indicates that the reactions increase to a certain

degree for a higher assumed strain rate of the concrete. Also in in these figures with the

exception of Figure 5.28 (a) and Figure 5.30 (a), the simulated reactions for an applied

load with varying durations are included.

87

Figure 5.26 Support reactions from simulations of beam type B12(5) with a peak

pressure of 15 MPa and varying durations.

Figure 5.27 Support reactions from simulations of beam types B7(5), B12(5) and

B27(5) subjected to a peak pressure of 15 MPa and varying durations.

0

200

400

600

800

1000

1200

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Re

acti

on

s (k

N)

Time (ms)

= 0.5 ms

= 1.0 ms

= 4.0 ms

= 8.0 ms

td

td

td

td

p = 15 MPa

B12(5)

0

200

400

600

800

1000

1200

1400

0.0 2.0 4.0 6.0 8.0 10.0

Re

acti

on

s (k

N)

Duration (ms)

B7(5)

B12(5)

B27(5)

p = 15 MPa

88

(a)

(b)

Figure 5.28 Support reactions from simulations and calculations using the design

manual according to FKR (2011) for beam types (a) B7(2) and (b) B7(5).

0

200

400

600

800

1000

1200

1400

1600

1800

2000

0 5 10 15 20 25 30 35

Re

acti

on

s (k

N)

Pressure (MPa)

FEA

FKR

B7(2)

휀̇ 1 s-1

휀̇ 5 s-1

td = 0.5 ms

0

500

1000

1500

2000

2500

3000

0 10 20 30 40 50

Re

acti

on

s (k

N)

Pressure (MPa)

FEA

FEA variation

FKR

B7(5)

휀̇ 5 s-1

td = 4 ms휀̇ 5 s-1

td = 1 & 4 ms휀̇ 1 s-1

휀̇ 1 s-1

td = 0.5 ms

89

(a)

(b)

Figure 5.29 Support reactions from simulations and calculations using the design

manual according to FKR (2011) for beam types (a) B12(2) and (b)

B12(5).

0

200

400

600

800

1000

1200

1400

0 2 4 6 8 10 12 14 16 18 20 22

Re

acti

on

s (k

N)

Pressure (MPa)

FEA

FEA variation

FKR

B12(2)

휀̇ 1 s-1

휀̇ 5 s-1

td = 1 & 4 ms

td = 0.5 ms

0

200

400

600

800

1000

1200

1400

1600

1800

0 5 10 15 20 25 30 35

Re

acti

on

s (k

N)

Pressure (MPa)

FEA

FEA variation

FKR

B12(5)

td = 1, 2 & 4 ms

휀̇ 5 s-1

휀̇ 1 s-1

td = 0.5 ms

90

(a)

(b)

Figure 5.30 Support reactions from simulations and calculations using the design

manual FKR (2011) for beam types (a) B27(2) and (b) B27(5).

0

100

200

300

400

500

600

700

800

900

1000

0 5 10 15 20 25

Re

acti

on

s (k

N)

Pressure (MPa)

FEAFEA variationFKR

B27(2)

휀̇ 5 s-1

휀̇ 1 & 10 s-1

B27(2)

휀̇ 5 s-1

휀̇ 1 & 10 s-1

td = 0.5 ms

0

200

400

600

800

1000

1200

0 5 10 15 20 25

Re

acti

on

s (k

N)

Pressure (MPa)

FEA

FEA variation

FKR

B27(5)

휀̇ 1 and 10 s-1

td = 1 and 4 ms

휀̇ 5 s-1

td = 0.5 ms

91

5.6 Parametric studies – shear capacity

Determining the shear capacity of reinforced concrete elements is important in order to

properly design such elements to resist blast loads. Determining the shear span is of

importance in the calculations of the shear capacity using the design manual FKR

(Swedish Fortifications Agency 2011). In order to do so, an evaluation of the shear

span and strains in the reinforcement are discussed in the two following sections. In the

subsequent sections, beam simulations serve as a basis for comparisons of different

expressions in the calculations of the flexural shear and direct shear capacities.

5.6.1 Evaluation of the shear span

The shear span in the simulations at different times was evaluated by considering the

von Mises stress distribution of the beam and locating the compressive zone of the

temporarily small beam as shown in Figure 5.31. The inclined compressive struts

between the loaded surface and the supports are clearly visible in this figure. The

distance from the face of the support to the centre of the compressive zone resembles

2a, in Figure 5.31, and twice this distance denotes the apparent shear slenderness L’/d

at that point in time. For each simulation, the measurements started off at 0.1 ms after

the load was applied and continued for each point in time with a time step of 0.1 ms

and ended at 0.5–0.6 ms. By performing this procedure in altogether 27 simulations for

beam types B7, B12 and B27 resulted in the diagram presented in Figure 5.32. The

growth of L’/d over time at a certain velocity is apparent in this figure. At

approximately 0.4–0.5 ms, the velocity reduces and the growth of L’/d levels off and

stops, which is due to large damage zones developing and arresting further growth. For

the B7 and B27 beams, a few simulations start off with a relatively high velocity up to

0.2 ms after which the growth of L’/d continuous at the same velocity as the other

simulations for each beam type. The reason for this initially higher velocity is the

evolution of damage zones in compression and tension of the beam. For beams

subjected to lower peak pressures, the damage zones develop at a slightly later point in

time. This results in an initially stiffer beam compared to beams subjected to higher

pressures. A similar behaviour is also shown for the B12 beams. The broken black lines

resemble the average response of each beam type. The amount of reinforcement for

each beam type does not appear to influence the behaviour at these early times.

92

Figure 5.31 Evaluation of the shear span using the von Mises stress distributions. The

simulations are shown at (a) 0.2 ms and (b) 0.4 ms after the load was

applied.

Figure 5.32 Evaluation of the apparent beam span to depth ratio L´/d at different

points in time for 9 of type B7, 10 of type B12 and 8 of type B27 beams.

2a

2a

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

L'/d

Time (ms)

B7

B12

B27

B7

B12

B27

(a)

(b)

93

The broken black lines in Figure 5.32 were used as reference in the subsequent

evaluation of the shear span and L’/d at varying loads presented in Figure 5.33–5.35.

These figures illustrate L’/d obtained from the calculations using Eq. (2.9–2.11) in

Section 2.4.1 and with Figure 5.32 as reference. The calculations using these equations

result in a specific value of L’/d, while in reality L’/d grows over time and the

simulations do not provide a specific value. Thus, for the purpose of this evaluation,

two methods were employed. Firstly, the point in time when initiation of the failure

zone occurs was used as an indication that L’/d is reached. For example, this shows

itself as a few elements reaches a damage level that resembles crushed concrete at the

supports. Another example is the initiation of a web shear crack that eventually causes

failure. Secondly, the point in time where the maximum support reactions is reached.

At that time, the cross section is evidently subjected to the maximum shear.

In Figure 5.33–5.35., these two methods were used and are denoted ‘Initiation of

failure’ and ‘FEA reactions’, respectively. The remaining curves in these figures

resemble the calculated values using Eq. (2.9–2.11).

According to Eq. (2.9–2.11), the shear span should decrease at an increased applied

pressure level, which is also the case using the two evaluation methods and presented

in Figure 5.33–5.34. The results for the B27 beams in Figure 5.35 show that L’/d

increase beyond a pressure level of 15 MPa. Overall, the results in these figures show

a similar trend but also show deviations between the calculated and the evaluated values

of L’/d. In some cases the calculated values using Eq. (2.9) and (2.11), correspond

relatively well to the results from the simulations, such as for the cases with the B7(5)

and B12(5) beams. For the remaining beams, the deviations were greater even if the

trends were similar. Also, using Eq. (2.11) where inertia is included (denoted ‘FKR Eq.

(2.11)’) resulted in similar values as using Eq. (2.9) as shown for beams B7(2), B12(2)

and B12(5). For beam B7(5), the Eq. (2.11) calculations resulted in greater L’/d, while

the opposite result was obtained for the B27 beams. It is also noted that for Beam B7(2)

in Figure 5.33 (a), the evaluation using the maximum reactions from the simulations

exhibited a different trend compared to the curves in the other diagrams. Instead of an

increasead L’/d at a reduced applied pressure, this curve appears to level off at pressures

of approximately 5–10 MPa.

94

(a)

(b)

Figure 5.33 Evaluation of the apparent beam span to depth ratio L´/d at different

applied peak pressures for the (a) B7(2) and (b) B7(5) beams.

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0 5 10 15 20 25 30 35 40

L´/d

Pressure (MPa)

Initiation of failure

FEA reactions

FKR Eq. (2.9)

FKR Eq. (2.10)

FKR Eq. (2.11)

B7(2)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0 5 10 15 20 25 30 35 40

L´/d

Pressure (MPa)

Initiation of failure

FEA reactions

FKR Eq. (2.9)

FKR Eq. (2.10)

FKR Eq. (2.11)

B7(5)

95

(a)

(b)

Figure 5.34 Evaluation of the apparent beam span to depth ratio L´/d at different

applied peak pressures for the (a) B12(2) and (b) B12(5) beams.

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

0 5 10 15 20 25 30

L´/d

Pressure (MPa)

Initiation of failure

FEA reactions

FKR Eq. (2.9)

FKR Eq. (2.10)

FKR Eq. (2.11)

B12(2)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

0 5 10 15 20 25 30

L´/d

Pressure (MPa)

Initiation of failure

FEA reactions

FKR Eq. (2.9)

FKR Eq. (2.10)

FKR Eq. (2.11)

B12(5)

96

(a)

(b)

Figure 5.35 Evaluation of the apparent beam span to depth ratio L´/d at different

applied peak pressures for the (a) B27(2) and (b) B27(5) beams.

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

0 5 10 15 20 25

L´/d

Pressure (MPa)

Initiation of failure

FEA reactions

FKR Eq. (2.9)

FKR Eq. (2.10)

FKR Eq. (2.11)

B27(2)

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

0 5 10 15 20 25

L´/d

Pressure (MPa)

Initiation of failure

FEA reactions

FKR Eq. (2.9)

FKR Eq. (2.10)

FKR Eq. (2.11)

B27(5)

97

5.6.2 Evaluation of reinforcement strains

The initial response illustrated in Figure 5.2 (Section 5.1), shows that bending moments

occur relatively close to the supports at early times. This may lead to yielding of the

reinforcement for sufficiently high loads. The plastic strains in the reinforcement for

simulations where the beams of types B7(2), B12(2) and B27(2) failed in direct shear

are presented in Figure 5.36. The B7(2) beam was subjected to a peak pressure of

30 MPa, and the B12(2) and B27(2) beams were subjected to 20 MPa. The load

duration was 0.5 ms for all beams. Figure 5.36 shows that the plastic strains are

localized in the area where the damage zone of crushed concrete occurs close to the

supports. The simulations of beams B12(2) and B27(2) also show that plastic strains in

the reinforcement develop in areas where flexural shear cracks initiated and propagates.

