Average reinforcement strain in reinforced
concrete structures loaded until failure
An experimental study on the effect of reduced interaction and
reinforcement ratio on plastic deformation capacity
Master’s thesis in Structural Engineering and Building Technology
ARGHAVAN NOZAD
MOA STEINER
DEPARTMENT OF ARCHITECTURE AND CIVIL ENGINEERING
CHALMERS UNIVERSITY OF TECHNOLOGY
Gothenburg, Sweden 2021
www.chalmers.se

Master’s thesis 2021
Average reinforcement strain in reinforced
concrete structures loaded until failure
An experimental study on the effect of reduced interaction and
reinforcement ratio on plastic deformation capacity
ARGHAVAN NOZAD
MOA STEINER
Department of Architecture and Civil Engineering
Division of Structural Engineering
Concrete Structures
Chalmers University of Technology
Gothenburg, Sweden 2021
Average reinforcement strain in reinforced concrete structures loaded until failure
An experimental study on the effect of reduced interaction and reinforcement ratio
on plastic deformation capacity
ARGHAVAN NOZAD, MOA STEINER
© ARGHAVAN NOZAD, MOA STEINER, 2021.
Master’s Thesis ACEX30
Department of Architecture and Civil Engineering
Division of Structural Engineering
Concrete Structures
Chalmers University of Technology
SE-412 96 Gothenburg
Telephone +46 31 772 1000
Cover: Schematic illustration of a deformed beam subjected to four-point bending
and the crack formation and propagation.
Typeset in LATEX, template by Magnus Gustaver
Gothenburg, Sweden 2021
Average reinforcement strain in reinforced concrete structures loaded until failure
An experimental study on the effect of reduced interaction and reinforcement ratio
on plastic deformation capacity
ARGHAVAN NOZAD
MOA STEINER
Department of Architecture and Civil Engineering
Division of Structural Engineering
Conrete structures
Chalmers University of Technology
Abstract
Reinforced concrete and its behavior have a significant role in structural engineer-
ing. Reinforced concrete structures are usually designed based on Eurocode for
static loading which is not sufficient in case of dynamic loading situations such as
explosion. For a structure subjected to impulse loading, plastic deformation capac-
ity is of higher importance rather than the load capacity. The regulations provided
by the Swedish Fortification Agency, FKR 2011, provides certain design regulation
to fulfill requirements for structures subjected to impulse load. According to these
regulations for which a revision is needed, average tensile strain of reinforcement at
failure is an effective factor representing plastic deformation capacity.
The aim of this project was to study the effect of reinforcement amount and de-
creased interaction between steel and concrete on reinforcement average strain and
plastic deformation capacity of reinforced concrete beams subjected to static loading.
Therefore, nine beams in three different categories were constructed and subjected
to a four-point bending test as: three beams with φ10 reinforcement, three beams
with φ12 reinforcement, and three beams with φ12 reinforcement with PVC tubes
placed in the middle of the reinforcement to decrease the bond. The number of bars
and the dimensions were the same for all beams. Before casting, the reinforcement
were marked every 50 mm and then remove after testing to measure the elongation
of the bars. Also some bars were 3D scanned as another method to measure the
elongation and to analyse the cross-sectional changes. During the tests, the struc-
tural response of beams was recorded using Digital Image Correlation (DIC).
Although there were some scatter in test results, in general it can be concluded
that beams with φ12 reinforcement provided a larger plastic deformation capacity
compared to beams with φ10 reinforcement. Also, application of PVC tubes to
reduce the bond between the concrete and the steel resulted in a larger plastic
deformation capacity and average reinforcement strain. The utilization rate for
strain values was estimated and the reinforcement bars with PVC tubes showed a
increase utilization rate.
Keywords: Reinforced concrete, plastic deformation capacity, average tensile strain,
bond interaction, DIC, four-point bending test, FKR.
, Architecture and Civil Engineering, Master’s Thesis ACEX30 i
Medeltöjning i armerade betongkonstruktioner belastade till brott
Försöksstudie på inverkan av reducerad vidhäftning och armeringsmängd med hän-
syn till den plastiska deformationsförmågan
ARGHAVAN NOZAD
MOA STEINER
Institutionen för arkitektur och samhällsbyggnadsteknik
Avdelningen för konstruktionsteknik
Betongbyggnad
Chalmers tekniska högskola
Sammanfattning
Armerad betong och dess beteende är av stor betydelse inom konstruktionsteknik.
Armerad betongkonstruktioner är vanligen dimensionerade enligt Eurokod för statisk
belastning vilket är otillräckligt vid dynamisk belastning såsom explosionslaster.
För en konstruktion belastad av en impulslast, är det högre krav på den plastisk de-
formationsförmågan jämfört med bärförmågan. Fortifikationsverket tillhandahåller
föreskrifter, FKR 2011, för att uppfylla de krav som ställs på konstruktioner utsatta
för impulslaster. Dessa föreskrifter är i behov av revidering och medeltöjningen av
armeringen vid brott är en viktigt egenskap gällnade den plastisk deformationsför-
mågan.
Syftet med avhandligen var att undersöka vilken effekt armeringsmängden och min-
skad vidhäftning mellan armeringsstålet och betongen har på medeltöjningen av
armeringen och den plastiska deformationsförmågan av armerade betongbalkar vid
statisk belastning. Således har nio balkar med tre olika armeringskonfigurationer
producerats och fyrpunktsböjningstests: tre balkar med φ10 armering, tre balkar
med φ12 armering och tre balkar med φ12 armering där mittdelen av armeringen
var belagd med PVC-rör för att minska vidhäftningen. Antalet armeringsstänger
och tvärsnittsdimensioner var konstant för alla balkar. Under provning användes
digital bildkorrelation (DIC) för att dokumentera och analysera balkens respons.
Innan gjutningen av balkarna märktes alla armeringsstänger varje 50 mm för att
sedan avlägsnas från betongen efter testning för att mäta förlängningen. Armerin-
gen var också 3D-skannad innan och efter testet som en annan metod för att mäta
förlängningen och analysera förändringen av armeringens tvärsnitt.
I allmänhet, balkarna med φ12 armering resulterade i en större plastisk deforma-
tionsförmåga jämfört med balkarna med φ10 armering. Likaså, användningen av
PVC-rör för att reducera vidhäftningen mellan betongen och armeringsstålet resul-
terade i större plastisk deformationskapacitet och en högre medeltöjning. Armerin-
gens utnyttjadegrad med hänsyn till de uppmätta töjningarna var också beräknad
och armeringsstängerna med PVC-rör medförde en ökad utnyttjandegrad.
Nyckelord: Armerad betong, plastisk deformationsförmåga, medeltöjning, vidhäft-
ning, DIC, fyrpunktsböjningstest, FKR.
ii , Architecture and Civil Engineering, Master’s Thesis ACEX30
Contents
Abstract i
Sammanfattning ii
Preface vii
Nomenclature viii
1 Introduction 1
1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Aim . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.3 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.4 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2 Material behaviour 3
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 Concrete . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2.1 Compressive strength . . . . . . . . . . . . . . . . . . . . . . . 4
2.2.2 Tensile strength . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2.3 Modulus of elasticity . . . . . . . . . . . . . . . . . . . . . . . 6
2.3 Reinforcing steel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.3.1 Strain hardening . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.3.2 Necking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3.3 Design simplifications . . . . . . . . . . . . . . . . . . . . . . . 8
2.4 Reinforced concrete . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.4.1 Structural response . . . . . . . . . . . . . . . . . . . . . . . . 9
2.4.2 Stages during loading . . . . . . . . . . . . . . . . . . . . . . . 9
2.4.3 Crack development . . . . . . . . . . . . . . . . . . . . . . . . 12
2.4.4 Crack formation . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.4.5 Cracking of flexural loaded beam . . . . . . . . . . . . . . . . 18
3 Plastic deformation capacity 21
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.2 Relation of plastic rotational capacity and
curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.2.1 Definition of curvature . . . . . . . . . . . . . . . . . . . . . . 22
3.2.2 Theoretical explanation for plastic rotation . . . . . . . . . . . 23
3.2.3 Ratio of Mu and plastic rotation . . . . . . . . . . . . . . . . . 25
My
3.3 Different methods for calculation of plastic rotational capacity . . . . 26
3.3.1 Plastic rotation estimation from test results . . . . . . . . . . 26
3.3.2 BK 25 method . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.3.3 Eurocode 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.3.4 German method (background of Eurocode 2) . . . . . . . . . . 32
3.3.5 Betonghandboken 1990 (ABC method) . . . . . . . . . . . . . 38
, Architecture and Civil Engineering, Master’s Thesis ACEX30 iii
4 Previous testing 41
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.2 Chalmers experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.3 KTH experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
5 Experimental procedure 45
5.1 General description . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
5.2 Manufacturing of concrete beams . . . . . . . . . . . . . . . . . . . . 45
5.2.1 Preparations before casting . . . . . . . . . . . . . . . . . . . 46
5.2.2 Casting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
5.2.3 After testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
5.3 Beam test setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
5.4 Digital image correlation . . . . . . . . . . . . . . . . . . . . . . . . . 50
6 Predictions 53
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
6.1.1 Ultimate load capacity . . . . . . . . . . . . . . . . . . . . . . 53
6.1.2 Load at yielding . . . . . . . . . . . . . . . . . . . . . . . . . . 54
7 Experimental results 57
7.1 General description . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
7.2 Material testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
7.2.1 Concrete . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
7.2.2 Reinforcement . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
7.3 Beam results, structural response . . . . . . . . . . . . . . . . . . . . 59
7.3.1 Beams with φ10 reinforcement . . . . . . . . . . . . . . . . . . 61
7.3.2 Beams with φ12 reinforcement . . . . . . . . . . . . . . . . . . 63
7.3.3 Beams with φ12 reinforcement and PVC tubes . . . . . . . . . 65
7.4 Beam results, strain measurements . . . . . . . . . . . . . . . . . . . 68
7.4.1 3D scanning . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
7.4.2 Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
7.4.3 Cross-sectional area from 3D scanning . . . . . . . . . . . . . 76
7.5 Determination of plastic rotation capacity . . . . . . . . . . . . . . . 78
7.6 Summary of the beam test results . . . . . . . . . . . . . . . . . . . . 81
8 Discussion 83
8.1 Structural response . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
8.2 Plastic reinforcement strain . . . . . . . . . . . . . . . . . . . . . . . 88
8.3 Test setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
9 Final remarks 93
9.1 General description . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
9.2 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
9.3 Further research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
References 97
A Experiment preparations A-1
iv , Architecture and Civil Engineering, Master’s Thesis ACEX30
B Material testing B-1
B.1 Concrete properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-1
B.1.1 Compressive test . . . . . . . . . . . . . . . . . . . . . . . . . B-1
B.1.2 Wedge splitting test . . . . . . . . . . . . . . . . . . . . . . . B-3
B.2 Steel reinforcement properties . . . . . . . . . . . . . . . . . . . . . . B-5
B.2.1 φ10 reinforcement . . . . . . . . . . . . . . . . . . . . . . . . . B-7
B.2.2 φ12 reinforcement . . . . . . . . . . . . . . . . . . . . . . . . . B-8
B.3 Second reinforcement testing . . . . . . . . . . . . . . . . . . . . . . . B-9
C Beam results, structural response C-1
C.1 Load-Displacement curves . . . . . . . . . . . . . . . . . . . . . . . . C-1
C.2 Symmetrical behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . C-7
C.3 Crack pattern . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-9
C.4 Final image of beam at end of test . . . . . . . . . . . . . . . . . . . C-12
C.5 Determination of plastic rotation capacity . . . . . . . . . . . . . . . C-15
C.6 Calculation of plastic rotation capacity based on BK 25 and German
method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-16
C.6.1 BK 25 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-16
C.6.2 German Method . . . . . . . . . . . . . . . . . . . . . . . . . C-17
D Plastic reinforcement strain D-1
D.1 Plastic strain measurements . . . . . . . . . . . . . . . . . . . . . . . D-1
D.2 3D scanning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D-11
D.2.1 3D model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D-11
D.2.2 3D scanning results . . . . . . . . . . . . . . . . . . . . . . . . D-12
D.3 Comparison of plastic strain results . . . . . . . . . . . . . . . . . . . D-16
D.4 Cross-sectional area from 3D scanning . . . . . . . . . . . . . . . . . D-19
, Architecture and Civil Engineering, Master’s Thesis ACEX30 v

