master of science thesis reinforceent ayot in oncrete ie
TRANSCRIPT
Master of Science Thesis
Reinforcement Layout in Concrete Pile FoundationsA study based on non-linear finite element analysis
MOHAMMAD MUSTAFA ANGAR
Stockholm, Sweden 2020www.kth.se
TRITA-ABE-MBT 2020:20418 ISBN: 978-91-7873-598-3
kth royal institute of technology
MOHAM
MAD M
USTAFA ANGAR Reinforcem
ent Layout in Concrete Pile FoundationsK
TH 2020
Reinforcement Layout in
Concrete Pile Foundations A study based on non-linear finite element analysis
MOHAMMAD MUSTAFA ANGAR
Master of Science Thesis
Stockholm, Sweden 2020
TRITA-ABE-MBT- 20418, Master Thesis
ISBN: 978-91-7873-598-3
KTH School of ABE
SE-100 44 Stockholm
Sweden
ยฉ Mohammad Mustafa Angar 2020
Royal Institute of Technology (KTH)
Department of Civil and Architectural Engineering
Division of Concrete Structures
i
Abstract
The main topic of this thesis concerns the behavior of concrete pile cap supported by four piles
with two varying positions of longitudinal reinforcements. The positions include top of piles
and bottom of the pile cap. For this purpose, non-linear finite element models of a pile cap were
created using software ATENA 3D. The goal was to observe which position of reinforcement
yields the higher bearing capacity and to observe the failure modes in the models.
To achieve the above goals, a short review of theoretical background concerning shear
phenomena was performed. This, in order to enhance the knowledge regarding shear stresses,
shear transfer mechanism, factors affecting shear capacity, modes of shear failure and relate
them to the behavior of pile cap. Furthermore, the calculation of shear resistance capacity based
on Eurocode 2 using strut and tie method and sectional approach is presented.
The numerical analysis started by creating four pile cap models in ATENA 3D. The difference
between the models being the position and ratio of longitudinal reinforcement. The purpose
behind the two reinforcement ratios was to observe the behavior of pile cap model in two cases:
a) when failure occurs prior to yielding of reinforcement; b) when failure occurs while
reinforcement is yielding. The models were then analyzed using software ATENA Studio.
The results revealed that placing the reinforcement on top of piles in case (a) increased the
capacity of the model by 23.5 % and in case (b) increased the capacity by 18.5 %. This because
the tensile stresses were found to be concentrated on top of piles rather than the bottom of the
pile cap. The final failure mode in the model with top reinforcement position was crushing of
the inclined compressive strut at the node beneath the column and in the model with bottom
reinforcement position, the splitting of the compressive strut due to tensile stresses developed
perpendicular to the inclined strut. The potential advantage of placing the reinforcement at the
bottom were a better crack control in serviceability limit state and a slightly less fragile failure
mode compared to the top position of reinforcement.
A parametric study was performed in the model as well to observe the effects of various
parameters on the results obtained. It was found that fracture energy had the most significant
effect on the results obtained.
Finally, a comparison between the results of numerical analysis and analytical design
approaches based on strut and tie method and sectional approach was performed. The
comparison revealed that the design values obtained based on strut and tie method for the model
were very conservative. In particular, the equation for the strength of inclined compressive strut
based on Eurocode 2 was very conservative.
iii
Sammanfattning
Det huvudsakliga รคmnet fรถr den hรคr examensarbetet handlar om beteendet hos pรฅlfundament
som stรถds av fyra pรฅlar med tvรฅ olika positioner av lรคngsgรฅende armering. Positionerna
inkluderar placering รถver pรฅlarna och placering i botten av fundamentet, dvs under
pรฅlavskรคrningsplanet. Fรถr detta รคndamรฅl skapades icke-linjรคra finita elementmodeller av en
pรฅlfundamentet med mjukvaran ATENA 3D. Mรฅlet var att observera vilket armeringslรคge som
ger den hรถgre bรคrkapaciteten och att identifiera brottmekanismen i modellerna.
Fรถr att uppnรฅ ovanstรฅende mรฅl utfรถrdes en kort genomgรฅng av den teoretisk bakgrunden
rรถrande skjuvningsfenomenet. Detta fรถr att fรถrbรคttra kunskapen om skjuvspรคnningar,
skjuvรถverfรถringsmekanism, faktorer som pรฅverkar skjuvkapacitet, skjuvbrott och att relatera
dem till beteendet av ett pรฅlfundament. Berรคkningar av skjuvkapaciteten baserad pรฅ Eurocode2
med hjรคlp av Srut and tie-metod och sektionsmetod presenteras.
Den numeriska analysen bรถrjade med att skapa fyra pรฅlfundament i ATENA 3D. Skillnaden
mellan modellerna รคr positionen och innehรฅllet av den lรคngsgรฅende armeringen. Syftet med tvรฅ
armeringsinnehรฅll var att observera beteendet av pรฅlfundamentet i tvรฅ fall: a) nรคr brott intrรคffar
innan armering plasticeras; b) nรคr brott intrรคffar medan armeringen plasticeras. Modellerna
analyserades sedan med hjรคlp av programvaran ATENA Studio.
Resultaten visade att placering av armeringen ovanpรฅ pรฅlarna i fall a) รถkade modellens kapacitet
med 23,5% och i fall (b) รถkade kapaciteten med 18,5%. Detta pรฅ grund av att dragspรคnningarna
visade sig vara koncentrerade pรฅ toppen av pรฅlarna snarare รคn pรฅ botten av pรฅlfundamentet. Det
slutliga brottet i modellen med armering รถver pรฅlarna var krossning av den lutande tryckstrรคvan
vid noden under pelaren. I modellen med armering i botten av fundamentet sprรคcktes
tryckstrรคvan pรฅ grund av dragspรคnningar vinkelrรคtt mot den lutande strรคvan. The potentiella
fรถrdelen med placering av armoring I botten av pรฅlfundamentet รคr ren bรคttre sprickkontroll och
en nรฅgot segare brottmod i jรคmfรถrelse med placering av armering รถver pรฅlarna.
En parametrisk studie genomfรถrdes ocksรฅ med modellen fรถr att observera effekterna av olika
parametrar pรฅ de erhรฅllna resultaten. Det visade sig att brottenergi hade den mest signifikanta
effekten pรฅ de erhรฅllna resultaten.
Slutligen genomfรถrdes en jรคmfรถrelse mellan resultaten frรฅn numerisk analys och analytiska
dimensioneringsmetoder baserade pรฅ fackverksmetoden och tvรคrsnittsmetoden. Jรคmfรถrelsen
avslรถjar att de kapaciteter som erhรถlls med fackverksmetoden var mycket konservativa. I
synnerhet var ekvationen fรถr kapaciteten hos det lutande tryckstrรคvorna baserad pรฅ Eurocode 2
mycket konservativa.
iv
Preface
In the name of God, the compassionate, the merciful
This report presents a masterโs thesis which was initiated in a joint effort between the division
of concrete structures at the KTH Royal Institute of Technology and Tyrรฉns AB in Stockholm,
Sweden. The completion of this thesis is the requirement for the last semester of a two-year
masterโs program in Civil and Architectural Engineering and the final chapter as part of my
Master of Science degree in Engineering.
First and foremost, I want to thank God, for his endless mercy kindness and favors. I have an
infinite gratitude towards my parents who have always supported and encouraged me in every
step of life.
I express my gratefulness and appreciation to my supervisor Adjunct professor Mikael
Hallgren, for his guidance throughout the completion of the thesis and his support in difficult
moments. I specially thank Pedro Studer Ferreira, the department manager at Tyrรฉns for
providing me the necessary means to complete the thesis and his guidance and motivation in
challenging times. I admire Ebrahim Zamani, my industrial supervisor at Tyrรฉns, for providing
me the required data and material which was necessary for the completion of this thesis
My utmost gratitude goes to all my instructors at KTH for their teachings and all the knowledge
I have gained throughout the masterโs program. Finally, I want to thank Swedish institute, a
government agency in Sweden for providing me a fully funded scholarship to complete a
masterโs degree program in Sweden.
Stockholm, May 2020
Mohammad Mustafa Angar
v
Contents
Abstract ...................................................................................................................................... i
Sammanfattning ...................................................................................................................... iii
Preface ...................................................................................................................................... iv
Introduction ......................................................................................................... 13
Background ........................................................................................................... 13
Problem statement ................................................................................................. 13
Objective ............................................................................................................... 14
Limitations and assumptions ................................................................................. 14
Outline of thesis ..................................................................................................... 14
Theoretical Background ..................................................................................... 15
Pile foundations in Sweden ................................................................................... 15
2.1.1 Piles .......................................................................................................... 15
2.1.2 Pile Caps ................................................................................................... 17
B and D regions in a structure ............................................................................... 19
History of strut and tie method .............................................................................. 21
Strut and tie method design based on European structural concrete code ............ 22
2.4.1 Definition ................................................................................................. 22
2.4.2 Struts ......................................................................................................... 23
2.4.3 Ties ........................................................................................................... 26
2.4.4 Nodes ........................................................................................................ 27
Shear ...................................................................................................................... 29
2.5.1 Shear force in a beam ............................................................................... 29
2.5.2 Shear cracks .............................................................................................. 31
2.5.3 Shear transfer mechanism ........................................................................ 32
2.5.4 Design according to EC2. ......................................................................... 35
Finite element analysis ........................................................................................ 37
vi
Theory of finite element ........................................................................................ 37
Non-linear finite element analysis ......................................................................... 38
3.2.1 Iterative procedure .................................................................................... 39
3.2.2 Crack opening laws and fracture mechanics ............................................ 40
3.2.3 Smeared crack approach ........................................................................... 41
3.2.4 Fixed crack model: ................................................................................... 41
3.2.5 Rotated crack model: ................................................................................ 42
3.2.6 Tensile behavior: ...................................................................................... 42
3.2.7 Fracture energy: ........................................................................................ 43
Modelling simplifications and assumptions in ATENA 3D ................................. 44
Material properties definitions .............................................................................. 46
3.4.1 Concrete material models ......................................................................... 46
3.4.2 Reinforcement model in ATENA ............................................................ 48
Boundary conditions and loads ............................................................................. 49
Mesh and elements ................................................................................................ 50
Non-linear iterative solvers ................................................................................... 51
Analysis and results in ATENA 3D ................................................................... 53
Models created ...................................................................................................... 53
Input data in model ................................................................................................ 55
4.2.1 Concrete ................................................................................................... 55
4.2.2 Reinforcement: ......................................................................................... 56
4.2.3 Interface: ................................................................................................... 56
4.2.4 Steel plate: ................................................................................................ 57
4.2.5 Input file: .................................................................................................. 57
Results: .................................................................................................................. 57
4.3.1 Load deflection response .......................................................................... 57
4.3.2 Crack pattern ............................................................................................ 59
4.3.3 Crack width and failure mode .................................................................. 61
4.3.4 Stress in concrete ...................................................................................... 63
4.3.5 Stress in reinforcement ............................................................................. 65
Parametric Study ................................................................................................ 69
Influence of mesh size ........................................................................................... 69
Influence of fracture energy .................................................................................. 72
Influence of Tensile Strength ................................................................................ 75
Influence of Compressive Strength ....................................................................... 78
vii
Influence of Modulus of Elasticity of piles ........................................................... 81
Results- Hand calculation ................................................................................... 83
Hand calculation based on strut and tie method and sectional approach .............. 83
6.1.1 Assumptions in design: ............................................................................ 83
6.1.2 Pile cap Geometry .................................................................................... 83
6.1.3 Reinforcement .......................................................................................... 84
6.1.4 Calculation based on strut and tie model: ................................................ 84
6.1.5 Calculation of pile cap based on beam theory: ........................................ 85
Discussion, conclusion and further research .................................................... 87
Optimizing the numerical model ........................................................................... 87
Comparison of numerical and analytical results ................................................... 88
Conclusion ............................................................................................................. 90
Further research: .................................................................................................... 91
Bibliography ........................................................................................................................... 93
A Design of pile cap based on strut and tie method ...................................................... 95
B Design of pile cap based on sectional approach
................................................................................................................................................ 105
viii
ix
Symbols Greek Symbols
ฯต1 Principle tensile strain
ฯต2 Principle compressive strain
โP Small load increment
ฮฑ Modulus of elasticity ratio
Shear reduction factor
ฮฒ* Angle between strut and tie in plan
ฮท Factor for effective strength
ฮธ Rotation angle
ฮธ* Angle between strut and tie in elevation
ฮป Factor for effective height of compressive zone
ฮผ Friction coefficient
Longitudinal reinforcement ratio
ฯlx Reinforcement ratio
ฯc Stress in concrete
ฯRd.max Maximum resistance of strut
ฯw Stress surrounding a crack
Compressive stress in concrete
ฯxy Shear stresses at a point
Total strain
Elastic strain
Plastic strain
Lower case Roman Symbols:
a Pile spacing factor
a/d Slenderness ration
av Shear plane
b Width of cross section
bpile Cross-sectional width of pile
bx Column cross section size in x direction
by Column cross section size in y direction
ca Displacement correction
d Effective depth
d1 Height of compressive zone
d2 Cover for reinforcement tie
e Eccentricity factor
ex Eccentricity in x direction
ey Eccentricity in y direction
fcd Concrete design strength
fck Characteristic compressive strength
fcm Mean concrete strength
fct Concrete tensile srain
x
ft Tensile stresses
fyd Design yield strength of reinforcement
h1 Lever arm in ST model A
h2 Lever arm in ST model B
k Size factor
kโ Safety factor
k1 National parameter
k2 National parameter
k3 National parameter
lb Development length of reinforcement
lmax Maximum allowed element size
n Converting factor
u0 Previous load step
v5 Version 5
w Crack width
wc Macro crack width
wk,max Maximum crack width
ฯw Stress surrounding a crack
Factor depending on national annexes relating to cracking of concrete strut
Shear stresses
Smallest cross-section width
Minimum resistance stress
Force
Displacement
Upper case Roman Symbols:
Ac Total area of column cross section
Ac.eff Area of concrete
Aci Equivalent column cross section
B Span width between piles
CA Total force in strut
Ch1 Force in strut in plan
Cz1 Force in strut in elevation
D Cohesion
Dmax Maximum gravel size
E Modulus of elasticity of concrete
E Modulus of elasticity
Ecm Mean modulus of elasticity
Gf Fracture energy
GPa Gega Pascal
H Pile cap height
Iy Moment of inertia
K Stiffness of the structure.