Even though plastic strains develop, these strains are not distributed across the entire

cross section of each bar. The central portion of each bar does not reach yielding at

0.6 ms after the load was applied as presented in Figure 5.37. This figure presents the

development of plastic strain over time for the cross section of the reinforcing bars as

marked by the broken lines in Figure 5.36. The curves in Figure 5.37 represent the

plastic strains in each element of that cross section. The distributions of plastic strains

indicate that the reinforcing bars are bending due to a transverse motion throughout the

beam depth. Such a transverse motion is also visible through the displacement of the

reinforcing bars. Furthermore, even though the reinforcement does not appear to reach

yielding, it is likely that the reinforcement reaches plastic strains throughout the cross

section of the bars as time progresses and the beam continues in a downwards motion.

The simulations of the DS1 test, see Section 5.1.2, show that the top reinforcing bars

reach plastic strains 0.3 ms after the load was applied as presented in Figure 5.38. This

figure shows that plastic strains mostly develop in the top bars at 0.5 ms after the load

was applied. Over time as the slab moves downwards and flexural cracks develop the

bottom reinforcement also reaches yielding. It is also likely that the top bars reach

relatively high plastic strains during the downward movement of the slab.

98

Figure 5.36 Plastic strains in the reinforcement 0.6 ms after the load was applied for

(a) B7(2), (b) B12(2) and (c) B27(2). The beams were subjected to 30

MPa, 20 MPa and 20 MPa, respectively, all with a duration of 0.5 ms.

The deformations are exaggerated with a factor 10. The broken lines

mark the cross section of the plastic strains presented in Figure 5.37.

(a) B7(2)

(b) B12(2)

(b) B27(2)

Flexurar shear

cracks

Flexurar shear

crack

99

(a)

(b)

Figure 5.37 The development of plastic strains over time in in each element of the

reinforcement marked with the broken lines in Figure 5.36 for (a) B7(2),

(b) B12(2) and (c) B27(2).

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.2 0.4 0.6 0.8 1 1.2

Pla

stic

str

ain

(%

)

Time (ms)

B7(2)

-0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0 0.2 0.4 0.6 0.8 1

Pla

stic

str

ain

(%

)

Time (ms)

B12(2)

100

(c)

Figure 5.37 continued.

Figure 5.38 Plastic strains in the reinforcement at one of the supports 0.5 ms after the

load was applied for the DS1 simulation. The deformations are

exaggerated with a factor of 10.

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0 0.2 0.4 0.6 0.8 1

Pla

stic

str

ain

(%

)

Time (ms)

B27(2)

101

5.6.3 Evaluation of flexural shear capacity

In order to evaluate the flexural shear capacity, a series of simulations using B12(2)

and B12(5) beams were performed with varying magnitudes of peak pressure. The

series were also divided into simulations with a 2.0 ms and 10 ms load duration. Even

though shear cracks may develop, a beam may still exhibit a flexural response mode.

This was observed in several tests where concrete beams were subjected to blast loads

and did not fail in shear even though shear cracks were present as shown in Paper II

and reported in Magnusson & Hallgren (2000). For a shear failure to occur, the beam

needs to be separated along a diagonal crack. In this separation the two beam parts

displace in relation to each other in a transverse motion, which may be detected by

analysing the plastic strains and deformations of the reinforcing bars. In a shear failure,

the reinforcement exhibits strains and deformations that reveal local bending along a

limited length similar to a dowel action, see Figure 5.39. Therefore, the distribution of

plastic strains and deformations were employed as the distinguishing factor in order to

determine if a shear failure occurred in the simulations. Beams where a shear crack

developed in the simulations without failing in shear did not show the same

characteristic displacement and local plastic strains in the reinforcement.

(a)

(b)

Figure 5.39 Shear failure of a B12(5) beam subjected to a peak pressure of 4000 kPa

and 2.0 ms duration. The figures illustrate (a) concrete cracking and (b)

plastic strains in the reinforcement. The deformations in (b) are

exaggerated with a factor of 5.

Location of

shear crack

102

Figure 5.40–5.41 present the results from the analyses at varying pressure levels, and

for a 2.0 ms and 10 ms load duration. The reactions from the simulations (denoted

‘FEA’ in the figures) are presented at the phase of the support and also at 1d and 2d

distance from the support, respectively. In most cases, the shear cracks developed at

approximately 2d from the supports. The calculations using the models in Eurocode 2

(Swedish Standards Institute 2005) and the draft revision of Eurocode 2 (CEN 2018)

are denoted ‘EN2’ and ‘Draft’, respectively. These calculations show that they

underpredict the shear strength of the beams for all cases in Figure 5.40–5.41. The

calculations using the FKR model (Swedish Fortifications Agency 2011) appear to

predict the shear failure relatively well, with respect to the shear at a distance of 2d

from the supports. This is valid for all cases except for that of beam B12(2) subjected

to a load duration of 10 ms in Figure 5.41 (a). The reason for this result appears to be

the combination of higher reactions in the simulations of the B12(5) beams at the same

applied load compared to the B12(2) beam reactions, and higher calculated shear

strength for the B12(2) beam.

In the model for calculating the shear capacity in FKR, the calculated shear slenderness

becomes important. The ratio q/p determines the magnitude of the shear span, see

Section 2.4.1, which means that reduced values of q results in smaller shear spans. This

is the case for the B12(2) beam and, consequently, the shear capacity increases in

relation to the B12(5) beam according to Section 2.4.1. Thus, the combination of

reduced reactions and an increased calculated shear capacity results in the curves

presented in Figure 5.41 (a). The same relations apply to the results in Figure 5.41 (b)

but the influence of the ratio q/p becomes less prominent at those pressure levels.

103

(a)

(b)

Figure 5.40 Reactions from simulations of B12(5) beams and calculated shear

capacities at different applied peak pressures, and a load duration of (a)

10 ms and (b) 2.0 ms.

0

50

100

150

200

250

0 500 1000 1500 2000 2500

R a

nd

VR

(kN

)

Pressure (kPa)

FEA FEA 1d FEA 2d FKR Draft EN2

Shear failure

EN2 VR

Draft VR

FKR VR

B12(5)td = 10 ms

0

50

100

150

200

250

300

350

400

1000 1500 2000 2500 3000 3500 4000 4500

R a

nd

VR

(kN

)

Pressure (kPa)

FEA FEA 1d FEA 2d FKR Draft EN2

Shear failure

EN2 VR

Draft VR

FKR VR

B12(5)td =2.0 ms

104

(a)

(b)

Figure 5.41 Reactions from simulations of B12(2) beams and calculated shear

capacities at different applied peak pressures, and a load duration of (a)

10 ms and (b) 2.0 ms.

0

20

40

60

80

100

120

140

700 800 900 1000 1100 1200 1300

R a

nd

VR

(kN

)

Pressure (kPa)

FEA

FEA 1d

FEA 2d

FKR

Draft

EN2

Shear failure

EN2 VR

Draft VR

FKR VR

B12(2)td = 10 ms

0

50

100

150

200

250

300

350

1500 2000 2500 3000 3500 4000

R a

nd

VR

(kN

)

Pressure (kPa)

FEA

FEA 1d

FEA 2d

FKR

Draft

EN2

Shear failure

EN2 VR

Draft VR

FKR VR

B12(2)td =2.0 ms

105

5.6.4 Evaluation of direct shear capacity

Simulations using beam types B7(5), B12(5) and B27(5) were used in the evaluation

of the direct shear capacity. Comparisons were made between the results of the

simulations and calculations of the direct shear capacity using the expressions

presented in Sections 2.4.1–2.4.2 Eq. (2.3–2.7), Eq. (2.21) and Eq. (2.23). Calculations

using these expressions were compared to the simulation results in Figure 5.42–5.44.

Calculations using Eq. (2.21) includes the dynamic compressive strength of concrete.

The simulations show variations in the strain rate within the area of the supports

between approximately 1–10 s-1. Based on Figure 3.3 in Section 3.1.1 for a compressive

strength of 45 MPa, the dynamic increase factor for concrete in compression at a strain

rate of 1.0 and 10 s-1 is 1.26 and 1.33, respectively. Thus, an average DIF of 1.3 was

used in the calculations of the direct shear capacity using Eq. (2.21). The abbreviations

‘UFC’, ‘Krauth.’ and ‘FKR’ in these figures refer to Eq. (2.17), (2.19) and (2.3–2.7),

respectively.

For beams B7(5) and B12(5), the figures illustrate that the calculations using Eq. (2.19)

appear to predict the direct shear capacity reasonably well. This is also the case for

Eq. (2.17) although these calculations appear to be more conservative. Using the

equations in FKR appear to give results that underpredict the direct shear capacity. In

the case of beam B27(5) in Figure 5.44, the results indicate that all the calculations

underpredicted the shear capacity with quite a large margin. The simulations show that

the direct shear failure develops due to concrete crushing (and possibly splitting) of the

compressive struts at the supports. Therefore, it was of interest to also include new

calculations that consider such a failure mode. Thus, in addition to the expressions

presented previously in this section, Eq. (2.3) in Section 2.4.1 was modified into

Eq. (5.7) and Eq. (5.8). Eq. (5.7) is the capacity of the compressive concrete strut at the

supports without any additional contribution from the reinforcement. In Eq. (5.8), the

same dynamic increase factor of 1.3 is employed as in Eq. (2.7) to account for the

average increase in concrete compressive strength at strain rates of 1.0 s-1 and 10 s-1.

(5.7)

(5.8)

Comparing the calculations using these expressions with the simulations provided the

results presented in Figure 5.45–5.46 for the B7(5) and B12(5) beams. Here, the

abbreviations ‘Krauth.’, ‘FKR concr.’ and ‘FKR concr. DIF’ refer to Eq. (2.19), (5.7)

and (5.8), respectively. In this case, all calculations appear to predict the direct shear

capacity reasonably well. It is interesting to note that the calculations using the

𝑉𝑐 = 0.25𝑓𝑐𝑘𝑏𝑑

𝑉𝑐 = 0.25 ∙ 1.3𝑓𝑐𝑘𝑏𝑑

106

‘Krauth.’ and the ‘FKR concr. DIF’ expressions ended up at the same shear capacity.

However, this would not be the case where other reinforcement ratios are used.

Figure 5.42 Reactions from simulations of B7(5) beams with varying peak pressures

and calculated direct shear capacities. Load durations of 0.5 and 4.0 ms

were used.