Preface
This master’s project is an experimental study about the effect of reinforcement
ratio and reduced interaction on plastic deformation capacity of reinforced concrete
beams subjected to static loading. The specimens were constructed and tested in
structural engineering laboratory of Chalmers University of Technology. The project
was a cooperation between Chalmers University of Technology and Norconsult AB
and financed by the Swedish Fortification Agency as a continuation of an ongoing
research to revise the FKR 2011 regulations.
We would like to express our appreciation to Morgan Johansson, our supportive
supervisor from Norconsult AB, for providing this great opportunity of having our
thesis with Norconsult and helping us obtain a deep knowledge through all stages
of our project. In addition, we are very grateful for all supports, guidance, and
encouragements from our examiner Joosef Leppänen. Thanks to both Morgan and
Joosef we experienced working on an interesting project which was well-planned,
instructive and under a stress-free situation. We appreciate the great assistance and
patience of Anders Karlsson with laboratory operation. Also, we are thankful to
Mathias Flansbjer, RISE, for the provided information and help with DIC procedure.
Arghavan Nozad, Moa Steiner, Gothenburg, June 2021
, Architecture and Civil Engineering, Master’s Thesis ACEX30 vii
viii , Architecture and Civil Engineering, Master’s Thesis ACEX30
Nomenclature
Greek letters
χ Curvature
χpl Plastic curvature
χu Curvature at ultimate state
χy Curvature at yielding state
∆r Length of which a local bond failure occurs in the concrete
λ Shear slenderness
ωs,crit Critical mechanical ratio of tensile reinforcement
ωs Mechanical ratio of reinforcement in tension
′
ωs Mechanical ratio of reinforcement in compression
ωv Mechanical ratio of stirrups
ρs Reinforcement ratio parameter
τb Bond stress
θpl,EC Plastic rotational capacity based on Definition in Eurocode 2
θpl Plastic rotational capacity based on Definition in BK 25 Method
εc1u Concrete strain limit at maximum load
εcc Concrete compressive strain
εcm Concrete mean strain
εct Concrete tensile strain
εcu Concrete ultimate strain
εcy Concrete strain at reinforcement yielding
εsu Reinforcement ultimate strain
ε∗su Reinforcement ultimate strain with simplified consideration of tension stiff-
ening
εsy Reinforcement strain at yielding
εs Reinforcement average strain over the equivalent plastic length
εuk Reinforcement ultimate strain at maximum load
Roman lowercase letters
a Distance between support and concentrated load
b Width of beam cross section
d Effective depth
fcc Concrete compressive strength
fck Characteristic concrete compressive strength
fcm Mean concrete compressive strength
fctk0,05 Lower characteristic tensile strength of concrete
fctk0,95 Upper characteristic tensile strength of concrete
fct Concrete tensile strength
fc Compressive cylinder strength of concrete
ftk Characteristic tensile strength of reinforcement
ftk Tensile strength of reinforcement
fu Ultimate strength of reinforcement
fyk Characteristic yield strength of reinforcement
, Architecture and Civil Engineering, Master’s Thesis ACEX30 ix
fy Yield strength of reinforcement
h Height of beam cross section
k Correction factor for plastic rotation
l Total length of beam
l0 Length between zero moment and plastic hinge
l1 The ratio of bending moment to shear force at support
lp Equivalent plastic hinge length
lt,max Maximum transmission length
lt Transmission length
ly Length of yielding region
q Uniformly distributed load
r Radius of curvature
s Distance of stirrups
sr,max Maximum crack spacing
sr,min Minimum crack spacing
sr Crack spacing
u Displacement
uel Elastic deformation
upl Plastic deformation
utot Total deformation
uu Ultimate deformation
x Distance to neutral axis from top edge (compressive block height)
xu Compressive block height at failure state
xy Compressive block height at yielding state
z Distance from gravity center
Roman uppercase letters
As Area of reinforcement in tension
′
As Area of reinforcement in compression
Av Area of reinforcement in shear
E Modulus of elasticity
Ecm Mean modulus of elasticity
Es Modulus of elasticity of steel
F External force
Fmax Maximum load
I Moment of inertia
Msup Bending moment at support
Mu Ultimate moment
My Yielding moment
R Internal resistance force
Vsup Shear force at support
Wi Internal work
x , Architecture and Civil Engineering, Master’s Thesis ACEX30
1 Introduction
1.1 Background
Reinforced concrete is a commonly used material for structural elements. Normally,
concrete structures are designed to withstand static loading and the design regu-
lations are made based on that fact. However, there are structures that can be
subjected to other types of loading. One example of this structure is protective
shelters which have to be designed not only to withstand static load but also e.g.
explosion loads. To be effective, the protective structures have high demand on its
plastic deformation capacity rather than large load capacity. Further, the response
of a structure subjected to impulse loading may have a different response compared
to structures subjected to only static loading. This means that a structure designed
to withstand static loading is not always as suitable to withstand the effect of an
explosion load.
Since protective structures have different demands compared to regular structures,
the design regulations from the Eurocodes are not sufficient. Therefore, a structure
subjected to impulse loading has to be designed according to certain design regula-
tions, e.g. FKR 2011 (Swedish Fortification Agency, 2011). Mentioned above, the
plastic deformation capacity of impulse loaded structures is crucial and, according
to FKR 2011, an important parameter to determine this capacity is the average re-
inforcement tensile strain at failure. This parameter depends on the reinforcement
configuration, mechanical properties of the materials and the interaction between
reinforcement and concrete. The current regulations are in need of update and the
average tensile strain is one parameter that needs to be revised.
1.2 Aim
The aim of this thesis is to provide a better understanding of structural behavior
of reinforced concrete structures loaded until failure with focus on reduced bond
between the reinforcing steel and the concrete and the average reinforcement strain.
This behavior is considered after yielding of reinforcement and at its rupture in order
to study the average reinforcement strain and plastic deformation capacity under
static loading.
The above mentioned data are needed as an update of input data for Design regula-
tions from Swedish Fortification Agency, used for impulse loaded structures – FKR
2011. The results from this project helps to calculate the average tensile reinforce-
ment strain based on material properties, reinforcement configuration and geometry.
It is important to consider that a structure exposed to static loading may have a dif-
ferent response compared with being exposed to an impulse load. Therefore, plastic
deformation capacity has a higher importance than load capacity in the latter type
of load case.
, Architecture and Civil Engineering, Master’s Thesis ACEX30 1
1.3 Methodology
The methodology was based on both literature survey and experimental testing.
This work was a continuation of Köseoğlu (2020), in which reinforced concrete prisms
loaded by a uniaxial tensile load until failure were studied. However, in the current
project, beams were examined under four-point bending tests until failure with focus
on the region between the point loads with constant moment.
The reinforcement bars were marked and went through 3D scanning before casting
and after tests for plastic strain measurements. Plastic tubes were used in some of
the test specimens to reduce the bond between steel and concrete. After casting
and during the four-point bending test the cracks and deformation developments
were monitored by a camera and digital image correlation (DIC) was used to to
measure deformations and calculate strain fields in the concrete. After testing, the
reinforcement was removed from the concrete in order to measure the elongation and
calculate the strain. The expected outputs were load-displacement relation, crack
pattern and plastic strain distribution in the reinforcement bars.
1.4 Limitations
The concrete material properties, specimen size and the number of bars were the
same for all specimen. The beams were limited to three types with difference either
in bar diameter or in bond interaction:
1. 2φ10 (without plastic tubes)
2. 2φ12 (without plastic tubes)
3. 2φ12 (with plastic tubes)
For this test series, three numbers of each beam type was studied. So the total
number of specimen was limited to nine beams. Plastic deformation along the bars
was determined in all beam specimens but only three beams (six bars) were further
studied using 3D scanning. Also, the analysis was limited to static loading with two
point loads.
2 , Architecture and Civil Engineering, Master’s Thesis ACEX30
2 Material behaviour
2.1 Introduction
In order to understand the structural behaviour and theoretical background of rein-
forced concrete beams, basic knowledge about the structural properties of plain con-
crete and reinforcing steel and their behaviour under loading is necessary. The com-
bination of the two materials makes it possible to utilize the compressive strength of
concrete and the tensile strength of reinforcement to form a high strength structure
(Engström, 2014). One important parameter in the reinforced concrete structures
is the bond between materials which has an influence on the transfer of stresses and
the structural behaviour as crack development, crack spacing and crack width.
2.2 Concrete
The mechanical behaviour of plain concrete is characterized by the difference in
tensile and compression strength where the tensile capacity is significantly lower
compared to the compressive capacity as indicated in the stress-strain relationship
curve in Figure 2.1 (Al-Emrani et al., 2013).
Figure 2.1 Stress-strain relationship for plain concrete under uniaxial loading
where fct corresponds to the tensile strength and fc the compression
strength (Jönsson & Stenseke, 2018).
From Figure 2.1, it can be observed by the steep slope at failure, that concrete is
a brittle material when it fails in tension. The failure in compression is partially
brittle but the behaviour depends on the concrete strength and the loading rate (Al-
Emrani et al., 2013). Concrete with lower strength classes present a larger strain at
failure and are therefore more ductile compare with higher strength classes which
have a more brittle failure.
, Architecture and Civil Engineering, Master’s Thesis ACEX30 3
The mechanical properties like tensile strength, deformation capacity and modulus
of elasticity are associated with uncertainties (Engström, 2014). Because of natural
scatter in the material, specimens made of the same concrete which are handle under
identical conditions give various results. Therefore, a frequency distribution can be
used to present results like mean values, standard variations and fractile values. An
example of a typical frequency distribution for the tensile strength of plain concrete
is presented in Figure 2.2.
Figure 2.2 Frequency distribution of the tensile strength with mean value fctm, low
and high characteristic value, fctk0,05 and fctk0,95 (Engström, 2014).
The low and high characteristic values, also called the fractile values, are defined by
the 5%-fractile and the 95%-fractile. The low fractile implies that 95% of all test
exceeds this value and for the high fractile, only 5% of the tests exceeds.
2.2.1 Compressive strength
The classification of concrete is done based on the compressive strength from stan-
dardised strength classes. The concrete compressive strength is influenced by dif-
ferent factor like size and shape of the specimen, curing, storage, handling and test
procedure. Therefore, a standardised procedure of the compressive test is stated in
the European Standard and according to Eurocode 2, the compressive strength is
defined as the strength determined at an age of 28 days.
The compressive strength is determined by applying an uniaxial force to a cylindri-
cal specimen with height 300 mm and diameter 150 mm (CEN, 2019). However, in
Sweden it is more common to determine the compressive strength of concrete with
cubes with a size of 150 mm. As mentioned above, the result of a compressive test
is depended on the testing method and therefore a cylindrical test and a cubic test
will give different concrete strengths. According to Engström (2014) the test with
a cubic specimens gives a mean strength approximately 1.2 time greater compared
to test preformed on cylindrical specimen. The difference in results are due to the
fact that a cube will have a more favourable stress state, due to confinement effects,
compared with a cylinder.
4 , Architecture and Civil Engineering, Master’s Thesis ACEX30
The relation between a cylinder’s and a cube’s compressive strength can be expressed
as
≈ fcm,cubefcm 1.2 (2.1)
where fcm is the mean strength of the compressive strength of concrete from a
cylindrical specimen and fcm,cube from a cubic specimen.
2.2.2 Tensile strength
The tensile strength of concrete can be determined by pure tensile tests, flexure or
splitting tests and the method used will influence the result. The tensile strength
increases with increasing compressive strength and according to Eurocode 2 (CEN,
2004), the tensile strength can be determined by known measured values of the
characteristic compressive strength. The mean tensile strength after 28 day and the
low and high fractile values can be estimated by the following relationships.
For strength classes ≤ C50/60
f 2/3ctm = 0.3(fck) (2.2)
For strength classes > C50/60 ( )
= 2 12 ln 1 + fck + 8fctm . 10 (2.3)
fctk0.05 = 0.70fctm (2.4)
fctk0.95 = 1.3fctm (2.5)
Where fck is the characteristic compressive strength in [MPa].
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2.2.3 Modulus of elasticity
A concrete specimen subjected to uniaxial compression, the non-linear stress-strain
relationship is shown in Figure 2.3. From the stress-strain curve, it can be observed
that the modulus of elasticity changes with the stress level, with higher stresses the
modulus of elasticity decreases. According to Eurocode 2 (CEN, 2004) the mean
modulus of elasticity Ecm can be calculated as the secant modulus between σs = 0
and σs = 0.4fcm.
Figure 2.3 Idealised stress-strain relationship for concrete under compression to
determine the mean value of the modulus of elasticity, Ecm by 40% of
the mean compressive strength, fcm (CEN, 2004).
The mean value of the modulus of elasticity can also be calculated as a relationship
to the concrete’s mean compressive strength
( )0.3
= 22 fcmEcm 10 (2.6)
2.3 Reinforcing steel
In contrast to concrete, the reinforcing steel has high tensile strength and a more
ductile behaviour. The classification of reinforcement is based on a large number of
properties like for instance, strength, fatigue strength, ductility class, size etc.
The structural response of the steel reinforcement can be examined by applying an
uniaxial tensile load to a reinforcement bar. The applied force and the elongation
of the specimen are measured and the steel stress can be calculated by dividing the
applied load by the cross-sectional area (Al-Emrani et al., 2013). This results is a
typical stress-strain relationship as is shown in Figure 2.4.
The stress-strain curve is characterised by four different stages: the elastic domain,
the yielding plateau, stain hardening up to maximum tensile stress and finally,
6 , Architecture and Civil Engineering, Master’s Thesis ACEX30
necking and rupture of the reinforcement. In the early stages of loading, during
low strains the behaviour of the reinforcement is characterised by elasticity with
elastic deformations. As the load increases, the strain increases and when the stress
reaches a point where the deformations in the material cannot longer be considered
as elastic. This point is called the yielding point and the level of stress is equal to
the yielding stress fy and the corresponding yielding strain εy.
Figure 2.4 Schematic stress-strain relationship for hot-rolled reinforcing steel,
modified from (Johansson & Laine, 2012).
As the yielding is reached, a plateau with constant stress and increasing strain can
be distinguished in the stress-strain curve. Past the yielding plateau, the elastic
behaviour has progressed to a plastic behaviour with permanent deformations in
the material. When the stress reaches the ultimate strength fu and the strain at
maximum stress εu, further increases of stress will induce dislocations in the material
and the steel bar begins to reach failure. The ratio between the ultimate strength
and the yielding strength is defined as
η = fu/fy (2.7)
and it is called hardening ratio which is an important parameter that influences the
material’s ductility.
2.3.1 Strain hardening
After yielding, the material response and the mechanical behaviour of the steel
reinforcement is altered. Dislocations or so called, point of defects are forming in the
lattice structure of the steel material (Al-Emrani et al., 2013). As the load increases,
the amount of dislocations increase and the steel begins to harden and strengthen
which means that a greater amount of force is required for further deformations. The
strain hardening can be distinguished in the stress-strain curve after the yielding
plateau and the it starts at the strain εh.
, Architecture and Civil Engineering, Master’s Thesis ACEX30 7
2.3.2 Necking
At the maximum load, where the strain is equal to εu and stress fu, the stress-
strain curve indicates a plateau and a decrease of the steel stress. The decrease
can be explained by locally accumulation of dislocations which is called necking.
The necking can be distinguished as a decrease in the cross-sectional area of the
specimens, see Figure B.7 in Appendix B. In the necking area, the local deformations
and reduced cross-sectional area under constant applied load will induce a local stress
increase according to the following equation:
F
σ = (2.8)
A
where σ is the internal steel stress, F is applied tensile force and A is the cross-
sectional area. When the necking is induced, the maximum capacity of the rein-
forcement bar and the failure load are reached. As a consequence of the reduced
cross-sectional area, less load is required to deform the steel reinforcement further
and necking is therefore an indication of near-failure.
2.3.3 Design simplifications
According to Eurocode 2 (CEN, 2004), a simplification of the stress-strain relation
with a inclined top branch as shown in Figure 2.5 can be assumed in the design. The
simplification disregards the yielding plateau and effects of necking. The ultimate
capacity defined in Eurocode 2 corresponds to the maximum stress fu and the strain
εu from Figure 2.4, thus the necking and the remaining capacity after necking occurs
is disregarded in the simplification. However, Eurocode 2 also allows for an even
more simplified relationship of the stress-strain curve with horizontal top branch
and maximum stress fy which disregard the hardening.
Figure 2.5 Simplified relationship of the stress-strain curve with a inclined top
branch for reinforcing steel where the ultimate strain corresponds to
the strain at maximum stress εu from Figure 2.4.
The modulus of elasticity Es is defined as the slope of the linear elastic part of the
stress-strain curve and calculated as the secant between the origin and the yielding
point.
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2.4 Reinforced concrete
When the plain concrete and the reinforcing steel are combined together it forms
a composite material with high compressive strength from the concrete and an in-
creased ductile behaviour and tensile strength from the reinforcement. The struc-
tural response and change of behaviour with respect to the bond between the mate-
rials are important knowledge for understanding the reinforced concrete’s behaviour
and the formation of cracks.
2.4.1 Structural response
Reinforced concrete as a combination of concrete and steel has a complex structural
behavior. This combination is due to compensate low tensile strength of concrete
and utilize its high compressive capacity. There are three simplified models, shown
in Figure 2.6, to explain the structural behavior of the composite material: linear
elastic, plastic and elastic-plastic response (Johansson & Laine, 2012).
Figure 2.6 Structural response and internal work Wi, assuming (a) linear elastic
response, (b) plastic response and (c) elastic-plastic response
(Johansson & Laine, 2012).
In the elastic model there is a linear stress-strain relation and the deformations
are not permanent which implies that the material returns to its original state in
case of unloading. For the plastic model the deformation is zero for stresses under
yield stress. Reaching the yield stress, deformation starts which is permanent and
the material can not return to its original state in case of unloading. The elastic-
plastic model is combination of linear elastic and ideal plastic responses. Before
reaching the yield stress the material has a linear elastic behavior and deformations
are reversible. Also, the material can not take more stress after the yielding stress
is reached so permanent deformations take place.
2.4.2 Stages during loading
Concrete’s poor performance in tension will induce cracks where the tensile stresses
exceeds the concrete tensile strength. In reinforced concrete, cracks is assumed in
the design in tensioned regions and reinforcing steel is used in order to distribute
, Architecture and Civil Engineering, Master’s Thesis ACEX30 9
cracks and limit crack widths (Al-Emrani et al., 2011). For uncracked concrete it
can be assumed that there is full interaction between the concrete and reinforce-
ment which means that the steel strain increases with the same amount as the
concrete strain. In order to get an effect of the reinforcement’s good performance
in tension, the concrete must crack so the steel strain can increase. Therefore, the
structural response of reinforced concrete is highly dependent on if the section is un-
cracked or cracked. Before cracks are initiated, the reinforced concrete has a linear
response which means that the strain increases linearly with the applied load and
the reinforcement steel is generally assumed to have a small influence of the stiffness.
When the section cracks, there will be a change in behaviour since the force that was
resisted by the tensile stresses in the concrete is suddenly carried by the reinforce-
ment. Consider a simply supported beam with one span, subjected to a uniformly
distributed load, q as shown in Figure 2.7. The structural behaviour of the beam
can be divided into three different stages: state I, state II and state III. As the load
increases, the deflection and structural response change, and those changes can be
described with the load-displacement curve. The load-displacement curve illustrat-
ing the three different stages together with the elastic-plastic response with bi-linear
relation, described in Figure 2.6 (blue curve).
Figure 2.7 The structural response of reinforced concrete beam with a uniformly
distributed load. The blue curve represent a simplified bi-linear
response of the structural behaviour. From (Jönsson & Stenseke,
2018), inspired by (Johansson & Laine, 2012).
State I: At this state the strain is equal in concrete and reinforcement, the con-
crete is uncracked and linear elastic material response is assumed. The influence of
reinforcement can be ignored as it has a small contribution to the stiffness.
State II: By increasing the load, tensile stress reaches concrete’s tensile capacity,
cracking takes place and stiffness decreases considerably. Linear elastic response is
considered for both steel and concrete but influence of concrete in the tensile zone
is neglected.
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State III: After reinforcements start yielding, the non-linear material response is
used for both steel and concrete and by load increment there will be failure either
in form of concrete crushing or steel rupture.
Tension siffening
When a section cracks, the concrete does not take any stress in the tensile zone
and all tensile stress is transferred into reinforcement. As a result the stiffness
undergoes a significant drop at the cracked section and the stiffness can only account
for reinforcement. However, in the global response of the beam, concrete between
cracked sections contributes to total stiffness of the beam due to the bond between
steel and concrete and this phenomena is called tension stiffening, see Figure 2.8.
This effect decreases by load increment and consequently by increasing the number
of cracks.
Figure 2.8 Sectional response with regard to tension stiffening in a concrete
member subjected to pure bending. Based on linear stress-strain
relationship for both concrete and steel, inspired by (Engström, 2015).
The effect of tension stiffening can be seen at the end of the prism in Figure 2.10 or
next to a crack in Figure 2.12. The concrete tensile stress at the ends or next to the
cracks gradually increases and becomes most active in the middle of the prism or
between the cracks. Later, when the reinforcement starts to yield, the bond between
the materials decrease and consequently, the stress in the concrete also decrease.
The effect of tension stiffening in the initial cracking stage was clearly seen in tests
made by Köseoğlu (2020). A concrete prisms with one centralized reinforcement bar
were loaded by a uniaxial tensile load until failure. The resulting force-strain curves
for three prism with φ16 reinforcement bars and the force-strain curve of a plain
φ16 bar are presented in Figure 2.9. At the initial stage, the prisms were stiffer
compared to the plain bar due to the tension stiffening effects where the concrete
contributes to the stiffness of the prisms. Towards the end of the loading, more
cracks forms, the crack width increase and less concrete contributes to the global
, Architecture and Civil Engineering, Master’s Thesis ACEX30 11
stiffness. The curves from the prisms are closer to the curve of the plain bar which
indicates that the effect of tension stiffening decreases and the reinforcement bar
mainly contributes to the global response of the prism.
Figure 2.9 Force-displacement curve for concrete prisms subjected to uniaxial
tension test (Köseoğlu, 2020).
As the number and width of cracks increase along the beam, the stiffness depends
on reinforcement. Larger amount of load will result in yielding of reinforcement
which starts at the section with largest stress. At this section the reinforcement
experiences more deformation than adjacent sections in which the reinforcing steel
has elastic response. A region with concentrated plastic rotation is called plastic
hinge (Johansson & Laine, 2012).
2.4.3 Crack development
A more detailed theory behind the cracking process can be explained by studying
the behaviour of a concrete prism with centric reinforcement subjected to a uniaxial
tensile load. At each end, the projecting reinforcement bar is loaded by a horizontal
tensile force that is less than the cracking load of the concrete, the resulting stress
distribution of such loaded prism is illustrated in Figure 2.10.
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Figure 2.10 Stress distribution of a reinforced concrete member loaded in tension
with a tensile force less than the cracking force (Engström, 2014).
As the reinforcement bar is loaded, the bond between the reinforcement and the
concrete is preventing the bar from slipping in relation to the concrete. The bond
is transferring the force from the reinforcement to the surrounding concrete which
leads to a decrease of stress in the reinforcement while the stress in the concrete
increases. The load transfer causes bond stresses called τb and the transfer takes
place within a distance called transmission length, lt. The bond stress varies along
the transmission length and the mean value of the bond stress τbm can be evaluated
as ∫ lt
= 0 τb(x)dxτbm (2.9)
lt
The stress transfer between the reinforcing steel and the surrounding concrete is
dependent on the surface properties of the reinforcement and the tensile strength of
the concrete. From empirical studies, considering k1 as a factor depending on the
surface of the reinforcement, it has been shown that the mean bond stress can be
estimated as
3
τbm = 2 · fct (2.10)· k1
The transmission is initiated at the ends of the specimen and continues towards
the mid-section until the stress in the steel and concrete is equal and no further
transmission occurs. When the stresses are equal, there is compatibility between
the deformation in the materials. However, this is not the case for the material
within the transmission length since the strain in the steel is higher compared to
the strain in the concrete and a local slip of the bar occurs when the bar elongates
more than the concrete.
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At the ends of the prism, the load is causing a local bond failure of the concrete’s
free edge which has a length ∆r, see Figure 2.11. The failure results in a zone
where no bond stress can be transferred from the reinforcement to the concrete and
therefore the stress in the bar is constant. According to Engström (2014), the local
cone failure can be approximated as two times the reinforcement bar’s diameter:
∆r ≈ 2φ (2.11)
Figure 2.11 Principle of local cone failure of the concrete near to the loaded end
(Engström, 2014).
2.4.4 Crack formation
As the load increases, the transmission length and the stress in the materials in-
crease. The increase continues until the tensile stress exceeds the tensile strength of
the concrete and a crack appears. The crack affects the bond between the materials
since no stresses can be transferred from the reinforcement to the concrete in the
cracked section.
Since the tensile stress in the concrete is higher in the mid-region compared to within
the transmission length, the crack can be initiated anywhere in the mid-region of
the specimen. When the first crack appears, the transmission length has reached its
maximum length, lt,max. No further increase of the transmission length can occur
since that requires an increase of the tensile stress, which in turn, is not possible
since no additional tensile force can be applied without further cracking. In theory,
when the tensile strength of the concrete is exceeded more cracks can appear with-
out an increase of the load. However, in practice a small increase of load is needed
since there is a slightly variation of tensile strength between sections due to normal
distributed scatter, illustrated in Figure 2.2.
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The cracking process can be described as shown in Figure 2.12 and after the first
crack appears, the concrete prism is divided into two parts which acts as two different
tensile members with the reinforcement loaded in their ends. The two members
behave similar to the originally uncracked concrete prisms described above. On each
side of the crack, the reinforcement tends to slip in relation to the concrete which
results in a development of a new transmission zone where the stress is transferred
from the loaded bar to the concrete. The distance between the two cracks cannot
be less than 2 · lt,max since the tensile stress within this distance cannot exceed the
concrete tensile strength and a new crack can not appear. On the other hand, if
the distance is more than 2 · lt,max the tensile stress can be build up again until the
concrete tensile strength is reached and a new crack can be initiated. The cracking
process will continue until the spacing between all cracks are smaller than 2·lt,max and
at that point, no further cracks can appear, and the crack formation is finalised. This
is called stabilized cracking stage. In theory, if the load increases, no further cracks
can occur but the steel stress is still increasing which results in an increased crack
width. Observations from Köseoğlu (2020) showed that further cracking appeared at
much later stages when the reinforcement was yielding. However, the initial cracking
prior to yielding was dominating the final crack pattern.
Figure 2.12 Concrete stress distribution during the cracking process in a
reinforced concrete member subjected to a tensile force higher than
the cracking load (Engström, 2014).
, Architecture and Civil Engineering, Master’s Thesis ACEX30 15
Bond-slip relation
As mentioned above, within the transmission length the steel stress is larger than
the concrete stress which cause elongation of the bar and a slip in relation to the
concrete. The bond stress has the maximum value at a cracked section and de-
ceases towards the opposite end of the transmission length. However, according to
Lundgren (1999), the transfer of bond stresses is highly effected when the reinforce-
ment bar yields. Due to the so called Poisson effect, the cross-sectional area of the
reinforcement decreases at yielding which reduces the stress transfer between the
reinforcement and concrete.
The relation between the bond stress and the slip has been studied experimen-
tally in pull-out tests and as a result a schematic relationship curve is presented
in Figure 2.13 where the slip is the relative difference in displacement between the
reinforcement bar and the concrete. The upper curve displays the bond-slip rela-
tion at pull-out and the dashed curve shows loss of bond due to yielding of the
reinforcement.
Figure 2.13 A schematic relationship between the bond stress and local slip from
pull-out tests and yielding of reinforcement. Figure inspired by (Plos
et al., 2021).
Crack spacing and crack width
The bond failure of the concrete near the ends of each cracked member is resulting in
zero transferred stress between the steel and concrete. As a consequence, the length
it takes to reach equilibrium between the concrete stress and the steel stress is equal
to the sum of the transmission length and the length of the concrete cone failure,
lt,max + ∆r. Therefore, the minimum crack distance sr,min, is equal to the distance
between the loaded end and the crack and cannot be less than lt,max+∆r, indicated
to the left hand side in Figure 2.14. The maximum crack distance sr,max, is the
distance between two cracks and cannot be more than 2(lt,max + ∆r) as shown to
the right in Figure 2.14. Thus, the crack spacing must fulfil the following condition
lt,max + ∆r ≤ sr ≤ 2 · lt,max + 2∆r (2.12)
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Figure 2.14 The minimum and maximum of the crack spacing after fully
developed crack formation (Engström, 2014).
With known bar diameter, concrete tensile strength and average bond stress the
minimum and maximum crack spacing can be calculated using Equations (2.13)
and (2.14) respectively.
1 fct φ
sr,min = ∆r + 4 · · (2.13)τbm ρ
= 2∆ + 1 · fct φsr,max r 2 · (2.14)τbm ρ
where ρ is the reinforcement amount described as the ratio between the steel area,
As and concrete cross-sectional area, Ac.
ρ = As (2.15)
Ac
In Eurocode 2 (CEN, 2004), the concrete area is determined as the effective concrete
area in tension that surrounds the reinforcement, Ac,eff . The height of the concrete
zone in tension hc,eff can be defined as
 2.5(h− d)
hc,eff = min (h− x)/3 (2.16)h/2
where h is the height of the cross section, d is the distance from the top of the cross
section to the center of gravity of the reinforcement layer and x is height of the
compressive zone.
According to Eurocode 2 (CEN, 2004), the characteristic crack width can be calcu-
lated by using the maximum crack spacing, sr,max multiplied with the mean strain
difference between the steel and concrete.
, Architecture and Civil Engineering, Master’s Thesis ACEX30 17
wk = sr,max(εsm − εcm) (2.17)
The characteristic crack width equation from Eurocode 2 is an approximation by
assuming perfect bond between the steel and concrete; i.e. no slip. However, a
more correct approach to determine the crack width is obtained by integrating the
difference between the steel strain εsx and the concrete strain εcx over the crack
spacing sr (Tammo, 2009). ∫
w = (εsx − εcx)dx (2.18)
sr
Factors influencing the crack width
An important parameter that effect’s the crack width is the steel stress and the
corresponding steel strain (Tammo, 2009). A weak bond between the materials re-
sults in larger crack spacing since less stresses are transfer from the reinforcement
bar to the surrounding concrete. The stress development is also effected by the
cross-sectional area of the steel bar. A larger reinforcement amount gives a decrease
of the steel stress and therefore a decrease in steel strain and as a result a smaller
strain difference between the steel and concrete, see Equation (2.17).
The configuration of the reinforcement bars is also affecting the crack width. The
bar diameter is important since the contact area between the steel and concrete
influence the transmission of bond stresses. The local slip of the bar on each side of
the crack affects the crack width. Therefore, a small bar diameter and small spacing
will increase the bond between the materials and reduce crack width, given that the
reinforcement area is constant.
2.4.5 Cracking of flexural loaded beam
The axially loaded concrete prism can be used to describe the cracking process of the
tensile zone of a beam loaded in bending. In Figure 2.15 a schematic explanation of
crack formation of a beam subjected to a bending moment is illustrated. According
to Al-Emrani et al. (2011), tests have shown that the crack distance decreases when
a structure is loaded by a bending moment compared to uniaxial tensile loading.
Further, the crack spacing is decreased with increasing curvature of the beam.
Figure 2.15 The principle of flexural loaded structure with tensile concrete stress
between two cracks (Al-Emrani et al., 2011).
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One important thing to mention is that the cracking process described above is valid
for cracking in SLS and is based on the approach used in Eurocode. However, in
this thesis the behaviour of the structure beyond the SLS is of interest. To describe
the structural behaviour at rupture of the reinforcement a theory based on plastic
deformation capacity is needed since the structure will have considerable plastic
deformation at rupture. Ansell and Svedbjörk (2000) refer to this stage as state IV.
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3 Plastic deformation capacity
3.1 Introduction
Reinforced concrete is a composite material and it can show either brittle or ductile
response depending on the reinforcement ratio and material properties. As a result
of ductility, plastic hinges are formed where reinforcement yields and stiffness finds
the lowest value along the beam at the place of plastic hinge. With change of stiff-
ness a moment redistribution takes place due to non-linear behavior of material and
increases load carrying capacity while deformations increase (Johansson & Laine,
2012).
To study structural behavior under impulse load the energy absorption capacity is
of great importance and according to Johansson and Laine (2012) plastic hinges
provide plastic redistribution of stresses and consequently, higher energy absorption
capacity. It is often assumed that a plastic hinge takes place in a concentrated point
while it distributes over a plastic region as shown in Figure 3.1.
Figure 3.1 Plastic hinge representation (Lozano & Makdesi, 2017).
With formation of plastic hinges as a consequence of ductile behavior from a struc-
ture subjected to loading, the structural element deforms. The resulting rotation is
interpreted as plastic rotation and the deformation caused by this rotation is plastic
deformation. According to Engström (2014) plastic rotational capacity is the rota-
tion of plastic hinge when the load increases from yielding to collapse. This concept
has a close connection to energy absorption capacity of structure as it is related to
ductility. Therefore, high plastic deformation capacity in a structure ensures higher
capacity of energy absorption.
, Architecture and Civil Engineering, Master’s Thesis ACEX30 21
3.2 Relation of plastic rotational capacity and
curvature
3.2.1 Definition of curvature
Considering the strain distribution of a reinforced concrete cross section, strain in
any section can be derived as (Engström, 2014)
1
εc(z) = εcm + · z (3.1)
r
Where εcm is mean strain or strain in center of gravity, r is curvature radius, 1 isr
curvature and z is distance from gravity center, see Figure 3.2.
Figure 3.2 Strain distribution in a reinforced concrete cross section (Jönsson &
Stenseke, 2018).
Denoting curvature as χ and considering a beam with constant curvature, where dϕ
is change of angle over element length and dx is element length along gravity axis,
it can be written
dx = r · dϕ ⇒ χ = 1 = dϕ (3.2)
r dx
Figure 3.3 Relation of curvature radius and deformation (Jönsson & Stenseke,
2018).
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For an infinitely small element, curvature can be derived as angle change per unit
length, where εcc is concrete compressive strain, εct is concrete tensile strain and εs
is steel strain.
χ = 1 = εcc = εcc + εct (3.3)
r x h
= 1 = εs = εcc + εsχ (3.4)
r d− x d
According to Jönsson and Stenseke (2018) Equations (3.3) and (3.4) can always be
used for states I, II, III and also for different material properties.
Figure 3.4 Geometry of cross section and strain distribution (Jönsson &
Stenseke, 2018).
3.2.2 Theoretical explanation for plastic rotation
According to Engström (2015) plastic rotation can be derived by integration of
plastic curvature over the length ly at which strain of steel goes above yielding
strain to ultimate strain. ∫
θpl = (χ(x)− χy)dx (3.5)
ly
Figure 3.5 shows moment-curvature diagram and two corresponding strain distribu-
tions for cross sections at yielding and ultimate states. Also, there is a simplified
bi-linear diagram based on elasto-plastic material response.
, Architecture and Civil Engineering, Master’s Thesis ACEX30 23
Figure 3.5 Moment curvature diagram for a concrete cross section reinforced at
the bottom and simplified bi-linear diagram (Jönsson & Stenseke,
2018).
When reinforcement starts yielding, concrete and steel strain can be denoted as
εcy and εsy respectively and the height of the compressive zone as xy. Therefore,