K0 Tangential stiffness
Knn Normal stiffness
Ktt Tangential stiffness
L Pile cap length
lp Fracture zone length
xi
M Bending moment force
Mcr Cracking moment
MEd Applied moment force
MPa Mega Pascal
Mx.Rd Moment resistance in x direction
My.Rd Moment resistance in y direction
N Normal force
Q Applied load on column
R Reaction force
S Pile span
St Static moment
T Force in the tie
T Force in Tie
Tm Torsional moment force
Vc Shear force in concrete
V Shear force
VEd. final Final applied shear force
Vs Shear force in steel
Shear resistance
Factor relating to loading case
Applied shear force
Abbreviations:
AASHTO American Association of State Highway and Transportation Officials
ACI American Concrete Institute
B-regions Bernoulli regions
CCC Compression -tension- tension
CCT Compression -compression- tension
CEB-FIP International federation for structural concrete
CSA Canadian Standards Association
CTT Compression -tension- tension
DOF Degree of freedom
D-regions Disturbed regions
EC2 Eurocode 2
FEM Finite element method
MC10 Model code 2010
MC90 Model code 1990
NFLEA Non-linear finite element analysis
ST Strut and tie
CHAPTER 1
13
Introduction
Background
Pile foundations are used to transfer the loads from superstructures to the firm ground. Piles can
be used individually to support loads or grouped and linked together with a reinforced
concrete cap. A pile cap is a thick concrete slab that rests on piles and distributes the load of
the structure into the piles. For calculating the forces and stresses in thick concrete members,
the beam theory is not applicable. Eurocode 2 suggests using strut and tie method where in the
structure the compressive stresses are carried by concrete strut and the tensile stresses are
carried by reinforcement tie. The analogy suggests that the tensile stresses are concentrated
horizontally over the top of piles and therefore, the reinforcement mesh is placed there.
However, a different position for reinforcement has been observed in pile caps while renovating
an old building in Stockholm. The building was built in 60โs and the reinforcement was
positioned further down at the bottom of the slab. The engineers at Tyrรฉns wanted to build more
stories on the existing building as part of a renovation project but were not sure about the
capacity of the pile cap. This raised curiosity for engineers in Tyrรฉns about the bearing capacity
and function of the pile cap when reinforcement is placed at the bottom of the cap.
Problem statement
Foundation slabs on ground are treated as a two-way slab where the reinforcement layer is
placed further down at the bottom of the slab. This, in order to achieve a higher lever arm for
an increased bending capacity and a better control over concrete cracking. Pile cap, on the other
hand is a structure with considerable dimensions in three directions with a very rigid behavior.
Eurocode 2 recommends strut and tie method for the calculation of a pile cap based on which
the reinforcement is placed on the top of piles. This placement, however, yields a lower bending
capacity based on sectional approach of calculating forces and a lower crack control due to a
very large concrete cover.
However, in the foundation of an older building in Stockholm, the reinforcement mesh is placed
at the bottom of the pile cap. The reason for this placement is unknown. Perhaps, it yields a
higher load bearing capacity due to higher lever arm and a better control over concrete cracking.
INTRODUCTION
14
Therefore, understanding the overall behavior of pile cap in top and bottom reinforcement
positions is the question of this mastersโ thesis.
Objective
The main objective of this thesis is to study the behavior of pile caps with two different
reinforcement positions in order to determine the ultimate load bearing capacity and the failure
mode that occurs in the pile cap models. To be able to achieve these objectives, the following
tasks were completed: first an extensive literature study was performed. The main areas studied
were; general theory of concrete, shear failure, strut and tie method, theory of non-linear finite
element analysis (NFLEA), and the theory manual for software (ATENA). Secondly, hand
calculations were performed for pile caps using strut and tie method and sectional approach.
Third, non-linear finite element (NLFE) models were created and analyzed using software
ATENA. Finally, the results for hand calculations were compared to results from software
ATENA and parametric studies in relation to certain parameters were performed.
Limitations and assumptions
The limitation of this thesis is mainly the non-linear finite element analysis (NLFEA). The
assumptions and simplifications include considering only vertical force on pile cap (absence of
lateral force and moment), absence of transverse shear reinforcement, equal stiffness for all
piles, and studying the pile cap separate from the rest of the structure.
Outline of thesis
This masterโs thesis consists of 7 chapters and the related appendices. Each chapter contain the
below contents
Chapter 2 - covers the theoretical background concerning the pile foundations, strut and tie
method, shear phenomena in concrete, shear resistance based on Eurocode 2.
Chapter 3 โ covers the theory regarding non-linear finite element method, material models
used in ATENA 3D and assumptions made when building the model.
Chapter 4 โ presents the results of the numerical analysis performed in ATENA Studio.
Chapter 5 โ presents the results of hand calculations based on strut and tie method and
numerical analysis
Chapter 6 โ presents the parametric studies regarding tensile strength, fracture energy,
compressive strength, and stiffness of piles.
Chapter 7 โ contains the comparison of analytical and numerical results.
CHAPTER 2
15
Theoretical Background
Pile foundations in Sweden
2.1.1 Piles
According to (Axelsson, 2016) the geological conditions in Sweden are most favorable for
using pile foundations. Almost all areas except the southern part (Skรฅne, รland and Gotland)
consists of very hard rock. A major part of this rock layer (approximately 75 %) is covered with
moraine or till which are very dense material. Overlaying the dense layers are loose soil
materials such as clay, sand, or silt. Depending on the soil profiles, variety of piles with different
mechanisms for function are available. Generally, according to their behavior, the piles are
divided into 3 groups (Axelsson, 2016).
A) End bearing piles:
End bearing piles have two types; driven to bed rock and drilled to bed rock (rock-
socket). Either kind of these piles are designed in two ways; a) dynamic pile load test
b) pile termination criteria.
B) Friction piles:
Friction piles are considered for cohesionless soils. They are mostly pre-cast concrete
piles which are driven into ground and the design is performed using pile load testing.
C) Cohesion piles:
Cohesion piles are considered for soft clays and are the only type of piles which are
designed based on calculations. For calculating in soft clays, the ฮฑ method is used in
which the undrained shear strength of clay is of very high importance (Axelsson,
2016).
According to Axlesson, the common pile types used in Sweden are listed in table
below:
THEORETICAL BACKGROUND
16
Table 2.1: The common pile types used in Sweden (Axelsson, 2016):
No. Percentage of usage Type of pile
1 60 % Driven pre-cast concrete piles
2 23 % Driven steel pipe piles
3 13 % Drilled steel piles
4 4 % Timber piles
5 <1 % Steel core piles
In this masterโs thesis, driven pre-cast concrete piles were used. The precast piles usually have
square cross section with dimensions (235 x 235 mm) and (270 x 270 mm). However, the
cross-section can be made up to the dimensions (400 x 400 mm) (Axelsson, 2016).
For this thesis, piles with dimension 300x300 mm are used.
Figure 2.1: Driven pre-cast concrete piles (Axelsson, 2016)
Figure 2.2: Pre-cast driven concrete piles (copied from KTH lecture notes course (Bjureland,
2017)
CHAPTER 2
17
In Sweden, the design and installation procedure for piles used for building and piles used for
infrastructure are different. For both cases, the guidance and instructions are present in the
below national annexes (Axelsson, 2016):
โข For infrastructure: VVFS 2004:43 (with changes in TRVFS 2011:12) provided by
Trafikverket (the Swedish Transport Administration).
โข For buildings: BFS 2015:6 EKS 11, provided by Boverket (The Swedish National
Board of Housing, Building and Planning).
The design of piles is not the included as part of this masterโs thesis. The reader is referred to
(Axelsson, 2016) and the national annexes for further information and design procedures.
2.1.2 Pile Caps
Pile caps are bulky structural concrete elements which has considerable dimension in three
directions. Their function is to transfer the load from superstructure (column or wall) over a
group of piles, and through piles to the solid ground. The construction procedure of a pile cap
is such that first the piles are driven into the ground using hammering or boring methods.
Figure 2.3: Piles being driven into the ground, (Bjureland, 2017)
THEORETICAL BACKGROUND
18
Afterwards, a layer of concrete is placed around the piles to even the surface of the ground
and provide a smooth base for placing the pile cap.
Figure 2.4: concrete base for the pile cap image copied from (Chantelot & Mathern, 2010)
Afterwards, the pile cap is placed on the top of piles. Pile caps can be prefabricated or cast in
place.
Figure 2.5: The pre-cast pile cap being supported by three piles, (Miller, 2020)
CHAPTER 2
19
The current design practices in Sweden for pile caps are based on Eurocode 2 which suggests
that pile caps ought to be treated as an area of discontinuity (D-region) for which the strut and
tie method is suitable. First the distinction between B and D regions in a structure is made and
later the strut and tie method is scrutinized in detail.
B and D regions in a structure
Parts of a reinforced concrete structure function either as B or D regions or both. B region refers
to the parts of a structure where the Bernoulli hypothesis of linear distribution of strain is valid.
For these regions, the state of internal stress is determined directly from sectional forces
(bending and torsional moments, shear and axial forces) (Figure 2.2). In contrast, D regions
refer to the parts in the structure where the linear distribution of strain is not valid. In fact, in a
disturbed (D) region, the distribution of the strain is significantly nonlinear. They are referred
as disturbed or discontinuity regions (Schlaich et al., 1987).
Examples of D regions include; areas near concentrated loads, corners, bends, openings,
footings and pile caps (Figure 2.4). To give a good representation of D regions, Saint-Venantโs
principle can be used which states that load effects in a certain point in a structure depends on
how far the point is located from the loading point. Due to complexity of stress distribution in
D regions, sectional approach does not yield accurate results. Therefore, a design method based
on lower bound approach called Strut and Tie is used (Schlaich et al., 1987).
THEORETICAL BACKGROUND
20
Figure 2.6: D- regions (shaded areas) recreated from (Schlaich et al., 1987)
CHAPTER 2
21
History of strut and tie method
The origin of (ST) method dates back to end of the 19th century where the reinforced concrete
design was in its infancy. In 1899, Wilhelm Ritter developed a truss model which represented
the internal state of compression and tension stresses in a reinforced concrete beam. The truss
consisted of struts and ties where the struts represented the compression stresses in concrete
and the tie represented the tensile stresses in reinforcement (Brown, 2005).
Figure 2.7: Ritterโs truss model recreated from (Brown, 2005)
In 1902, Ritterโs truss model was refined by Mรถrch who suggested that the discrete diagonal
forces used in Ritterโs truss should be replaced with a continuous field of diagonal compression.
(Brown, 2005) (Figure 2.6).
Figure 2.8: Ritterโs truss refined by Mรถrch copied from (Brown, 2005)
The truss model was studied by Talbot (1909) who found that the model ignored the tensile
strength of concrete which is an important factor in shear resistance. Later, Richart (1927)
further studied the truss model and developed a method for shear design of the beam. This
method considers separately the effect of reinforcement (Vs) and the concrete (Vc) with regard
to shear resistance and sums them up to find the total shear resistance (Vc+Vs) (Brown, 2005)
Nevertheless, the truss model was limited to concrete beams. In 1987, Marti and Schlaich
through a gradual work extended the truss model to a strut and tie model applied to all types of
concrete structures. Their concept was that a complex structure could be simply divided into
regions of continuity and dis-continuity. Then, using basic tools and techniques, the design is
performed based on the behavior model of a structure. In other words, Marti and Schlaich
introduced a design concept which is consistent for all types of structures (Brown, 2005).
After this work, the strut and tie method started to appear in several codes as an accepted design
approach. First, it appeared in 1984 in Canadian standard (CSA, 1984) for shear design of D-
THEORETICAL BACKGROUND
22
regions. Afterwards, (AASHTO, 1989) included it in its specifications for segmental guide in
1989 and in bridge design specifications in 1994. The next was CEB-FIP Model code to include
strut and tie method in 1990 as an alternative to analyze the problems in D-regions. In 2002,
ACI building code (US structural concrete code) embraced strut and tie method and modified
parts of the code to make room for the use of (ST) method. (Brown, 2005). In 2004, Eurocode
2 embraced (ST) method in two sections of analysis and design (section 5.6.4 and section 6.5
subsequently).
Strut and tie method design based on European
structural concrete code
2.4.1 Definition
Strut-and-tie (ST) is a method based on lower bound theory of plasticity used to design
reinforced concrete structures which are in D- regions. Strut and tie (ST) method considers all
load effects (M, N, V, Tm) simultaneously and reduces complicated states of stress in a structure
to a number of simple stress paths. Each stress path is represented with truss members loaded
with uniaxial stress (compression or tension) parallel to the axis of stress path. The compressed
truss members are called struts and the tensioned members are called ties. The point where
struts and/or ties intersect is called nodes. The collection of struts and ties and nodes is called a
strut and tie model (Brown, 2005).
Figure 2.9: Complex stress state in deep beams simplified as strut and tie model recreated
from (Schlaich et al., 1987)
CHAPTER 2
23
In a (ST) model, the forces are in the form of pure tension or compression and can be determined
using laws of static if the model is statically determinate. After determining the forces in the
strut and tie model, only the stresses within the struts, ties and nodes are compared to allowable
stresses. Meanwhile, reinforcements could be provided to resist the tensile stresses in portions
of the model influenced with tensile stresses or to add additional strength and confinement
required by the ties (Brown, 2005).