Figure 5.43 Reactions from simulations of B12(5) beams with varying peak pressures

and calculated direct shear capacities. A load duration of 0.5 ms was

used.

0

200

400

600

800

1000

1200

1400

8 10 12 14 16

R a

nd

VR

(kN

)

p (MPa)

FEA

UFC

Krauth.

FKR

B7(5)Direct shear

td = 4.0 ms

0

200

400

600

800

1000

1200

2 4 6 8 10 12 14 16

R a

nd

VR

(kN

)

p (MPa)

FEA

UFC

Krauth.

FKR

B12(5)

Direct shear

107

Figure 5.44 Reactions from simulations of B27(5) beams with varying peak pressures

and calculated direct shear capacities. A load duration of 0.5 ms was

used.

Figure 5.45 Reactions from simulations of B7(5) beams with varying peak pressures

and modified calculations of the direct shear capacities. Load durations

of 0.5 and 4.0 ms were used.

0

200

400

600

800

1000

5 10 15 20 25

R a

nd

VR

(kN

)

p (MPa)

FEA

UFC

Krauth.

FKR

B27(5)Direct shear

0

200

400

600

800

1000

1200

1400

8 10 12 14 16

R a

nd

VR

(kN

)

p (MPa)

FEA

Krauth.

FKR concr.

FKR concr. DIF

B7(5)Direct shear

td = 4.0 ms

108

Figure 5.46 Reactions from simulations of B12(5) beams with varying peak pressures

and modified calculations of the direct shear capacities. A load duration

of 0.5 ms was used.

0

200

400

600

800

1000

1200

2 4 6 8 10 12 14 16

R a

nd

VR

(kN

)

p (MPa)

FEA

Krauth.

FKR concr.

FKR concr. DIF

B12(5)

Direct shear

109

Chapter 6

Summary of appended papers

6.1 Paper I: Fibre reinforced concrete beams

subjected to air blast loading

Magnusson, J.

Nordic Concrete Research (2006), 35, pp. 18-34.

This paper involves testing of steel fibre reinforced concrete (SFRC) beams subjected

to static and blast loads. Unreinforced concrete in tension is characterized by a brittle

failure but by adding steel fibres to the matrix, the ductility can be improved

considerably. The fibres provide stress transfer across cracks that form under tensile

stresses, which may arrest further crack propagation. However, as a concrete specimen

is subjected to a continuously increasing tensile load, a major crack will eventually

form and, after the peak load has been reached, crack opening will follow under the

unloading phase. During this phase the fibres across the crack are either pulled out of

the matrix or pulled to rupture. Fibre pull-out characterizes a ductile failure of the

concrete specimen. The purpose of the work in Paper I was to investigate the

mechanical behaviour of SFRC beams subjected to various blast loads.

A total of 40 beams were tested, of which 22 beams were subjected to blast loading and

18 to static loading. Three concrete grades were used with concrete strengths of 36

MPa, 97 MPa and 189 MPa. The beams were reinforced with Dramix hooked-end steel

fibres with a volume fraction of 1.0 %. Half of the beams contained 30 mm long fibres

and the other half were reinforced with fibres of 60 mm length. The static beam tests

were performed with four point loads and the blast tests were performed in a shock tube

with an explosive charge at a 10 m distance from the beam. At this distance, the blast

load can be considered as a uniformly distributed load across the beam surface.

The tests show that, for beams of the two lower concrete grades, the failure mechanism

was by fibre pull-out, while a combination of fibre pull-out and fibre ruptures were

observed for beams of the highest grade. In the static tests, beams with the two lowest

concrete grades and with long fibres exhibited greater strain hardening compared to

110

those containing short fibres. However, for beams with the highest concrete grade,

beams with short fibres obtained the greatest strain hardening. The probable cause for

this may be the larger amount of fibre ruptures for the long fibres that was observed.

For beams of the highest concrete grade, an increased number of fibre ruptures in the

blast tests in relation to the static tests were observed. In these beams, the blast tests

also revealed an increased number of fibre ruptures in the beams with long fibres

compared to beams with short fibres. Furthermore, the load capacity of the beams

increased in the blast tests compared to the corresponding static tests, see Figure 6.1.

The main reason may be due to strain rate effects. Both the static and the dynamic load

capacity also increased with an increased concrete strength.

Even though SFRC has many benefits, the traditional way of reinforcing concrete

structural elements with steel bars is still preferred in the design of blast resistant

structures. The presence of steel bars enables the element to deform and absorb the

dynamic load in a controlled manner. Previous research by Magnusson & Hallgren

(2000) indicate that a flexural shear failure could be prevented by adding steel fibres.

Another investigation by Magnusson & Hallgren (2003) also indicate that the

introduction of steel fibres contribute to increased ductility of the concrete in the

compression zone of the beam. The increased ductility appears to contribute to an

enhanced residual strength in the post-peak stage of a statically loaded beam.

Figure 6.1 Mean load capacities in the tests. The numerals above the bars indicate

the ratio between the dynamic and static load capacity. The beam types

resemble: S=short fibres, L=long fibres, numeral=concrete grade

(MPa).

111

6.2 Paper II: Air-blast-loaded, high-strength

concrete beams. Part I: Experimental

investigation

Magnusson, J., Hallgren, M. & Ansell, A.

Magazine of Concrete Research (2010), 62 (2), pp. 127-136.

This paper presents a summary of tests on 49 reinforced concrete beams that were

performed using both static and blast loads. Altogether 38 beams were subjected to a

variety of blast loads and the remaining 11 beams were tested with static loads as

reference. The tests series consisted of 11 beam types, each type having individual

concrete grade and amount of reinforcement. The purpose of the investigations was to

analyse the structural behaviour of reinforced beams of high strength concrete (HSC),

i.e. concrete with a compressive strength exceeding 80 MPa. The static tests were

performed with four point loads and the blast tests were performed in a shock tube with

an explosive charge at a 10 m distance from the beam. The blast can be considered as

a uniformly distributed load across the beam surface at this distance.

For beams with the same amount of reinforcement, the static tests showed that the load

capacity and stiffness increased with an increase in concrete strength. Increasing the

concrete strength of beams with the same amount of reinforcement reduces the

mechanical ratio of reinforcement. As this ratio was reduced, an increase in

deformation capacity was observed in the static tests. The blast tests show that the

dynamic load capacity is larger for beams of all concrete grades compared to the

corresponding static load capacity. The results also show that beams failing in flexure

in the static tests could fail in flexural shear when subjected to blast loads. However,

such a shear failure could be prevented for beams of the same concrete grade by

reducing the tensile reinforcement and thereby reducing the load capacity and stiffness

of the beam. In this case the beam failed in flexure due to the reduction of shear forces

in the beam.

In the blast tests, beams containing steel fibres failed in flexure while similar beams

without fibres failed in shear. Hence, the increased ductility, and possibly the increased

tensile strength, of the concrete matrix in tension prevented a shear crack to develop

and propagate. The results from the static tests also show that steel fibres contribute to

an enhanced ductility of the concrete in the compression zone. This enhances the

ductility of the entire beam response and allows for increased deflection capabilities.

112

6.3 Paper III: Air-blast-loaded, high-strength

concrete beams. Part II: Numerical non-linear

analyses

Magnusson, J., Ansell, A. & Hansson, H.

Magazine of Concrete Research (2010), 62 (4), pp. 235-242.

Numerical simulation enables detailed analyses of the structural response of reinforced

concrete elements subjected to blast loads. This paper presents such analyses using the

software Ansys Autodyn version 5.0 with the experimental results presented in Paper

II as reference. The RHT material model was employed for calculations of the stress-

strain states of the concrete and a damage model, which describes the damage evolution

and strength reduction due to increasing plastic strains. A linear and a bi-linear function

was used for the crack softening of concrete in tension. The Johnson & Cook

constitutive model was used for description of the stress states of the reinforcing steel.

The bond between concrete and reinforcement was modelled using a bond-slip

relationship according to CEB (1993). Simulations were performed with full bond and

bond-slip in order to evaluate its effect. Strain rate effects were included for concrete

and reinforcement but not for the bond-slip model.

Overall, the simulations showed the ability to analyse beams of varying concrete

strengths and content of reinforcement subjected to blast loads with good accuracy. The

simulations show some noticeable effects when considering with or without strain rate

effects of the reinforcement. Employing strain rate effects results in a slightly stiffer

response with reduced deflections. This is in agreement with a corresponding static

case with an increased yield strength of the beams.

The simulations demonstrates the ability to correctly predict the failure mode of beams

of varying concrete grades subjected to blast loads. Comparing simulations with a

linear and a bi-linear crack softening in tension does not seem to affect the crack

patterns noticeably. Using a bond relationship between the concrete and reinforcing

bars results in increased deflections and a reduced amount of flexural cracks. In this

case the location of the shear cracks changed such that the crack appeared at a larger

distance to the supports in the direction towards midspan. A similar change in location

of the shear cracks was observed in simulations where the strain rate effects of the

reinforcement was excluded.

113

6.4 Paper IV: Shear in concrete structures

subjected to dynamic loads

Magnusson, J., Hallgren, M. & Ansell, A.

Structural Concrete (2014), 15(1), pp. 55-65.

This paper presents a review of the literature dealing with shear in reinforced concrete

elements subjected to dynamic loads such as explosions and impacts. The review

focused on parameters that control shear and, for this reason, the initial response was

also highlighted. In dynamic events, high stresses and strains can occur locally in the

structure for short periods of time. The effects of structural wave propagation, strain

rate effects and dynamic load characteristics therefore need to be considered in shear

analyses. Elastic analyses using Euler-Bernoulli beam theory illustrates the effects of

flexural wave motions and build-up of shear close to the supports soon after a

distributed dynamic load has been applied. The results of the review concluded that

shear in concrete elements depends on load characteristics and structural parameters.

Load characteristics that were found to typically contribute to shear are peak load and

rise time. Typical characteristics of impulsive loads are high pressures, small rise times

and short durations, which is the reason such loads contribute to large shear forces in

the element. The load duration was reported in Ross (1983) as not having a significant

influence on direct shear. However, the load duration may have some influence on

flexural shear since this mode occurs at a much later time. Structural parameters

important to shear were concluded to be element resistance and stiffness, span-to-

effective depth (L/d) ratio and strain rate effects. Higher stiffness and resistance

contribute to larger shear forces in the element compared to a softer element with a

lower resistance. Strain rate effects in the concrete and reinforcing steel also contribute

to stiffer elements.

Arch action in the shear span will always be present in a concrete element, which

distributes a portion of the load directly to the supports. Soon after the load has been

applied, shear forces and bending moments will occur in the vicinity of the supports

while the remaining beam is straight and subjected to a rigid body motion. The element

may therefore be regarded as temporarily responding with an apparently low shear

slenderness, which may contribute to an enhanced shear strength. Wave propagation

effects over time will change the shear distribution, eventually becoming similar to that

of quasi-static loading, and the apparent shear slenderness will increase. Thus, the

enhancements in shear strength should decrease as the response progresses.