curvature at the beginning of plas(tic)deformation can be derived as
= 1 = εcy = εsyχy (3.6)
r y xy d− xy
For failure, depending on concrete crushing or reinforcement rupture the equations
for curvature can vary ( )
Concrete crushing: χu =
1 = εcu = εs (3.7)
r u xu d− xu
(1)= = εcc = εsuReinforcement failure: χu (3.8)r u xu d− xu
The plastic curvature can be estim(at)ed as1 ( ) ( )
χpl = =
1 − 1 (3.9)
( r pl r u r y
= 1
)
= εs εsy εs − εsyχpl − ≈ (3.10)
r pl d− xu d− xy d− xu
Both reinforcement strain and plastic curvature vary between yielding state and
failure state. Therefore, plastic rotation can be calculated by integration of plastic
curvature for all sections along ly (Jön(sson & Sten)seke, 2018)∫ ly
= εs(x)− εsyθpl dx (3.11)
0 d− xu
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According to Engström (2015) although Equation (3.11) is theoretically correct and
it provides a basic theoretical definition it cannot be used for calculation of real
plastic rotation as practically this parameter is dependent on several factors such
as material properties, amount of reinforcement and geometry, loading, cracking,
tension stiffening and bond slip, definition of yielding curvature and ultimate cur-
vature.
3.2.3 Ratio of MuM and plastic rotationy
Ultimate and yielding values have a significant role in estimation of plastic rotational
capacity. In theoretical methods the ratio between ultimate momentMu and yielding
moment My is a decisive factor influencing plastic rotational capacity and length of
plastic hinge. This ratio is based on moment distribution which depends on load
application and boundary conditions. In Figure 3.6 three different load cases are
shown for a simply supported beam and case (a) is the only case that Eurocode
2 estimates the rotational capacity for. The parameter ly is the length of yielding
region which increases from three-point loading to uniform load case showing that
the plastic rotational capacity also increases by increment of Mu/My.
Figure 3.6 Moment diagrams for three different load cases for a simply supported
beam.
From the ratio between ultimate moment to yielding moment Mu/My the ratio be-
tween ultimate load to yielding load fu/fy can be derived. Larger values of Mu/My
lead to larger fu/fy ratio and higher capacity of plastic rotation in a structure.
Table 3.1 provides the values of εsu and fu/fy for reinforcement classes A, B and C
showing that ductility increases as a result of larger ultimate strain and fu/fy ratio.
Table 3.1 Values of fu/fy and εsu for reinforcement classes A, B, and C.
Reinforcement class fu/fy εsu
A 1.05 25‰
B 1.08 50‰
C 1.15 75‰
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3.3 Different methods for calculation of plastic
rotational capacity
Plastic deformation capacity is a complex phenomena and the results for calculation
of plastic rotational capacity have a wide variety due to its dependence on several
parameters and as a result different methods have been introduced to estimate this
parameter.
3.3.1 Plastic rotation estimation from test results
Definition of yielding and ultimate values is an important step for calculation of
plastic rotation. Eurocode 2 (CEN, 2004) defines the ultimate deformation value
corresponding to ultimate load in critical section but ultimate deformation is often
not presented in a load-displacement curve from test results as the structure experi-
ences considerable deformations after reaching the maximum load values. In order to
obtain a more robust definition, Betonghandboken (1990) suggests a method which
is based on an utilized value considering elastic deformation and ultimate deforma-
tion corresponding to 95% of Fmax. This approximation estimates plastic rotational
capacity based on relationship between load and deformation.
Figure 3.7 The value of 0.95 · Fmax in load-deformation curve (Johansson &
Laine, 2012).
Therefore, upl can be defined as:
upl = uu − uel (3.12)
where uu represents the total deformation of the beam at 95% of maximum load on
the descending branch and uel indicates the elastic deformation at the same load
level. From this, for the cases illustrated in Figure 3.7, the plastic rotational capacity
can be calculated as (Johansson & Laine, 2012)
= uplθpl (3.13)
l0
26 , Architecture and Civil Engineering, Master’s Thesis ACEX30
the parameter l0 is the length between zero moment and plastic hinge which is
dependant on loading type and boundary conditions.
Figure 3.8 Length of l0 for two different types of loading and boundary conditions.
Parameter a is the distance between support and the point at which concentrated
load is applied. As shown is Figure 3.9 for a special case which is studied in this
project and is simply supported, subjected to four-point bending, and the distance
between loads is equal to the distance between each support and load, it can be said
that a = l0 = l/3 and the plastic rotation can be determined as
θpl =
3 · upl (3.14)
l
Figure 3.9 Simply supported beam subjected to four-point bending.
3.3.2 BK 25 method
FKR 2011 is a set of certain design regulations from the Swedish Fortification Agency
which is based on BK 25 Method. According to Swedish Fortifications Agency,
plastic rotational capacity for a reinforced concrete beam subjected to impulse load
can be estimated considering a simply supported beam with uniformly distributed
load using BK 25 method (Johansson & Laine, 2012).
, Architecture and Civil Engineering, Master’s Thesis ACEX30 27
Figure 3.10 Geometry definitions of a simply supported beam based on Johansson
and Laine (2012).
In this method the length of the equivalent plastic hinge, lp with constant curvature
is defined as
lp = 0.5 · d+ 0.15 · l (3.15)
Which is directly used in calculation of plastic rotational capacity, θpl
= lpθpl (3.16)
r
Total plastic deformation caused by plastic hinge at mid-span is
θpl · l
u = 2 (3.17)
for a cross section at ultimate moment, curvature is defined according to
Equation (3.7)
= 1 = εcu = εs = Mχu (3.18)
r x d− x EI
For a beam with tensile reinforcement at bottom layer, considering yielding of rein-
forcement, the compressive block height can be solved as
= ωs · dx 0 8 (3.19).
where ωs is mechanical ratio of tensile reinforcement and ρs is reinforcement ratio
parameter defined as
= Asρs (3.20)
b · d
= · fyωs ρs (3.21)
fc
ωs,crit is a limit for mechanical ratio that specifies the failure mode. If (ωs > ωs,crit)
the failure is due to concrete crushing, otherwise, reinforcement rupture is the reason
of failure.
28 , Architecture and Civil Engineering, Master’s Thesis ACEX30
= 0.8 · εcuωs,crit
εcu +
(3.22)
εsu
Finally, the plastic rotational capacity in the span can be obtained considering the
failure mode.
Failure due to concrete crushing (ωs > ωs,crit)
= lp = 0.8 · εcuθpl · (0.5 · d+ 0 15 · ) =
0.4 · εcu l
. l (1 + 0.3 · ) (3.23)
r ωs · d ωs d
Failure due to reinforcement rupture (ωs < ωs,crit)
= lp = 0.8 · εsu · (0 5 · + 0 15 · ) = 0.4 · εsu lθpl . d . l
r d · (0.8− ωs) (0 8 )
(1 + 0.3 · ) (3.24)
. − ωs d
Also, considering fixed supports for the same beam at Figure 3.10, the equivalent
plastic hinge takes a shorter value compared to a simply supported beam.
lp = 0.5 · d+ 0.1 · l1 (3.25)
where l1 can be described as the ratio of bending moment to shear force at support
= Msupl1 (3.26)
Vsup
If Msup = Mfield this ratio can be derived as
l1 = 0.125 · l (3.27)
Therefore, plastic rotational capacity for a beam with fixed supports can be derived
as
Failure due to concrete crushing (ωs > ωs,crit)
= lp = 0.8 · εcu (0 5 · + 0 1 · ) = 0.4 · εcu (1 + 0 2 · l1θpl . d . l1 . ) (3.28)
r ωs · d ωs d
Failure due to reinforcement rupture (ωs > ωs,crit)
= lp = 0.8 · εsuθpl (0 8 )(0.5 · d+ 0
0.4 · εsu l1
.15 · l1) = (0 8 )(1 + 0.2 · ) (3.29)r d · . − ωs . − ωs d
, Architecture and Civil Engineering, Master’s Thesis ACEX30 29
3.3.3 Eurocode 2
Definition of plastic rotational capacity in Eurocode 2
According to Eurocode 2 (CEN, 2004) plastic rotational capacity, θpl,EC , for a con-
tinuous beam is defined as the rotation over an inner support. Also, the length of
rotation is assumed 1.2 times of support section depth.
Figure 3.11 Plastic rotational capacity based on Eurocode 2.
For an equivalent simply supported beam with concentrated load and a length cor-
responding to zero moments over the support of mentioned continuous beam, the
rotation of mid-span is approximately equal to the same value of θpl,EC (Latte, 1999).
Figure 3.12 Simply supported beam equivalent to a continuous beam for plastic
rotation capacity calculation (Lozano & Makdesi, 2017).
According to Eurocode 2, the plastic rotation capacity in the middle of this equiv-
alent beam is twice the rotation at its support which corresponds to the value of
plastic rotation presented by BK 25.
30 , Architecture and Civil Engineering, Master’s Thesis ACEX30
Figure 3.13 The relation between plastic rotational capacity and support rotation
for equivalent simply supported beam based on Lozano and Makdesi
(2017).
Plastic rotational capacity and x/d ratio
Eurocode 2 provides a simplified procedure in which there is a relation between
plastic rotational capacity and x/d depending on reinforcement class and concrete
class (CEN, 2004). For this approach the following conditions must be checked for
plastic hinge regions
For concrete classes less than or equal to C50/60:
xu ≤ 0.45 (3.30)
d
For concrete classes greater than or equal to C55/67:
xu ≤ 0.35 (3.31)
d
A correction factor kλ which depends on shear slenderness should be multiplied to
θpl,EC
√
λ
kλ = 3 (3.32)
θ∗pl,EC = kλ · θpl,EC (3.33)
l0 denotes the distance between sections with zero moment and maximum moment
after redistribution, d is effective depth and λ is shear slenderness defined as
l0
λ = (3.34)
d
The significant difference in Figure 3.14 shows how more ductile reinforcement pro-
vide larger plastic rotational capacity.
, Architecture and Civil Engineering, Master’s Thesis ACEX30 31
Figure 3.14 Allowable rotation of reinforced concrete sections for different
concrete classes, and Class B and C steel reinforcement. Valid for
shear slenderness λ=3. Modified from CEN (2004).
When reinforcement rupture failure mode is decisive, the steel bars use their strain
capacity completely and θpl,EC increases by increasing x/d ratio. While for concrete
crushing failure mode θpl,EC decreases by increasing x/d ratio as concrete crushing
happens when steel strain capacity has not been fully utilized.
As shown in Table 3.1 reinforcement bars in class C have larger value of fu/fy and
a larger ultimate strain which leads to a higher rotational capacity. It is also ob-
served in Figure 3.14 that plastic rotational capacity is considerably larger for class
C reinforcement with more ductile behaviour.
3.3.4 German method (background of Eurocode 2)
According to Zilch and Zehetmaier (2010) and CEB (1998) there is an earlier numer-
ical model provided by a German approach called "Stuttgart method". This method
considering an equation for calculation of plastic rotation in which plastic rotation
for an internal support of a continuous beam under distributed load is the same as
plastic rotation for mid-section of a simply supported beam under point load and
with a length corresponding to distance between zero-moment points as shown in
Figure 3.15. This approach is a base for what is presented in Eurocode 2.
32 , Architecture and Civil Engineering, Master’s Thesis ACEX30
Figure 3.15 Plastic rotation of an internal support in continuous beam under
distributed load equivalent to plastic rotation of midspan in a simply
supported beam under concentrated load (Zilch & Zehetmaier, 2010).
In this method, the moment-curvature diagram is estimated for a confined and an
unconfined model based on provided geometry and stress-strain curves for steel and
concrete taking Euler–Bernoulli assumption so plane sections remain plane.
Figure 3.16 Moment-Curvature model from Stuttgart method (CEB, 1998).
From moment-curvature relationship, tensile force-curvature and as a result, curva-
ture in cracks can be anticipated as shown in Figure 3.17. Where dash line represents
confined model not considering cracks.
, Architecture and Civil Engineering, Master’s Thesis ACEX30 33
Figure 3.17 Integration of curvature from Stuttgart model (CEB, 1998).
CEB (1998) defines plastic rotation as the difference between rotation at yielding
of reinforcement and rotation at ultimate load. Also, Eurocode 2 is based on this
definition.
Figure 3.18 Schematic definition of plastic deformation based on Eurocode 2 and
CEB (1998).
In addition, Stuttgart method introduces the relation between plastic rotation and
percentage of reinforcement. This approach is also used in Eurocode 2 to provide
the graph in Figure 3.14. Although, it is not clearly mentioned in Eurocode 2,
but according to CEB (1998), Zilch and Zehetmaier (2010) and DAfStb (2010)
Figure 3.14 is based on fu/fy ratio.
34 , Architecture and Civil Engineering, Master’s Thesis ACEX30
Figure 3.19 Relation between rotation and percentage of reinforcement from
Stuttgart model (CEB, 1998).
Finally, Zilch and Zehetmaier (2010) gives the following equation based on Stuttgart
model to calculate plastic rotational capacity.√
ε∗su − εsy λθ 3pl,EC = βn · βs · 1 x · 3 · 10 [mrad] (3.35)− d
d
where  ( ) x 0.20.28 · β · dcd · εuk (steel failure)d
ε∗su = min ( ) 2 ( ( ) ) (3.36)x 3 x −11.75 · d · 1− d · εc1u (concrete failure)d d
Different parameters included in equation of plastic rotation capacity are defined as
βn = 22.5 (3.37)
= (1− fykβs ) (3.38)
( fuk
= −0.0035
)3
βcd (3.39)
εc1u
While, ε∗su is steel ultimate strain with simplified consideration of tension stiffen-
ing, εuk is steel ultimate strain at maximum load and εsy is steel ultimate strain at
yielding. εc1u denotes concrete strain limit at maximum load, fyk represents yielding
strength of reinforcement and ftk is tensile strength of reinforcement.
Johansson et al. (2021) provides a study on effect of fu/fy ratio and εsu on plas-
tic rotational capacity based on the background of Eurocode. From this study,
, Architecture and Civil Engineering, Master’s Thesis ACEX30 35
Figure 3.20 shows the relation between plastic rotational capacity and x/d ratio for
reinforced concrete specimen with steel classes A, B, C, Ks 40 (1), and Ks 40 (2)1.
The reinforcement classifications Ks 40 (1), and Ks 40 (2) are from old Swedish
regulations which were used until 1995. It is obvious that reinforcement Ks 40 is
noticeably better than reinforcements A, B, and C.
Figure 3.20 Relation between plastic rotation and x/d ratio for reinforced
concrete specimen with steel classes A, B, C, ks40(1), and ks40(2).
According to Johansson et al. (2021) having a constant value for the ratio of fu/fy
leads to higher plastic rotation capacity for a bar with higher value of ultimate
strain. Also, larger constant value of fu/fy results in larger plastic rotation capacity.
Similarly, keeping εsu constant shows higher plastic rotation capacity for a bar with
larger ductility and larger constant value of ultimate strain provides larger plastic
rotation capacity.
1The properties of Ks 40 was never defined and are estimated by Johansson et al. (2021)
36 , Architecture and Civil Engineering, Master’s Thesis ACEX30
Figure 3.21 Relation between plastic rotation and x/d ratio for reinforced
concrete specimen with steel classes A, B, C, ks40(1), and ks40(2) by
keeping fu/fy constant.
Figure 3.22 Relation between plastic rotation and x/d ratio for reinforced
concrete specimen with steel classes A, B, C, ks40(1), and ks40(2) by
keeping εsu constant.
, Architecture and Civil Engineering, Master’s Thesis ACEX30 37
3.3.5 Betonghandboken 1990 (ABC method)
This is an empirical method to calculate plastic rotation for a reinforced concrete
beam under static loading and on one side of critical cross section provided by
Betonghandboken 1980 (Lozano & Makdesi, 2017). As mentioned in Section 3.3.1,
Betonghandboken considers the plastic rotations corresponding to 95% of ultimate
load and in order to calculate it, three parameters are used considering different
factors affecting plastic rotational capacity.
θpl,95% = A ·B · C · 10−3 (3.40)
Factor A is derived based on amount of tensile, compression and transverse rein-
forcement with strength values from Swedish concrete code
A = 1 + 0.6 · ωv + 1.7 ·
′ − 1 4 · ωsωs . ≥ 0.05 (3.41)ωb
ωv denotes mechanical ratio of stirrups,
′
ωs is mechanical ratio of reinforcement in
compression and ωs is mechanical ratio of reinforcement in tension
Av fy
ωv = · (3.42)
b · s fct
′
′ = As · fyωs (3.43)b · d fcc
= As · fyωs (3.44)
b · d fcc
Parameters ′Av, As and As are areas of reinforcement in shear, compression and
tension, respectively. fy is strength value of reinforcement. fct denotes concrete
tensile strength and fcc is concrete compressive strength. Parameter b is width of
cross section, s is distance of stirrups and d is effective height. Considering ultimate
strain of concrete as εcu = 3.5‰ and reinforcement yield strain as εsy = fy/Es
= 0.8 · εcuωb + (3.45)εcu εsy
For measured values of strength, the following equation can be used for A
A = 1 + 1.3 · ωv + 3 ·
′
ωs − 5 · ωs (0.05 ≤ A ≤ 2.30) (3.46)
Factor B is derived based on shape of the stress-strain diagram for reinforcement
and has different values depending on type and grade of bars. The reinforcement
classification in Table 3.2 is from old Swedish regulations which were used until 1995.
38 , Architecture and Civil Engineering, Master’s Thesis ACEX30
Table 3.2 Values for factor B. εp is the strain value for prestressing steel
(Jönsson & Stenseke, 2018)
Type of reinforcement B Max A.B
Ks 60, Ks 40*, Ss 26, Ss 26S 1.0 1.7
Ks 60S, Ks 40S ( 0.8 ) 1.1
Cold-worked steel with εsu ≥ 3% and f /f ≥ 1.1 0.6 · 1− 0.7 · εpt y 0.5εsu
*NOTE: Values for Ks 60 and Ks 40 can be used if εsu ≥ 8% and ft/fy ≥ 1.4. The
letter S in Ks 60S and Ks 40S indicates that the reinforcement can be welded and
it should be considered that ductile properties will be reduced after welding.
Factor C is derived based on the placement of plastic hinge and can be calculated
as
Csup = 10 ·
l0,sup (3.47)
d
l0,field
Cfield = 7 · (3.48)
d
while l0,sup and l0,field are defined as shown in Figure 3.8.
ABC method and BK 25 use the same definition of plastic rotation. BK 25 studies a
beam under uniformly distributed load with fixed supports while Eurocode 2 consid-
ers a simply-supported beam with point load. With a similar approach to the idea
presented in Figure 3.12, fixed supports on a beam can be taken as internal support
in a continuous beam. Considering an equal simply supported beam subjected to a
point load in midspan, the rotation values from BK 25 and ABC method can be the
same value as the support rotation in Figure 3.13. Therefore, in order to compare
the results from BK 25 or ABC method with Eurocode 2, values from Eurocode 2
must be multiplied by a factor 0.5.
, Architecture and Civil Engineering, Master’s Thesis ACEX30 39
40 , Architecture and Civil Engineering, Master’s Thesis ACEX30
4 Previous testing
4.1 Introduction
This chapter provides a summary from two sets of studies on plastic deformation
capacity of reinforced concrete elements which were performed at Chalmers Uni-
versity of Technology 2020 and KTH from 2000 to 2005. These projects are also
used as reference and study background for this thesis. Hence, a short description
of mentioned projects and their results are provided.
4.2 Chalmers experiment
This project is a continuation of a BSc thesis made by Köseoğlu (2020). The project
was meant to study the average reinforcement strain and how the strain is affected
by altered parameters. The experiments were performed under static axial tensile
loading as a simplification to study the average strain of reinforcement bars i order
to estimate the plastic rotational capacity.
Eighteen reinforced concrete prisms with one centralized reinforcement bar were
subjected to static tensile load until failure, see Figure 4.1. Difference in size of
reinforcement bars (φ16, φ12, φ10) and implementing PVC tube segments around
some of the φ12 bars with different configurations to reduce the bond resulted in
different strain values. Strain fields, crack widths and total displacements were
studied for each prism using data from DIC.
Figure 4.1 Reinforced concrete prism geometry and cross section (Köseoğlu,
2020).
, Architecture and Civil Engineering, Master’s Thesis ACEX30 41
The test results showed that with increased bar diameter the average strain value
also increased, more cracks were observed and influence of concrete between cracks
decreased. Prisms with PVC tubes showed fewer cracks compared with prisms with
full interaction. Also, it was expected that the plastic deformation capacity increases
by reduction of interaction between steel and concrete while, the results showed
the reverse and lower values of strains and total displacements were obtained with
increasing PVC volumes. How the average reinforcement strain varies with applied
tensile force for the different bar sizes with and without plastic tubes is presented
in Figure 4.2.
Figure 4.2 Results for all reinforced concrete prisms. Red curves represent φ12
prisms without PVC and orange curves represent φ12 prisms with
PVC (Köseoğlu, 2020).
The average strains at rupture in the reinforcement bars without PVC tubes were
9% in the φ16, 7% in φ12 and 6% in φ10. The average reinforcement strain in the
bars with PVC tubes was approximately equal to 6.5%, compared with the 7% for
the φ12 prisms without PVC tubes. However, it was predicted that the results will
be different in beams due to different reinforcement ratio and configuration and also
uneven concrete cover.
4.3 KTH experiments
Previous research about impulse loaded structures and plastic deformation capacity
have been preformed at KTH university in Stockholm. A series of static and dy-
namic tests on concrete slab strips have been performed on behalf of the Swedish
Armed force between the years 2000-2005.
Today’s modern reinforcement has a decreased ratio between the yielding strength
and the ultimate strength (Ansell & Svedbjörk, 2000). Therefore, the aim of the test
series was to compare the plastic behaviour and plastic rotational capacity between
concrete members with the new reinforcement, B500BT and the previous used Ks40
42 , Architecture and Civil Engineering, Master’s Thesis ACEX30
and Ps50. The test set-up from KTH is similar to the one used in this thesis with a
four-point bending test, shown in Figure 4.3. During the test, the applied load was
measured and the corresponding deformation was registered.
Figure 4.3 Illustration of the test set-up from the four-point bending test of a slab
strip from test conducted at KTH.
Before the casting, the reinforcement bars were marked every 50 mm and after the
tests, the bottom reinforcement was removed and the strain difference between two
marks was evaluated. The results were presented in bar charts and an example of
the strain distribution of on of the slab strip with reinforcement quality B500BT is
presented in Figure 4.4.
Figure 4.4 Example of results from the measured strain distribution in the bottom
reinforcement. The first diagram is the strain difference in the
reinforcement bar to the left and the second diagram for the
reinforcement bar to the right (Ansell & Svedbjörk, 2000).
, Architecture and Civil Engineering, Master’s Thesis ACEX30 43
Johansson and Laine (2012) summarized the results from the test series at KTH for
simply supported slab strips with varying cross sections, reinforcement amount and
reinforcement quality, see Table 4.1.
Table 4.1 Summary of the average strain results from tests conducted at KTH
(Johansson & Laine, 2012).
44 , Architecture and Civil Engineering, Master’s Thesis ACEX30
5 Experimental procedure
5.1 General description
Reinforced concrete beams were manufactured and tested to study the parame-
ters that effects the average reinforcement strain and plastic deformation capacity.
Nine concrete beams were constructed as well as one test beam. The length of the
beams were equal to 2.8 m and a cross-sectional height and width of 200 mm, see
Figure 5.1. The beams were reinforced with two steel reinforcement bars with a
nominal diameter of 10 mm or 12 mm, respectively. In order to eliminate the in-
fluence of top reinforcement and stirrups on plastic deformation capacity, the bars
are just placed at the bottom layer of the beams. After casting and curing, the
beams were subjected to a four-point bending test until failure. The manufacturing
of the beams and performing of tests were conducted in the Structural engineering
laboratory at Chalmers University of Technology. A more detailed description of
the preparations can be found in Appendix A.
Figure 5.1 Reinforced concrete beam geometry and cross section.
5.2 Manufacturing of concrete beams
As mentioned, in total ten beams were manufactured and tested of which one beam
was a test beam. Three different beam types were constructed, see Table 5.1. All
three beam types had the same dimensions but different reinforcement amount, φ10
or φ12 and one beam type with the φ12 reinforcement had low-friction PVC tubing.
The test beam, beam number 10, was made by the research engineer to be used
as a test beam or an additional beam for unpredicted situation. However, the φ10
reinforcement bars in beam number 10 were unmarked and could not be used for
strain measurements and the reinforcement quality was unknown.
, Architecture and Civil Engineering, Master’s Thesis ACEX30 45
Table 5.1 Naming of beams based on used reinforcement type and PVC tubes.
Name Bar diameter Bar label Configuration
Beam 1 10 A,B No PVC
Beam 2 10 A,B No PVC
Beam 3 10 A,B No PVC
Beam 101 10 A,B No PVC
Beam 4 12 A,B No PVC
Beam 5 12 A,B No PVC
Beam 6 12 A,B No PVC
Beam 7 12 A,B 4x100 PVC
Beam 8 12 A,B 4x100 PVC
Beam 9 12 A,B 4x100 PVC
1 Additional test beam.
5.2.1 Preparations before casting
The preparation started with cutting the reinforcement bars in length of 2.79 m to
fit the beams and cut 6 samples from each bar diameter for tensile strength testing.
To be able to measure the elongation of the reinforcement and calculate the plastic
reinforcement strain all bars were marked each 50 mm using a hammer and an awl.
Also, the middle one meter of the bars was marked using zip ties.
From each type of beam, two bars were scanned with 3D scanning tools to make a
3D model of the reinforcement bar. The bars for beams 1, 4 and 7 were scanned
both before casting and after testing as another tool to measure the elongation and
estimate the plastic strains. The 3D models were also used to analyse the change of
cross sectional-area due to yielding.
After 3D scanning, PVC tube segments with length of 100 mm and spacing 200 mm
were glued at the 1 m length in the middle of the reinforcement bars of the beams
with plastic tubes, see Figure 5.2. The intention with the plastic tubes was to reduce
the interaction and the transfer of bond stresses between the reinforcement and the
concrete to increase the plastic deformation capacity of the beam.
46 , Architecture and Civil Engineering, Master’s Thesis ACEX30
Figure 5.2 Configuration of the PVC tubes.
Before the reinforcement was placed inside the moulds, the moulds were oiled to
facilitate the cleaning afterwards. Spacers were nailed inside the forms to have
concrete cover equal to 35 mm, see Figure 5.3.
Figure 5.3 Geometry of the cross section and concrete cover.
5.2.2 Casting
Concrete with the strength class C40/50 was ordered and delivered to the site by
a concrete mixing truck and the fresh concrete was poured from the truck into the
forms. Concrete cubes for material testing were also casted to use for compressive
strength tests and wedge-splitting tests (WST). A vibrator was used to eliminate
air in the concrete. The beams were watered before covered by a plastic sheet to
harden and the cubes were submerged in water until the material testing.
After demoulding, the beams were prepared for the digital image correlation (DIC).
A DIC camera was used during the beam test to document the development of cracks
and the deformation of the beam. The software require a high contrast pattern and
therefore the side towards the camera was painted with a white backdrop. After
drying, a black speckle pattern was added to the white backdrop with a brush.
5.2.3 After testing
In order to measure the plastic strain of the reinforcement, the bars was removed
from the concrete beams after testing. A cut was sawn along each reinforcement
bar and a sledgehammer was used to remove the concrete. A schematic illustration
of the cutting process is shown in Figure 5.4.
, Architecture and Civil Engineering, Master’s Thesis ACEX30 47
Figure 5.4 A schematic illustration of the process to remove the reinforcement
bars from the concrete. The dashed lines represent the position of the
saw cuts and the arrows hit with the sledgehammer.
After the reinforcement was removed from the concrete the elongation of the bars was
measured by measuring the distance between the marks on the reinforcement bars.
The 3D scanned bars was scanned again and the concrete residuals was removed prior
to the 3D scanning by sandblasting the bars. To fit the bars into the sandblasting
machine, the middle one meter was cut with 100 mm margin at each side. Also,
reinforcement specimens was cut at each end of the reinforcement bars to be used
for a second reinforcement tensile strength test.
5.3 Beam test setup
The load rig was prepared for a four-point bending test with two point loads. Be-
tween the two point loads, a tensile zone with constant moment can be distinguished
and it is within this tensile zone, the first cracks were expected to appear, see
Figure 5.5.
Figure 5.5 Schematic illustration of the moment distribution.
The loading was deformation controlled by a hydraulic jack connected to the steel
loading beam with two roller supports. Steel plates and thin wooden plates were
placed between the roller supports and the concrete beam. The beam was placed
on a pinned support to the left and a roller support to the right, the test setup is
presented in Figure 5.6. A load cell and a displacement sensor were connected to the
48 , Architecture and Civil Engineering, Master’s Thesis ACEX30
hydraulic jack to measure applied load and midspan deflection. At the beginning of
the loading session, the beam was loaded with an initial load equal to 8 kN and then
unloaded to 2 kN to avoid effect of settlements in the load-deformation results. After
the unloading, the loading continued according to the predetermined loading rate
until failure. The loading was stopped when the applied load decreased drastically
due to rupture of reinforcement or crushing of concrete. Also, for some beams the
test was stopped before failure because the loading device got in contact with the
beam.
Figure 5.6 Test setup. Example from beam number 4.
The measuring sequence had a limited number of captured images and to prevent a
restart of the program during a test, adjustment of the load rates was done during
testing. The loading was divided into two phases, the first phase with a lower speed
to capture the changes in the force-displacement curve more carefully and allow for a
slower crack development. The second phase, with a higher load rate, was initiated
at a certain deformation which are presented together with the load rates for each
beam in Table 5.2.
, Architecture and Civil Engineering, Master’s Thesis ACEX30 49
Table 5.2 Adjustments of the loading rates.
Beam Load rate Deformation
1, 5 1 mm/min < 10 mm
4 mm/min
8 2 mm/min < 20 mm
4 mm/min
2, 3, 4, 6, 7, 9, 10 2 mm/min < 20 mm
5 mm/min
As mentioned in the description of the beam test setup, loading plates were placed
under the point loads. For beams 1, 5 and 8 the size of the loading plates was equal
to 100x200 mm. To increased the capacity and prevent eccentricity of the loading
points due to movements of the concrete beam, the size of the plates was increase
to 200x200 mm. An illustration of the eccentricity problem due to movements of
the beam is shown in Figure 5.7.
Figure 5.7 Eccentricity of loading point for beam 8 with small loading plate.
5.4 Digital image correlation
DIC is a non-interferometric method and is as a powerful analysis tool used for
experimental measurements and the method has been used to capture the behaviour
of the beam under loading. Two cameras were used to record data that resulted
in 3D models and strain fields of the beam surface. The cameras use the white
and black speckle pattern to analyse the deformation of the painted surface during
testing. The cameras recorded the data from the 1 m in the middle of the beams, see
Figure 5.8.
50 , Architecture and Civil Engineering, Master’s Thesis ACEX30
Figure 5.8 Area captured by the DIC camera.
The frequency of the DIC camera was set to 0.5 Hz which implies that one pic-
ture is captures every two seconds. However, the frequency was changed for beam
6, 7 and 9 where the first 600 s had the frequency 0.5 Hz and between 600 s and
1656 s the frequency was set to 0.25 Hz and the remaining running time, the fre-
quency was again 0.5 Hz. The adjustment of the camera frequency was due to a
limitation of total number of pictures that can be captured with the DIC camera
before a restart of the sequence is needed. The frames captured by the DIC cameras
were processed in the software GOM Correlate to analyse the deformation and crack
development of the beam during testing.
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52 , Architecture and Civil Engineering, Master’s Thesis ACEX30
6 Predictions
6.1 Introduction
Predictions of the load capacity were made to estimate the ultimate load and the
load at yielding. The material properties used for the calculations are results from
the material testing presented in Section 7.2. The concrete compressive strength was
approximated by interpolating the results from the compressive strength tests pre-
formed at 28 days and at 40 days after casting to estimate to the strength at 33 days.
The average values from the material testing have been used in the calculations, see
Table 7.1 for concrete compressive strength and Table 7.3 for reinforcement prop-
erties. The method used to estimated the load capacity is based on the procedures
presented in (Al-Emrani et al., 2011) and (Al-Emrani et al., 2013).
6.1.1 Ultimate load capacity
The ultimate load was determined using the stress block method. The ultimate
concrete strength is assumed to be reached and is equal to εcu = 3.5‰ and the
reinforcement is yielding. A linear strain distribution and a parabolic relation of
stress distribution in the compressive zone are also assumed, see Figure 6.1.
Figure 6.1 Schematic illustration of the stress block method to determine the
ultimate moment capacity.
Equilibrium condition: α · fc · b · x = fsy · As (6.1)
Ultimate moment: Mu = α · fc · b · x(d− β · xu) (6.2)
Equation (6.1) was used to calculate the height of the compressive zone, xu and
then inserted in Equation (6.2) to calculate the ultimate bending moment. The
coefficients α and β is called stress block factors and are stated in (Al-Emrani et al.,
2013) and have the values
, Architecture and Civil Engineering, Master’s Thesis ACEX30 53
α = 0.810 (6.3)
β = 0.416 (6.4)
To consider the dead weight of the beam, the ultimate moment was reduced by the
moment due to dead weight to calculate the remaining moment capacity that the
beam can carry in the ultimate limit state
2
Mq = Mu −Mg = −
g · l
Mu 8 (6.5)
The corresponding load can be calculated by dividing the reduced moment by the
distance a, which is the length from the support to the point load equal to L/3 then
multiply the expression with two to obtain the total load from the two point loads,
see Equation (6.6). The estimated ultimate moment including dead weight and load
capacity is presented in Table 6.1.
Mq
Ftot,q = 2 · (6.6)
a
Table 6.1 Predicted ultimate load capacities when dead weight is considered, for
beams with reinforcement φ10 and φ12, respectively.
Beam Ultimate moment, Mq [kNm] Ultimate applied load, Ftot,q [kN]
φ10 13.6 34.0
φ12 18.6 46.6
6.1.2 Load at yielding
The stress block method was also used to calculate the capacity at yielding. The
calculations are similar to the calculations at the ultimate limit state but with a
triangular stress block, see Figure 6.2.
Figure 6.2 Schematic illustration of the triangular stress block to determine the
yielding moment.
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The height of the compressive zone, xII is determined for the cross section in state
II where α is the ratio between the modulus of elasticity of the reinforcement and
concrete.
· · xIIb xII 2 = α · As(d− xII) (6.7)
Equilibrium condition:
σcc
2 · b · xII = σs · As (6.8)
The stress in the concrete at yielding is unknown and the stress in the steel is
equal to the yielding strength fy. The yielding moment is calculated with moment
equilibrium around the concrete compressive resultant and then reduced by the dead
weight moment, see Equation (6.9). The corresponding load capacity at yielding is
calculated according to Equation (6.6) and the results are presented in Table 6.2.
My = σs · As( −
xII
d 3 )−Mg (6.9)
Table 6.2 Predicted yielding moment and yielding load when dead weight is
considered, for beams with reinforcement φ10 and φ12, respectively.
Beam Yield moment, My [kNm] Yield load, Ftot,y [kN]
φ10 13.0 32.4
φ12 17.7 44.4
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56 , Architecture and Civil Engineering, Master’s Thesis ACEX30
7 Experimental results
7.1 General description
The results from the material testing, bending tests, strain measurements and de-
termination of plastic rotation capacity are presented within this chapter. Firstly,
the material testing of the concrete and reinforcement are treated in order to get
necessarily background information of the materials. Then the results from the
bending tests are presented in terms of structural response under loading and crack
formation. Also, beam results with respect to plastic reinforcement strain from mea-
surements and 3D scanning are presented. At last, determination of plastic rotation
capacity and a summary of the concrete beam results are included.
7.2 Material testing
To obtain necessary background information about the materials, testing of both
plain concrete and steel reinforcement was performed. Compressive strength tests
and wedge-splitting test (WST) were performed to determine the concrete properties
and tensile tests of the reinforcement were performed twice, both before and after
beam testing to determine the steel properties.
7.2.1 Concrete
Compressive strength test and wedge-splitting test were performed on the concrete in
order to obtain material properties. Therefore, three concrete cubes were subjected
to compressive test at 28 days after casting. Also, the test was repeated for three
concrete cubes at the age of 40 days. Wedge-Splitting test was performed on three
cubic specimens to reach the fracture energy of concrete. The detailed test results
are provided in Appendix B and the results from the compressive tests in presented
in Table 7.1 where the cylindrical compressive strength was approximated using
Equation (2.1) and the result from the WST in Table 7.2.
Table 7.1 Average compressive strength.
Test type Cube strength [MPa] Cylinder strength [MPa]
28-days compressive strength 64.8 54.0
40-days compressive strength 70.4 58.7
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Table 7.2 Average values from the WST results.
Accumulated GF Maximum splitting load Fsp Maximum CMOD
[Nm/m2] [kN] [mm]
151 5.79 2.13
7.2.2 Reinforcement
Six reinforcement specimens for each bar diameter was subjected to a tensile test to
provide material properties. The stress-strain curves are plotted for respective rein-
forcement diameter in Figures 7.1 and 7.2 and a summary of the average material
properties is presented in Table 7.3. Complete result from the twelve reinforce-
ment specimens and more information about the tensile test setup are presented in
Appendix B.
Figure 7.1 Stress-strain curves for six φ10 reinforcement specimens.
58 , Architecture and Civil Engineering, Master’s Thesis ACEX30
Figure 7.2 Stress-strain curves for six φ12 reinforcement specimens.
With results from the tensile tests, material data as modulus of elasticity, yield-
ing strength, ultimate strength, tensile strain at maximum stress and the yield to
ultimate tensile strength ratio were determined. The average properties for each
respective reinforcement diameter are presented in Table 7.3.
Table 7.3 Average modulus of elasticity, strain at ultimate stress, yielding
strength, ultimate tensile strength and fy/fu-ratio.
φ Es εsu fy fu fy/fu
[mm] [GPa] [‰] [MPa] [MPa] [-]
10 198 64 564 656 1.16
12 191 90 535 622 1.18
7.3 Beam results, structural response
In this section the results are presented from the four-point bending tests on ten rein-
forced concrete beams which were introduced in detail in Section 5.2.1. The results
are illustrated and compared in form of load-displacement curves, displacement-
width curves for cracks, displacement-time curves for loading points, and crack pat-
tern figures.
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The values in load-displacement curves are provided from hydraulic jack while the
other values are provided from DIC results. In order to facilitate the comparison,
the first part of loaBde-admis 4p lacement curves including initial loadBinegama n5d unloading are
omit2t00ed and the curves are adjusted to have the20s0ame starting point.
150 150
Test10r0esults are presented and processed in thre1e00categories φ10 reinforcement, φ12
reinforcement, and φ12 reinforcement with PVC tubes. From each category, the
load-5d0isplacement curve of all beams is provided50in one graph. The displacement-
width0 curves for cracks together with correspond0 ing load-displacement curve and
crack pa0ttern fig2u0r0e0 for one40b00eam is p60r0e0sented in the0 report r2e0p00resentin4g00t0he gene6r0a00l
structural behaviorTiomfe,e ta[sc]h group. All other curves and figurTeims e,c ta[ns] be found in
Appendix C.
Beam 6 Beam 7 
200 200
Symmetrical structural behavior and its effect on structural response is studied
for e15a0ch beam. It is observed that some beam1s50have more symmetrical behavior
such10a0s beam 8 compered to beam 9 which ha1s00an unsymmetrical behaviour, see
Figure 7.3. Displacement-time curve under each loading point and the point between
50 50
loading points (see Figure C.11) are provided in these graphs. All relevant curves
are pr0ovided in Appendix C. 0
0 2000 4000 6000 0 2000 4000 6000
Time, t [s] Time, t [s]
Beam 8 Beam 9 
200 200
150 150
100 100
50 50
0 0
0 2000 4000 6000 0 2000 4000 6000
Time, t [s] Time, t [s]
  