Figure 2.10: Reinforcement position in a strut and tie model for a deep beam recreated from
(Schlaich et al., 1987)
2.4.2 Struts
The compressive forces in a Strut and tie model are carried by the struts. The bearing capacity
of a strut is influenced from the multi axial state of stress that a strut goes through. Meaning the
capacity of strut increases with transverse compressive stresses because of triaxial compression
and decreases with presence of transverse tensile stresses. For each of the cases, the
corresponding design strength are presented below:
A) Stronger strut where with no transverse tensile stresses are present:
๐Rd.max = fcd 2.1
Where;
๐Rd.max- design strength
THEORETICAL BACKGROUND
24
Figure 2.11: Recreated form (Eurocode2, 2004)
B) Strut with lower strength where the transverse tensile stresses are present:
๐Rd.max = kโฒ โ vโฒ โ fcd 2.2
๐โฒ = 0.6 2.3
vโฒ = 1 โfck
250
2.4
Where;
๐Rd.max - design strength
k'- is a safety coefficient to cover for worst case condition for multiaxial stress in strut.
vโฒ - factor based on national annex
The reduction in the compressive strut is difficult to quantify because it primarily depends on
the direction of the tensile stresses in the strut. The worst-case scenario is when the tensile
stresses are not perpendicular to the strut in which case the compressive force is carried in shear
across the cracks. (AASHTO, 1989) relates the compression in the strut to the principle tensile
strain and its direction. However, it is not always practical to know about principle strain in the
structure. Therefore, the Eurocode 2 uses the most conservative values for the design strength
of struts that covers all situations (Hendy & Smith, 2007).
Figure 2.12: Recreated from (Eurocode2, 2004)
CHAPTER 2
25
According to the shape of stress field, the struts are classified into three types. The prism, the
fan shaped, and the bottle shaped.
The prism is the simplest forms of struts which has a uniform cross section over its length. An
example for such a case is the compressive stress block of a beam in a section of constant
moment (Figure 2.13 a) (Brown, 2005).
The second is the fan shaped struts (compression fun). (Figure 2.11 c). A fan shaped develops
when the stresses flow radially from a large area to a much smaller one. An example for such a
case is when large distributed loads flow into supports. Within a fan shaped strut, there are no
tensile stresses because the forces are co-linear (Figure 2.13 c) (Brown, 2005).
Third is the bottle shaped struts (Figure 2.13 b). These struts are characterized with the stresses
that are not confined to a portion of a structural element. These struts are formed when the force
is applied to a small zone and the stresses disperse as they flow through the member. The
dispersed stresses form an angle to the axis of the strut. The angled stresses have two
components. To counteract the lateral component of the inclined compression stress, a tensile
force is developed. To model a bottle shaped truss, a number of struts and ties are required to
compensate for the tensile force (Brown, 2005).
THEORETICAL BACKGROUND
26
Figure 2.13: Types of struts recreated from (Brown, 2005)
2.4.3 Ties
The tensile forces in a strut and tie model are carried by ties. The position of the ties coincides
with the central gravity axis of reinforcements therefore ties have very simple Geometry. The
design force for the ties in ULS, considering the bars are anchored, is the yield strength ๐๐ฆ๐ of
the steel (Hendy & Smith, 2007):
๐ โค ๐๐ฆ๐ โ ๐ด๐ 2.5
Where:
T- Tension force in the model
๐๐ฆ๐- Yield strength of steel
๐ด๐ โ Cross-sectional area of steel reinforcement
Reinforcement ties may be discrete or smeared. For example, in case of pile caps, the
reinforcement tie placed at the bottom of pile cap is discrete whereas the transverse
reinforcement place in the web due to budging of the strut is called smeared. If smeared, the tie
should be distributed over the length of the tension zone (Figure 2.14) (Brown, 2005).
CHAPTER 2
27
Figure 2.14: Compression field with smeared reinforcement in regions of partial and full
discontinuity recreated from (Eurocode2, 2004).
2.4.4 Nodes
A node is a simplified idealization of reality which represents the areas where struts and ties
intersect. In a strut and tie model, a node represents a point where an abrupt change in direction
of forces occur. In reality, this change occurs over a certain length and width. Considering this
fact, if any of either strut or tie components represent concentrated stress field, the node is called
singular (or concentrated). On the other hand, if the struts representing wide concrete stress
field and/or ties representing a number of closely distributed bars intersect each other, the node
is called smeared (or continuous) (Schlaich et al., 1987).
In case of pile cap, the nodes which are on top of piles or the nodes located directly under the
column are singular nodes, all the rest are smeared nodes. The smeared nodes are not critical
and do not impose any problems. Whereas the singular nodes are places of stress concentration
in concrete and need to be checked. There are three types of singular nodes, CCC-node, CCT-
node, and CTT- node (C stands for compression and T for Tension) (Figure 2.15).
THEORETICAL BACKGROUND
28
T
T
C
C
T
CC
C
C
CCC-node CCT-node CTT-node
Figure 2.15: Types of singular nodes
Based on (Eurocode2, 2004), the maximum allowable stress ๐๐ ๐.๐๐๐ฅ in nodes are determined
using below equations:
a) All members in a node are in compression (CCC-node)
๐๐ ๐.๐๐๐ฅ = ๐1 ๐ฃโฒ ๐๐๐ 2.6
where;
k1 โ nationally determined parameter whichโs value is 1.0
vโ โ nationally determined parameter whichโs value is recommended to be:
๐ฃโฒ = 1 โ๐๐๐
250
b) One member in a node in tension, others in compression (CCT-node)
๐๐ ๐.๐๐๐ฅ = ๐2 ๐ฃโฒ ๐๐๐
2.7
where;
k2 โ nationally determined parameter whichโs recommended value is 0.85
c) Two members in a node in Tension formed by bent bar, others in compression
(CTT-node)
๐๐ ๐.๐๐๐ฅ = ๐3 ๐ฃโฒ ๐๐๐ 2.8
๐๐ ๐.๐๐๐ฅ = ๐3 ๐ฃโฒ ๐๐๐
where;
k3 โ nationally determined parameter whose recommended value is 0.75
CHAPTER 2
29
Shear
If an arbitrary plane is passing through a body, the force which acts along this plane is called
shearing force ( (Nash & Potter, 2010). Generally, two types of shear failure are differentiated:
linear shear and punching shear. Concerning linear shear, there are other subtypes including
diagonal tension or compression shear for short shear spans and flexural shear for longer shear
spans. For a pile cap, which is a thick concrete structure with a small shear span, usually the
linear shear is dominant failure mode. Punching shear, however, occurs if the pile cap is very
slender in which a concrete cone separates from the slab under the concentrated column load.
Linear shear and punching shear are also known as one way and two ways shears. For the
purpose of this thesis, linear shear is studied in depth in order to understand the behavior of
concrete in shear failure. Punching shear, on the other hand, is not studied because the pile caps
considered in this thesis have higher thickness and punching is not a problem.
2.5.1 Shear force in a beam
In a simply supported beam in uncracked state, shear forces are induced due to the variation of
moment forces along its length. This variation results into principle stresses which are inclined
to the natural axis (Figure 2.16).
Figure 2.16: Principle stresses in an un-cracked concrete beam- red lines representing
tension and blue lines representing compression recreated from (Chantelot & Mathern, 2010)
The shear stress according to theory of elasticity becomes maximum at the neutral axis level
and zero at the surfaces of the beam. On the contrary, the normal forces become maximum at
the top and bottom surface level and zero at the neutral axis level (Figure 2.17).
THEORETICAL BACKGROUND
30
Figure 2.17: Shear and normal stresses in a rectangular cross-section based on theory of
elasticity recreated from (Chantelot & Mathern, 2010).
However, a simplified representation of the above shear and normal stress distribution which
is based on beam theory is generally accepted (Figure 2.18). The beam theory is based on:
a) Saint-Venant principle: the condition of stress in a given point located away from the
load application point depends only on the resultant of moment and forces in that
point.
b) Bernoulli hypotheses: even after deformation, plain cross section remains plain.
Figure 2.18: Shear and normal stresses in a rectangular cross-section based on beam theory
recreated from (Chantelot & Mathern, 2010)
Based on the above distribution of stresses, the shear stresses ฯxy at distance y from the
neutral axis, can be found using:
๐๐ฅ๐ฆ =๐๐ฅ โ ๐๐ฆ
๐ โ ๐ผ๐ฆ
2.9
CHAPTER 2
31
where;
Vx - is the applied shear force
Sy - is the static moment with respect to the neutral plane
Iy - is the moment of inertia with respect to neutral plane and
b- is the width of the cross-section.
The direction of principal normal and shear stresses in any point in the beam is determined
by the angle ฮธ which is found using Mohrโs circle.
Figure 2.19: a) Strains in an arbitrary point in a beam recreated from (Chantelot & Mathern,
2010) b) Direction of strains based on Mohrโs circle recreated from (Chantelot & Mathern,
2010)
In Figure (2.19);
ฯต1 - principle tensile strain.
ฯต2 - principle compressive strain.
ฮธ โ is the angle determining direction of principle compression
Figure 2.18; represents the Mohrโs circle.
2.5.2 Shear cracks
In Figure 2.16, the tensile strains in the beam are shown as red tensors. As soon as the strains
exceed the deformation limit of concrete, cracks start to appear in the beam. The first crack
THEORETICAL BACKGROUND
32
appears in locations where cracking moment Mcr is reached first. These cracks are vertical with
ฮธ close to 90 degrees as shown in (Figure 2.19). With additional load steps, new cracks are
formed near support which some of them propagate in the compression zone and bend in the
direction of the load. The direction of these cracks depends on the value of ฮธ (Figure 2.19).
Mainly, two types of shear failure occur in the beams; a) flexural shear failure and b) web shear
failure. A flexural shear failure occurs when a diagonal crack initiates from reinforcement level
and propagates towards the compression zone and flattens out. This is the dominant failure
mode for beams with normal reinforcement loaded in bending (Ansell & Hallgren et al., 2017)
(Figure 2.20).
Figure 2.20: Types of shear cracks in a beam recreated from (Engstrรถm, 2004)
On the other hand, web shear failure can be caused due to compressive stresses or tensile
stresses in beams. The failure due to tensile stresses occurs when a beam is subjected to very
high shear force. In this case, the principle tensile stresses in the middle of the beam in the
vicinity of neutral axis becomes greater than the tensile strength of concrete resulting into
diagonal cracks. These cracks are typical to pre-tensioned beams. (Ansell & Hallgren et al.,
2017)
The web shear failure due to compressive stresses occur in members with either high shear
reinforcement or stocky structures such as deep beams or pile caps. For determining the forces
in these structures, the truss analogy or the strut and tie method is used. Based on these methods,
the shear force is taken by the compressive struts and failure occurs when the compressive
stresses exceed the compressive strength of concrete.
The web shear failure is difficult to observe because the failure could occur inside the web of
the beam. Another characteristic of web shear failure is that it does not occur in combination
with any flexural cracks. In other word, no flexural cracks are present in the beam.
2.5.3 Shear transfer mechanism
Shear resistance in a structural element could be enhanced with transverse reinforcement
(stirrups). However, a beam without shear reinforcement too has a certain degree of shear
CHAPTER 2
33
resistance capacity. In a beam, the flexural shear capacity is built by; a) shear stresses in
uncracked concrete; b) aggregate interlock mechanism; c) residual tensile stress d) dowel action
caused by longitudinal reinforcing bars (Yang, 2014). Other mechanisms affecting shear
resistance in reinforced concrete elements are; e) shear slenderness (arch action); f) concrete
strength; g) reinforcement content; h) cross-section height; i) longitudinal reinforcement bond
(Ansell & Hallgren et al., 2017).
Figure 2.21: Shear transfer mechanisms recreated from (Yang, 2014)
a) Shear stresses in uncracked concrete zone:
When the cracks appear in concrete beam, the reinforcement takes the whole tensile forces.
However, there remains uncracked concrete parts between two adjacent cracks. This part is
called lamella and it functions as an uncracked beam in elastic condition and can take certain
amount of shear stresses (Ansell & Hallgren et al., 2017).
b) Aggregate interlock mechanism:
Concrete is a inhomogeneous material made from several materials with variety of sizes.
Whenever an inclined crack is propagated, the concrete surfaces on the two side of the crack
are not plane. In contrary, they are very rough and having irregular shape held together by
longitudinal reinforcement. The connectivity between the two surfaces in a crack creates a
friction which contributes in taking shear stresses. The amount of shear stresses that can be
taken by aggregate interlock mechanism depends on the gravel size and how wide has the crack
opened (the more the longitudinal reinforcement, the less crack opening) (Yang, 2014).
c) Residual tensile strength:
The cracked concrete can still carry certain amount of tensile stresses given that the crack width
is around (0.1mm). The tensile ties which are formed across the cracks and can carry shear
stresses (Yang, 2014).
THEORETICAL BACKGROUND
34
d) Dowel action
In a beam, just before the tensile cracks starts to appear, the concrete is taking the limiting
tensile stresses (ft). As the crack appears, the reinforcement takes the tensile stresses and bends.
At this point the adjacent concrete parts want to slip over the reinforcement in vertical direction
but it is resisted by the longitudinal bars. This resistance in vertical direction is called dowel
force and it transfers a certain amount of shear stresses.
e) Influence of shear slenderness - arch action:
(Leonhardt & Walther, 1962) conducted a series of tests on a number of simply supported
beams with same cross-sectional properties loaded in shear with two concentrated forces. The
only parameter varying in the beams was the slenderness ratio (a/d), where a is the distance of
support from the load application point and d is the effective depth. All beams were loaded until
failure. The beams with (๐๐โ โฅ 3), showed almost the same shear failure load. Beams with
(๐๐โ = 7 ๐๐ 8) showed a flexural failure due to high bending moment. And beams with (๐
๐โ โฅ
2.5) showed a higher resistance in shear with decreasing ๐ ๐โ (Ansell & Hallgren et al., 2017).