114

6.5 Paper V: Numerical analyses of dynamic shear

in concrete structures subjected to distributed

loads

Magnusson, J., Hallgren, M., Malm, R. & Ansell, A.

Submitted to Engineering Structures (2019)

The purpose of this paper was to analyse reinforced concrete beams subjected to

extreme dynamic loads with the use of numerical simulations. The shear mechanism

during dynamic loading is not yet fully understood, especially in the case for direct

shear failures. Therefore, the analyses focused on direct shear failure modes at early

structural response times due to uniformly distributed pressure loads. The software

Abaqus/Explicit 6.11 was used for modelling and simulations of reinforced concrete

beams subjected to varying blast loads. In this work, three types of beams with different

depths of the cross section were modelled. The amount of reinforcement was also

varied for each cross section. The Concrete Damaged Plasticity (CDP) model was used

for the calculations of the stress-strain states of the concrete and a damage model, which

describes the damage evolution and strength reduction of the concrete due to increasing

plastic strains. A bi-linear tension softening model was employed. The bond between

concrete and reinforcement was modelled using a bond-slip relationship that was based

on Magnusson (2000, Chalmers University of Technology) and fib (2012).

The analyses show that the dynamic direct shear mode appears to be a combination of

bending moment and shear in a deep beam response and is therefore different from the

static direct shear mode. Dynamic direct shear appears to be initiated by concrete

crushing in the vicinity of the supports. Once the failure zone has evolved throughout

the depth of the element, a sliding motion commences that separates the element along

a near-vertical plane. The direct shear mode may also involve splitting of the concrete

or the formation of a web shear crack near the supports. In a case where the failure is

caused by web shear, the failure planes appear to have a smaller inclination to the

horizontal plane. The simulations also indicate that a deeper beam is more susceptible

to dynamic shear failures compared to a slender beam, and that a higher reinforcement

content causes the beam to become more susceptible to shear compared to a

corresponding beam with less reinforcement. These findings are in agreement with

beams subjected to static loads. The analyses also show that both the blast pressure and

the duration of the load influence the development of a direct shear failure.

115

Chapter 7

Discussion

7.1 General

The initial beam deflections during the early response of the beam are discussed in

Section 5.1 and also in Paper IV and V. Similar curves of the initial beam deflections

as in Paper IV were generated from the simulations using solid elements. The initial

response in the simulations was verified by mapping the beam deflections at equidistant

points along the beam span over time for beams B7(2), B12(2) and B27(2). The

distance between each point was 40 mm and the deflections were plotted for every

0.1 ms as shown in Figure 7.1. Corresponding simulations with the use of an elastic

Timoshenko beam were carried out as reference as shown in the figure. The

Timoshenko beam theory also includes the effects of shear deformations and rotary

inertia that become increasingly significant for deep beams and for dynamic loads when

higher vibration modes are excited. These calculations were analysed in Abaqus using

linear beam elements of 38 mm length and the same dimensions, elastic modulus and

load as in the solid-element simulations. The diagram representing the B12(2) beam is

also included in Paper V. The deflected shapes at different times exhibit a similar

pattern as those obtained in Paper IV. Soon after the dynamic load has been applied to

the beam surface, a wave will propagate through the thickness of the beam and the

entire beam will be accelerated. As soon as the pressure wave reflects against the rear

surface of the beam, bending moments and shear forces develop at the supports while

the remaining beam will be subjected to a rigid body motion. This condition is reflected

in the deflected curves at 0.1 ms in Figure 7.1. Structural wave motions over time will

cause the beam to deflect in its fundamental motion of vibration. In the figure, the

beams B7(2) and B12(2) appear to deflect in their fundamental mode at approximately

0.4 ms and 0.5 ms, respectively. Beam B27(2) reaches this deflection mode at times

beyond 0.6 ms. Thus, it is evident that deeper beams reach their fundamental vibration

mode earlier than slimmer beams. Furthermore, the deflected shapes of the elastic

Timoshenko beam simulations agree well with the simulations using solid element

models even though there are slightly increasing deviations over time between the two

types of simulations. The results of the simulations show that concrete cracking and

crushing need a certain time to develop into specific failure zones (e.g. see Figure 5.18),

116

and therefore the beam initially exhibits an approximately elastic response. As the

concrete damage evolves over time, there is a continuous change in beam stiffness and

the deviations in deflections compared to the Timoshenko simulations grow.

Figure 7.1 Calculated deflected shapes at different times for beam types (top) B7(2),

(middle) B12(2) and (bottom) B27(2) subjected to a uniformly distributed

load of 10 MPa and a duration of 0.5 ms. Each curve represents a time

difference of 0.1 ms.

-2.0

-1.8

-1.6

-1.4

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

De

fle

ctio

n (

mm

)

x/L

FEA Timoshenko

FEA solid elem.

B7(2)

-3.5

-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

De

fle

ctio

n (

mm

)

x/L

FEA Timoshenko

FEA solid elem.

B12(2)

117

Figure 7.1 continued.

The initial response soon after the load has been applied shows that the concentration

of bending moments and shear within a narrow region of the beam in the vicinity of the

supports may initially subject the cross section to large stresses, see Figure 5.2. This is

supported by blast tests on concrete beams where relatively large reactions were

registered before any noticeable deflections were measured (Magnusson, 2007). Thus,

at a sufficiently intense load, the cross section of the element may reach its ultimate

strength and fail. Specifically the shear forces appear to reach relatively large values

compared to the evolution of bending moments at an early time, see Figure 5.3.

However, as the beam depth is reduced, the point in time where the bending moments

and shear forces reach their capacities become rather similar, which may cause a

slender beam to respond in flexure rather than shear as shown in Figure 5.4. The

moment and shear curves in this figure are normalized to the calculated moment and

direct shear capacities. The results in Figure 5.44 indicate that the direct shear capacity

from the simulations appears to be significantly larger for the B27 beam compared to

the calculated capacity. Thus, assuming a direct shear capacity of 700 kN, as indicated

in the simulation (according to Figure 5.44) results in moment and shear curves as

presented in Figure 7.2. In this case a flexural mode is predicted to dominate the

response of the beam. This may be regarded as the effect that deeper beams are more

susceptible to a shear response compared to slender beams in similarity to a static

loading case, as stated in Paper V. This is known from static tests and analyses as

previously discussed in Section 2.2.

-6.0

-5.0

-4.0

-3.0

-2.0

-1.0

0.0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

De

fle

ctio

n (

mm

)

x/L

FEA Timoshenko

FEA solid elem.

B27(2)

118

Figure 7.2 Simulation of the bending moment and shear forces (at supports) over

time normalized to their ultimate capacity using a Timoshenko beam

under a uniformly distributed load of 15 MPa and duration of 4.0 ms.

In the simulations performed herein, the concrete strength properties in compression

and tension were increased due to an assumed constant strain rate in the beam. This is

a simplification since, naturally, the strain rates will change throughout the beam

depending on both location and over time. Due to this, specific simulations were

performed in order to verify the effects of this approach on the structural response in

terms of support reactions and failure modes, see Appendix A. These simulations show

that the chosen strain rate enhancement of the concrete strength resulted in certain

variations in the reactions. The resulting compressive and tensile damage also show

certain variations depending on the chosen strain rate. However, the failure mode did

not change even though the variations in strength properties for different strain rate was

considerable. The simulation of the B27 beam using a strain rate of 30 s-1 did change

the failure mode, but on the other hand, this high strain rate did not occur in the

simulation. The results of these simulations indicate that the influence on the results by

choosing this approach is within reasonable limits.

7.2 Failure in dynamic shear

It is well known in a statically loaded concrete element that a certain deflection occurs

with initiation and propagation of flexural cracks prior to the formation of flexural shear

cracks. The simulations presented herein show that such shear failures follows the same

sequence of events for elements subjected to blast loads, see Figure 5.10. The same

0

0.25

0.5

0.75

1

1.25

1.5

1.75

2

0 0.025 0.05 0.075 0.1 0.125 0.15 0.175 0.2

M/M

u ;

V/V

u

Time (ms)

M

V

B27 beam

119

results was obtained using a Ansys Autodyn and the RHT concrete model as presented

in Paper IV. Once the diagonal crack has formed, the beam could be regarded as failing

but this may not be the case. However, the element needs to possess a certain amount

of remaining kinetic energy to be able to drive the element to a shear failure as the

beam separates along the shear crack. In the simulations, this was the criterion for

evaluating whether a shear failure occurred, see Section 5.6.3. The simulations indicate

that shear cracks may form but the beam did not fail and were therefore able to resist

an increased load. The shear crack appears to initiate approximately 2 ms after the load

was applied, and the shear failure occurred at around 4–5 ms, see Figure 5.10 and

Paper IV. Referring back to the discussion of initial response in Section 7.1, flexural

shear typically occurs at a point in time when the element is responding according to

its fundamental flexural mode.

Tests on roof slab subjected to intense dynamic loads showed that several of the slabs

failed along vertical or near vertical failure planes at the supports in a direct shear mode

as reported by Slawson (1984). The simulations of these tests presented in Section 5.3

indicate that severe crushing occurs throughout the entire depth of the compressive

struts at the supports caused the failure of the slabs along inclined failure planes, see

Figure 5.16. The simulations sequence in Figure 5.18 shows that the crushing damage

of the struts originate in the interior corner of the adjacent supporting wall and

propagates upwards over time. This damage evolution has a strong resemblance to the

shear failure of a deep beam subjected to static loading. As discussed in Section 2.2,

static tests have shown that the failure was caused by crushing of the compressive struts

close to the supports, possibly in combination with splitting of the concrete in the struts.

The compressive strut is clearly visible in Figure 5.31 and also presented in Paper V.

In these figures, the concentration of relatively high stress levels in the region above

the support is notable. It is therefore expected that crushing of the concrete originates

from this region. Also, the compression zone at the top of the beam indicates that the

response has evolved and generated a temporarily small beam. The temporary shear

slenderness L’/d of the beam in Figure 5.31 (a) and (b) are estimated to approximately

2 and 3, respectively. Thus, the initial response and stress distribution support the

presence of a deep beam response during the initial stages of response. The evolution

of the shear failure in Figure 5.18 indicates that crushing of the compressive struts

commence at approximately 0.1–0.2 ms. Thus, this failure mode occurs during the

initial response and the temporarily small beams occur in a similar fashion as shown in

Figure 5.31. Figure 5.21–5.22 present more simulations with crushed concrete struts

and tension cracks for different types of beams. The direct shear mode may also involve

the formation of a web shear crack near the supports. In a case where the failure is due

to web shear, the failure planes appear to have a larger inclination to the vertical, see

also Paper V. In all, the analyses show that the dynamic direct shear mode is a

120

combination of bending moment and shear in the vicinity of the supports. Therefore,

the element may be regarded as temporarily responding in a deep beam response. The

final step of the failure process is a sliding motion that occurs after the struts are

completely crushed throughout the beam depth, and the element separates along the

failure planes.