F igur
 e 7.3 Displacement-time curve under loading points and mid-point for beams
 8 and 9 with φ12 reinforcement. The data for mid-point, point 1, and
 point 2 is shown by green dash-line, orange, and blue line, respectively
 (see Figure C.11).
A table of results is also included for each category at which the force, maximum dis-
placement, and displacement for maximum force values are obtained from hydraulic
jack. Load level at which first crack appears, displacement of mid-point, number
of cracks, average crack width, and total crack width values are provided from DIC
results. Symmetrical behavior is also studied based on DIC results. The number of
cracks are counted at yielding and cracks which are formed after yielding are not
considered in measurements. The crack width was measured at the level of rein-
forcements and when the load reached the maximum value. All crack measurements
are performed at the region with constant moment between loading points.
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Displacement, u [mm] Displacement, u [mm] Displacement, u [mm]
Displacement, u [mm] Displacement, u [mm] Displacement, u [mm]
7.3.1 Beams with φ10 reinforcement
Beams 1, 2, 3, and 10 contain φ10 reinforcement bars. However, reinforcement bars
in beam 10 are not identical to φ10 bars in other three beams. Also, beam 10 with
a symmetrical behavior shows a higher deformation capacity under same four-point
bending test and other three beams have a more similar behavior with smaller de-
formation capacity.
Beams 1, 2, and 3 failed due to rupture of reinforcement. Beam 10 did not reach
failure and the test was stopped because of test setup limitations when the structure
was close to failure and the load was not increasing anymore. Although there was a
damage observed in concrete on the top surface of beam 10, the exact failure mode
cannot be predicted for it.
Beam 1 
1      2    3      4          5          6           7 
 