(Figure 2.21)
Figure 2.22: Beam loaded with two concentrated loads tested for shear slenderness recreated
from (Ansell & Hallgren et al., 2017)
The reason why the short beams have better shear resistance capacity is due to arch action. The
redistribution of forces in the beam occurs after a shear crack appears. For the beams with
(๐๐โ โฅ 1 ๐๐ 1.5), the whole force is transferred through arc action and stress in reinforcement
is constant. The only concern is that the reinforcements need to be anchored. Arching action is
especially pronounced for beams, slabs and foundations with short shear spans. The failure
mode is then usually shear compression at the supports or in the web. (Ansell & Hallgren et al.,
2017).
f) Influence of concrete strength:
In a beam without shear reinforcement, it is considered that the shear forces are carried through
concrete. And thus the higher the strength of concrete the better resistance in shear. However,
it is natural that tensile strength plays a more important role since shear is related to tensile
cracking in concrete. Eurocode 2 presents an accurate model which relates the compressive
CHAPTER 2
35
strength and shear resistance of concrete. According to the model, the compressive strength
raised to the power of 1/3 gives an accurate prediction of shear strength (Ansell & Hallgren et
al., 2017).
g) Influence of reinforcement content:
The positive influence that the longitudinal reinforcement content has in flexural shear
resistance can be described in three points (Ansell & Hallgren et al., 2017):
a) The compression zone area helps in resisting the shear force. More reinforcement
content increases the height of compression zone which increases the shear resistance
capability.
b) It resists the diagonal crack opening.
c) The dowel force i.e. resistance in vertical direction is increased.
h) Influence of cross-section height:
Considering the equation:
๐๐ฃ = ๐๐ โ ๐โ 2.10
the shear resistance of a cross section with increasing height is supposed to increase, - but in
the contrary, it decreases. Based on statistical data and fracture mechanics, the higher the height
of a concrete specimen, the lesser strength it has. Furthermore, regarding flexural shear failure,
with increased beam height the wider cracks propagate due to decreased friction in shear links
of the concrete material. It can be concluded that the shear resistance decreases with increasing
cross section height (Ansell & Hallgren et al., 2017).
i) Influence of bond in flexural reinforcement:
Considering the load carrying capacity in a beam, the smooth re-enforcement gives a higher
load bearing capacity in shear. However, the smooth reinforcements need to be anchored and
the failure mode in this way is very brittle with a few very large cracks. With ribbed
reinforcement, the failure load is lower, but the cracks formed are very fine. In practice,
however, too many fine cracks are preferred compared to a few large cracks. In addition, ribbed
reinforcement bars donโt require to be anchored as well (Ansell & Hallgren et al., 2017).
2.5.4 Design according to EC2.
Flexural shear failure: The shear resistance of members with flexural cracks without shear
reinforcement is found using the empirical equations (2.10). The equation considers various
mechanisms that contribute to shear resistance.
๐๐ ๐,๐ = {[๐ถ๐ ๐,๐ โ ๐. (100 โ ๐๐ โ ๐๐๐)^1
3] + ๐1 โ ๐๐๐} ๐๐ค โ ๐ โฅ (๐ฃ๐๐๐ + ๐1 โ ๐๐๐)๐๐ค โ ๐
2.11
THEORETICAL BACKGROUND
36
๐ถ๐ ๐,๐ =0.18
๐พ๐ is a coefficient dependent on loading case
๐ = 1 + โ200๐โ โค 2.0 (k) is the size factor (d) is the effective depth in (mm)
d- is the effective depth of slab in (mm)
๐๐ =๐ด๐ ๐
๐๐คโ๐โค 0.02 is the longitudinal reinforcement ratio
๐ด๐ ๐ is the area of tensile reinforcement extending โฅ ๐๐๐ + ๐
๐๐ค is the smallest width of cross-section in (mm).
๐๐๐ concrete characteristic compressive strength (cylinder) in (MPa).
๐1= 0.15 set as per (BFS, 2011)
๐๐๐ is the compressive stress in concrete from axial load in (MPa).
๐ฃ๐๐๐ = 0.035 โ ๐3
2โ โ ๐๐๐
12โ is the minimum resistance stress in (MPa).
The minimum shear resistance where no longitudinal reinforcement is present and the shear
resistance is provided by concrete, can be found according to:
๐๐ ๐,๐ = ๐ฃ๐๐๐ โ ๐๐ค โ ๐ 2.12
For calculating the shear stresses in pile cap, the smallest shear resistance width ๐๐ค should be
carefully selected. For loads close to support, a reduction factor ฮฒ is considered:
๐ฝ = {
๐๐ฃ
2๐, ๐๐๐ 0.5 โ ๐ โค ๐๐ฃ โค 2 โ ๐
0.5 โ ๐
2 โ ๐, ๐๐๐ 0.5 โ ๐
}
2.13
Web shear failure: For the web shear failure due to transverse tensile stresses in the web, the
shear capacity is found using equation:
๐๐ ๐,๐ =๐ผ โ ๐๐ค
๐โ โ๐๐๐ก๐
2 + ๐๐๐ โ ๐๐๐ก๐ 2.14
For the web shear failure due to crushing of compressive strut in the middle of the member,
the shear resistance according to EC2 is;
๐๐ธ๐ โค 0.5 โ ๐ฃ1 โ ๐๐๐ โ ๐๐ค โ ๐ 2.15
where v1 is a factor of reduction due to cracks in concrete;
๐ฃ1 = 0.6 (1 โ๐๐๐
250)
2.16
CHAPTER 3
37
Finite element analysis
Theory of finite element
The finite element method (FEM) or finite element analysis is a numerical method for obtaining
an approximate solution to any given field problem which are mathematically described by
differential equations or integral expressions (Cook et al., 2002). When comparing different
numerical methods, FEM has numerous advantages over other numerical methods namely, no
restrictions for geometry, loading and boundary conditions. Due to versatility of FEM, one can
combine various components with different mechanical behavior (bar, beam, shell, cable, ..etc.)
in one general FEM model (Cook et al., 2002).
In FEM, a FE-model representing a structure is created by first simplifying the real structure to
a mathematical model. Through discretization, the mathematical model is converted to a FE-
model which consists of small elements. These small elements are connected at points called
nodes. Each node has a certain degrees of freedom (DOF) in the form of displacement and
rotation. The specific ways that the elements are arranged is called mesh. The process of
creating an FE-model can be seen in (Figure 3.1).
Figure 3.1: Process of simplification and discretization of a structure into a FEM model
recreated from (Cook et al., 2002)
FINITE ELEMENT ANALYSIS
38
According to (Malm, 2016) a finite element model is created by;
a) defining geometry of the structure and discretization
b) assigning material properties
c) adding loadings, boundary conditions, and prescribed deformations
Finally, an appropriate mesh is assigned to the model corresponding to the response of the
structure.
FEA is used for solving both linear and non-linear problems. Nevertheless, this masterโs thesis
is based on non-linear finite element analysis (NLFEA) using ATENA. Therefore, the theory
henceforward will be focused on describing features relating to NLFEA.
Non-linear finite element analysis
For many practical design matters, linear models provide satisfactory results. In a linear
analysis, the parameters such as geometry and material properties are set as constant. For
instance, in case of the basic load, displacement and stiffness equation.
[๐พ]{๐} = {๐น}
The displacement is linearly related to load through the constant stiffness matrix. However,
there are certain cases, where these parameters become functions of the model. In this case the
analysis becomes non-linear. Other examples of non-linear behavior in the realm of structural
analysis are; local buckling, yielding, creep, opening of cracks etc. (Cook et al., 2002).
According to (Cook et al., 2002), there are three types of non-linearity:
Material non-linearity, where material properties in a model are functions of state of stress
and strain e.g. nonlinear elasticity, plasticity and creep
Contact nonlinearity, where gap between adjacent parts may open or close, the contact area
between parts changes as the contact force changes, or there is sliding contact accompanied
with frictional forces.
Geometric nonlinearity, where the geometrical deformation is large enough that equilibrium
equations must be written with respect to deformed structural capacity.
For the specific study concerning this thesis, the vector load{R}and the stiffness matrix [K]
become the functions of {d}displacements. In this case the relationship between load and
displacement is not linear and the displacement cannot be determined immediately from load
and stiffness. Meanwhile the displacement needs to be found using an iterative procedure. A
procedure where the stiffness matrix and the load both need to be iterated.
CHAPTER 3
39
3.2.1 Iterative procedure
In a linear analysis, the load {๐น} and displacement {๐} are linearly related through a constant
stiffness matrix [๐พ]. Therefore, it is possible obtain {๐} directly. In a non-linear analysis,
however, the equation cannot be solved directly. The reason for this is that the displacement is
not proportional to the load (stiffness matrix is not constant); thereby, an iterative procedure
must be used to obtain the solution. In other words, the final load is divided into small
increments and gradually increased up to final load level. Consider the load level P and the
small load increment โP. To determine the nonlinear response of the structure, the tangential
stiffness K0 is used. K0 is based on stiffness of the structure at the previous load step (u0).
Through extrapolation, a displacement correction (ca) is found which is then used to update the
displacement of the structure from u0 to ua. ua is used to find the corresponding load level Ia. By
subtracting Ia from the final load P, the residual load for the iteration is found. (Malm, 2016)
(Figure 3.2).
Figure 3.2: Iteration of an increment in nonlinear finite element analysis recreated from
(Malm, 2016)
If the residual force is equal to zero, the load level P would co-inside with the load-displacement
curve and the equilibrium would be satisfied. However, in a non-linear analysis the residual
force is never zero. Mostly the goal of the iterative procedure is to achieve a predefined
tolerance criterion (0.5 % for instance). If the value for Ra is smaller than the tolerance level,
the equilibrium is satisfied. However, if the value for Ra is bigger than the tolerance criteria, the
procedure is performed again (Malm, 2016).
FINITE ELEMENT ANALYSIS
40
3.2.2 Crack opening laws and fracture mechanics
Crack opening in concrete is usually explained by the laws of fracture mechanics. According
to fracture mechanics, three failure modes can occur in concrete;
Mode I tensile
Mode II shear
Mode III tear
Failure mode I is the only failure that occurs in practice. i.e even the shear failure starts as a
tensile failure where the maximum principle stresses in the concrete becomes higher than the
tensile strength of concrete. Figure 3.3 illustrates the mechanism for stress distribution near
the tip of a tensile crack. As can be seen in the figure, the total crack length consists of macro
crack length a0 and fracture zone length . Moreover, the crack width is shown as (w), and
macro crack width as (wc). A macro crack is visible by eye and has a width of โฅ 0.1 mm.
Before a crack reaches the width of 0.1 mm, it is considered as micro crack and is located in
the fracture zone. The stress (ฯw) is zero at the transition point between macro crack and
fracture process zone and maximum (ฯw,max= ft) at the crack tip (Malm, 2016).
Figure 3.3: Stress distribution in crack tip for failure mode I, recreated from (Hillerborg et.
al, 1976)
CHAPTER 3
41
3.2.3 Smeared crack approach
There are two ways to describe the cracking phenomena in concrete; discrete crack approach
and smeared crack approach. In discrete crack approach, the crack position is known
beforehand and therefore an interface element is introduced in the part where the crack is
expected to appear. In smeared approach, however, the position of the crack is not known and
therefore the crack is smeared over the whole element (Malm, 2016). Since its very difficult
to model the formation of cracks in a large structure such as pile caps, therefore only the
smeared approach is studied for this masterโs thesis.
In smeared approach, the behavior of uncracked concrete and the behavior of crack are
illustrated by one element. Therefore, the strain in an element is the result of elastic part
(uncracked concrete) and nonlinear part of crack opening (Malm, 2016).
๐๐ก๐๐ก๐๐ = ๐๐๐๐๐ ๐ก๐๐ + ๐๐๐๐๐๐.
Considering this, two different models based on smeared approach are differentiated: fixed
crack model and rotated crack model. For each of the models, the cracks are formed when the
maximum principle stress exceeds the tensile strength of concrete. However, the results for each
model are different (Cervenka et al., 2018).
3.2.4 Fixed crack model:
In the fixed crack model, the initial direction of a crack is the same as maximum principle
stresses and does not change after further loading. And since the direction of the crack does not
change with the change in principal stress direction, this would give rise to shear stresses at the
surface of the crack. However, if orthogonal change in the direction of stresses occur, secondary
cracks would give rise at the same integration point. In a 3D problem, a maximum of 3 cracks
can propagate in the same integration point (Malm, 2016).
Figure 3.4: Fixed crack model recreated from (Malm, 2016)
FINITE ELEMENT ANALYSIS
42
3.2.5 Rotated crack model:
In the rotated crack model, the direction of crack will always change according to the direction
of maximum principle tensile stress. Therefore, shear forces at the surface of the crack does not
arise i.e. the crack will rotate and arrange its direction normal to principal stresses. The only
stresses present in the plane of a crack would be maximum tensile stresses and maximum
compressive stresses. (Malm, 2016)
Figure 3.5: Rotated crack model recreated from (Malm, 2016).
3.2.6 Tensile behavior:
The tensile behavior of concrete is explained by the uniaxial tests and consist of two stages: the
linear elastic part and non-linear part. At the beginning of loading, micro cracks appear due to
poor bond between cement and concrete. The cracking process continues until the concrete
tensile strength limit fct is reached due to loading. Before reaching fct the response of the
concrete is linear i.e. if the loading is removed, the concrete will have a small number of cracks
and small residual deformation. However, if the maximum stress level fct is passed, the stiffness
reduces and cracks growth goes out of control and with constant maximum stress, many cracks
appear until they are connected and form a macro crack which is stress free. As can be seen in
figure (3.6), the elastic part of the curve is represented with ฯ-ฮต (stress-strain) whereas the non-
linear (descending) part is defined with ฯ-w (stress-crack opening displacement) (Malm, 2016).