7.3 Support reactions

7.3.1 Flexural shear

The support reactions from tests and in simulations represent the shear forces at the

supports and are compared to the capacity of the element in a shear design and the

ability to predict the reactions as accurately as possible is therefore vital. A model for

calculating the reactions of a concrete beam subjected to a uniformly distributed blast

load is described by Eq. (2.24) in Section 2.4.3. This expression depends on the

maximum applied load level and the maximum resistance. The simulations of B12(2)

and B12(5) beams show that the general shape of the support reactions change

depending on the combination of pressure level duration of the load. Figure 5.23–5.24

shows that the peak reactions appear at a much earlier point in time for a duration of

2.0 ms compared to the corresponding reactions for a load duration of 10 ms. In the

former case, the maximum reactions appear at approximately 0.4 ms, while the

reactions in the latter case appear at approximately 2–3 ms. At 0.4 ms, the beam has

not yet deflected in its fundamental mode as shown in Figure 7.1 and, ideally, Eq. (2.24)

should be modified to better account for a different deflected shape. However, the

deflected shape is not entirely different from the fundamental mode and this relatively

small deviation does not appear to affect the calculated reactions in a negative manner.

To a certain degree, the calculated reactions appear to underestimate the reactions from

the simulations, Figure 5.23–5.24. The high-frequency oscillations that occurred in the

simulations but not in the tests may need to be taken into account when comparing the

calculated reactions. A probable cause for these oscillations is the ideal model of the

concrete beams without imperfections such as existing microcracks in a real beam. If

these oscillations do not occur in a real structural element, the simulations may

overpredict the actual reactions to a certain degree.

The duration of each reaction pulse appears to control shear failure. This becomes

clearer when comparing the reactions that caused shear failure in Figure 5.25 (a). The

impulse of these reactions was evaluated by numerical integration of the reaction-time

curves in Figure 5.24 (a) and (b) and presented in Figure 5.25 (b). Two curves in this

121

figure represent the impulse that caused shear failure, while the two other curves

represent the impulse without a failure. These latter curves are below the values of the

former curves, which indicates that a certain reaction impulse is necessary to cause

shear failure.

7.3.2 Direct shear

In order to evaluate the reactions for loads that caused a direct shear mode, a series of

simulations were performed using the B7(5), B12(5) and B27(5) beam types subjected

to blast loads of varying peak pressures and durations. The results from these

simulations show that the support reactions depend on both the amplitude and the load

duration of the applied load. The beam stiffness appears to also influence the reactions

such that greater reactions were obtained for deeper beams compared to slimmer

beams. The support reactions were calculated with the use of Eq. (2.24) in a similar

manner as discussed in Section 7.3.1. One may question this approach because the

deflected shape at the early point in time where the maximum reactions occur is very

different from the fundamental flexural mode. Figure 5.26 shows that the maximum

support reactions occur at approximately 0.3 ms for the B12(2) beam type, while the

maximum reactions occurred at approximately 0.35 ms and 0.2 ms for beam types

B7(2) and B27(2). Thus, ideally, Eq. (5.6) should be modified to better account for this

different deflected shape because the distribution of inertial forces are different from

these in the assumption of a fundamental mode of vibration. However, this does not

appear to have affected the calculated reactions in a negative manner. In fact, the

calculated reactions corresponded relatively well to the reactions in the simulations.

Even though the deflected shapes initially deviate from the fundamental mode, the

inertia forces are not completely different in these two cases. Thus, Eq. (5.6) appears

to provide a relatively accurate approximation of the reactions also for impulsive loads.

However, this may not be the case for beams with longer spans because the initial

distribution of inertia forces could to a larger degree deviate from those in a

fundamental mode than for the case with shorter spans.

7.4 Shear capacity

7.4.1 Shear span

The discussion in Sections 5.1 and 7.1–7.2, and in Paper IV and V, shows that there

exists an initial response with the temporary moment and shear distributions of short

beams. It is therefore reasonable to believe that such a temporarily short beam has the

122

ability to transfer larger shear loads prior to failure than a beam with a larger shear

slenderness. This enhancement in shear capacity is time dependent due to structural

wave effects, such that the initial positive effects on the shear capacity will gradually

diminish over time until the fundamental mode of vibration is reached. Such initial

behaviour is of interest to use when considering impulsive loads with large amplitudes

and of short duration. A model that accounts for the effects of such time-dependent

shear capacity was originally included in Publikation 25 (1973) and has also been

incorporated in the design manual FKR. The derivation of the equations used in this

manual is presented in Appendix C. An attempt was made in Section 5.6.1 to evaluate

the equations for determining the shear span of the element based on simulations. The

evaluation exhibited a similar trend of the shear span at an increasing pressure level.

However, there were deviations present that do not establish a foundation to draw any

general conclusions.

7.4.2 Plastic strain in the reinforcement

The bending moment that occur relatively close to the supports soon after the load has

been applied may lead to yielding of the reinforcement for sufficiently high loads. Such

yielding is assumed in the model for calculating the shear slenderness in FKR, see

Appendix C. However, the simulations of the B7(2), B12(2) and B27(2) beams indicate

that the reinforcing bars did not entirely reach plastic strains throughout the cross

sections at the point in time when the direct shear failure was developed. According to

the model in FKR, a case where the stresses in the reinforcement are assumed to stay

below yielding would result in a reduced shear slenderness and an increased shear

capacity. Vecchio & Collins (1988) showed that a beam section fails at lower loads in

cases where high moment and shear occur simultaneously at one location. Such a

location may typically be close to the supports in a direct shear failure. Further

investigations would be necessary to analyse the bending moments that occur close to

the supports soon after the load has been applied.

7.4.3 Flexural shear capacity

In the evaluation of the flexural shear capacity in the simulations, the shear cracks

developed at approximately 2d from the supports in most cases. The calculations using

Eurocode 2 (Swedish Standards Institute 2005) and the draft revision of Eurocode 2

(prEN 1992-1-1:2018, CEN 2018) show that these underpredict the dynamic shear

strength of the beams for all cases as shown in Figure 5.40–5.41, see Section 5.6.3. On

the other hand, the calculations using the FKR model appear to predict the shear failure

123

in a better way. There appears to be a need for further investigations to draw more

general conclusions.

Furthermore, tests on concrete beams subjected to blast loads have shown that beams

containing steel fibre reinforcement failed in flexure while similar beams without fibres

failed in shear as reported in Paper II. This is likely the result of an increased tensile

strength and fracture energy in tension of fibre reinforced concrete, see also Paper I. In

this work it was noted that longer fibres were favourable in terms of toughness of the

beams and load capacity. However, for an increased concrete strength, the tests indicate

that the positive effects of the long fibres are reduced. This is likely due to the observed

fibre fractures that occurred in beams containing long fibres.

7.4.4 Direct shear capacity

The results of the evaluation of the direct shear capacity showed that the used models

were able to fairly well predict the shear capacity. The FKR model appears to give

more conservative results compared to the two other models used (Krauthammer et al.

1986; Department of Defense 2008). The model by Krauthammer predicted the direct

shear failures in the simulations relatively well. The simulations show that the direct

shear failure develops due to concrete crushing (and possibly splitting) of the

compressive struts at the supports. Therefore, part of the FKR model was used to only

consider the failure mode of concrete crushing, see Eq. (5.7) in Section 5.6.4. This

equation was extended to also include strain rate effects of concrete in compression

using a dynamic increase factor (DIF) of 1.3 as shown in Eq. (5.8). This latest

modification resulted in an improved prediction with respect to the results of the

simulations. Apart from the results of the simulations, the use of a DIF is also supported

by the work presented by Krauthammer et al. (1986) and has also been observed in

tests on shear keys as reported by French et al. (2017). Also in this work, the

observation was made that the addition of steel fibres enhanced the direct shear

capacity.

FKR (2011) puts a limit on the allowed reinforcement content to a maximum of 0.5 %,

and it should be noted that the reinforcement content were 0.59 % and 1.47 % for the

beam sections employed in the simulations. Thereby both beam types exceeding the

allowed reinforcement limit. It is therefore of interest to further analyse the flexural

and direct shear capacity with beams of lower reinforcement ratios. Such analyses may

also include varying beam spans. Furthermore, in FKR (2011) the design shear force

at each support occurs at the location of the shear span and is therefore given to be half

the calculated support reactions. However, the analyses performed within the work in

this thesis shows that a direct shear failure occurs due to crushing and splitting of the

124

compressive struts at the supports where the entire reactions arise. It is therefore

advisable to change the design shear force such that the full support reactions are used.

In the case of flexural shear, the design shear force should be taken at the distance d

from the support, which follows the same principle as stated in Eurocode 2 (Swedish

Standards Institute 2005).

125

Chapter 8

Conclusions and further research

8.1 Conclusions

Reinforced concrete elements subjected to an explosion at close range may fail in shear

due to the high intensity of the load. Shear is a brittle failure mode that limits the ability

of the element to deform and respond in flexure. The work in this thesis involved

experimental and theoretical investigations using the finite element method in order to

analyse the different aspects on dynamic shear. Several research questions were

specified in order to direct the work into specific key areas of research as outlined in

Section 1.2. With reference to these questions, the conclusions of the work in this thesis

are stated in the following.

Soon after the dynamic load has been applied, relatively large shear forces and

bending moments may occur within a narrow region of each support. The

magnitude of the load is apparently a key parameter in the initiation and

evolution of shear failures. An increased pressure resulted in an increased shear

force at the supports. According to the literature, the rise time to peak pressure

of the applied load is of importance. However, specific analyses of the influence

of the rise time were not part of the work in this thesis.

Several aspects of the configuration of the concrete element influences the risk

of shear failures. For instance, a stiff beam generally attracts larger shear forces

compared to a beam with lower stiffness. Additional stiffness of a beam may be

due to an increased amount of flexural reinforcement or an increased depth. The

simulations indicate that a deeper beam is more susceptible to dynamic shear

failures compared to a slender beam, and that a higher reinforcement content

may cause the beam to become more susceptible to a shear response compared

to a similar beam with less reinforcement. A possible size effect, which is a

well-known effect in static loading cases, in combination with the enhancement

of shear forces due to the stiffness may contribute to the susceptibility to shear

126

failures of deeper structural elements. Thus, the aspects that affect shear in static

loading cases also appear to be valid in dynamic events.