Figure 7.4 YLieoldaidng- doif srpeilnafocrecmemeenntt curves for beams with φ10Mraexiinmfuomr cloeamd ent.
 
 
Beam 2 
 1     2      3      4             5    6      7    8 
 
Yielding of reinforcement Maximum load 
 
 
Figure 7.5 Crack pattern of beam 2 at yielding and maximum load.
Beam 3 
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  1       2           3           4     5          6   
 
Yielding of reinforcement Maximum load 
 
 
Beam 10 
1         2  3       4    5           6          7 
 
Yielding of reinforcement Maximum load 
 
 
 
 
 
 
Figure 7.6 Load-displacement curve of beam 2 and displacement-width curves for
cracks.
Table 7.4 Summary of testing results for beams with φ10 reinforcement.
Beam no. 1 2 3 10
Fmax [kN] 36.4 36.4 37.1 37.3
Fcr [kN] 10.8 10.0 11.0 10.4
u(Fmax) [mm] 81 93 78 141
umax [mm] 107 106 89 155
umid [mm] 122 121 101 187
Failure1 R R R -
Symmetrical behavior2 US US US S
Number of cracks 7 8 6 7
Average crack width [mm] 3.2 3.5 2.7 6.1
Total width of cracks [mm] 22 28 16 43
Loading plate3 S L L L
1 "R", "C", and "-" stand for reinforcement rupture, concrete
crushing, and not reaching failure, respectively.
2 "S" and "US" stand for Symmetrical and asymmetrical behavior,
respectively.
3 "L" and "S" stand for large and small loading plate, respectively.
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7.3.2 Beams with φ12 reinforcement
Beams 4, 5, and 6 contain φ12 reinforcement bars. Beams 4 and 5 show symmetrical
behavior compared to beam 6 according to Appendix C.2. However, beam 4 shows a
considerable higher deformation capacity. Also, beam 5 is tested with small loading
plate which led to eccentricity of loading point.
 
Beam 4 
1       2      3    4     5     6       7       8 
 
Yielding of reinforcement Maximum load 
 
 
Beam 5 
Figure1 7   . 7      2L   o  a3 d   - d  4i s  p5l  a  c 6e  m    e7n   t   c8 urves for beams with φ12 reinforcement.
 
Yielding of reinforcement Maximum load 
All beams in this group failed due to concrete crushing and all obtained eight cracks
between l oading points at the stage of reinforcement yielding.
Beam 6  
1       2       3   4   5    6         7      8 
 
Yielding of reinforcement Maximum load 
 
 
Figure 7.8 Crack pattern of beam 6 at yielding and maximum load.
 
 
 
 
 
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Figure 7.9 Load-displacement curve of beam 6 and displacement-width curves for
cracks.
Table 7.5 Summary of testing results for beams with φ12 reinforcement.
Beam no. 4 5 6
Fmax [kN] 50.6 47.8 48.0
F 1cr [kN] - 10.5 10.5
u(Fmax) [mm] 160 103 126
umax [mm] 170 117 136
umid [mm] 197 133 159
Failure2 C C C
Symmetrical behavior3 S S US
Number of cracks 8 8 8
Average crack width [mm] 4.6 2.6 3.4
Total width of cracks [mm] 37 21 28
Loading plate4 L S L
1 The data for beam 4 is missing due to delayed DIC recording.
2 "R", "C", and "-" stand for reinforcement rupture, concrete
crushing, and not reaching failure, respectively.
3 "S" and "US" stand for Symmetrical and asymmetrical behavior,
respectively.
4 "L" and "S" stand for large and small loading plate, respectively.
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7.3.3 Beams with φ12 reinforcement and PVC tubes
Beams 7, 8, and 9 contain φ12 reinforcement bars with PVC tubes. Beam 8 with
small loading plates shows symmetrical behavior compared to beams 7 and 9 and
it has the lowest deformation capacity in this group. Beams 7 and 9 did not reach
failure as the beams got connected to the steel loading beam after deformation, see
Figure 7.10, and it was not possible for the test to be continued. However, both
beams were expected to be close to failure according to their load-displacement
curves, see Figure 7.11.
Figure 7.10 Connection of beam 7 and loading device before failure.
Figure 7.11 Load-displacement curves for beams with φ12 reinforcement and PVC
tubes.
As for the φ12 beams without PVC tubes, concrete crushing was the cause of fail-
ure. All beams 7, 8, and 9 have seven cracks between loading points at the stage of
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Beam 7 
1         2       3       4          5       6      7      
 
Yielding of reinforcement Maximum load 
 
reinforcem ent yielding.
Beam 8 
1          2       3     4     5       6        7      
 
Yielding of reinforcement Maximum load 
 
Figure 7.1 2 Crack pattern of beam 8 at yielding and maximum load.
Beam 9 
1     2            3   4       5            6       7      
 
Yielding of reinforcement Maximum load 
 
 
 
 
 
 
 
 
 
 
 
Figure 7.1 3 Load-displacement curve of beam 8 and displacement-width curves for
 cracks.
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Table 7.6 Summary of testing results for beams with φ12 reinforcement and PVC
tubes.
Beam no. 7 8 9
Fmax [kN] 48.2 50.2 48.9
Fcr [kN] 8.0 9.3 8.7
u(Fmax) [mm] 165 149 132
umax [mm] 165 155 169
umid [mm] 186 179 191
Failure1 - C -
Symmetrical behavior2 US S US
Number of cracks 7 7 7
Average crack width [mm] 5.8 5.3 6.1
Total width of cracks [mm] 41 37 42
Loading plate3 L S L
1 "R", "C", and "-" stand for reinforcement rupture, concrete
crushing, and not reaching failure, respectively.
2 "S" and "US" stand for Symmetrical and asymmetrical behavior,
respectively.
3 "L" and "S" stand for large and small loading plate, respectively.
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7.4 Beam results, strain measurements
The results from the strain measurements are presented in bar charts for every
50 mm mark. Each figure includes strain measurements for each reinforcement bar
and the average plastic strain for the two bars which corresponds to the average
plastic reinforcement strain of the beam. The crack pattern figure from the DIC
is also included to show the connection between the crack pattern and the plastic
strain. The crack pattern figure is placed in relation to the average reinforcement
strain of the beam in order to match the loading points.
Since the strains close to the ends of the beam are equal to zero, the presented
results are limited to the coordinates 600 mm to 1800 mm. The loading points
are marked with a red circle and a rupture of the reinforcement is denoted with
*. The result from the reinforcement tensile strength test shows that the average
ultimate strain for the φ10 bars is limited to 64 ‰ and 90 ‰ for the φ12 bars, see
Table 7.3. Therefore, in the calculations to determine an average strain, the indi-
vidual strain value in a given section is also limited to 64 ‰ for φ10 bars and to 90
‰ for φ12 bars, hence all strains above that limit is set to equal to 64 ‰ or 90 ‰
in the calculations. However, if the measured strain exceeds this value, the strain
value is marked in a text box alongside the bar in the bar chart.
In the bar chart with the average reinforcement strain of the beam, the total average
plastic strain, εave is presented as a horizontal line. This value is calculated by taking
the average of all strains within the region with measured strains. One example of
the strain measurement result, from the reinforcement in beam 1 is presented in
Figure 7.14. The results from the other beams are presented in Appendix D.
68 , Architecture and Civil Engineering, Master’s Thesis ACEX30
 
Beam 1 - Bar A Beam 1 - Bar B
*
60 80 ‰ 220 ‰
*
60 120 ‰ 180 ‰ 
40 40
20 20
0 0
600 800 1000 1200 1400 1600 1800 600 800 1000 1200 1400 1600 1800
Length [mm] Length [mm]
  
 
Reinforcement Beam 1
*
60 100 ‰ 200 ‰
50
40
eave = 25.2 ‰
30
20
10
0
600 800 1000 1200 1400 1600 1800
Length [mm]
 
 
 
 
 
 
 
Figure 7.14 Plastic reinforcement strain measurements for each reinforcement bar
(upper charts), the average strain of the two bars (middle chart) and
the crack pattern figure from DIC for beam 1.
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Strain [‰]
Strain [‰]
Strain [‰]
A summary of the average plastic reinforcement strain results is presented in
Table 7.7 and as average values of each beam type in Table 7.8 and Table 7.9.
Due to asymmetric loading of some of the beams, the average plastic strain is also
calculated to the left and to the right side of the middle. If the reinforcement rup-
tured, the measured strain is marked with *, both for the full beam and to the left
or right side of the middle depending on where the rupture was located.
The average plastic reinforcement strain was also calculated without the limit of
64 ‰ or 90 ‰, presented in parentheses. The results show that it is mainly the φ10
bars that are effected by the strain limit since they had measured strains greater
than the 64 ‰. The beams that are not effected by the limit and have the same
average plastic strain with and without strain limits are not presented. Note that the
average plastic strain within a ruptured area was approximated by assembly the two
ruptured parts together to measure the elongation. Therefore, the measured plastic
strain is an approximation of the distance between the to marks. Also note that the
theoretical measured strain at rupture is included in the average strain calculation
and the maximum measured plastic strain presented in Table 7.7 is without the
strain limits.
Table 7.7 Average plastic reinforcement strain from measured elongation. The
presented strains are the maximum average strain, the average value of
the full beam and the average of the left and right part respectively. The
average plastic strains in parentheses is calculated without the strain
limits.
Beam no. Maximum strain Full beam To the left To the right
[‰] [‰] [‰] [‰]
1 200* 25.2* (34.2) 31.6* (47.3) 16.7
2 200* 24.3* (30.5) 32.6* (43.1) 13.0
3 190* 19.2* (25.2) 21.8* (31.5) 15.6
4 110 43.3 (44.2) 40.0 47.5 (49.2)
5 75 30.9 26.7 32.7
6 80 37.6 40.8 34.4
7 80 42.1 51.9 30.8
8 80 42.2 36.5 49.1
9 90 39.2 50.8 26.2
* Ruptured bar.
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Due to unsymmetrical loading, the measured strains for most of the beams were
larger at one side. Therefore, the average strain for each beam type is presented in
two different ways. In Table 7.8, the average strain is based on the result from the
full beam from Table 7.7. Whilst in Table 7.9, the average strain of each beam type
is based on the average strain to the left or to the right, where the highest strain at
each side of the beams is used in the average strain calculation of the beam type. To
compare the measured strains with the strain at ultimate stress from the material
testing, the utilization rate is calculated by
εave
η = (7.1)
εu
where εave is equal to the average strain presented in Tables 7.8 and 7.9 and εu the
strain at ultimate stress from the material testing presented in Table 7.3. The strain
at ultimate stress was equal to 64 ‰ for the φ10 reinforcement and 90 ‰ for the
φ12 reinforcement.
Table 7.8 A summary of the average reinforcement strain for each beam type,
based on the average value from the full beam and the utilization rate
with respect to the ultimate strain from material testing.
Beam type Average strain Utilization rate
[‰] [%]
φ10 22.9* (30.0) 36
φ12 37.3 (37.6) 41
φ12 PVC 41.2 46
* Ruptured bar.
Table 7.9 A summary of the average reinforcement strain for each beam type,
based on the average value to the left or to the right side and the
utilization rate with respect to the ultimate strain from material testing.
Beam type Average strain Utilization rate
[‰] [%]
φ10 28.7* (40.6) 45
φ12 40.3 (40.9) 45
φ12 PVC 50.6 56
* Ruptured bar.
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7.4.1 3D scanning
The plastic strain results from the 3D scanning are presented in the same way as the
plastic strain measurements. The plastic strain between coordinate
1650 mm and 1700 mm is missing due to an unfortunate placing of the zip-tie
which made the 1700 mm mark invisible. Data that was missing from the scanning
is illustrated with a red cross. For the reinforcement in beam 1, the resolution of the
3D model made the marks less visible at the second scanning and therefore some
data went missing. One example of the 3D strain measurement result, from the re-
inforcement in beam 1 is presented in Figure 7.15. The results from the other beams
are presented in Appendix D. Note that the same interval in the bar charts is used
as for measurements made by hand even though the results from the 3D scanning
were between the coordinate 700 mm to 1700 mm. From the strain measurement
results from beam 1, bar B, see Figure 7.15 is can be seen that the strains between
the coordinates 1200-1400 mm are somewhat higher compared to the results from
the manually made measurements, see Figure 7.14.
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Beam 1 - Bar A Beam 1 - Bar B
* * 
60 79 ‰ 220 ‰ 60 81 ‰ 800 ‰ 85 ‰ 67 ‰
40 40
20 20
0 0
600 800 1000 1200 1400 1600 1800 600 800 1000 1200 1400 1600 1800
Length [mm] Length [mm]
  
 
Reinforcement Beam 1
*
60 80 ‰ 510 ‰
50
eave = 33.8 ‰
40
30
20
10
0
600 800 1000 1200 1400 1600 1800
Length [mm]
 