CHAPTER 3
43
Figure 3.6: Crack propagation in concrete at uniaxial tensile loading recreated from (Malm,
2016)
3.2.7 Fracture energy:
Crack opening displacement (w) is related to a factor called fracture energy (Gf) which is
represented by the area under curve of the descending part. Basically, the fracture energy (Gf)
is defined as the amount of energy necessary to create one unit area of a crack (Hallgren, 1996).
For a concrete with known material composition, the value for fracture energy (Gf) is normally
determined from uniaxial tension testing. Factors which has influence in the fracture energy
are; maximum aggregate size, concrete age and water cement (w/c) ratio (MC10, 2012).
However, if testing is not available, the value for Gf according to (MC10, 2012) for a normal
strength concrete is determined from equation:
๐บ๐ = 73 โ ๐๐๐0.18 3.1
Where:
fcm is the concrete mean compressive strength in (MPa)
In the earlier version of CEB/FIP Model Code (MC90, 1990), the fracture energy was related
to largest aggregate size and concrete grade as shown in Table 3.1 and equation (Malm, 2016).
Table 3.1: Fracture energy for different concrete grades and aggregate sizes as per (MC90,
1990), (Malm, 2016):
Gf (N/mm)
Dmax C12 C20 C30 C40 C50 C60 C70 C80
8 40 50 65 70 85 95 105 115
16 50 60 75 90 105 115 125 135
32 60 80 95 115 130 145 160 175
FINITE ELEMENT ANALYSIS
44
In addition, based on the descending part of the stress-crack width curve shown in Figure 3.6
and with the curve shape coefficient used in ATENA, the fracture energy (Gf ) is directly related
to the critical crack width (wc) through below equation:
๐ค๐ = 5.14 โ๐บ๐
๐๐ก
3.2
Equation 3.2 presents the value for exponential crack opening curve. Linear and bilinear curves
could also be used to define crack opening (Malm, 2016):
For linear:
๐ค๐ = 2 โ๐บ๐
๐๐ก
3.3
For bilinear:
๐ค๐ = 3.6 โ๐บ๐
๐๐ก
3.4
An increase in fracture energy (Gf) or a decrease in tensile strength (ft) increases the value for
critical crack width (wc). The higher the value for (wc), the more energy is needed to propagate
macro crack. The crack width is not directly dependent on wc. However, a larger wc will give
more ductility as the crack will be able to carry stress for a larger deformation.
Modelling simplifications and assumptions in
ATENA 3D
After reviewing the drawings of the old building in Stockholm, it was found that there were a
variety of pile foundations with different sizes and dimensions used. Therefore, it was decided
to select a single pile cap model by referring to engineering codes and handbooks. Finally, the
dimensions of the models were adjusted according to engineering handbook by (Reynolds &
Steedman) (detailed calculations are in section 6.1).
For modelling, two software were used; ATENA engineering 3D (v.5) and ATENA studio
(v.5). ATENA engineering was used in the pre-processing stage where the material properties,
geometry, element types, loading, boundary conditions and mesh were introduced. Later, the
model was run using ATENA studio where the analysis was completed, and the required results
were extracted.
CHAPTER 3
45
The following assumptions and simplifications were considered while creating the model:
a) Only vertical compressive force affects the pile cap i.e. lateral forces and moments
were not considered.
b) The stiffness of all the piles supporting the pile cap are equal and therefore the force
from column is equally divided between the four piles. Considering this assumption,
the pile cap is double symmetric i.e. only one fourth of the pile cap was modelled.
(Figure 3.8)
c) The pile cap was studied in isolation from structure and soil beneath, i.e. only short
column and short pile were considered for the model.
d) No transverse shear reinforcements were provided.
Figure 3.7: Full pile cap model created in ATENA 3D
Figure 3.8: Symmetric model created in ATENA 3D
FINITE ELEMENT ANALYSIS
46
Material properties definitions
To define material properties in ATENA 3D, there are three approaches:
a) direct; where one can select from the available material models in ATENA and also make
changes to parameters if required.
b) properties from file; where one can use material models created earlier.
c) properties from catalogue, where materials are defined based on standards, mainly EC2.
For the models created for the purpose of this thesis, all the materials were defined using direct
approach. In general, five materials were defined for the model: concrete, steel, reinforcement,
contact and springs.
3.4.1 Concrete material models
In the model created in ATENA, all the concrete parts are modelled with non-linear material
properties. The non-linear concrete material is named as CC3DNonLinCementitious which is
based on a fracture-plastic model. The significance of the fracture-plastic model is that it
captures the tensile and compressive behavior of the concrete at the same time and therefore
can simulate crushing, splitting, crack opening and closure very well.
Uniaxial failure criterion
To characterize the uniaxial behavior of concrete, the material CC3DNonLinCementitious in
ATENA 3D, uses the uniaxial stress-strain law.
Figure 3.9: Uniaxial stress-strain used by CC3DNonLinCementitious in ATENA. recreated
from (Cervenka et al., 2018)
CHAPTER 3
47
Bi-axial failure criterion
The bi-axial state of stresses is different than that of uniaxial and is presented by a failure
envelop. The yielding of material occurs when the state of stresses reaches the boundary of the
envelope. To depict the bi-axial behavior, the material model in ATENA
(CC3DNonLinCementitious), uses the failure criteria suggested by Kupfer in (Kupfer et al.,
1969).
Figure 3.10: Biaxial failure criteria represented by an envelope recreated from (Cervenka et
al., 2018)
As it can be seen from the envelope, the compressive strength increases in case of bi-axial
compression and the in the mix state of stresses (compression and tension) the strength reduces.
The confinement effect in concrete is the reason for increase in case of bi-axial compression
and is said to increase the strength up to 16 % (Malm, 2016).
Triaxial failure criterion
In the tri-axial state of stresses, the concrete compressive strength increases considerably more
compared to bi-axial state (Malm, 2016). To employ the triaxial failure criterion, the model in
ATENA (CC3DNonLinCementitious) utilizes separate models for cracking and crushing in
concrete. The cracking is presented by Rankine fracturing model where the strains and stresses
in a structure are adapted to the direction of material. Crushing is presented by plasticity model
whichโs failure criterion is based on work by (Menetrey & William, 1995). The surfaces in
triaxial state of stresses resemble the shape of a cone and are related to eccentricity factor (e).
The value of (e) can vary between 0.5 (more circular edges) up to 1 (more triangular edges)
(Menetrey & William, 1995).
FINITE ELEMENT ANALYSIS
48
Figure 3.11: Triaxial state of stresses recreated and modified from (Menetrey & William,
1995)
3.4.2 Reinforcement model in ATENA
There are two approaches to model reinforcement in ATENA 3D; smeared approach and
discrete approach. In smeared approach, the reinforcement ratio is smeared over an element
whereas in discrete approach the reinforcement bars are introduced into the model. For the
models in the thesis, the discrete approach was used. The behavioral properties of the
reinforcement was selected to be bi-linear which corresponds to elastic-perfectly plastic
response. In addition, for further simplification, a perfect bond and anchorage of reinforcement
was assumed. This is normally sufficient for a ULS analysis, provide that the failure does not
depend on the anchorage of the bars. However, it is possible to model the bond between rebar
and concrete with a bond-slip.
Figure 3.12: Elastic perfectly plastic material response of reinforcement corresponding to bi-
linear material properties in ATENA 3D
CHAPTER 3
49
Boundary conditions and loads
The pile cap was modelled in ATENA in such a way that represented a universal frame
consisting of hydraulic jack applying load on the pile cap. Steel plates were used both on
loading area (column top) and supports (bottom of pile). Regarding loading, two methods are
usually used in numerical analysis to define loading; a) load controlled; b) displacement
controlled. For the model analyzed, the latter (displacement controlled) method was used. This
was done on purpose to get a representation of the failure load and post failure response.
Therefore, instead of column force, a prescribed deformation was assigned on the top of column
which increased gradually. Corresponding to the prescribed displacement, ATENA calculated
a reaction force.
Other boundary conditions included symmetry and supports. As discussed previously, the pile
cap model was double symmetric. Therefore, only one quarter of the pile cap was modelled
with fixed boundary condition in two direction x and y respectively. To actualize this in
ATENA, surface supports were used on the face of pile cap, column, and loading plate in two
directions. The other benefit of using symmetry boundary is that you get a stable model where
rotation of the structure does not occur after loading.
The last boundary conditions included support at the bottom of concrete pile which resembles
hard soil layer. In the model, the bottom of the pile was assumed to be fully supported. The
behavior of the pile were, however, stiffer then they are on the site. The reason for a stiff
behavior is because short piles were assumed to reduce the analysis- time. This certainly had
effects on the behavior of the pile cap.
a) b)
Figure 3.13 a) symmetry and support boundary conditions in ATENA 3D
b) prescribed deformation as support in Z direction
FINITE ELEMENT ANALYSIS
50
The details about boundary conditions applied in the model are listed in table (3.2).
Table 3.2: Boundary conditions used in the model:
Structural
element
Boundary
condition
applied
Direction
X Y Z
Pile Cap Symmetry Fixed Fixed Free
Pile Support Free Free Fixed
Column Symmetry Fixed Fixed Free
Mesh and elements
When introducing elements in ATENA 3D, concrete and reinforcement bars are introduced
separately. When meshing is concerned, only the concrete part (solid element) is meshed. No
mesh is defined for the reinforcement bars in the pre-processing stage. As soon as the analysis
starts, the bar elements are counted as embedded elements within the mesh of solid elements.
Therefore, mesh was only introduced for the solid elements in the model. ATENA 3D has three
types of elements; brick, tetrahedron and pyramid (Cervenka et al., 2018).
Since the geometry for the pile foundation model is simple i.e. does not contain any
irregularities such as opening or refinement, therefore, only brick shaped elements were used.
Figure 3.14: Meshed model with brick elements: a) 10 cm mesh size; b) 5 cm mesh size
CHAPTER 3
51
Another factor related to meshing is the mesh size. The mesh size affects the results directly,
i.e. if the mesh sizes are smaller (more elements are used), the results would be of more quality
(better results). However, the mesh size is also related to analysis time, i.e. the more elements,
the more the analysis time. To find an acceptable mesh size for the model a mesh convergence
analysis is performed where three mesh sizes are examined on a model. After obtaining and
studying the results, if the results for two mesh sizes are close enough the larger mesh size is
accepted. Care must be taken, if the mesh size is very large, the model might behave very stiff.
Therefore, the maximum element size in the model must be able to capture the fracture zone
process which can be found using the below equation (Malm, 2016):
๐๐๐๐ฅ โค๐ธ โ ๐บ๐
๐๐ก2
3.5
Mesh convergence analysis is performed in section 4.3
Non-linear iterative solvers
In a non-linear analysis, the solutions cannot be obtained directly. An iterative procedure must
be used to solve the equations (Malm, 2016). ATENA presents two methods to solve non-linear
equations; a) Standard arc length method; b) Standard Newton-Raphson method. It is also
possible to change the solution parameters and define a new parameter based on requirements.
In the arc-length method the displacement and load both are iterated while the solution path is
kept constant. This method is more general when compared to Newton-Raphson, but it is not
useful for all cases i.e. in case of the body force it changes the weight of the structure.
In Newton-Raphson method, a tolerance limit for satisfaction of equilibrium solution is defined.
For this tolerance limit, a constant load increment is introduced for which the displacement is
iterated until equilibrium is reached. Therefore, it is beneficial in cases where the load value
must be met. For a displacement-controlled analysis, Newton-Raphson method is suggested by
(Cervenka et al., 2018). Therefore, for all modelling stages in this masterโs thesis, the Newton-
Raphson method was used. The limitation of this method is though that it is not possible to
capture any possible snap-back behaviour.
FINITE ELEMENT ANALYSIS
52
CHAPTER 4
53
Analysis and results in ATENA 3D
Models created
At first, four models of the pile were created. The models were identical with the only
difference between them being the position of reinforcement and the ratio of reinforcement.
The details of the models are presented in Table (4.1).
Table 4.1: Identification of models with varying reinforcement position and reinforcement
ratio
Model name: Model A Model B Model C Model D
Position of
reinforcement
Top Bottom Top Bottom
Reinforcement area in
each direction of pile
cap
35.6 cm2 35.6 cm2 21.5 cm2 21.5 cm2
ANALYSIS AND RESULTS IN ATENA 3D
54
Figure 4.1: Reinforcement plan and layout in models A and C (placement of reinforcement at
the top of piles)
Figure 4.2: Reinforcement plan and layout in models B and D (placement of reinforcement at
the bottom of the pile cap)
CHAPTER 4
55
Input data in model
4.2.1 Concrete
Since the aim in a non-linear analysis is to observe the failure of the model, therefore, the mean
values for strength of concrete materials were considered.
Table 4.2:Values for concrete material properties:
Material property Unit Pile cap Piles and column
Poissonโs ratio (vโ) - 0.2 0.2
Modulus of elasticity Ecm GPa 33 37
Tensile strength (Ft) MPa 2.7 3.7
Compressive strength (fck. cube) MPa 38 60
Compressive strength (Fck) MPa 30.4 50
Mean Strength (Fcm) MPa 38 58
Fracture energy (Gf) MN/m 67.8 92
Aggregate size: M 0.02 0.02
All the input values in ATENA presented in Table (4.2) are calculated based on Eurocode,
except the values for tensile strength (ft) and fracture energy (Gf). For tensile strength, the
Eurocode 2 uses the following equation:
๐๐ก = 0.3 ๐๐๐2/3
4.1
Whereas ATENA relates the tensile strength to concrete cube strength (fcu):
๐๐ก = 0.24๐๐๐ข2/3
4.2
For fracture energy (Gf) ATENA uses an equation recommended by (E. VOS, 1983) which
relates the fracture energy to concrete tensile strength:
๐บ๐ = 0.000025 ๐๐ก 4.3
Whereas the (MC90, 1990)and 2010 have different criterion for fracture energy as presented
earlier. (MC90, 1990) relates the fracture energy to maximum aggregate size as presented in
Table 3.1 and (MC10, 2012) relates it to concrete compressive strength as seen in equation 3.1.