Dynamic flexural shear is shown to follow the same sequence of events as in

the case with a static load. The simulations show that the shear crack originates

from a flexural crack and propagates into a diagonal crack, at approximately

1–2 beam depths from the supports. Both tests on concrete beams and

simulations show that a flexural shear crack may form but does not necessarily

lead to a shear failure. The element also needs to have a sufficient kinetic energy

to be able to fully develop a shear failure. The dynamic direct shear mode

typically occurs soon after the load has been applied at an early stage of the

structural response and appears to be due to the combination of bending moment

and shear. It is shown that dynamic direct shear appears to follow the same

sequence of events as in a static case of shear in a deep beam. Such response

may cause concrete crushing and possibly splitting of the compressive struts at

the supports. The dynamic direct shear mechanism is therefore different from

the static direct shear mode. The dynamic direct shear mode may also involve

the formation of a web shear crack near the supports.

The calculations using FKR (Swedish Fortifications Agency) appears to

provide a relatively accurate approximation of the reactions for moderate blast

loads that may cause flexural shear. This is also the case for more impulsive

loads of higher intensity that may cause a direct shear failure. The simulations

show that the reactions depend on the applied load level, beam stiffness and

also, to a certain degree, on the duration of the load. The analyses also show

that the combination of peak support reactions and duration control the

evolution of a flexural shear failure. Thus, the structural element may be able

to resist a relatively large reaction with a short duration compared to the case

with a reduced reaction but with longer duration. Therefore, calculations of the

maximum reactions alone may be misleading without also considering the

duration. In FKR, it is prescribed to use half the calculated reactions as the

design shear force for direct shear. However, based on the analyses herein, it is

recommended to instead use the entire reaction since the direct shear failure

develops at the face of the supports. In the case of flexural shear, it is suggested

that the design shear force at a distance equal to the effective beam depth (d )

away from the support should be used. This is in accordance with the European

design rules for concrete structures Eurocode 2 (Swedish Standards Institute

2005).

Calculations of the capacity in flexural shear using Eurocode 2 (Swedish

Standards Institute 2005) and the draft revision of Eurocode 2 (CEN 2018)

127

show that these appear to underpredict the shear strength of beams in dynamic

events. On the other hand, the calculations using the FKR model appear to

predict the shear capacity in a better way. In calculations of the direct shear

capacity, the FKR model appears to result in conservative results. A

modification of this model to also include strain rate effects of the concrete was

made to better reflect the actual failure with crushing of the compressive struts

at the supports. This modification resulted in an improved prediction with

respect to the results of the simulations. Another model by Krauthammer et al.

(1986) also appeared to predict the direct shear capacity relatively well. A third

model (Department of Defense 2008) resulted in underprediction of the direct

shear capacity.

Furthermore, tests on concrete beams containing steel fibres and subjected to blast

loads were observed to fail in flexure, while similar beams without fibres failed in

flexural shear. Long fibres appear to be favourable in terms of an enhanced load

capacity and toughness of a beam. However, for an increased concrete strength, the

positive effects of the long fibres are reduced due to an increased number of fibre

ruptures across flexural cracks. Even though shear may be prevented by the inclusion

of fibres in the concrete, it is apparent that the shear capacity can not rely on fibres as

shear reinforcement. Instead, transverse reinforcement such as stirrups is necessary to

enhance the strength for flexural shear. However, stirrups will not be effective in

preventing direct shear failures.

8.2 Further research

The work in this thesis is based both on testing and on numerical simulations and it is

of interest in future research to further analyse the evolution of flexural shear and direct

shear failures. Such analyses could include variations in parameters such as support

conditions, longer spans at varying element depths, applied pressures and load

durations. Also, the work herein is entirely based on the assumption of a uniformly

applied load. However, an explosion at a relatively close range gives rise to an uneven

load distribution across the span of the element where one support may be subjected to

substantially larger shear forces compared to that of the other support. For this reason,

an uneven load distribution is of interest to include in further research.

All suggestions of further research should involve numerical simulations and an

experimental program. It may be of interest to perform the simulations with another

software and material models for comparisons. Future work should also include

128

analysing the models in FKR for calculations of the shear capacity and compare these

to tests and simulations. In this context, continued analyses of the dynamic shear span

and its influence on the shear capacity during the initial response appears to be of

importance. Furthermore, calculations of the maximum reactions only may be

misleading without also considering the duration of the reactions. Thus, the influence

of maximum reactions and duration on the evolution of shear failures should also be

included in future research.

A certain portion of the results of the simulations were compared to experiments that

were conducted within the scope of this thesis but also compared to other experiments.

The simulations were, however, extended to a variation of geometries and

reinforcement contents in parametric studies. In future research, it is of interest to

include experimental investigations with the same configurations. In addition, the

analyses in this thesis are based on simply supported beams without end-restraints or

axial loads. It is therefore of interest to further analyse dynamic shear using end-

restraints and axial loads. Such work may also include analyses on slabs supported on

all four sides.

129

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137

Appendix A

Parametric study of the strain rate

Separate simulations were performed with the beam types B7(5), B12(5) and B27(5)

in order to verify the dependence on used material strength data in compression and

tension due to different strain rates. In the simulations of flexural shear failures, the

material data for strain rates 0.1, 1.0 and 10 s-1 were used. Corresponding simulations

of direct shear failures used material data for strain rates 1.0, 10 and 30 s-1.

The concrete damage in compression and tension from the simulations are presented in

Figures A1–A4. In the first figure, the applied load was limited to produce flexural

shear failure, while the load was increased to produce direct shear failures in the

remaining figures. Figure A1 shows that varying materialdata for substantially different

strain rates did not change the failure mode. The shear cracks developed closer to the

supports for the case with 0.1 s-1. The results in Figures A2–A3, show that varying the

material properties between 1.0 s-1 and 30 s-1 did not change the failure mode in direct

shear. Crushing of the compressive struts appeared although even though the crushed

zones were reduced for increasd strength properties. The tension cracks also show

similarities for different strain-rate properties. However, the simulations with the B27

beam show a distinct difference in failure modes. In a case with material properties for

1.0 s-1, a direct shear failure was obtained with crushing of the compressive struts,

whereas the failure mode may be interpreted as a flexure shear failures in simulations

using material properties for higher strain rates. Thus, the material properties appears

to play a more significant role for B27 beams compared to the B12 and B7 beams.

138

(a) Concrete properties for a strain rate of 0.1 s-1.

(b) Concrete properties for a strain rate of 10 s-1.

Figure A1. Simulations of B12(5) beams subjected to a uniform pressure of 2.0 MPa

with a duration of 10 ms. Damage in compression (left) and tension

(right). Plots at 3.0 ms after the load was applied.

(a) Concrete properties for a strain rate of 1.0 s-1.

(b) Concrete properties for a strain rate of 10 s-1.

(c) Concrete properties for a strain rate of 30 s-1.

Figure A2. Simulations of B7(5) beams subjected to a uniform pressure of 20 MPa

with a duration of 1.0 ms. Damage in compression (left) and tension

(right). Plots at 0.7 ms after the load was applied.

139

(a) Concrete properties for a strain rate of 1.0 s-1.

(b) Concrete properties for a strain rate of 10 s-1.

(c) Concrete properties for a strain rate of 30 s-1.

Figure A3. Simulations of B12(5) beams subjected to a uniform pressure of 20 MPa

with a duration of 1.0 ms. Damage in compression (left) and tension

(right). Plots at 0.7 ms after the load was applied.

(a) Concrete properties for a strain rate of 1.0 s-1.

(b) Concrete properties for a strain rate of 10 s-1.

(c) Concrete properties for a strain rate of 30 s-1.

Figure A4. Simulations of B27(5) beams subjected to a uniform pressure of 15 MPa

with a duration of 1.0 ms. Damage in compression (left) and tension

(right). Plots at 0.7 ms after the load was applied.

The support reactions from the simulations are presented in Figures A5–A8 for the

corresponding cases shown in Figures A1–A4. Figure A5, that corresponds to a flexural

shear failure, shows that the support reactions are barely affected by different strain

rates. A probable reason is that the reactions depend on the development of tensile

stresses in the reinforcement and not in the concrete. However, the reactions are

affected to a higher degree for the case with direct shear failures in Figures A6–A8.

This is expected because a direct shear mode depends to a large degree on the

compressive struts that develop at the supports.

140

Figure A5. Support reactions from simulations with B12(5) beams at varying strain

rates.

Figure A6. Support reactions from simulations with B7(5) beams at varying strain

rates.

0

50

100

150

200

250

0 2 4 6 8 10 12

Re

acti

on

s (k

N)

Time (ms)

Strain rate 0.1 s-1

Strain rate 1.0 s-1

Strain rate 10 s-1

B12(5)p = 2.0 MPatd = 10 ms

s-1

s-1

s-1

0

200

400

600

800

1000

1200

1400

1600

1800

0 0.2 0.4 0.6 0.8 1 1.2

Re

acti

on

s (k

N)

Time (ms)

Strain rate 1 s-1

Strain rate 10 s-1

Strain rate 30 s-1

B7(5)p = 20 MPatd = 1.0 ms

s-1

s-1

s-1

141

Figure A7. Support reactions from simulations with B12(5) beams at varying strain

rates.

Figure A8. Support reactions from simulations with B27(5) beams at varying strain

rates.

0

200

400

600

800

1000

1200

1400

1600

0 0.2 0.4 0.6 0.8 1 1.2

Re

acti

on

s (k

N)

Time (ms)

Strain rate 1 s-1

Strain rate 10 s-1

Strain rate 30 s-1

B12(5)p = 20 MPatd = 1.0 ms

s-1

s-1

s-1

0

100

200

300

400

500

600

700

800

900

0 0.2 0.4 0.6 0.8 1 1.2

Re

acti

on

s (k

N)

Time (ms)

Strain rate 1 s-1

Strain rate 10 s-1

Strain rate 30 s-1

B27(5)p = 15 MPatd = 1.0 ms

s-1

s-1

s-1

142

143

Appendix B

Material data

B1 Concrete in compression

The relation between stresses and strains for the initial hardening up to the uniaxial

compressive strength fcm was calculated using (fib 2010):

for c ≤ c1 (B.1)

where the parameters are explained in Figure B1. The concrete strain at fcm is calculated

according to (CEB 1993):

(B.2)

Červenka et al. (2018) describes a model with a fictitious compression plane based on

the assumption that compression failure is localized in a plane normal to the

compressive principal stress, where all compressive displacements wd occur. These

displacements are assumed to be independent on the size of the structure, which is

supported by experiments reported by van Mier (1986), see Figure B2. From these

experiments, a value for wd of 0.5 mm was determined for normal strength concrete.