Figure 7.15 Plastic reinforcement strain from 3D scanning for each reinforcement
bar (upper charts), the average strain of the two bars (middle chart)
and the crack pattern figure from DIC for beam 1.
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Strain [‰]
Strain [‰]
Strain [‰]
A summary of the plastic reinforcement strain results from 3D scanning is presented
in Table 7.10. In the same way as in Table 7.7, the average plastic strain is also
calculated to the left and to the right side of the middle to indicate asymmetric
loading. Note that the theoretically measured strain at rupture is included in the
calculations without strain limit, result presented in parenthesis. For beam 1, the
difficulty in assembling the ruptured parts together resulted in a large elongation
and therefore high measured strain.
Table 7.10 Average plastic reinforcement strain from 3D scanning. The presented
strains are the average maximum strain value, average strain of the
full beam and the average of the left and right part respectively.
Beam no. Maximum strain Full beam To the left To the right
[‰] [‰] [‰] [‰]
1 510* 33.8* (64.6) 36.2* (93.3) 28.3
4 110 49.1 43.3 56.2
7 105 50.5 (52.0) 55.5 (58.2) 40.3
* Ruptured bar.
7.4.2 Comparison
The comparison of the plastic strain results from measurements and 3D scanning
is done by comparing the average strain from the reinforcement bars in each beam
and the results are presented in bar charts see Figures 7.16-7.18. Note that 1 mm
difference in elongation results in strain difference equals to 20 ‰, which means that
the difference in strain for a 50 mm long mark compare to a 51 mm long mark is
equal to 20 ‰. The effect of the unexpected high strains from bar B in beam 1 can
also be seen when comparing the two methods in Figure 7.16. A closer comparison
of the strain between each mark is presented in Appendix D.
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Figure 7.16 Comparison of the plastic reinforcement strain results by 3D
scanning and measurements for beam 1.
Figure 7.17 Comparison of the plastic reinforcement strain results by 3D
scanning and measurements for beam 4.
Figure 7.18 Comparison of the plastic reinforcement strain results by 3D
scanning and measurements for beam 7.
, Architecture and Civil Engineering, Master’s Thesis ACEX30 75
In Table 7.11, a comparison of the average strain from the two methods is presented.
The results considered the average strain between the coordinates 700 mm and 1700
mm and are based on the maximum measured strain to the left or to the right side
for beams 1, 4 and 7.
Table 7.11 Comparison of the average strain results from 3D scanning and
manually measurements.
Beam no. 3D scanning Measurements
[‰] [‰]
1 36.2* 31.6*
4 56.2 47.5
7 55.4 51.9
* Ruptured bar.
7.4.3 Cross-sectional area from 3D scanning
As mentioned in Section 5.2.1, the results from 3D scanning of reinforcement in
beams 1, 4, and 7 before and after testing, are used to study the change of cross-
sectional area due to yielding. For each bar the data are provided for the part
with largest strain value. However, for bar B in beam 1 and bar B in beam 4 the
data was missing for the part with largest strain and the comparison is therefore
made for the part next to it with the second largest strain value. Figure 7.19
shows the cross-sectional area for a part from coordinate 800 mm to 850 mm in
bar A of beam 1 and the considerable drop in the orange curve is due to rupture of
reinforcement. The regular fluctuation of each curve is because of ribs along the bar.
Other graphs are provided in Appendix D. Also, the reduction of area is calculated at
three sections for each part: at the beginning, middle, and the end and is presented at
Table 7.12.
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Figure 7.19 Cross-sectional area of bar A from beam 1 for the part with
coordinate from 800 mm to 850 mm, before and after testing.
Table 7.12 Cross-sectional area and comparison before and after testing for 3
sections at a part with largest strain value from each bar of beams 1, 4,
and 7.
Part Area 1 [mm2] Area 2 [mm2] Reduction [%]
S1 S2 S3 S1 S2 S3 S1 S2 S3
Beam 1-A (800-850)* 80.1 79.7 78.6 78.5 72.5 55.8 2 8.9 28.9
Beam 1-B (750-800) 80.6 78.1 80.1 70.6 76.9 79.9 12.5 1.6 0.3
Beam 4-A (1550-1600) 114.1 112 113.8 100.6 98.67 98.3 11.8 11.9 13.6
Beam 4-B (1500-1550) 111.4 114 111.7 108.9 107.9 101.1 2.2 5.4 9.5
Beam 7-A (800-850) 112.1 114.4 11.6 101.9 102 101.2 9.1 10.8 9.3
Beam 7-A (800-850) 113.5 110.7 112.1 103.4 101.2 102.2 8.9 8.6 8.8
* Ruptured bar.
, Architecture and Civil Engineering, Master’s Thesis ACEX30 77
7.5 Determination of plastic rotation capacity
Plastic rotation capacity is calculated from test results for each beam correspond-
ing to Fmax which is the maximum load and 95% of Fmax, using the method in
Section 3.3.1 and Equation (3.14). Plastic deformation is estimated from load-
displacement curve as shown in Figure 7.20. Notations uel and uu represent elastic
and ultimate displacement for a certain load, respectively. The results for plastic
rotational capacities are provided in Table 7.13. The stiffness value at state II is
derived using a calculation software as 2.7 kN/mm for beams with φ10 and 3.6
kN/mm for beams with φ12.
Figure 7.20 Calculation of plastic displacement from load-displacement curve.
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Table 7.13 Values of plastic rotation capacity for all tested beams corresponding
to Fmax and 95%Fmax.
Beam no. θpl,95% [mrad] θpl,100% [mrad]
1 109 82
2 107 93
3 89 76
10 171 152
4 188 175
5 124 106
6 146 133
7 182 180
8 170 162
9 177 139
Plastic rotational capacity for all beams has a larger value being calculated based
on 95% of maximum load. For beams 7 and 9 the value of θpl,95% is even expected
to be larger as these beams have not failed and uu,95% reaches a larger value if beam
fails.
Although the equations from BK 25 and German Method which are provided in
Sections 3.3.4 and 3.3.2 are based on different loading conditions, they are used to
calculate plastic rotational capacity to compare with the values from test results.
Detailed calculations are presented in Appendix C. Bk 25 calculates θpl for a beam
subjected to uniformly distributed load while German method considers concen-
trated point load in the middle of beam. Comparing the values of plastic rotational
capacity from results of four-point bending test in Table 7.13 with results from BK
25 and German method in Table 7.14, it is observed that the structural behavior is
more similar to uniformly distributed loading condition and in this case using the
results from German method leads to underestimation of θpl. As the provided solu-
tion in Eurocode 2 is based on German method, it also can be said that Eurocode
2 underestimates θpl for the loading condition used in this project.
Table 7.14 Values of θpl from BK 25 and German method.
Bar size θpl,German [mrad] θpl,BK25 [mrad]
φ10 16 68
φ12 22 143
, Architecture and Civil Engineering, Master’s Thesis ACEX30 79
Figure 7.21 Plastic rotational capacity of each beam corresponding to 100% Fmax
and 95% Fmax.
80 , Architecture and Civil Engineering, Master’s Thesis ACEX30
7.6 Summary of the beam test results
A summary of the maximum load, maximum deformation, average plastic reinforce-
ment strain and utilization rate when considered the maximum average strain to the
left or to the right side of the beam and plastic rotational capacity for each beam is
presented in Table 7.15.
Table 7.15 Summary of test results.
Name Fmax u(Fmax) umax εave Utilization θpl,95% θpl,100%
[kN] [mm] [mm] [‰] [%] [mrad] [mrad]
Beam 1 36.4 81 107 31.6* 50 109 82
Beam 2 36.4 93 106 32.6* 51 107 93
Beam 3 37.1 78 89 21.8* 34 89 76
Beam 10 37.3 141 155 - - 171 152
Beam 4 50.6 160 170 47.5 53 183 175
Beam 5 47.8 103 117 32.7 36 124 106
Beam 6 48.0 126 136 40.8 45 146 133
Beam 7 48.2 165 165 51.9 58 182 180
Beam 8 50.2 149 155 49.1 54 170 162
Beam 9 48.9 132 169 50.8 56 177 139
* Ruptured bar.
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82 , Architecture and Civil Engineering, Master’s Thesis ACEX30
8 Discussion
The discussion includes general observations and interpretations of the results and
the chapter is divided into three parts, the first part regarding the results from
the beam tests and the second part the plastic reinforcement strain. The last part
includes discussion about the test setup and how the results can be applicable on
impulse loaded structures.
8.1 Structural response
From the beam test results it can be observed that the beams with φ 10 reinforce-
ment have lower capacity for both maximum load and plastic deformation compared
to beams with φ 12 reinforcement, see Figure 8.1. Theoretically, the final failure
should have occurred in the middle of the beam because of the dead weight. How-
ever, due to reasons as asymmetric loading, natural scatter in the material and
locally confinement effects of loading plates, most of the failures were located closer
to one of the point loads.
Figure 8.1 Load-displacement curve of all tested beams.
, Architecture and Civil Engineering, Master’s Thesis ACEX30 83
In general, the beams with φ12 reinforcement and PVC tubes showed larger plastic
deformation capacity compared with the beams with φ12 reinforcement without
PVC tubes, see Figure 8.2.
Figure 8.2 Load-displacement curve of beams with φ 12 reinforcement.
However, beams 10 and 4 provided larger deformation capacity compared to other
beams in their group. The difference in behavior of the beams in each group can
somehow be justified based on differences in material properties, test setup and test
application. Symmetrical deformation and larger loading plates are both expected
to have had positive effect on the ultimate capacity of the tested beams. Larger
loading plates increased the confinement effect and lead to locally higher concrete
strength and ductility in regions close to the loading plates.
The results from the tensile strength test of the reinforcement in beam 10, see
Table B.9, indicates that the reinforcement quality differ from the one used in beam
1, 2 and 3. The results did not show a significant difference in total tensile capacity
but higher fu/fy ratio. The larger deformation capacity of beam 10 can be due to
its symmetrical behavior while beams 1, 2, and 3 had a more unsymmetrical defor-
mation.
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Beam 4 provides a considerable large capacity even compared to beams with PVC
tubes. This beam has large loading plates which lead to larger capacity than
beam 5 with small loading plates. It is also expected for this beam to have larger
capacity than beam 6 as it has a more symmetrical behavior while the response of
beam 6 is unsymmetrical.
The average values of load-displacement for beams in each group represent larger
capacity for beams with PVC tubes than beams without. The φ12 beams also show
larger capacity than the φ10 beams as expected. As the number of data from results
are not the same for different beams and beams do not fail at the same displacement,
it is not possible to continue the average curve after failing of the first beam in a
group. The average value of final deformation capacity is therefore shown by dashed
lines in Figure 8.3.
Figure 8.3 Average load-displacement curve of each group.
Beams with φ10 reinforcement failed due to reinforcement rupture and beams with
φ12 reinforcement failed because of concrete crushing. This behaviour can also be
reasoned according to the concept presented in Figures 3.14 and 3.19. In
Figure 8.4 the value of θpl,EC is calculated based on German method, see
Section 3.3.4, for beams with material and geometry properties of tested beams.
The blue line and dash-line show the curve for the relation between plastic rotation
and x/d ratio for class C reinforcement. The ratio of x/d is calculated in Appendix
C for beams with material and geometry properties of tested beams with φ10 and
φ12 reinforcement. The ratio of x/d for a tested beam with φ10 reinforcement is
0.061 and for a tested beam with φ12 reinforcement is 0.083 which are shown by
the yellow points. The position of these yellow points indicates that the type of
, Architecture and Civil Engineering, Master’s Thesis ACEX30 85
failure observed in the tests can be anticipated from these curves. Although the
concept in Eurocode 2 and German method is not meant to be used to estimate θpl
for four-point bending test, but it can still provide help to predict the failure mode.
Figure 8.4 Average load-displacement curve of each group.
In case of plastic rotation capacity from test results, the φ12 beams with PVC tubes
show larger capacity than beams without tubes and the φ12 beams provide larger
capacity than φ10 beams. Although beam 10 in first group and beam 4 in second
group are exceptions, comparison of average plastic rotational capacity can prove
the significant positive effect of PVC tubes.
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Figure 8.5 Average plastic rotational capacity for each group.
Plastic tubes prevent stress transition between reinforcement and concrete so, the
reinforcement can elongate more inside the PVC tube as it is not restricted by con-
crete. Consequently, the number of cracks decrease by using PVC tubes but the
crack width increases as there is larger plastic deformation.
Figure 8.6 Average values of crack width, total width of cracks , and number of
cracks for each group.
, Architecture and Civil Engineering, Master’s Thesis ACEX30 87
8.2 Plastic reinforcement strain
From Section 8.1 it can be concluded that the PVC tubes had a positive effect on
the average plastic strain. By decreasing the interaction and the bond between
the reinforcement and concrete, the plastic deformation capacity increased. The
beams with plastic tubes generally provided a higher average strain and utilization
rate compared with the beams without tubes. The φ12 bars had higher measured
strains compared to the φ10 bars. However, since the result from the material test-
ing showed the the strain at ultimate stress for the φ12 reinforcement was equal to
90 ‰ and 64 ‰ for φ10, see Table 7.3, they both resulted in similar utilization
rates. The utilization rate is an important parameter with respect to the plastic de-
formation capacity. To utilize the plastic deformation capacity of the reinforcement,
the average strains should be similar to the strain at ultimate stress of the material
in order to increase the deformation capacity of the beams.
The result with similar utilization rates for the different reinforcement diameters
were not expected. The φ12 reinforcement was expected to result in a higher uti-
lization rate compared to the φ10 reinforcement since an increased bar diameter
results in larger reinforcement ratio with more cracks and therefore higher strains
and higher load capacity which reduce the effect of tension stiffening. One possible
reason to the similar utilization rates could be that the results from the material
testing of the φ10 bar showed that the strain at ultimate stress was equal to 64 ‰
which is a lower result compared to the recommended value for class C reinforce-
ment, equal to 75 ‰, see Table 3.1.
How the calculations are performed to estimate the average reinforcement strains
highly affects the results. From Table 7.7 it can be seen that the average calculations
of the full beam, to the left or to the right side of the beam were different depending
on which values were considered. In the summary table, see Table 7.15 the average
strains and utilization rates are estimated based on the maximum measured strain
to the left or to the right side of the beam since this value is more reasonable to
used when defining the capacity of the reinforcement. From Table 7.7, it can also be
seen that there are some scatter in the results. Beam 3 showed a considerable lower
average strain compared to the other φ10 beams and the same for beam 5 which also
had a lower average strain compared to the other beams of the same type. Beam
4 provided a higher average strain compared to the other beams of same type and
the results from the second tensile strength test of the reinforcement indicates a
somewhat higher fu/fy-ratio compared to the other reinforcement specimens, see
Table B.10. However, the results from the strain measurements corresponds well
to the results from the beams tests. As mentioned, beam 4 had large measured
strains and large deformation capacity, see Table 7.15, and beam 3 had the lowest
measured strain and the lowest deformation capacity which shows that the there is
a good correlation between the results.
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The effect of plastic tubes on the strain measurements was expected to result in
constant strain where the tubes were placed since no interaction with the concrete
results in no stress transfer and an evenly distributed elongation. To see the effect
with constant strain under the tubes in the bar charts from the strain measurement
result, the position of the tube in relation to the marks is important, see Figure 8.7.
From Figures D.7 - D.9 and Figure D.17 it can be seen that the expected results with
constant strain were more or less achieved for most of the bars. A small difference
between the measured strains at the position of the tube can be explained by the
precision of the measurements since the accuracy highly effects the resulting strain
and as mentioned in Section 7.4.2, 1 mm elongations corresponds to 20 ‰ strain.
Another factor that might influence the results is the friction between the materials
which may result in a bond between the materials where it is expected to be no
interaction. At the location of some tubes, e.g. beam 7, see Figure D.7 at the third
tube (coordinate 1300 and 1350), the strain was not constant. However, the results
from the 3D scanning of beam 7, Figure D.17 showed a smaller difference between
the measured strains which indicates that the non constant strain from the manually
made measurements is caused by an error in the measurement.
Figure 8.7 Placing of tube in relation to the marks which is expected to result in
constant measured strain.
During this master’s project, the 3D scanning has been used as an alternative method
to measure the elongation and calculate the reinforcement strain rather then esti-
mate the average reinforcement strain and utilization rates since only three beams
were scanned. The 3D model was used in a similar way as for manually measure-
ments, by measuring the elongation between the marks. The results from the two
different methods were compared and they resulted in similar plastic strain, see
Figures 7.16 - 7.18. However, some exceptions with deviating results due to loss of
data in the 3D model. As long as there is no data loss in the 3D scanning, the 3D
model enables a more accurate measuring tool since the accuracy of the measure-
ments with the measuring tape were limited to approximately 0.5 mm and 1 mm
elongation corresponds to 20 ‰ measured plastic strain.
, Architecture and Civil Engineering, Master’s Thesis ACEX30 89
Also, the 3D model of the bars made it possible to study the change in cross-
sectional area due to yielding. The results showed that the cross section decreased
approximately 8%-13% for the part with largest measured strain. At rupture, the
cross-sectional reduction was even higher. Due to missing data from two of the bars,
the part that was next into the part with largest strain had to be used in the analysis
and the results show a somewhat lower reduction compared to the bars where the
part with largest strain was used.
8.3 Test setup
As mentioned in the introduction, the FKR 2011 (Swedish Fortification Agency,
2011) provides design regulations to structures subjected to not only static loading
but also to withstand impulse loads such as explosions. An explosion load can be
described or simplified by a uniformly distributed load. However, due to difficulties
to simulate a uniformly distributed load, a test setup with two point loads was chosen
in this master’s project as a simplification to analyse the structural behaviour of
reinforced concrete structures loaded until failure. The difference in terms of plastic
deformation and formation of plastic hinges between a uniformly distributed load
and point loads are described in Section 3.2.3. From Figure 3.6, it can be seen that
a four-point bending test setup results in similar length of yielding region ly as for
a uniformly distributed load. In case of impulse loaded structures, the application
of PVC tubes would have been performed in a similar way as in this project. To
get the desired effect of the plastic tubes, the tubes should be placed in the region
where the plastic hinge is expected to form. In Figure 8.8 a schematic illustration
of the formation of plastic hinges is shown. The location of the plastic hinges in this
project is a simplification whereas in reality the hinges is distributed over an area,
see Figure 3.1.
90 , Architecture and Civil Engineering, Master’s Thesis ACEX30
Figure 8.8 The concept of plastic tubes with respect to the formation of plastic
hinges and the difference between two point loads and a uniformly
distributed load.
Although the loading situations are different from each other, the concept of placing
PVC tubes in the region with plastic hinges are formed to increase the deformation
capacity can still be applied.
The configuration of the plastic tubes is one parameter that effects the reinforcement
strain and the deformation capacity. Test results from Köseoğlu (2020) showed that
the PVC configuration with 8x50mm was the arrangement which resulted in the
largest deformation capacity. The test also included PVC tubes with configuration
4x100, but with a spacing equal to 100 mm instead of 200 mm which was used
in this project. The different spacing, results in different crack development and
reinforcement strain. A larger spacing means a longer transmission length which
allows the stresses to be transferred from the reinforcement to the concrete. This
implies that the configuration used might not be the most optimal one. Therefore,
an optimization of the PVC tube configuration might be effective to further increase
the reinforcement strain and the deformation capacity.
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92 , Architecture and Civil Engineering, Master’s Thesis ACEX30
9 Final remarks
9.1 General description
The aim of the thesis project was to increase the knowledge about the structural re-
sponse of reinforced concrete beams with focus on the average plastic reinforcement
strain at failure to increase the plastic deformation capacity. Experimental studies
in terms of static four-point bending tests were preformed in order to investigate
how the average plastic strain can be affected by the reinforcement amount and
reduced bond between the reinforcing steel and the concrete.
At first, a literature survey was done to deepen the knowledge about the structural
behaviour of reinforced concrete structures. Background information about plain
concrete, reinforcing steel and reinforced concrete was treated with focus on the
structural response and change of behaviour with respect to the bond between the
materials, which is an important factor of reinforced concrete’s behaviour and the
formation of cracks. To describe the structural behaviour at failure, a theory based
on plastic deformation capacity is needed since the structure will have considerable
plastic deformation. The literature study also includes different methods to deter-
mine the plastic deformation capacity.
Ten concrete beams with the same dimensions, different reinforcement amount and
bond between the concrete and reinforcement were manufactured. The plain con-
crete and reinforcement was tested to obtain necessary background information
about the material properties. Static four-point bending tests were performed until
rupture of reinforcement or crushing of concrete. DIC, measurements and 3D scan-
ning were used to determine load-displacement curves, crack pattern and plastic
reinforcement strain.
9.2 Conclusions
In general, the beams with φ12 reinforcement provided higher capacity with re-
spect to ultimate load and plastic deformation compared with the beams with φ10
reinforcement and the effect of PVC tubes resulted in larger plastic deformations.
During testing, the importance of symmetric loading on the ultimate capacity of the
beam was observed. The beams with symmetrical deformation showed an increased
plastic deformation compared to the beams with unsymmetrical deformation.
Moreover, two different failure modes were observed, the beams with φ10 reinforce-
ment were more likely to fail due to rupture of reinforcement and φ12 beams because
of crushing of concrete. The main factors influencing the structural behaviour and
location of rupture were asymmetric loading, natural scatter in the material and
locally confinement effects in the concrete close to the loading plates.
, Architecture and Civil Engineering, Master’s Thesis ACEX30 93
Comparing the values of plastic rotational capacity from test results with results
from BK 25 and Eurocode, it was observed that BK 25 provides a good estimation
of θpl for a simply supported beam under four-point bending test while the results
from Eurocode 2 showed a large difference. However, the concept of Eurocode for
anticipation of failure mode considering the ratio of x/d showed the same outcome
compared to failure modes in tests.
The results from the plastic strain measurements showed similar utilization rates for
the φ10 and φ12, equal to 45 ‰. The PVC tubes resulted in positive effects with
increased average reinforcement strain and utilization rate equals 56 ‰ . The results
from the strain measurements and the 3D scanning were similar with some deviating
results due to loss of data in the 3D model. However, in case of no data loss, the
3D model enables a more accurate tool to measure the elongation since the accu-
racy of the measuring tape is limited to distinguish elongations greater than 0.5 mm.
Overall, the test results show that application of PVC tubes has a positive effect
of the average reinforcement strain with an increased utilization rate and plastic
deformation capacity. The reduced bond resulted in fewer cracks and larger average
crack width which had a positive effect on the average reinforcement strain and thus
the plastic rotation and deformation capacity of the beams. Also, DIC method is
considered as a powerful tool in this type of experimental study to monitor and
record the structural behavior during testing.
9.3 Further research
In this test series, the beams were subjected to two point loads to obtain a zone
within the loads with constant moment. However, an improvement of the test setup
would have been to increase the number of point loads to simulate a loading sit-
uation more similar to an uniformly distributed load; i.e that of an impulse load
from an explosion. A load rig with larger space would allow for larger deformations
and prevent the session to be interrupted before failure. For some of the beams,
the session stopped because the steel loading beam got in contact with the concrete
beam due to large deformations. Also the movements of the supports, both the
roller supports at the point loads and the supports of the beam were crucial during
testing. To increase the capacity, the size of the loading plates were changed and the
plate size’s effect on the structural response and ultimate capacity was discussed.
However, further investigation on the load plate size effect requires non-linear FE
analysis.
Furthermore, the PVC volume’s effect on the reinforcement strain is also of interest
in further research, by testing different PVC tube configurations and volumes and
application of PVC tubes on critical regions to find the most optimal configuration
and increase the reinforcement strain and the deformation capacity. An addition to
the laboratory testing, a FE-analysis can be performed to investigate the interac-
tion between the materials and how the bond effects the reinforcement strain and
the deformation capacity. Also, a more complex reinforcement configuration or in-
94 , Architecture and Civil Engineering, Master’s Thesis ACEX30
creased reinforcement ratio with e.g. reinforcement cages, stirrups or reinforcement
in multiple layers can result in an increased effect of the reduced bond’s influence on
the reinforcement strain. The advantage with two reinforcement bar in one layer at
the bottom is the simpler manufacturing process but a more complex reinforcement
configuration could be of interest in further studies.
In conclusion, more point loads would help the loading situation to be more similar
an impulse load and a load rig which allow for larger deformations would ensure the
beams to reach failure without stopping the loading session. An optimized or differ-
ent PVC configurations together with a more complex reinforcement configuration
and different reinforcement ratios to display the influence on the average reinforce-
ment strain and plastic deformation capacity might be of interest in further research.
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96 , Architecture and Civil Engineering, Master’s Thesis ACEX30
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och beredskap.
Jönsson, J., & Stenseke, A. (2018). Concrete beams subjected to repeated drop-weight
impact and static load (Master’s thesis). Chalmers University of Technology.
Köseoğlu, E. (2020). Average reinforcement strain in reinforced concrete structures
loaded until failure. (Bachelor’s thesis). Chalmers University of Technology.
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thesis).
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A Experiment preparations
The preparation started with cutting the reinforcements in length 2.79 m for beams
and six samples from each size of reinforcement for tensile testing. The cut rein-
forcement bars were marked every 50 mm by using a hammer and an awl to prepare
for the plastic strain measurement. Also, each bar was marked with a letter to keep
the different reinforcement bars apart and to know which bar that correspond to a
certain beam. The middle 1 m of the reinforcement bar was marked using zip ties.
Figure A.1 The reinforcement was cut to fit into the beam.
From each type of beam, (beam 1, 4 and 7) two bars were scanned with 3D scanning
tools to be used for comparison with the plastic strain calculations. The reinforce-
ment bar was placed in a frame build for the 3D scanning, see Figure A.2. The
software VXelement was used to create the 3D model of the bars, see Figure A.3
and the data was exported as a point cloud mesh to measure the elongation of the
bars and the cross-sectional change due to yielding. Further information about the
3D scanning can be found in Appendix D.
Figure A.2 The setup for the 3D scanning, which shows the frame, 3D scanner
and placement of the reinforcement.
, Architecture and Civil Engineering, Master’s Thesis ACEX30 A-1
Figure A.3 The 3D scanning model in the software VXelements.
After the 3D scanning, PVC tube segments with length of 100 mm and spacing
200 mm were glued at the middle 1 m of the reinforcement bars used in the three
beams with PVC tubes.
The forms were oiled to simplify the cleaning after casting and then the reinforcement
bars were placed inside the forms. Spacers were nailed in the from in order to have
35 mm concrete cover. Figure A.4 shows an example of the reinforcement configu-
ration with and without plastic tubes.
Figure A.4 Example of the two casting forms. The form to the left with plastic
tubes and to the right without plastic tubes.
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Figure A.5 Preparations of the forms before casting. The ten closest forms were
prepared for this thesis.
Figure A.6 Preparations of the forms before casting. The forms to the left were
prepared for this thesis.
The concrete was delivered to the site by a concrete mixing truck and the fresh
concrete was poured from the truck into the forms, see Figure A.7 and A.8 . Also,
concrete cubes for material testing were casted. A vibrator was used to eliminate
air in the concrete. The beams were watered before covered by a plastic sheet to
harden, see Figure A.9 and the cubes were submerged in water until the material
testing.
, Architecture and Civil Engineering, Master’s Thesis ACEX30 A-3
Figure A.7 The fresh concrete was poured into the forms from the concrete mixer
truck.
Figure A.8 Casting of the concrete beams.
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Figure A.9 The beams were stored under plastic sheet to cure.
After hardening of concrete, the moulds were removed and the beams were painted
with a white background and black speckle pattern to prepare for the DIC recording.
A DIC camera was used during the bending test to document the development of
cracks and the deformation of the beam.
Figure A.10 The beams were painted with a white backdrop to prepare for the
DIC.
, Architecture and Civil Engineering, Master’s Thesis ACEX30 A-5
Figure A.11 The white beams were painted with a black speckle pattern to prepare
for the DIC.
The last part was to prepare the loading rig with two point loads, see Figure A.12
and the DIC camera for recording of crack development and deformation of the
beam, see Figure A.13.
Figure A.12 The static four-point bending test setup.
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Figure A.13 The static four-point bending test setup with the DIC camera.
, Architecture and Civil Engineering, Master’s Thesis ACEX30 A-7