The effect of different values of fracture energy are studied in the models.
ANALYSIS AND RESULTS IN ATENA 3D
56
4.2.2 Reinforcement:
Reinforcement is defined using bilinear law and the following properties are introduced in
ATENA:
Table 4.3: Values for reinforcement material properties
Material property Unit No of bars in each direction
14 bars in each direction
Elasticity Modulus (Es) (MPa) 210000
Yield strength (fy) (MPa) 500
The connection between the reinforcement and the surrounding concrete was accepted as
perfect i.e. no bond slips were defined.
4.2.3 Interface:
To reduce the over stiff behavior of the numerical model, an interface material was introduced
at the contacts between (column-pile cap) and (pile-pile cap). In ATENA 3D, this material is
called 3D interface and is defined by two types of parameters; a) physical parameters; b)
stiffness parameters. The physical parameters relate to the physical properties of interface such
as; friction, cohesion and tensile strength. The recommended value in a compression only
support for these parameters based on (Cervenka et al., 2018) are:
Tensile strength (ft) = Tensile strength of weaker material at the contact
Friction coefficient (ฮผ) = 0.5
Cohesion (D) = (ฮผ) * (ft)
The stiffness parameters (Knn) and (Ktt) are only for numerical purposes. Each of them has two
sets of values: basic and minimal. The basic value represents closed state (rigid connection) and
the minimal value represents open contact. The recommended values for basic (initial)
stiffnesses according to (Cervenka et al., 2018) can be found using equations:
๐พ๐๐ =๐ธ
๐ก ๐พ๐ก๐ก =
๐บ
๐ก
4.4
Where:
E- is the elastic modulus of weaker material
G- is the shear modulus of weaker material
t- is the thickness of the interface element which is assumed as 0.02 m
The values for residual (minimal) normal and shear stresses are estimated as (initial stiffness *
0.001) (Cervenka et al., 2018).
CHAPTER 4
57
Table 4.4: 3D interface material properties
Material property Unit 3D Interface
Normal Stiffness (Knn) MN/m3 3000000
Tangential stiffness:(Ktt) MN/m3 3000000
Tensile strength (ft) MPa 2.7
Cohesion (D) MPa 1.35
Friction coefficient (ฮผ): - 0.5
Min. normal stiffness (Knn,min) MN/m3 3000
Min. tangential stiffness (Ktt,min) MN/m3 3000
4.2.4 Steel plate:
The steel plates used at the loading and support were defined as linear elastic material with the
below properties:
Table 4.5: Steel plate material properties
Material property Unit Value
Elasticity modulus (Es) MPa 210000
Poissonโs ratio - 0.3
4.2.5 Input file:
After the pre-processing stage was competed in ATENA 3D, the model was saved as an input
file for ATENA Studio. After slight modifications in the input file based on recommendations
by (Cervenka et al., 2018), the analysis was run in ATENA studio.
Results:
4.3.1 Load deflection response
The load displacement diagram is a graphical representation of a collection of points which
correspond to certain load and displacement level. One can observe the response of a structure
with the increasing load level and extract the ultimate failure load and maximum deflection. As
stated previously, all the analysis performed were deformation controlled. A specific attribute
of a deformation-controlled analysis is that there is a drop in the load displacement diagram
after the concrete is cracked. As it can be seen in the load displacement diagram igure 4.3), the
load drop is clearly visible in all the curves. The displacement increment corresponding to each
load step was (0.1mm) in uncracked concrete and (0.02 mm) for the steps where the concrete
was cracked. This, to improve the convergence in the results obtained. Further on, the load in
(load โ displacement) diagram is factored by 4. This because only one-fourth of the pile cap
was modelled using symmetry boundary condition.
ANALYSIS AND RESULTS IN ATENA 3D
58
As can be seen in the load displacement diagram for the four models, in the beginning load
steps, all the curves follow the same path. This is the part where the concrete is uncracked
(elastic region). After the first visible flexural crack appears in pile cap midspan, the inclination
angle of the curves change. How much the curve flattens depends on the bending stiffness of
the individual model. So, it is fair to say that in the beginning load steps, the pile cap models
functioned as a two-way slab. Therefore, Model B has the highest bending stiffness due to
higher reinforcement ratio and higher lever arm. Next to model B is the model A which has the
higher reinforcement ratio. In the same way model D with higher lever arm and model C with
the lower reinforcement and lower lever arm (Figure 4.3).
Figure 4.3: Load displacement diagram for Models A, B, C and D
After the bending cracks appeared, the load displacement curves still follow a smooth path until
the shear cracks appear in the models (circle points on the curves). At this point the first load
drop occurs in the load displacement diagram. A redistribution of stresses happens due to
cracking of concrete and the stresses and strains in reinforcement grow rapidly.
0
1000
2000
3000
4000
5000
6000
0,0 0,5 1,0 1,5 2,0 2,5
Lo
ad (
kN
)
Displacement (mm)
Comparison between top and bottom reinforcement positions
Model A-top reinforcement high ratio
Model B-bottom reinforcement high ratio
Model C-top reinforcement low ratio
Model D-bottom reinforcement low ratio
CHAPTER 4
59
The curves after cracked concrete stage follow an uneven (changing) path. In model A and C
where the reinforcement is placed based on strut and tie method, the pile cap carries a high level
of load until failure. Whereas the models B and D (bottom reinforcement position), the structure
could not carry much after redistribution of stresses and failed at a lower load level. The
difference in load bearing capacity is 1 Mega Newton (23.5 % increase) when reinforcement is
placed on top of pile (Model A) in comparison to (Model B). The same way, in the models with
lower reinforcement ratio, the difference is 0.73 Mega Newton (18.5 % increase) when the
reinforcement is placed on the top (Model C) in comparison to model (D). The failure mode in
models A and C looks more abrupt due to sudden drop of the curve compared to models B and
D where some ductility is observed at the end of load displacement curve.
4.3.2 Crack pattern
Since the analysis was displacement controlled, the crack patterns were studied at increasing
midspan deflection. Table 4.6 presents the crack pattern at increasing percentage of max
deflection of (25 %, 50%, 75%, and 100%). The corresponding maximum crack width (wk, max)
is depicted on the images as well as presented in Table 4.6.
In all the four models (A, B, C and D), the first crack appeared in the mid span at the bottom of
the pile cap (25 % of midspan deflection). At this point, in models A and C (top reinforcement
position) more cracks appeared in comparison to models C and D in the beginning load steps
The second series of cracks that appeared in all models were very steep shear cracks. The cracks
start in the middle of the compressive strut and propagate along the struts towards the nodes.
However, an attribute related to the cracking phenomena in concrete is that when the concrete
cracks the material seeks for new equilibrium. This equilibrium is provided by the
reinforcement. Therefore, the crack pattern has a tendency to propagate in the direction where
the reinforcement is placed. Considering this attribute, in models A and C which are built based
on strut and tie method, the cracks propagated towards the node on top of pile cap. And in
models B and D, the cracks grew not towards the node, but towards bottom of the pile cap
(where the reinforcement is placed).
After the mid-deflection is increased beyond 50 %, the crack propagation and crack width in
models A and C grew steadily along the compressive strut. Until the cracks reached the superior
nodal zone (node below column). The failure occurred in the form of concrete crushing at the
superior node. In models (B and D) however, new shear cracks appeared at the face of the pile
cap which run rapidly across the pile cap at the reinforcement level. With increasing load, the
crack width increased quickly with the widest crack located in the midspan at the reinforcement
level. However, it is not the cracks at the reinforcement level that caused the ultimate failure,
rather it is the tensile stresses perpendicular to the inclined compressive strut inside the pile cap
that caused the failure. The tensile stresses started to appear a number of load steps prior to the
failure, in the middle of the compressive strut, and propagated towards the far corner resulting
in splitting cracks which split the pile cap in two halves.
ANALYSIS AND RESULTS IN ATENA 3D
60
Table 4.6: Crack pattern in models A, B, C and D at increasing max midspan deflection:
Crack pattern propagation at increasing midspan max deflection (deflection at peak
load)
Crack pattern at 25 %
of maxmid-deflection
Crack filter wc โฅ
0.01mm
Crack pattern at 50 %
of max mid-
deflection Crack
filter wc โฅ 0.1mm
Crack pattern at 75
% of max mid-
deflection Crack
filter wc โฅ 0.1mm
Crack pattern at failure
Crack filter wc โฅ
0.1mm
Figure 4.4: Crack propagation in MODEL A (top placed reinforcement and high reinf.
ratio)
Figure 4.5: Crack propagation in MODEL B (bottom placed reinforcement and high reinf.
ratio)
CHAPTER 4
61
Figure 4.6: Crack propagation in MODEL C (top placed reinforcement and low reinf. ratio)
Figure 4.7: Crack propagation in MODEL D (bottom placed reinforcement and low reinf.
ratio)
4.3.3 Crack width and failure mode
The crack width can be studied at two stages; a) before the shear cracks appear and b) after the
shear cracks appear. In models with top reinforcement layout, the crack width grows steadily
in both stages (before redistribution of stresses and after redistribution). However, in models
with bottom position of reinforcement, the crack width before the redistribution of stresses is
very low. This explains the positive impact of reinforcement in crack control and serviceability
performance
After the shear cracks appeared (at 50% of midspan deflection at peak load) the crack width
grew rapidly. As it can be seen in (Table 4.7) the crack width grew more than three times until
the failure load.
ANALYSIS AND RESULTS IN ATENA 3D
62
Table 4.7: Crack width at increasing midspan deflection and corresponding load
Crack width at increasing midspan deflection and corresponding load Percentage of
midspan
deflection
25 % of max
midspan
deflection
50 % ofmax
midspan
deflection
75 % of max
midspan
deflection
100 % of max
midspan
deflection
MODEL A (top position and high ratio)
Crack width
(mm)
0.10 0.50 0.80 0.87
Load (kN) 2865 3601 4497 5267
MODEL B (bottom position and high ratio)
Crack
width(mm)
0.02 0.46 1.1 1.70
Load (kN) 2702 3489 3786 4265
MODEL C (top position and low ratio)
Crack width
(mm)
0.17 0.56 0.79 1
Load (kN) 2713 3436 4071 4665
MODEL D (bottom position and low ratio)
Crack width
(mm)
0.02 0.47 1.2 2
Load (kN) 2693 3255 3583 3938
To better understand the positive influence of placing the reinforcement at the bottom of the
pile cap, the crack widths are compared in models before the appearance of shear cracks. Table
(4.8) shows that the crack width in model A is significantly higher compared to model B at the
same load level. In the same way the crack width in model C is much higher compared to model
D at the same load level. This shows the better serviceability limit state performance of placing
the reinforcement at the bottom of pile cap.
Table 4.8: The comparison of crack widths in models A, B, C and D at the serviceability limit
state
Parameter Model A
(top position
and high ratio)
Model B
(bottom position
and high ratio)
Model C
(top position
and low ratio)
Model D
(bottom position
and low ratio)
Load level 3855 3855 3338 3338
Crack width 0.65 0.122 0.576 0.125
Concerning the failure mode, the dominant failure modes in a pile cap are splitting of the
concrete along the inclined compressive struts due to tensile stresses and crushing of the
compressive strut due to compressive stresses.
CHAPTER 4
63
The failure in models A and C occurs due to crushing of the concrete at the superior node (node
beneath the column) (Figure 4.8 a). The crushing occurs when the plastic strain in the node
exceeds the limiting plastic strain defined for the concrete (0.00096). The principle compressive
stresses at failure are higher than the compressive strength of concrete, but due to multiaxial
state of stresses in the node, the resistance is enhanced.
The failure mode in models B and D however is the splitting of the concrete due to tensile
stresses perpendicular to the compressive strut. The cracks propagate from the midspan to the
corners causing the concrete to split in two halves (figure 4.8 b).
a) Failure mode in models A and C
(top reinforcement position)
b) Failure mode in models B and D
(bottom reinforcement position)
Figure 4.8: Pile cap failure modes
4.3.4 Stress in concrete
In theory part it was discussed that a compressive strut can have three shapes; prism, fan-shaped
and bottle shaped. In the models with both top and bottom and reinforcement positions, full
bottle shaped compressive struts were developed (figure 4.9 a and b). The stresses flow from a
small area (column- pile cap connection) and disperse as they flow through pile cap and then
flow back into a small area (pile - pile cap connection).
ANALYSIS AND RESULTS IN ATENA 3D
64
Figure 4.9: Principle compressive stress flow path in concrete with a) top reinforcement
layout b) bottom reinforcement layout
The stresses were highest close to the corners of the nodes (regions surrounded by circles)
which indicates the high possibility of crushing of the concrete in these locations in nodal zones.
Perpendicular to the compressive stress in the struts, tensile stresses also developed which
causes the splitting of the concrete (Figure 4.10 a and b). To counteract these stresses, usually
a transverse reinforcement mesh (similar to a cage) is provided in the web of the pile cap.
CHAPTER 4
65
Figure 4.10: Principle tensile stress flow path in concrete with a) top reinforcement layout b)
bottom reinforcement layout
In models A and C, the failure occurred due to compressive stresses at the superior node (figure
4.9). In model B and D, failure occurred due to tensile stresses perpendicular to the compressive
stresses (Figure 4.10).
4.3.5 Stress in reinforcement
To study the effect of reinforcement in the pile cap model, two reinforcement ratios were
examined. This was managed by increasing the bar diameter from 14mm to 18mm, rather than
adding new bars. As stated previously, the reinforcement was modelled using bi-linear material
properties where the stress after yield limit is constant 500 MPa. The strain at yielding is: fy/Es=
0.0025 = 0.25 %.