Based on the specimen size, the plastic displacement wd can be calculated according to

Červenka et al. (2018):

(B.3)

where Ld denotes the specimen length. The calculations of the softening branch of the

stress strain curve will thereby depend on the strain in the element of the FEM model.

Using Ld = leq, the expression of the compressive stresses at increasing strains becomes:

for c > c1 (B.4)

𝜎𝑐 =𝐸𝑐𝑖

휀𝑐𝑓𝑐𝑚

− 휀𝑐휀𝑐1

2

1 + 𝐸𝑐𝑖휀𝑐1𝑓𝑐𝑚

− 2 휀𝑐휀𝑐1

𝑓𝑐𝑚

휀𝑐1 = 0.0007 𝑓𝑐𝑚 0.31

𝜎𝑐 = 1−𝑙𝑒𝑞𝑤𝑑

휀𝑐 − 휀𝑐1 𝑓𝑐𝑚

휀𝑑 = 휀𝑐 +𝑤𝑑𝐿𝑑

144

Figure B1 Schematic representation of the stress-strain relation for uniaxial

compression. From fib (2012).

Figure B2 Softening displacement relation in compression. From Červenka et al.

(2018).

145

B2 Concrete properties

The concrete properties were adjusted to fit available data on the specimens in the

experiments. The following parameters were used for all types of concrete.

Eccentricity = 0.1

fb0/fc0 = 1.16

Kc = 0.667

Viscosity parameter = 10-7

= 2400 kg /m3

= 0.2

B1.1 Flexural shear failures of concrete beams

This section refers to simulations of the B40-D3 and B40-D4 beam tests.

Static properties:

fc = 45 MPa

fct = 4.1 MPa

Ec = 34 GPa

GF = 130 N/m

Dilation angle = 45°

Concrete properties at a strain rate of 1 s-1 and 4 mm mesh.

Compression Tension

c (MPa) c,in (m/m) dc c (MPa) w (mm) dt

5.7 0 0.000 5.96 0 0

27.2 9.01E-05 0.042 1.99 0.0151 0.027

39.0 0.000226 0.072 0.06 0.0904 0.847

47.9 0.000425 0.106

An average value of the elastic modulus in

tension and compression was used:

Ec = 46.6 GPa.

53.8 0.000692 0.147

56.7 0.00103 0.195

56.7 0.00163 0.277

55.6 0.00405 0.494

53.5 0.00870 0.685

51.2 0.0138 0.782

47.6 0.0218 0.860

41.2 0.0360 0.921

26.3 0.0690 0.972

146

Concrete properties at a strain rate of 1 s-1 and 2 mm mesh.

Compression Tension

c (MPa) c,in (m/m) dc c (MPa) w (mm) dt

5.7 0 0 5.96 0 0

27.2 9.01E-05 0.042 1.99 0.0151 0.053

39.0 0.000226 0.072 0.06 0.0904 0.917

47.9 0.000425 0.106

An average value of the elastic modulus in

tension and compression was used:

Ec = 46.6 GPa.

53.8 0.000692 0.147

56.7 0.00103 0.195

56.8 0.00163 0.277

56.3 0.00404 0.490

55.2 0.00866 0.677

54.1 0.0137 0.772

52.3 0.0217 0.848

49.1 0.0358 0.907

41.6 0.0687 0.957

B1.2 Direct shear failures of roof slab

This section refers to simulations of the DS1 and DS4 tests.

Static properties:

fc = 27 MPa

fct = 2.4 MPa

Ec = 30 GPa

GF = 52 N/m

Dilation angle = 30°

147

Concrete properties at a strain rate of 1 s-1 and 4 mm mesh.

Compression Tension

c (MPa) c,in (m/m) dc c (MPa) w (mm) dt

3.9 0 0 4.10 0 0

21.9 0.000160 0.082 1.37 0.00879 0.020

29.6 0.000369 0.132 0.04 0.05250 0.800

34.7 0.000643 0.184

An average value of the elastic modulus in

tension and compression was used:

Ec = 41.1 GPa.

37.7 0.000970 0.238

38.9 0.00144 0.311

38.8 0.00174 0.353

37.9 0.00456 0.594

36.5 0.00910 0.752

35.0 0.0141 0.831

32.5 0.0222 0.893

28.1 0.0363 0.940

17.9 0.0694 0.979

B1.3 Parametric studies of concrete beams

This section refers to the parametric studies of concrete beams.

Static properties:

fc = 45 MPa

fct = 4.1 MPa

Ec = 34 GPa

GF = 130 N/m

Dilation angle = 45°

148

Concrete properties at a strain rate of 0.1 s-1.

Compression Tension

c (MPa) c,in (m/m) dc c (MPa) w (mm) dt

5.4 0 0 5.60 0 0

22.6 6.28E-05 0.034 1.87 0.0156 0.026

34.5 0.000178 0.061 0.06 0.0929 0.843

43.8 0.000357 0.093

An average value of the elastic modulus in

tension and compression was used:

Ec = 41.1 GPa.

50.2 0.000604 0.132

53.6 0.000924 0.179

53.9 0.00142 0.249

52.7 0.00414 0.497

50.8 0.00869 0.683

48.6 0.0137 0.781

45.2 0.0218 0.859

39.1 0.0360 0.921

24.6 0.0699 0.973

Concrete properties at a strain rate of 1 s-1.

Compression Tension

c (MPa) c,in (m/m) dc c (MPa) w (mm) dt

5.7 0 0.000 5.96 0 0

27.2 9.01E-05 0.042 1.99 0.0151 0.027

39.0 0.000226 0.072 0.06 0.0904 0.847

47.9 0.000425 0.106

An average value of the elastic modulus in

tension and compression was used:

Ec = 46.6 GPa.

53.8 0.000692 0.147

56.7 0.00103 0.195

56.7 0.00163 0.277

55.6 0.00405 0.494

53.5 0.00870 0.685

51.2 0.0138 0.782

47.6 0.0218 0.860

41.2 0.0360 0.921

26.3 0.0690 0.972

149

Concrete properties at a strain rate of 5 s-1.

Compression Tension

c (MPa) c,in (m/m) dc c (MPa) w (mm) dt

5.9 0 0 10.18 0 0

28.2 9.30E-05 0.044 3.39 0.0106 0.012

40.4 0.000232 0.074 0.10 0.0633 0.703

49.5 0.000434 0.109

An average value of the elastic modulus in

tension and compression was used:

Ec = 48.6 GPa.

55.7 0.000702 0.150

58.7 0.00104 0.198

58.9 0.00163 0.279

57.7 0.00426 0.507

55.6 0.00880 0.688

53.2 0.0139 0.784

49.4 0.0219 0.861

42.8 0.0361 0.922

27.3 0.0692 0.972

Concrete properties at a strain rate of 10 s-1.

Compression Tension

c (MPa) c,in (m/m) dc c (MPa) w (mm) dt

6.0 0 0 12.83 0 0

28.7 9.43E-05 0.045 4.28 0.0101 0.009

41.0 0.000234 0.075 0.13 0.0604 0.646

50.3 0.000438 0.110

An average value of the elastic modulus in

tension and compression was used:

Ec = 49.3 GPa.

56.5 0.000706 0.151

59.9 0.00133 0.240

59.6 0.00224 0.348

58.6 0.00426 0.508

56.4 0.00881 0.689

54.0 0.0139 0.785

50.2 0.0219 0.861

43.5 0.0361 0.922

27.8 0.0692 0.973

150

Concrete properties at a strain rate of 30 s-1.

Compression Tension

c (MPa) c,in (m/m) dc c (MPa) w (mm) dt

6.1 0 0 18.51 0 0

29.4 9.63E-05 0.046 6.17 0.0094 0.006

42.0 0.000238 0.077 0.19 0.0562 0.548

51.4 0.000444 0.112

An average value of the elastic modulus in

tension and compression was used:

Ec = 50.7 GPa.

57.8 0.000713 0.153

61.4 0.00134 0.242

61.0 0.00225 0.350

60.0 0.00427 0.510

57.8 0.00881 0.690

55.3 0.0139 0.785

51.4 0.0219 0.862

44.5 0.0361 0.922

28.4 0.0692 0.973

B1.4 Interface properties

This section refers to the properties of the bond-slip model employed between the

reinforcing bars and concrete. The plastic strain is based on a brick element of 4 mm

length.

= 2400 kg /m3

= 0.2

E = 30 GPa

c (MPa) p (m/m)

1.0 0

2.3 0.0032

3.1 0.0079

3.6 0.013

4.4 0.027

5.4 0.042

6.3 0.072

2.5 0.3

2.5 1.2

151

B2 Reinforcement properties

B2.1 Flexural shear failures of concrete beams

This section refers to simulations of the B40-D3 and B40-D4 beam tests.

Static �̇� = 0.001 s-1 �̇� = 0.01 s-1

T (MPa) p (m/m) T (MPa) p (m/m) T (MPa) p (m/m)

600 0 623 0 646 0

601 0.0010 623 0.0010 647 0.0010

613 0.0180 636 0.0180 660 0.0180

657 0.0296 680 0.0296 703 0.0296

686 0.0392 708 0.0392 730 0.0392

727 0.0583 746 0.0583 765 0.0583

752 0.0770 767 0.0770 782 0.0770

768 0.0953 779 0.0953 790 0.0953

780 0.1133 786 0.1133 793 0.1133

�̇� = 0.1 s-1 �̇� = 1 s-1 �̇� = 10 s-1

T (MPa) p (m/m) T (MPa) p (m/m) T (MPa) p (m/m)

670 0 695 0 722 0

671 0.0010 696 0.0010 722 0.0010

685 0.0180 710 0.0180 737 0.0180

727 0.0296 752 0.0296 778 0.0296

753 0.0392 777 0.0392 801 0.0392

784 0.0583 805 0.0583 826 0.0583

798 0.0770 814 0.0770 830 0.0770

801 0.0953 815 0.0953 831 0.0953

802 0.1133 816 0.1133 832 0.1133

152

B2.2 Direct shear failures of roof slabs

This section refers to simulations of the DS1 and DS4 tests.