B Material testing
In order to obtain necessary background information about the materials, differ-
ent material tests were performed. A more detailed description of the test setups,
performance and results are presented in the following appendix.
B.1 Concrete properties
To study the structural behaviour of the concrete beams, material testing is nec-
essary to gain knowledge about the concrete properties. Two different tests were
conducted, one to determine the compressive strength and one to determine the
fracture energy.
B.1.1 Compressive test
The concrete compressive strength test was performed at two different occasions.
The first one to determine the 28-days strength and the second one after 40 days.
The test was performed and the compressive strength was determined according
to CEN (2019). The setup of the compressive test is presented in Figure B.1 and
examples of failure modes are shown in Figure B.2.
Figure B.1 Test setup for the compressive strength test.
, Architecture and Civil Engineering, Master’s Thesis ACEX30 B-1
Figure B.2 Example of failure modes.
The applied force and compressive strength for each specimen and the mean strength
of the 28 days and 40 day, respectively are presented in Table B.1 and Table
B.2. The average density of the cubes was equal to 2370 kg/m3 at 28 days and
2445 kg/m3 at 40 days.
Table B.1 28-days compressive strength test results.
Cube no. Applied force [kN] Strength [MPa]
1 1472.3 65.4
2 1391.2 61.8
3 1508.7 67.1
Mean strength 1457.4 64.8
Table B.2 40-days compressive strength test results.
Cube no. Applied force [kN] Strength [MPa]
1 1621.4 72.1
2 1588.8 70.6
3 1539.5 68.4
Mean strength 1583.2 70.4
The results can be used to calculate the cylindrical strength according to Equation
(2.1) and then the modulus of elasticity can be estimate with Equation (2.6). In
Table B.3, the estimated cylindrical strength and modulus of elasticity is presented
for the days where the compressive tests were performed and the results are used to
interpolate the concrete properties at 33 days which correspond to the day in the
middle of the week when the beam test were preformed and used as input data for
the predictions in Chapter 6.
B-2 , Architecture and Civil Engineering, Master’s Thesis ACEX30
Table B.3 The estimated cylindrical strength and modulus of elasticity.
Age [days] Cylindrical strength [MPa] Modulus of elasticity [GPa]
28 54.0 36.5
33 56.0* 36.9*
40 58.7 37.4
* Values based on interpolation.
B.1.2 Wedge splitting test
A wedge-splitting test was performed according to the recommendations given in
(Löfgren et al., 2004) in order to predict the fracture energy GF. The setup of the
testing machine can be seen in Figure B.3.
Figure B.3 Test setup for the WST.
A groove in the specimen was made during casting and starter notch was sawed with
a thickness of 4 mm, see a1 and a2 in Figure B.4. Before the test was conducted, the
three cubes were measured according to Figure B.4 and the dimensions are presented
in Table B.4.
Figure B.4 Dimensions of the testing cube.
, Architecture and Civil Engineering, Master’s Thesis ACEX30 B-3
Table B.4 Dimensions of the different cubes with notations indicated in B.4.
Cube no. weight [kg] h1 [mm] h2 [mm] l1 [mm] l2 [mm] A [mm2]
1 7.823 71.5 73.5 149.0 150.0 10839
2 7.852 72.5 74.5 148.0 150.0 10952
3 7.779 73.0 74.5 148.5 149.0 10970
The fracture energy is calculated as the area under the splitting load-CMOD di-
agram, see Table B.5. A summery of the fracture energy, maximum splitting
force and the maximum CMOD for each of the three specimens are presented in
Table B.5.
Figure B.5 Splitting load-CMOD diagram.
Table B.5 Summary of the WST results.
Cube no. Accumulated GF Maximum splitting load Fsp Maximum CMOD
[Nm/m2] [kN] [mm]
1 145 5.87 1.65
2 134 5.32 2.08
3 173 6.17 2.67
B-4 , Architecture and Civil Engineering, Master’s Thesis ACEX30
B.2 Steel reinforcement properties
The reinforcement had the quality class C and the steel reinforcement properties
were determined by a tensile test. Six bars of each type of reinforcement were tested
to determine the tensile strength. However, since the reinforcement was removed
from the beams after the four-point bending test a second tensile test was performed
by cutting one specimen from each reinforcement bar. Also, the reinforcement type
in beam number 10 was unknown and needed to be determined.
• The length of the specimens were 410 mm
• The distance between the clamps on the tensile machine was set to 298 mm
• The loading speed was 5 mm/min up to 3 mm of elongation, followed by
120 mm/min until failure
An extensometer was connected to the specimens in order to get a detailed mea-
surement of the elongation, see Figure B.6. The loading was stopped to remove the
extensometer prior to failure which resulted in a drop in the stress-strain curve.
, Architecture and Civil Engineering, Master’s Thesis ACEX30 B-5
Figure B.6 Steel reinforcement test setup with the extensometer.
Figure B.7 Example of steel reinforcement necking under tensile strength test.
B-6 , Architecture and Civil Engineering, Master’s Thesis ACEX30
B.2.1 φ10 reinforcement
The results from the first reinforcement test are presented in a stress-strain curve,
see Figure B.8. The modulus of elasticity, strain at ultimate stress, yielding strength,
ultimate strength and the fu/fy ratio are presented in Table B.6.
Figure B.8 Stress-strain curve for six φ10 steel reinforcement bars.
Table B.6 Modulus of elasticity, strain at ultimate stress, yielding strength,
ultimate strength and fu/fy ratio for six φ10 together with average
values.
Bar sample Es εsu fy fu fu/fy
[GPa] [‰] [MPa] [MPa] [-]
1 205 63 570 659 1.16
2 220 67 563 653 1.16
3 189 70 551 654 1.19
4 203 60 555 643 1.16
5 187 65 571 664 1.16
6 188 57 576 664 1.15
Average 198 64 564 656 1.16
, Architecture and Civil Engineering, Master’s Thesis ACEX30 B-7
B.2.2 φ12 reinforcement
The results from the first reinforcement test are presented in a stress-strain curve,
see Figure B.9. The modulus of elasticity, strain at ultimate strength, yielding
strength, ultimate strength and the fu/fy ratio are presented in Table B.7.
Figure B.9 Stress-strain curve for six φ12 steel reinforcement bars.
Table B.7 Modulus of elasticity, strain at ultimate stress, yielding strength,
ultimate strength and fu/fy ratio for six φ12 together with average
values.
Bar sample Es εsu fy fu fu/fy
[GPa] [‰] [MPa] [MPa] [-]
1 188 93 523 626 1.20
2 194 87 558 652 1.17
3 189 89 536 631 1.18
4 197 90 534 637 1.19
5 192 92 523 620 1.19
6 187 90 533 631 1.18
Average 191 90 535 633 1.18
B-8 , Architecture and Civil Engineering, Master’s Thesis ACEX30
B.3 Second reinforcement testing
The removal of reinforcement from the concrete beams after the four-point bending
test made a second reinforcement test possible. Since the plastic strain close to
the ends was expected to be equal to zero, specimens were cut from the end of the
reinforcement bars. One specimen from each reinforcement bar was tested with the
same procedure as the first tensile strength test. For beam number 10, the test
beam, the reinforcement type was unknown and therefore in total four test was
performed with specimens from each side of the reinforcement bars. The specimens
for the second testing were cut to the same length as the ones from the first testing
and the samples were taken from the end of the reinforcement bars. In total, six
φ10 bars, four φ10 bars from the test beam and twelve φ12 bars were tested and the
results are presented in Tables B.8 - B.10.
Table B.8 Modulus of elasticity, strain at ultimate stress, yielding strength,
ultimate strength and fu/fy ratio for six φ10 together with average
values.
Beam no Bar sample Es εsu fy fu fu/fy
[GPa] [‰] [MPa] [MPa] [-]
1 A 195 72 567 660 1.16
B 201 71 570 660 1.16
2 A 206 72 567 659 1.16
B 205 68 573 664 1.16
3 A 187 65 584 673 1.15
B 189 70 566 657 1.16
Average 197 70 571 662 1.16
Table B.9 Modulus of elasticity, strain at ultimate stress, yielding strength,
ultimate strength and fu/fy ratio for four φ10 bars from beam 10
together with average values.
Bar sample Es εsu fy fu fu/fy
[GPa] [‰] [MPa] [MPa] [-]
A-1 203 70 544 643 1.18
A-2 202 73 561 660 1.18
B-1 195 69 553 651 1.18
B-2 191 74 556 655 1.18
Average 198 72 553 652 1.18
, Architecture and Civil Engineering, Master’s Thesis ACEX30 B-9
Table B.10 Modulus of elasticity, strain at ultimate stress, yielding strength,
ultimate strength and fu/fy ratio for twelve φ12 together with average
values.
Beam no Bar sample Es εsu fy fu fu/fy
[GPa] [‰] [MPa] [MPa] [-]
4 A 165 86 527 631 1.20
B 199 90 540 635 1.17
5 A 205 93 530 625 1.18
B 198 93 520 619 1.19
6 A 193 93 526 625 1.19
B 195 92 524 620 1.18
7 A 202 90 523 622 1.19
B 192 93 517 616 1.19
8 A 194 89 526 621 1.18
B 203 87 544 630 1.16
9 A 198 90 536 627 1.17
B 196 93 524 619 1.18
Average 195 91 528 624 1.18
The results from the second testing indicates that the reinforcement bars in beam
10 were of a different type compared to the φ10 bars in the other beams. Therefore,
the results from beam number 10 were treated separately from the other φ10 beams.
B-10 , Architecture and Civil Engineering, Master’s Thesis ACEX30
C Beam results, structural response
C.1 Load-Displacement curves
Figure C.1 Load-displacement curve for beam 1 (φ10, without PVC tubes) and
displacement-width curves for cracks. The values of crack width are
measured at the level of reinforcements.
, Architecture and Civil Engineering, Master’s Thesis ACEX30 C-1
Figure C.2 Load-displacement curve for beam 2 (φ10, without PVC tubes) and
displacement-width curves for cracks. The values of crack width are
measured at the level of reinforcements.
Figure C.3 Load-displacement curve for beam 3 (φ10, without PVC tubes) and
displacement-width curves for cracks. The values of crack width are
measured at the level of reinforcements.
C-2 , Architecture and Civil Engineering, Master’s Thesis ACEX30
Figure C.4 Load-displacement curve for beam 10 (φ10, without PVC tubes) and
displacement-width curves for cracks. The values of crack width are
measured at the level of reinforcements.
Figure C.5 Load-displacement curve for beam 4 (φ12, without PVC tubes) and
displacement-width curves for cracks. The values of crack width are
measured at the level of reinforcements.
, Architecture and Civil Engineering, Master’s Thesis ACEX30 C-3
Figure C.6 Load-displacement curve for beam 5 (φ12, without PVC tubes) and
displacement-width curves for cracks. The values of crack width are
measured at the level of reinforcements.
Figure C.7 Load-displacement curve for beam 6 (φ12, without PVC tubes) and
displacement-width curves for cracks. The values of crack width are
measured at the level of reinforcements.
C-4 , Architecture and Civil Engineering, Master’s Thesis ACEX30
Figure C.8 Load-displacement curve for beam 7 (φ12, with PVC tubes) and
displacement-width curves for cracks. The values of crack width are
measured at the level of reinforcements.
Figure C.9 Load-displacement curve for beam 8 (φ12, with PVC tubes) and
displacement-width curves for cracks. The values of crack width are
measured at the level of reinforcements.
, Architecture and Civil Engineering, Master’s Thesis ACEX30 C-5
Figure C.10 Load-displacement curve for beam 9 (φ12, with PVC tubes) and
displacement-width curves for cracks. The values of crack width are
measured at the level of reinforcements.
C-6 , Architecture and Civil Engineering, Master’s Thesis ACEX30
C.2 Symmetrical behavior
Figure C.11 Illustration of mid-point, point 1, and point 2 for which
displacement-time curves are provided.
 
Beam 1 Beam 2 
200 200
150 150
100 100
50 50
0 0
0 1000 2000 3000 0 1000 2000 3000
Time, t [s] Time, t [s]
Beam 3 Beam 10 
200 200
150 150
100 100
50 50
0 0
0 1000 2000 3000 0 1000 2000 3000
Time, t [s] Time, t [s]
  
 
 
Figure C.12 Displacement-time curve und er loading points and mid-point for
 
beams with φ10 reinforcemen t. The data for mid-point, point 1, and
point 2 is shown by green da  sh-line, blue, and orange line,
respectively.   
 
 
 
 
 
 
 
 
 
 
 
 
, Architecture and Civil Engineering, Master’s Thesis ACEX30 C-7
Displacement, u [mm] Displacement, u [mm]
Displacement, u [mm] Displacement, u [mm]
Beam 4 Beam 5 
200 200
150 150
100 100
50 50
0 0
0 2000 4000 6000 0 2000 4000 6000
Time, t [s] Time, t [s]
Beam 6 Beam 7 
200 200
150 150
100 100
50 50
0 0
0 2000 4000 6000 0 2000 4000 6000
Time, t [s] Time, t [s]
Beam 8 Beam 9 
200 200
150 150
100 100
50 50
0 0
0 2000 4000 6000 0 2000 4000 6000
Time, t [s] Time, t [s]
  
 
F igure C.13 Displacement-time curve under loading points and mid-point for
 beams with φ12 reinforcement. The data for mid-point, point 1, and
 
 point 2 is shown by green dash-line, blue, and orange line,
respectively.
C-8 , Architecture and Civil Engineering, Master’s Thesis ACEX30
Displacement, u [mm] Displacement, u [mm] Displacement, u [mm]
Displacement, u [mm] Displacement, u [mm] Displacement, u [mm]
C.3 Crack pattern
Beam 1 
1      2    3      4          5          6           7 
 
Yielding of reinforcement Maximum load 
 
 
Beam 2 
 1     2      3      4             5    6      7    8 
 
Yielding of reinforcement Maximum load 
 
 
Beam 3 
  1       2           3           4     5          6   
 
Yielding of reinforcement Maximum load 
 
 
Beam 10 
1         2  3       4    5           6          7 
 
Yielding of reinforcement Maximum load 
 
 
Figure C.14 Crack pattern of beams with φ10 reinforcement.
 
 
 
 
, Architecture and Civil Engineering, Master’s Thesis ACEX30 C-9
 
Beam 4 
1       2      3    4     5     6       7       8 
 
Yielding of reinforcement Maximum load 
 
 
Beam 5 
1           2     3       4   5     6      7      8 
 
Yielding of reinforcement Maximum load 
 
 
Beam 6  
1       2       3   4   5    6         7      8 
 
Yielding of reinforcement Maximum load 
 
Figure C. 15 Crack pattern of beams with φ12 reinforcement.
 
 
 
 
 
 
C-10 , Architecture and Civil Engineering, Master’s Thesis ACEX30
Beam 7 
1         2       3       4          5       6      7      
 
Yielding of reinforcement Maximum load 
 
 
Beam 8 
1          2       3     4     5       6        7      
 
Yielding of reinforcement Maximum load 
 
 
Beam 9 
1     2            3   4       5            6       7      
 
Yielding of reinforcement Maximum load 
 
Figure C. 16 Crack pattern of beams with φ12 reinforcement and PVC tubes.
 
 
 
 
 
 
 
 
 
 
 
, Architecture and Civil Engineering, Master’s Thesis ACEX30 C-11
C.4 Final image of beam at end of test
Figure C.17 Image of beam 1 at end of test.
Figure C.18 Image of beam 2 at end of test.
Figure C.19 Image of beam 3 at end of test.
Figure C.20 Image of beam 4 at end of test.
C-12 , Architecture and Civil Engineering, Master’s Thesis ACEX30
Figure C.21 Image of beam 5 at end of test.
Figure C.22 Image of beam 6 at end of test.
Figure C.23 Image of beam 7 at end of test.
Figure C.24 Image of beam 8 at end of test.
Figure C.25 Image of beam 9 at end of test.
, Architecture and Civil Engineering, Master’s Thesis ACEX30 C-13
Figure C.26 Image of beam 10 at end of test.
C-14 , Architecture and Civil Engineering, Master’s Thesis ACEX30
C.5 Determination of plastic rotation capacity
Table C.1 contains plastic rotational capacity and its relevant values for all tested
beams. Parameters uel, uu, and upl are estimated from load-displacement graph of
each beam as shown in schematic Figure 7.20. The parameter l0 is equal to 800
mm for all beams. Plastic rotational capacity is calculated based on equations in
Section 3.3.1.
Table C.1 Estimated parameter of plastic rotation capacity for all tested beams
corresponding to Fmax and 95%Fmax. Displacement values are in [mm]
and rotation values are in [mrad].
Beam no. uel,95% uu,95% upl,95% θpl,95% uel,100% uu,100% upl,100% θpl,100%
1 16.3 103.4 87.1 109 15.5 81.1 65.6 82
2 17.9 103.4 85.5 107 18.7 93.0 74.3 93
3 16.8 87.7 70.9 89 17.5 78.0 60.5 76
10 18.0 154.5 136.5 171 19.3 140.6 121.3 152
4 19.5 170.0 150.5 188 20.5 160.5 140.0 175
5 17.5 116.7 99.2 124 18.8 103.4 84.6 106
6 19.0 135.5 116.5 146 20.0 126.3 106.3 133
7 19.0 164.6 145.6 182 20.6 164.6 144.0 180
8 18.5 154.5 136.0 170 19.4 149.2 129.8 162
9 20.0 161.5 141.5 177 21.0 131.8 110.8 139
, Architecture and Civil Engineering, Master’s Thesis ACEX30 C-15
C.6 Calculation of plastic rotation capacity based
on BK 25 and German method
C.6.1 BK 25
BK 25
b≔0.2 m h≔b=0.2 m c≔40 mm d≔h-c=0.16 m
l≔2.4 m εcu≔0.35% fc≔56 MPa
For beams with ϕ10 reinforcement:
fy≔564 MPa fu≔656 MPa φ≔10 mm εsu≔2.29%
π ⋅φ2 As
As≔2⋅――=⎛⎝1.571 ⋅10
-4⎠⎞ m2 ρs≔――=0.005
4 b ⋅d
fu 0.4 ⋅εsu ⎛ l ⎞
ωs≔ρs ⋅―=0.058 θpl≔―――⋅⎜1+0.3 ⋅―⎟=0.068 rad
fc 0.8-ωs ⎝ d ⎠
For beams with ϕ10 reinforcement:
fy≔535 MPa fu≔622 MPa φ≔12 mm εsu≔3.73%
π ⋅φ2 As
A ⎛s≔2⋅――=⎝2.262 ⋅10
-4⎞⎠ m2 ρs≔――=0.007
4 b ⋅d
fy 0.4 ⋅εcu ⎛ l ⎞
ωs≔ρs ⋅―=0.068 θpl≔―――⋅⎜1+0.3 ⋅―⎟=0.143 rad
fc 0.8 ωs ⎝ d ⎠
C-16 , Architecture and Civil Engineering, Master’s Thesis ACEX30
C.6.2 German Method
Germ an m ethod
⎛ 0.0035 ⎞3 l0
βn≔22.5 βcd≔⎜―――⎟ =1 l0≔0.8 m λ≔―=5 αR≔0.81
⎝ εcu ⎠ d
For beams with ϕ10 reinforcement:
fy≔564 MPa fu≔656 MPa φ≔10 mm Es≔198 GPa εsu≔6.4%
π ⋅φ2
A ≔2⋅――=⎝⎛1.571 ⋅10-4⎞
fy fy
s ⎠ m
2 εsy≔―=0.003 βs≔1-―=0.14
4 Es fu
fy ⋅As x
x≔―――=0.01 m ―=0.061
b ⋅αR ⋅fc d
2
0.2 ―
⎛ x ⎞ ⎛ ⎞ 3x ⎛ ⎛x ⎞-1⎞
εsu.s≔0.28 ⋅ ⎜βcd ⋅―⎟ ⋅εsu=0.01 εsu.c≔-1.75 ⋅ ⎜―⎟ ⋅⎜1-⎜―⎟ ⎟ ⋅εcu=0.015
⎝ d ⎠ ⎝d ⎠ ⎝ ⎝d ⎠ ⎠
εsu.star≔min ⎝⎛ε ⎞su.s ,εsu.c⎠=0.01
εsu.star-εsy ‾λ‾ θpl.EC
θpl.EC≔βn ⋅βs ⋅――――⋅ ―=0.032 rad θpl≔――=0.016 rad
x 3 2
1-―
d
, Architecture and Civil Engineering, Master’s Thesis ACEX30 C-17
For beams with ϕ12 reinforcement:
fy≔535 MPa fu≔622 MPa φ≔12 mm Es≔191 GPa εsu≔9.0%
π ⋅φ2 fy fy
A ≔2⋅――=⎛s ⎝2.262 ⋅10
-4⎠⎞ m2 εsy≔―=0.003 βs≔1-―=0.14
4 Es fu
fy ⋅A≔ s
x
x ―――=0.013 m ―=0.083
b ⋅αR ⋅fc d
2
0.2 ―
⎛ ⎞ ⎛ ⎞ 3x x ⎛ ⎛x ⎞-1⎞
εsu.s≔0.28 ⋅ ⎜βcd ⋅―⎟ ⋅εsu=0.015 εsu.c≔-1.75 ⋅ ⎜―⎟ ⋅⎜1-⎜―⎟ ⎟ ⋅εcu=0.013
⎝ d ⎠ ⎝d ⎠ ⎝ ⎝d ⎠ ⎠
εsu.star≔min ⎝⎛εsu.s ,ε ⎞su.c⎠=0.013
εsu.star-εsy ‾λ‾ θpl.EC
θpl.EC≔βn ⋅βs ⋅――――⋅ ―=0.045 rad θpl≔――=0.022 rad
x 3 2
1-―
d
C-18 , Architecture and Civil Engineering, Master’s Thesis ACEX30
D Plastic reinforcement strain
D.1 Plastic strain measurements
The results from the strain measurements are presented in Figures D.1- D.9. An
explanation of the notations in the following bar charts is presented in Section 7.4
together with a summary of the average plastic strain results. For the beams with
PVC tubes, Figures D.7- D.9, the position of the tubes is marked with a shaded
area.
 