The plastic strain values could be obtained from the results as the reinforcements were yielding.
The values for stress and strain for an increasing midspan deflection are presented in (Table
4.9).
In the beginning of load steps (25 % of midspan deflection), the models A and C (top
reinforcement position), carry higher stresses. But as soon as the redistribution of stresses occur
ANALYSIS AND RESULTS IN ATENA 3D
66
due to shear cracks, the reinforcement stress and strain in models B and D (bottom
reinforcement position) increase at a much faster rate compared to models A and C. Table 4.8
shows an increase of 100 % in stress and strain from mid span deflection of 25 % to 50 %.
Table 4.9: Reinforcement stress and strain at increasing mid-span deflection
Reinforcement strain relation with mid-span deflection Percentage
of max
midspan
deflection
25 % of max
midspan
deflection
50 % of max
midspan
deflection
75 % of max
midspan
deflection
100 % of max
midspan
deflection
Model: Stress
(MPa)
Strain
(%)
Stress
(MPa)
Strain
(%)
Stress
(MPa)
Strain
(%)
Stress
(MPa)
Strain
(%)
Model A 91.7 0.044 258 0.123 351 0.167 439 0.21
Model B 27.4 0.013 279 0.133 356 0.17 455 0.22
Model C 205 0.098 404 0.19 500 0.26 500 0.55
Model D 37 0.018 392 0.19 500 0.28 500 0.39
Another interesting aspect observed in models C and D (low reinforcement ratio), is that the
reinforcement already reached the yield limit at 75 % of mid-span deflection. Whereas models
with a higher reinforcement ratio (models A and B) the failure occurs before reinforcement
reaches the yield limit (Table 4.9).
Concerning strain, the maximum strain was observed in bars located in the vicinity of pile tops
in models A and C. It indicates the concentration of tensile stresses on pile tops as suggested
by the strut and tie method (Figure 4.10). In models, with reinforcement placed at the bottom
of pile cap, only the bars located around the piles reached the yield limit prior to failure. All the
other bars were well below the yield limit. This reveals that even though the placement of bars
at the bottom contributes to crack control in serviceability limit state, it ultimately does not
contribute much to carrying the tensile stresses in the pile cap.
Finally, one of the most important aspects related to deep beams is the anchorage of
reinforcements. In all models, the anchorage of bars did not seem to be a problem. As it can be
seen in Figure 4.10, the end of the reinforcement bars (located at the borders of pile caps) are
carrying compressive stresses. This reveals that the bars are under compression at the ends
which contributes to the anchorage of bars. Further on, separate models were created in ATENA
where longitudinal reinforcement bars were bent-up towards the top of pile cap to increase the
anchorage length. The results of the analysis of models show no difference with initial models
and therefore confirm the absence of problems related to anchorage.
In conclusion, increasing the reinforcement bar diameters from 14 mm to 18mm in models with
top reinforcement layout increased the ultimate failure load about 602 kN (13%). Whereas
increasing the bar diameters in models with bottom reinforcement layout increased the ultimate
load about 327 kN (8%) i.e. lower than the increasement for top reinforcement layout. This
illustrates the efficiency of placing the reinforcement bars on the top piles.
CHAPTER 4
67
a) Model A (top position of reinforcement and high ratio)
b) Model B (bottom position of reinforcement and high ratio)
c) Model C (top position of reinforcement and low ratio)
d) Model D (bottom position of reinforcement and low ratio)
Figure 4.10: Strain in reinforcement at failure load
ANALYSIS AND RESULTS IN ATENA 3D
68
CHAPTER 5
69
Parametric Study
In order to study the effects of different parameters in the non-linear analysis, the results for
model A (as a default model) is compared with the results of same model with either increased
or decreased particular parameter.
Influence of mesh size
A mesh convergence analysis is performed to observe the effect of mesh size on the results of
the analysis. This is particularly important in ATENA because it is based on smeared approach
where a very dense meshed model could behave overly stiff. In total, three mesh sizes (10 cm,
5 cm and 4 cm) were examined on a single pile cap model (model A). All other parameters
were kept constant and the load displacement diagram, crack patterns and crack width were
measured. After measurement, it was revealed that the results are slightly mesh dependent.
PARAMETRIC STUDY
70
Figure 5.1: Load displacement diagram for mesh convergence
Concerning the load displacement diagram, the curves for different mesh sizes follow each
other until the first shear crack has appeared. Afterwards, the path followed by the curves are
agitated and unique. The mesh with 5 cm element size showed slightly more ultimate failure
load and more midspan deflection. Following that, mesh sizes 10 cm and 4cm in sequence. As
stiffness is concerned, the smallest mesh size (4cm) showed a slightly higher stiffness than the
other two mesh sizes. This due to stress locking effect in smeared cracking approach. Another
interesting aspect is that the load-drop in the curve which is an attribute of deformation-
controlled analysis is most clearly visible in model with mesh size of 5 cm. At the failure load,
the curve for mesh sizes 5 cm and 4 cm both fail abruptly where the 10 cm mesh size shows
slightly more ductility at the end of the curve.
0
1000
2000
3000
4000
5000
6000
0,0 0,5 1,0 1,5 2,0 2,5
Lo
ad (
kN
)
Displacement (mm)
Mesh convergence analysis
Mesh size 5cm
Mesh size 10 cm
Mesh size 4cm
CHAPTER 5
71
a) Crack propagation in
model with 5 cm
mesh size
b) Crack propagation in
model with 10 cm
mesh size
c) Crack propagation in
model with 4 cm
mesh size
Figure 5.2: The number of cracks in models with varying mesh sizes
Concerning number of cracks at failure, the model with smallest mesh size (4 cm) had more
cracking density and more cracking area covered compared to two other mesh sizes. However,
where crack propagation is concerned, in model with 5 cm mesh size, a large number of wide
cracks run not only in the vicinity of compressive strut but also at the reinforcement level. In
model with 10 cm, more cracks were observed at the bottom of the pile cap.
a) Mesh size 10 cm
Crack filter wc โฅ
0.1mm
b) Mesh size 5cm
Crack filter wc โฅ
0.1mm
c) Mesh size 4cm
Crack filter wc โฅ
0.1mm
Figure 5.3: The crack pattern and crack width at failure in the models with varying mesh
Regarding crack width, there is a very slight difference between the models which is negligible.
The analysis time for the model with mesh size 10cm was completed in 36 minutes. The mesh
size 5cm was completed in 391 minutes and mesh size 4 cm was completed in 1591 minutes.
The analysis time for the mesh size 4cm increased by 307 % in comparison to model with mesh
size 5 cm. Meanwhile, the amount of occupied disk space in computer also increased greatly as
the size of mesh decreased.
Finally, the model with mesh size 5 cm was selected because it exhibited the wide cracks along
the compressive strut and the reinforcement level and also had much less analysis time
compared to model with the smallest mesh size.
PARAMETRIC STUDY
72
Influence of fracture energy
As discussed in the theory part, the fracture energy (Gf) is defined as the amount of energy
necessary to create one unit area of a crack (Hallgren, 1996). It is accepted that increasing the
value for fracture energy increases the deformation and reduces the crack width. Therefore to
observe the effects of fracture energy on the model, model A with default fracture energy value
calculated according (E. VOS, 1983), is compared to models with modified fracture energy
values according to (MC90, 1990), and (MC10, 2012)
In the original model A, the default value for fracture energy used by ATENA 3D is calculated
with the equation by (E. VOS, 1983)
๐บ๐ = 0.000025 ๐๐ก= 67.82 N/m 5.1
The calculation of fracture energy based on (MC90, 1990)is related to the maximum aggregate
size in the concrete. In Sweden, the largest aggregate size used in concrete mix is up to 65 mm.
In the model however, a maximum aggregate size 32 mm is used. The corresponding value for
fracture energy to the aggregate size is taken from the table in (MC90, 1990):
๐บ๐ = 95 N/m 5.2
This value accounts for an increase of 40 % in the fracture energy of the default value obtained
with the equation by (E. VOS, 1983). The third value for (Gf) is calculated based on equation
recommended by (MC10, 2012) which increases the value for fracture energy by 107 %.
๐บ๐ = 73 โ ๐๐๐0.18 = 140.5๐/๐ 5.3
All other parameters were kept constant and the effect of fracture energy was studied. The
results reveal that the overall stiffness of the structure is increased with increased fracture
energy, i.e. the failure load and the mid-span deformation at failure increased. Modifying the
fracture energy according to (MC10, 2012), increased the ultimate failure load up to 1
Meganewton. In the same way, modifying the fracture energy according to (MC90, 1990),
increased the ultimate failure load up to 0.8 Meganewton. The load displacement diagram
shows that with increased fracture energy, the curves become smoothened. The load drop after
the redistribution of stresses due to propagation of shear cracks is no longer visible in the curves
(figure 5.4).
CHAPTER 5
73
Figure 5.4: Load displacement diagram for varying fracture energy
The results reveal an overall increase in number of cracks with increased fracture energy. The
cracks are however more concentrated rather than being spread over the volume.
a) Crack propagation in
model A with
Gf=67.8N/m
b) Crack propagation in
model with Gf=95
N/m
c) Crack propagation in
model with
Gf=140.5 N/m
Figure 5.5: The crack propagation at the increased fracture energy
0
1000
2000
3000
4000
5000
6000
7000
0,0 0,5 1,0 1,5 2,0 2,5
Lo
ad (
kN
)
Displacement (mm)
Modification of fracture energy
MODEL A
Modified fractureenergy according toMC90
Modified fractureenergy according toMC10
PARAMETRIC STUDY
74
On the other hand, the crack width decreased by 14 % when the fracture energy according to
(MC90, 1990) was used and reduced by 21% when the fracture energy was considered
according (MC10, 2012) (figure 5.6).
a) Crack pattern and
maximum crack width
in model A with
Gf=67.8N/m
Crack filter wc โฅ
0.1mm
b) Crack pattern and
maximum crack
width in model with
Gf=95 N/m
Crack filter wc โฅ
0.1mm
c) Crack pattern and
maximum crack
width in model with
Gf=140.5 N/m
Crack filter wc โฅ
0.1mm
Figure 5.6: Crack pattern and crack width in models with varying fracture energy (Gf)
The reason that the modelsโ stiffness increased, crack widthโs decreased, and the behavior
becomes more ductile is due to the value of critical crack width (wc). The critical crack width
is directly related to fracture energy as presented in equation (3.2)
If in equation (3.2), only the value for fracture energy is increased, the value for critical crack
width (wc) increases greatly which means that more deformation is required to propagate a
macro crack. Achieving a concrete mix with a very high fracture energy without compromising
the other strength parameters is challenging. Nevertheless, changing the types of aggregate, size
of aggregate and adding fiber reinforcement are believed to increase the fracture energy.
CHAPTER 5
75
a) Compressive stresses
in model with
Gf=67.8N/m
b) Compressive stresses
in model with Gf=95
N/m
c) Compressive stresses
in model with
Gf=140.5 N/m
Figure 5.7: Principle compressive stresses in pile cap with varying fracture energy (Gf)
Influence of Tensile Strength
As discussed earlier, the dominant failure modes in a pile cap are crushing of concrete due to
compressive stresses along the inclined compressive struts and splitting of concrete due to
tensile stresses perpendicular to compressive stresses along the strut. Since the failure mode for
the models with reinforcement placed on top has been due to the crushing of concrete along the
struts, therefore the modelsโ behavior is studied at reduced tensile strength to observe if the
failure mode changes. This is performed by reducing the tensile strength at two stages: 25 %
and 50 %. As a result, model A (with default value of 2.7 MPa tensile strength) is compared to
models with 2.10 MPa and 1.35 MPa tensile strengths.
The results reveal that the ultimate bearing capacity at 25 % reduction of concrete tensile
strength did not change. At 50 % of reduction, the ultimate bearing capacity reduced 13% and
the midspan deflection also decreased. The noticeable part in the load displacement curve is
that the curves became smoother with reduced tensile strength. Even with 25 % of tensile
strength reduction, the stiffness increased slightly i.e. when the tensile failure mode was not
dominant, reducing the tensile strength resulted into increased stiffness of the model. This
points to the high influence that the critical crack width had on the models.
PARAMETRIC STUDY
76
Figure 5.8: Load displacement diagram for models with varying tensile strength
Concerning cracking phenomena, as the tensile strength was reduced, the number of cracks
increased. The propagation of the cracks, however, became concentrated over certain volumes
compared to model A where cracks were more scattered (Figure 5.9). The crack width
decreased a bit (Figure 5.10).
a) Crack propagation in
model A with
concrete with tensile
strength ft = 2.7 MPa
b) Crack propagation in
model with concrete
with tensile strength
ft = 2.01 MPa
c) Crack propagation
in concrete with
tensile strength ft =
1.35 MPa
Figure 5.9: Crack propagation in models with varying tensile strength
0
1000
2000
3000
4000
5000
6000
0,0 0,5 1,0 1,5 2,0 2,5
Lo
ad (
kN
)
Displacement (mm)
Load displacement diagramReduction of tensile strength
MODEL A
25 % of tensile strengthreduction
50 % of tensile strengthreduction
CHAPTER 5
77
a) Crack pattern and
maximum crack
width in model A
with ft= 2.7 MPa
Crack filter wc โฅ
0.1mm
b) Crack pattern and
maximum crack width
in model with ft= 2.02
MPa
Crack filter wc โฅ
0.1mm
c) Crack pattern and
maximum crack
width in model
with ft=1.35 MPa
Crack filter wc โฅ
0.1mm
Figure 5.10: Crack pattern and maximum crack width for models with varying tensile strength
The reason for the smaller crack width and the ductile behavior of model is again due to value
of critical crack width(wc). The same way as the increased fracture energy, the decreased tensile
strength, increases the value for (wc) in equation 3.2. Therefore, the structure behaves in ductile
manner i.e. more deformation is required to propagate a macro crack.