Static �̇� = 0.001 s-1 �̇� = 0.01 s-1

T (MPa) p (m/m) T (MPa) p (m/m) T (MPa) p (m/m)

434 0 467 0 503 0

434 0.0010 468 0.0010 504 0.0010

437 0.0078 471 0.0078 507 0.0078

508 0.0149 546 0.0149 582 0.0149

574 0.0247 610 0.0247 649 0.0247

645 0.0392 678 0.0392 714 0.0392

696 0.0535 724 0.0535 754 0.0535

733 0.0677 754 0.0677 777 0.0677

754 0.0816 768 0.0816 782 0.0816

772 0.0933 778 0.0933 785 0.0933

�̇� = 0.1 s-1 �̇� = 1 s-1 �̇� = 10 s-1

T (MPa) p (m/m) T (MPa) p (m/m) T (MPa) p (m/m)

542 0 583 0 628 0

542 0.0010 584 0.0010 628 0.0010

546 0.0078 588 0.0078 633 0.0078

624 0.0149 669 0.0149 717 0.0149

691 0.0247 736 0.0247 785 0.0247

752 0.0392 793 0.0392 837 0.0392

786 0.0535 821 0.0535 857 0.0535

801 0.0677 826 0.0677 858 0.0677

802 0.0816 827 0.0816 859 0.0816

803 0.0933 828 0.0933 860 0.0933

153

B2.3 Parametric studies of concrete beams

This section refers to the parametric studies of concrete beams.

Static �̇� = 0.001 s-1 �̇� = 0.01 s-1

T (MPa) p (m/m) T (MPa) p (m/m) T (MPa) p (m/m)

500 0 530 0 563

501 0.0010 531 0.0010 563 0.0010

515 0.0223 547 0.0223 580 0.0223

528 0.0296 559 0.0296 591 0.0296

561 0.0392 590 0.0392 622 0.0392

603 0.0583 630 0.0583 658 0.0583

633 0.0770 655 0.0770 679 0.0770

653 0.0953 671 0.0953 690 0.0953

671 0.1133 684 0.1133 696 0.1133

�̇� = 0.1 s-1 �̇� = 1 s-1 �̇� = 10 s-1

T (MPa) p (m/m) T (MPa) p (m/m) T (MPa) p (m/m)

597 0 633 0 672 0

598 0.0010 634 0.0010 673 0.0010

615 0.0223 653 0.0223 693 0.0223

625 0.0296 662 0.0296 700 0.0296

655 0.0392 691 0.0392 728 0.0392

688 0.0583 719 0.0583 752 0.0583

704 0.0770 730 0.0770 757 0.0770

709 0.0953 731 0.0953 758 0.0953

710 0.1133 732 0.1133 759 0.1133

154

B3 Applied dynamic loads

The applied loads in the simulations were adjusted as picewise linear fits to the

measured loads in the tests. The loads used in the simulations are presented below.

B3.1 Flexural shear failures of concrete beams

This section refers to simulations of the B40-D3 and B40-D4 beam tests.

B40-D3 B40-D4

Time (ms) Pressure (kPa) Time (ms) Pressure (kPa)

0 0 0 0

0.14 765 0.070 1170

2.53 375 1.97 610

2.80 485 2.87 710

3.31 480 4.77 410

3.88 380 5.42 500

7.78 200 7.47 250

9.78 200 8.17 380

10.58 120 9.47 200

29.28 0 22.77 0

B3.2 Direct shear failures of roof slab

This section refers to simulations of the DS1 and DS4 tests.

DS1 DS4

Time (ms) Pressure (MPa) Time (ms) Pressure (MPa)

0 0 0 0

0.036 22 0.074 22.7

0.0741 24.5 0.17 19

0.16 21 0.23 18

0.296 21.5 0.32 19

0.37 17.5 0.495 5.5

0.444 11.5 0.605 0

0.494 5.5

0.623 0

155

Appendix C

Derivation of the shear span

C1 Introduction

A brief review of the derivation of the shear span used in the design manual Swedish

Fortifications Agency (2011) is given in these sections. The idea of a specific shear

span in dynamic events is used in the design of the shear capacity of reinforced concrete

elements subjected to dynamic loads to account for an increase in shear strength due to

a temporarily short shear span. The derivation is based on notes from one of the authors

of the design manual (G. Svedbjörk) and considers a simply supported beam subjected

to a uniformly distributed blast load. Soon after the load has been applied, limited

deformations occur in the vicinity of each support and where the central portion of the

beam exhibits a rigid body motion, see also Figure 7.1 and Paper IV. At this point in

time, xm denotes the distance from the support to the location where the maximum

bending moment develops. The design shear force Vd is assumed to occur somewhere

between the same distance. Furthermore, the shear capacity is assumed to be a function

according to:

(C1)

where k denotes a constant, and x and d denote the shear span and effective depth of

the element. Define a function that describes the ratio between the shear force V0 and

the shear capacity of the cross section such that:

(C2)

The next step is to find the distance x where the ratio between the maximum shear force

and the shear capacity at that location is minimal. Hence, the location where the cross

section is subjected to the largest ratio between shear force and shear capacity. This

distance is calculated by derivation of Eq. (C2) with respect to x and finding the

maximum:

(C3)

𝑓(𝑥) =𝑘𝑥𝑑⁄

𝑔 =𝑉0 ∙ 1 −

𝑥𝑥𝑚⁄

𝑓(𝑥)=𝑉0 ∙ 1−

𝑥𝑥𝑚⁄ ∙ 𝑥 𝑑⁄

𝑘=

𝑉0

𝑘 ∙ 𝑑 𝑥 −

𝑥2

𝑥𝑚

𝑑𝑔

𝑑𝑥=

𝑉0

𝑘 ∙ 𝑑∙ 1−

2 ∙ 𝑥

𝑥𝑚 = 0

156

Solving for x results in:

(C4)

Thus, the shear span is:

(C5)

Thus, Eq. (C5) states that the shear span is always half the distance between the support

and the location of the maximum bending moment. Due to this, Swedish Fortifications

Agency (2011) states that the design shear force is taken as half the support reaction.

C2 The shear span of a simply supported beam

As a first assumption, the static force equilibrium is only considered without the

influence of inertia forces, see Figure C1. At a distance greater than xm, the beam is

considered as straight without any bending moments and shear forces. Take moments

about the support:

(C6)

It is assumed that the ultimate bending moment is reached, thus:

(C7)

where q and L is the load capacity and the span, respectively. This results in:

(C8)

Solving for xm/L the expression below is obtained:

(C9)

Thus, using Eq. (C5), the expression for calculating the shear span becomes:

𝑥 =𝑥𝑚

2

𝑎𝜏 =𝑥𝑚2

𝑝 𝑡 ∙ 𝑥𝑚 ∙𝑥𝑚2−𝑀 𝑡 = 0

𝑀 =𝑞 ∙ 𝐿2

8

𝑝 ∙𝑥𝑚

2

2=𝑞 ∙ 𝐿2

8

𝑥𝑚𝐿

= 1

4∙𝑞

𝑝= 0,5 ∙

𝑞

𝑝

157

(C10)

In Swedish Fortifications Agency (2011), an extra factor of 0.025 is added to Eq. (C10)

to ensure that a does not approach zero. Thus, the final expression becomes:

(C11)

Figure C1 Simply supported beam subjected to a uniformly distributed load without

inertia forces.

When considering the influence of inertia forces, the dynamic equilibrium in Figure C2

needs to be considered. The same conditions are assumed as for the static case above.

The straight central part of the beam, without bending moments and shear, is moving

in the y-direction downwards as a solid. According to Newton’s second law of motion:

(C12)

where and h denote the mass density and the beam depth, respectively.

Assuming a deflection and a corresponding inertia force distribution according to

Figure C2 results in the resulting inertia force:

(C13)

Taking moments about the support and assuming that the ultimate bending moment is

reached:

(C14)

p(t)

V1(t)

M(t)

xm

𝑎𝜏𝐿

= 0,25 ∙ 𝑞

𝑝

𝑎𝜏𝐿

= 0,025 + 0,25 ∙ 𝑞

𝑝

𝑝 = 𝜌 ∙ ℎ ∙𝑑2𝑦

𝑑𝑡2

𝐼 =𝑝 ∙ 𝑥𝑚

2

𝑝 ∙ 𝑥𝑚 ∙𝑥𝑚2−

1

2∙ 𝑝 ∙ 𝑥𝑚 ∙

2

3∙ 𝑥𝑚 −

𝑞 ∙ 𝐿2

8= 0

158

Solving for xm/L the expression below is obtained:

(C15)

Thus, the expression for calculating the shear span becomes:

(C16)

Figure C2 Simply supported beam subjected to a uniformly distributed load

including inertia forces.

C3 The shear span of a fixed beam

The same conditions as in Section C2 is applied for a fixed beam without the influence

of inertia forces, see Figure C3. Take moments about the support:

(C17)

It is assumed that the ultimate bending moment is reached at the support and at xm,

which results in:

(C18)

Solving for xm/L the expression below is obtained:

(C19)

Thus, the expression for calculating the shear span becomes:

p(t)

V1(t)

M(t)

I(t) 𝜌 ∙ ℎ ∙ 𝑦

xm

𝑥𝑚𝐿

= 3

4∙𝑞

𝑝≈ 0,866 ∙

𝑞

𝑝

𝑎𝜏𝐿≈ 0,43 ∙

𝑞

𝑝

𝑝 𝑡 ∙ 𝑥𝑚 ∙𝑥𝑚2−𝑀 𝑡 −𝑀 𝑡 = 0

𝑝 ∙𝑥𝑚

2

2=𝑞 ∙ 𝐿2

4

𝑥𝑚𝐿

= 1

2∙𝑞

𝑝

159

(C20)

In Swedish Fortifications Agency (2011), an extra factor of 0.01 is added to Eq. (C20)

to ensure that a does not approach zero. Thus, the final expression becomes:

(C21)

Figure C3 Fixed beam subjected to a uniformly distributed load without inertia

forces.

The same conditions as in Section C2 is applied for a fixed beam where the influence

of inertia forces is included, see Figure C4. Taking moments about the support and

assuming that the ultimate bending moment is reached at the support and at xm:

(C22)

Solving for xm/L the expression below is obtained:

(C23)

Thus, the expression for calculating the shear span becomes:

(C24)

p(t)

V1(t)

M(t)

xm

M(t)

𝑎𝜏𝐿

= 1

8∙𝑞

𝑝≈ 0,35 ∙

𝑞

𝑝

𝑎𝜏𝐿

= 0,01 + 0,35 ∙ 𝑞

𝑝

𝑝 ∙ 𝑥𝑚 ∙𝑥𝑚2−

1

2∙ 𝑝 ∙ 𝑥𝑚 ∙

2

3∙ 𝑥𝑚 −

𝑞 ∙ 𝐿2

4= 0

𝑥𝑚𝐿

= 3

2∙𝑞

𝑝

𝑎𝜏𝐿

= 3

8∙𝑞

𝑝≈ 0.61 ∙

𝑞

𝑝

160

Figure C4 Fixed beam subjected to a uniformly distributed load including inertia

forces.

V1(t)

M(t)

I(t) 𝜌 ∙ ℎ ∙ 𝑦

xm

M(t)

p(t)