Beam 1 - Bar A Beam 1 - Bar B
60 80 ‰
*220 ‰ 60 120 ‰ *180 ‰ 
40 40
20 20
0 0
600 800 1000 1200 1400 1600 1800 600 800 1000 1200 1400 1600 1800
Length [mm] Length [mm]
  
 
Reinforcement Beam 1
*
60 100 ‰ 200 ‰
50
40
eave = 25.2 ‰
30
20
10
0
600 800 1000 1200 1400 1600 1800
Length [mm]
 
 
 
 
 
 
 
Figure D.1 Plastic reinforcement strain measurements for each reinforcement bar
(upper charts), the average strain of the two bars (middle chart) and
the crack pattern figure from DIC for beam 1.
, Architecture and Civil Engineering, Master’s Thesis ACEX30 D-1
Strain [‰]
Strain [‰]
Strain [‰]
 
180 ‰ * Beam 2 - Bar A 220 ‰ Beam 2 - Bar B*
60  60
40 40
20 20
0 0
600 800 1000 1200 1400 1600 1800 600 800 1000 1200 1400 1600 1800
Length [mm] Length [mm]
  
 
* Reinforcement Beam 2
60 200 ‰ 
50
40
eave = 24.3 ‰
30
20
10
0
600 800 1000 1200 1400 1600 1800
Length [mm]
 
Figure D .2 Plastic reinforcement strain measurements for each reinforcement bar
 (upper charts), the average strain of the two bars (middle chart) and
 the crack pattern figure from DIC for beam 2.
D-2 , Architecture and Civil Engineering, Master’s Thesis ACEX30
Strain [‰]
Strain [‰]
Strain [‰]
 
140 ‰ Beam 3 - Bar A 240 ‰ Beam 3 - Bar B*
60 80 ‰ 60
40 40
20 20
0 0
600 800 1000 1200 1400 1600 1800 600 800 1000 1200 1400 1600 1800
Length [mm] Length [mm]
  
 
* Reinforcement Beam 3
60 190 ‰
50
40
eave = 19.2 ‰30
20
10
0
600 800 1000 1200 1400 1600 1800
Length [mm]
 
 
F igure D.3 Plastic reinforcement strain measurements for each reinforcement bar
(upper charts), the average strain of the two bars (middle chart) and
the crack pattern figure from DIC for beam 3.
, Architecture and Civil Engineering, Master’s Thesis ACEX30 D-3
Strain [‰]
Strain [‰]
Strain [‰]
 
Beam 4 - Bar A Beam 4 - Bar B
90 120 ‰ 90 100 ‰
60 60
30 30
0 0
600 800 1000 1200 1400 1600 1800 600 800 1000 1200 1400 1600 1800
Length [mm] Length [mm]
  
 
Reinforcement Beam 4
90
110 ‰  
80  
70 eave = 43.3 ‰ 
60
50
40
30
20
10
0
600 800 1000 1200 1400 1600 1800
Length [mm]
 
F  igure D.4 Plastic reinforcement strain measurements for each reinforcement bar
 (upper charts), the average strain of the two bars (middle chart) and
the crack pattern figure from DIC for beam 4.
D-4 , Architecture and Civil Engineering, Master’s Thesis ACEX30
Strain [‰]
Strain [‰]
Strain [‰]
 
Beam 5 - Bar A Beam 5 - Bar B
90 90
60 60
30 30
0 0
600 800 1000 1200 1400 1600 1800 600 800 1000 1200 1400 1600 1800
Length [mm] Length [mm]
  
 
Reinforcement Beam 5
90
80
70
60
50 eave = 30.9 ‰ 
40
30
20
10
0
600 800 1000 1200 1400 1600 1800
Length [mm]
 
 
 
Figure D  .5 Plastic reinforcement strain measurements for each reinforcement bar
 (upper charts), the average strain of the two bars (middle chart) and
the crack pattern figure from DIC for beam 5.
, Architecture and Civil Engineering, Master’s Thesis ACEX30 D-5
Strain [‰]
Strain [‰]
Strain [‰]
 
Beam 6 - Bar A Beam 6 - Bar B
90 90
60 60
30 30
0 0
600 800 1000 1200 1400 1600 1800 600 800 1000 1200 1400 1600 1800
Length [mm] Length [mm]
  
 
Reinforcement Beam 6
90
80
70
60
e
50 ave 
= 37.6 ‰
40
30
20
10
0
600 800 1000 1200 1400 1600 1800
Length [mm]
 
 
 
 
Figure D.6 Plastic reinforcement strain measurements for each reinforcement bar
(upper charts), the average strain of the two bars (middle chart) and
the crack pattern figure from DIC for beam 6.
D-6 , Architecture and Civil Engineering, Master’s Thesis ACEX30
Strain [‰]
Strain [‰]
Strain [‰]
 
Beam 7 - Bar A Beam 7 - Bar B
90 90
60 60
30 30
0 0
600 800 1000 1200 1400 1600 1800 600 800 1000 1200 1400 1600 1800
Length [mm] Length [mm]
  
 
Reinforcement Beam 7
90
80
70
60 eave = 42.1 ‰
50
40
30
20
10
0
600 800 1000 1200 1400 1600 1800
Length [mm]  
 
Figure D .7 Plastic reinforcement strain measurements for each reinforcement bar
 
 (upper charts), the average strain of the two bars (middle chart) and
the crack pattern figure from DIC for beam 7.
, Architecture and Civil Engineering, Master’s Thesis ACEX30 D-7
Strain [‰]
Strain [‰]
Strain [‰]
 
Beam 8 - Bar A Beam 8 - Bar B
90 90
60 60
30 30
0 0
600 800 1000 1200 1400 1600 1800 600 800 1000 1200 1400 1600 1800
Length [mm] Length [mm]
  
 
Reinforcement Beam 8
90
80
70
60 eave = 42.2 ‰ 
50
40
30
20
10
0
600 800 1000 1200 1400 1600 1800
Length [mm]
 
 
 Figure D.8 Plastic reinforcement strain measurements for each reinforcement bar
 (upper charts), the average strain of the two bars (middle chart) and
 
 the crack pattern figure from DIC for beam 8.
 
 
 
D-8 , Architecture and Civil Engineering, Master’s Thesis ACEX30
Strain [‰]
Strain [‰]
Strain [‰]
 
Beam 9 - Bar A Beam 9 - Bar B
90 90
60 60
30 30
0 0
600 800 1000 1200 1400 1600 1800 600 800 1000 1200 1400 1600 1800
Length [mm] Length [mm]
  
 
Reinforcement Beam 9
90
80
70
60 eave = 39.2 ‰ 
50
40
30
20
10
0
600 800 1000 1200 1400 1600 1800
Length [mm]
 
 
 Figure D.9 Plastic reinforcement strain measurements for each reinforcement bar
 
 (upper charts), the average strain of the two bars (middle chart) and
the crack pattern figure from DIC for beam 9.
, Architecture and Civil Engineering, Master’s Thesis ACEX30 D-9
Strain [‰]
Strain [‰]
Strain [‰]
Table D.1 Calculated strains for each reinforcement bar from elongation
measurements. The x-coordinates are presented in mm and the strains
in ‰.
Beam 1 Beam 2 Beam 3 Beam 4 Beam 5 Beam 6 Beam 7 Beam 8 Beam 9
x A B A B A B A B A B A B A B A B A B
600 0 0 0 20 20 0 40 50 0 0 40 30 20 20 40 20 40 20
650 0 0 40 40 0 20 20 30 20 60 0 20 60 60 40 40 40 40
700 20 0 0 0 20 0 40 0 40 20 40 30 80 80 20 40 100 80
750 80 120 20 30 0 0 0 20 20 0 40 40 80 80 20 20 60 60
800 220 180 40 30 20 20 60 60 40 40 20 40 80 80 0 20 80 90
850 20 40 180 220 140 240 40 20 0 20 80 80 70 60 40 60 40 70
900 60 60 60 50 80 0 40 40 60 20 40 40 50 60 40 20 40 60
950 60 40 60 10 20 40 40 60 20 60 40 60 20 40 40 20 60 40
1000 0 0 40 60 40 20 60 40 40 20 40 40 40 40 60 60 20 60
1050 40 40 20 20 20 0 20 40 20 40 20 60 60 40 40 40 60 50
1100 0 20 20 40 0 20 40 80 20 20 40 0 40 40 40 60 60 30
1150 0 0 40 40 20 40 60 40 40 20 40 80 40 40 40 40 40 40
1200 0 40 0 40 40 0 40 60 0 0 60 40 50 20 60 30 0 40
1250 40 10 0 40 0 40 40 40 40 20 20 40 10 40 20 50 40 0
1300 20 10 60 0 40 0 60 40 20 40 40 20 20 20 60 40 20 40
1350 20 0 20 20 20 0 60 40 40 20 40 80 80 60 40 40 60 20
1400 20 40 0 0 20 20 40 60 20 40 0 20 50 60 60 60 20 60
1450 0 0 20 0 20 20 40 60 50 60 60 40 20 20 40 40 60 40
1500 20 40 20 20 0 0 60 60 70 80 40 0 40 40 40 20 0 0
1550 20 0 0 0 0 20 120 100 40 20 20 40 30 20 60 80 60 40
1600 20 0 0 0 20 20 80 80 60 60 20 40 20 20 80 80 40 40
1650 0 0 20 0 0 0 20 0 0 20 0 0 20 40 60 80 40 40
1700 0 0 0 0 0 0 0 0 0 20 0 0 20 20 0 40 0 0
1750 0 0 0 0 0 0 40 40 0 0 0 0 20 0 0 0 0 0
1800 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 20
D-10 , Architecture and Civil Engineering, Master’s Thesis ACEX30
D.2 3D scanning
The middle one meter of in total six bars were 3D scanned before the casting and
after testing removed from the concrete and scanned again to measure the elongation.
Data was extracted for every 50 mm segment as a point cloud mesh to evaluate the
length of each bar segment. The procedure was done both for the first and second
scanning. The first part of the appendix treats the 3D model and the second part
the results from the scanning.
D.2.1 3D model
Before the second scanning, the bars were sandblasted to remove concrete residuals.
However, the sandblasting made the surface more shiny which effected the resolution
of the scanning since shiny surfaces are more difficult to scan and the marks were
less visible, see Figure D.10 and Figure D.11. Those effects were more severe for the
φ10 bars where some of the marks were not visible at all.
Figure D.10 Example of marks in the 3D model before casting for a φ12 bar.
Figure D.11 Example of marks in the 3D model after casting and sandblasting for
a φ12 bar.
Figure D.13 and Figure D.12 show examples of the 3D models for a φ10 bar and a φ12
bar, respectively, at the first and second scanning. Both φ10 bars were ruptured and
steel plates were used as supports, and to capture the fractured area the scanning
was made twice with two different configurations of the steel plates, see Figure D.14.
, Architecture and Civil Engineering, Master’s Thesis ACEX30 D-11
Figure D.12 Example of a φ12 bar 3D model before and after casting.
Figure D.13 Example of a φ10 bar 3D model before and after casting.
Figure D.14 The scanning of the rupture bars was made twice with two different
configurations of supporting steel plates to scan the fractured area.
D.2.2 3D scanning results
The plastic strain results from the 3D scanning are presented in Figures D.15 - D.17.
The results are presented as bar charts in the same way as the plastic strain mea-
surements. An explanation of the notations in the following bar charts is presented
in Section 7.4 together with a summary of the average plastic strain results from
the 3D scanned bars.
D-12 , Architecture and Civil Engineering, Master’s Thesis ACEX30
 
Beam 1 - Bar A Beam 1 - Bar B
* * 
60 79 ‰ 220 ‰ 60 81 ‰ 800 ‰ 85 ‰ 67 ‰
40 40
20 20
0 0
600 800 1000 1200 1400 1600 1800 600 800 1000 1200 1400 1600 1800
Length [mm] Length [mm]
  
 
Reinforcement Beam 1
*
60 80 ‰ 510 ‰
50
eave = 33.8 ‰
40
30
20
10
0
600 800 1000 1200 1400 1600 1800
Length [mm]
 
Figure D.15 Plastic reinforcement strain measurements for each reinforcement
bar (upper charts), the average strain of the two bars (middle chart)
and the crack pattern figure from DIC for beam 1.
, Architecture and Civil Engineering, Master’s Thesis ACEX30 D-13
Strain [‰]
Strain [‰]
Strain [‰]
 
Beam 4 - Bar A Beam 4 - Bar B
90 145 ‰ 9091 ‰ 114 ‰ 
60 60
30 30
0 0
600 800 1000 1200 1400 1600 1800 600 800 1000 1200 1400 1600 1800
Length [mm] Length [mm]
  
 
Reinforcement Beam 4
90
129 ‰  
80  
70 eave = 49.1 ‰ 
60
50
40
30
20
10
0
600 800 1000 1200 1400 1600 1800
Length [mm]
 
Figure D.16 Plastic reinforcement strain measurements for each reinforcement
bar (upper charts), the average strain of the two bars (middle chart)
and the crack pattern figure from DIC for beam 4.
D-14 , Architecture and Civil Engineering, Master’s Thesis ACEX30
Strain [‰]
Strain [‰]
Strain [‰]
 
112 ‰ Beam 7 - Bar A Beam 7 - Bar B90 109 ‰ 135 ‰ 158 ‰ 90 149 ‰
60 60
30 30
0 0
600 800 1000 1200 1400 1600 1800 600 800 1000 1200 1400 1600 1800
Length [mm] Length [mm]
  
 
98 ‰ Reinforcement Beam 7
90
98 ‰ 104 ‰ 105 ‰
80
70 eave = 50.4 ‰ 
60
50
40
30
20
10
0
600 800 1000 1200 1400 1600 1800
Length [mm]
 
 
 
Figure D .17 Plastic reinforcement strain measurements for each reinforcement
 
 bar (upper charts), the average strain of the two bars (middle chart)
 and the crack pattern figure from DIC for beam 7.
 
 
 
, Architecture and Civil Engineering, Master’s Thesis ACEX30 D-15
Strain [‰]
Strain [‰]
Strain [‰]
D.3 Comparison of plastic strain results
A more detailed comparison of the plastic strain results are presented in
Tables D.2-D.4. The results are presented without the 64 ‰ and 90 ‰-limit. See
Section 7.4 for comparison of results as bar charts. For beam 1, due to difficulties in
fitting the ruptured parts together between the coordinates 800 mm and 850 mm,
the elongation and plastic strain value from the 3D scanning are greater.
Table D.2 Comparison of the plastic strain values from 3D scanning and
measuring and the difference in measured elongation between the
methods for beam 1.
Starting coordinate 3D scanning Measurements Difference
[‰] [‰] [mm]
700 7 10 3
750 80 120 40
800 510 180 330
850 43 30 13
900 - 60 -
950 52 50 2
1000 0 0 0
1050 46 40 6
1100 5 10 5
1150 - 0 -
1200 8 20 12
1250 46 25 21
1300 32 15 17
1350 34 10 24
1400 42 30 12
1450 11 0 11
1500 17 30 13
1550 35 10 25
1600 - 10 -
1650 - 0 -
D-16 , Architecture and Civil Engineering, Master’s Thesis ACEX30
Table D.3 Comparison of the plastic strain values from 3D scanning and
measuring and the difference in measured elongation between the
methods for beam 4.
Starting coordinate 3D scanning Measurements Difference
[‰] [‰] [mm]
700 29 20 9
750 27 10 17
800 72 60 12
850 25 30 5
900 51 40 11
950 50 50 0
1000 52 50 2
1050 34 30 4
1100 40 60 20
1150 48 50 2
1200 50 50 0
1250 40 40 0
1300 54 50 4
1350 31 50 19
1400 60 50 10
1450 60 50 10
1500 54 60 10
1550 90 90 0
1600 68 80 12
1650 - 10 -
, Architecture and Civil Engineering, Master’s Thesis ACEX30 D-17
Table D.4 Comparison of the plastic strain values from 3D scanning and
measuring and the difference in measured elongation between the
methods for beam 7.
Starting coordinate 3D scanning Measurements Difference
[‰] [‰] [mm]
700 90 80 10
750 90 80 10
800 90 80 10
850 62 65 3
900 50 55 5
950 29 30 1
1000 52 40 12
1050 41 50 9
1100 34 40 6
1150 46 40 6
1200 26 35 9
1250 22 25 3
1300 38 20 18
1350 43 70 27
1400 37 55 18
1450 42 20 22
1500 25 40 15
1550 90 25 65
1600 52 20 32
1650 - 30 -
D-18 , Architecture and Civil Engineering, Master’s Thesis ACEX30
D.4 Cross-sectional area from 3D scanning
The cross section analysis was performed according to a provided MATLAB-file.
The blue curve in Figure D.18 represent the cross section of the reinforcement before
testing and the orange after testing. The part with the largest strain value was used
in the analysis except from bar B in beam 1 and and bar B in beam 4 due loss of
data. The reinforcement part of beam 1, bar A was ruptured which can be seen in
the results as a drastic decrease of the cross-sectional area.
 
 
Beam1-A (800-850 mm) Beam1-B (750-800 mm) 
90 90
75 75
60 60
45 45
30 30
0 10 20 30 40 50 0 10 20 30 40 50
Coordinate along part [mm] Coordinate along part [mm]
  
  
Beam4-A (1550-1600 mm) Beam4-A (1550-1600 mm) 
140 140
125 125
110 110
95 95
80 80
0 10 20 30 40 50 0 10 20 30 40 50
Coordinate along part [mm] Coordinate along part [mm]
  
  
Beam7-A (800-850 mm) Beam7-B (800-850 mm) 
140 140
125 125
110 110
95 95
80 80
0 10 20 30 40 50 0 10 20 30 40 50
Coordinate along part [mm] Coordinate along part [mm]
  
 
 
Figure D .18 Cross-sectional area before and after testing of a part with largest
 
 strain value for each bar of beams 1, 4, and 7. The values before and
after testing is shown by blue (top) and orange (bottom) line,
respectively.
, Architecture and Civil Engineering, Master’s Thesis ACEX30 D-19
Cross-sectional area,  A [mm2] Cross-sectional area,  A [mm2] Cross-sectional area,  A [mm2]
Cross-sectional area,  A [mm2] Cross-sectional area,  A [mm2] Cross-sectional area,  A [mm2]
DEPARTMENT OF ARCHITECTURE AND CIVIL ENGINEERING
CHALMERS UNIVERSITY OF TECHNOLOGY
Gothenburg, Sweden
www.chalmers.se