PARAMETRIC STUDY
78
a) Compressive stresses in
model A with ft= 2.7
MPa
b) Compressive
stresses in model
with ft= 2.02 MPa
c) Compressive
stresses in model
with ft= 1.35 MPa
Figure 5.11: Principle compressive stresses in pile cap with varying tensile resistance
Influence of Compressive Strength
To observe the effect of compressive strength, model A was analyzed with an increased and
decreased concrete compressive strength. The default mean compressive strength of model A
(38 MPa) was increased by 30 % (to 49 MPa) and decreased by 30 % (to 26 MPA). All other
parameters relating to concrete material properties were kept constant. The results revealed that
with the increased or decreased compressive strength, the failure load and midspan deflection
increased and decreased. The load displacement curves for varying compressive strength follow
the exact same path unlike the variation of fracture energy i.e. in the variation of fracture energy
the curves smoothened. Furthermore, the compressive strength also had an effect over the
stiffness of the models. After the redistribution of stresses occurred, the models with higher
compressive strength showed increased stiffness.
CHAPTER 5
79
Figure 5.12: Load displacement diagram for models with varying concrete compressive
strength
Concerning the number of cracks and the propagation area, there was a direct relationship with
the compressive strength of concrete. The number of cracks and the crack propagation volume
increased as the compressive strength of concrete increased and decreased as the compressive
strength decreased.
a) Crack propagation in
model A with
compressive strength
fcu = 38 MPa
b) Crack propagation in
model with
compressive strength
fcu = 26 MPa
c) Crack propagation in
concrete
compressive strength
fcu = 42 MPa
Figure 5.13: Crack propagation is models with varying concrete compressive strength
0
1000
2000
3000
4000
5000
6000
0,0 0,5 1,0 1,5 2,0 2,5
Lo
ad (
kN
)
Displacement (mm)
Varying compressive strength (fck mean)
Fcm=38 MPa
Fcm=26 Mpa
Fcm=49 MPa
PARAMETRIC STUDY
80
Concerning the crack width, the difference between the models were very small. Therefore, the
compressive strength plays not much role in crack width. Rather, it was the fracture energy and
the tensile strength that had high influence over the crack width in the model.
a) Crack pattern and
maximum crack width
in model A with fcu=
38 MPa
Crack filter wc โฅ
0.1mm
b) Crack pattern and
maximum crack
width in model with
fcu= 26 MPa
Crack filter wc โฅ
0.1mm
c) Crack pattern and
maximum crack
width in model
with fcu=42 MPa
Crack filter wc โฅ
0.1mm
Figure 5.14: Crack width in models with varying compressive strength
CHAPTER 5
81
a) Compressive
stresses in model A
with fcu = 38 MPa
b) Compressive stresses
in model with fcu =
26 MPa
c) Compressive
stresses in model
with fcu= 42
MPa
Figure 5.15: Compressive stresses at failure in models with varying concrete compressive
strength
Influence of Modulus of Elasticity of piles
A study was made to study the effect of the stiffness of the piles on the pile cap behavior. In
analytical calculation based on strut and tie method, the assumption is that each pile carries the
same amount of load, i.e. all the piles have the same stiffness. As a result, the structure is
accepted as statically determinate and the forces in strut and tie are calculated based on basic
laws of static. However, this assumption seems to be conservative since the stiffness of a pile
is dependent on the geological conditions underneath the ground which differs from situation
to situation.
To understand the effect of pilesโ stiffness, model A with non-linear concrete material
properties was compared with models in which piles were modeled with linear elastic material
properties. This was done by performing three separate analysis with varying modulus of
elasticity for piles (30, 40 and 50 GPa). The results reveal that the behavior of the model is
dependent on the stiffness of each individual pile. The load displacement curves show that the
curves follow almost the same path but slight difference. The model with the same modulus of
elasticity for pile as model A, showed slightly higher stiffness and less deflection and same load
bearing capacity. The model with lower modulus of elasticity (30 GPa), showed the highest
bearing capacity and midspan deflection. And model with the highest stiffness for piles (50
GPa), showed the highest stiffness but lower bearing capacity and deflection. It was concluded
that as the stiffness of the piles increases, the modelsโ behavior becomes more brittle.
PARAMETRIC STUDY
82
Figure 5.16: The load displacement diagram for piles with varying stiffness
The number of cracks and propagation volume increased with increasing pile stiffness, but
crack width remained the same.
a) Crack propagation in
model with Ecm=30
GPa
b) Crack propagation in
model with Ecm=40
GPa
c) Crack propagation
in model with
Ecm=50 GPa
Figure 5.17: Crack propagation with increased pile stiffness
0
1000
2000
3000
4000
5000
6000
0,0 0,5 1,0 1,5 2,0 2,5
Lo
ad (
kN
)
Displacement (mm)
Varying pile stiffness
30 GPa
40 GPa
50 GPa
Model A
CHAPTER 6
83
Results- Hand calculation
Hand calculation based on strut and tie method
and sectional approach
6.1.1 Assumptions in design:
1- The pile cap is a rigid block that equally distributes the forces over the piles
2- The connection between piles and pile cap are hinged i.e. no bending moment is transferred
3- Only short piles are assumed i.e. the distribution of stresses and displacements are planar.
6.1.2 Pile cap Geometry
(Whittle & Beattie, 1972) recommend a relationship between the span of the pile cap and the
cross-sectional width of the pile:
๐ = ๐ โ ๐๐๐๐๐ 6.1
Where:
a=3 pile spacing factor recommended value
bpile cross sectional width of pile
Since there is possibility for deviations in driving of piles, the handbook by Reynolds C.E,
and Steedman J.C. recommends the pile cap size to be 300 mm wider (150 mm on each sides)
to accommodate this deviation.
RESULTS- HAND CALCULATION
84
๐ฟ = ((๐ + 1) โ ๐๐๐๐๐) + 0.4๐ 6.2
The thickness of the pile cap is selected based on below requirements:
1. That it is adequate to resist the shear force without shear reinforcement.
2. That enough bond length could be accommodated for the longitudinal reinforcement.
3. That it should not be less than 300mm
The pile cap must be rigid enough so that the forces are equally distributed over the piles. To
increase the rigidity one can increase the reinforcement ratio or the concrete height.
Nevertheless, due to cost, it is preferred to increase the height of the pile cap. A minimum of
500 mm height is generally recommended. However, the handbook by Reynolds C.E, and
Steedman J.C. recommends the below equations:
๐๐ ๐๐๐๐๐ โค 0.55๐
๐ป = 2 โ ๐๐๐๐๐ + 0.1๐
6.3
๐๐ ๐๐๐๐๐ > 0.55๐
๐ป =1
3โ (8 โ ๐๐๐๐๐ + 0.6๐)
6.4
6.1.3 Reinforcement
The reinforcement in pile cap is determined either by strut and tie method or by the beam
method where the critical sections are designed for bending and shear force. The minimum
reinforcement recommended by the CP 110 code of practice is:
As = 0.15 % b โ d 6.5
Note: usually when the forces are higher, a reinforcement cage is designed for the pile cap
consisting of stirrups and horizontal reinforcements. This is not the subject of this masterโs
thesis.
6.1.4 Calculation based on strut and tie model:
For this masterโs thesis, the calculation based on strut and tie method is based on a combined
model. In combined model, the force in reinforcement concrete column is distributed between
the reinforcement part and the concrete part. This is done to utilize the reinforcement part in the
column and to reduce the stress concentration on node beneath the column (the CCC node).
The combined model is a combination of two models; model A and model B (sketch of the
models, calculations and results are found in Appendix A). In model A, the amount of force
CHAPTER 6
85
transferred through concrete part of column into pile cap is considered. In model B, the force
transferred by the reinforcement part of column into pile cap is considered. Finally, the two
models are combined resulting into final model. After finding the forces in struts and ties in the
final model, the following components are checked against failure:
1. Check of longitudinal reinforcement.
2. Check of nodes.
3. Check of struts.
6.1.5 Calculation of pile cap based on beam theory:
In calculation based on beam theory, a pile cap is treated as a two-way slab for which two
calculations steps are completed: a) calculation of bending moment b) calculation of shear
force.
a) The applied moment at the face of pile cap is calculated the same as moment in a simply
supported slab. The section (EC 2-1-1/clause 5.3.2.2) presents two values from which the
maximum value is taken as applied sectional moment. The resistance of the section against the
applied moment is calculated with simple beam analysis. For verification of failure, the applied
moment is compared with the sectionsโ moment resistance.
b) The shear resistance of the section is calculated based on equation 2.10 (shear resistance of
section without shear reinforcement). The maximum shear force is found by treating the model
as a simply supported beam supported by columns. After the maximum shear force is found it
is reduced with the factor . The final applied shear force equals:
๐๐ธ๐.๐๐๐๐๐ = ๐๐ธ๐ โ ๐ฝ 6.6
๐ฝ =๐๐ฃ
2 โ ๐ 6.7
av- is the shear plane which is the horizontal distance between the face of the column and the
face of the pile plus 20% of the diameter of the pile.
Table 6.1: Results of design bearing capacity based on strut and tie method and sectional
method
Strut and tie method Sectional approach
Reinforcement
ratio
Ultimate
load
Failure
mode
Yield
load
Reinforcement
ratio
Failure
load
Failure
mode
Yield
load
14 ฯ 14 in 2
directions
2970 Crushing
of strut
6540 14 ฯ 14 in 2
directions
3605 shear 4460
14 ฯ 18 in 2
directions
2970 Crushing
of strut
10800 14 ฯ 18 in 2
directions
4200 shear 7150
Note: Since the edges of the pile are located closer than 2D to the surface of the column, most
of the load goes directly to the piles. A punching shear check is only necessary if the pile are
located more than 2D away from the column face.
RESULTS- HAND CALCULATION
86
Detailed calculation based on strut and tie method is presented in Appendix A of this report.
Detailed calculation based on sectional approach is presented in Appendix B of this report.
CHAPTER 7
87
Discussion, conclusion and further
research
Optimizing the numerical model
One of the important aspects related to the analysis in ATENA 3D is the convergence issue. As
the concrete is cracked, the convergence becomes aggravated. The possibility that the
convergence criteria is not satisfied is a common hassle. This is the case specially in load steps
near failure of the structure. Before the final models were created, many modifications were
made to optimize the models so that the convergence criteria is satisfied.
The first models created in ATENA 3D were full models of a pile cap supported by four piles.
In these models, the piles were modelled as non-linear springs. However, when running the
model, the analysis had difficulty finding convergence or the convergence error was too high
that the results were unreliable. To improve the convergence, a symmetry model of pile cap
was created. It resulted in a stable model and slightly improved convergence compared to full
model but still the displacement in the model after load was erected was very high and
unrealistic. Therefore, it was decided to avoid non-linear springs and model the piles as short
columns supported vertically on steel plates- representing an ideal lab condition i.e. a universal
testing frame. After the models were run, the obtained failure mode and displacements seemed
more accurate, but the convergence problem was still very high.
The next step to improve convergence was to decrease the displacement increment in each step.
Initially, the displacement increment used was (0.1mm). A total midspan deformation of 2.5
mm was erected on the loading plate in 25 load steps. After the stage where the concrete began
to crack, the displacement increment was further decreased to 20 % of initial value to (0.02mm).
This step which is recommended by (Cervenka et al., 2018), improved the convergence greatly.
In the models with reinforcement placed on the top of piles, the analysis was completed
successfully, and no convergence problems were witnessed. But in models with reinforcement
at the bottom of the cap, the analysis was stopped due to convergence criteria not being satisfied.
Fortunately, it was at the stage where the failure had already occurred in the structure.
DISCUSSION, CONCLUSION AND FURTHER RESEARCH
88
The final attempt to optimize the convergence was to increase the number of iterations. The
standard Newton-Raphson method has a total iteration limit of 40. This number of iterations
was increased to 80 by modifying the standard Newton-Raphson method. Performing the
analysis with increased iterations slightly improved the convergence.
Comparison of numerical and analytical results
In this section, the expected bearing capacity which is determined with design equations in
Eurocode 2 is compared with the failure load of the numerical models created in ATENA 3D.
After comparison, it can be commented on how much safety margin is included in the design
of pile cap based on equations in Eurocode 2.
The calculation method for the individual pile caps were selected based on the position of the
reinforcement i.e. pile cap models with top reinforcement position were calculated with strut
and tie method and models with bottom reinforcement position were calculated based on
sectional approach. (detailed calculations are presented in Annex A and B). The results
revealed a big difference in comparison between design failure load based on strut and tie
method and the failure load obtained from numerical models. In design based on sectional
approach, however, the difference was not so high. The design results are presented together
with results from numerical models in the table 7.1.
Table 7.1: Comparison between hand calculations and numerical analysis
Method Model A Model C Model B Model D
NLFEA 5267 4665 4265 3938
Failure mode: crushing of
concrete compressive strut at
the superior node
Failure mode: splitting due to
tensile stresses perpendicular to
compressive strut
Strut and tie method 2970 2970 - -
Failure mode: Crushing of concrete strut
Sectional approach - - 4200 3605
Failure mode: Shear failure
Concerning design based on strut and tie method, when the reinforcement is not yielding, the
safety margin of (1.77) was found to be effective. When the reinforcement yields, the safety is
(1.57).
One possible reason for such a big safety margin is because of the very conservative equation
in Eurocode for the resistance of the compressive strut. As it can be seen in equation 2.2, the
value for reduction factor for the resistance of the strut is kโ=0.6. The large reduction factor for
the strutsโ resistance is because it considers all multi-axial states in a strut and the corresponding
cracking which affects the strength of strut. In addition, the equation does not consider the value
for tensile strain and does not differentiate between transversely reinforced and unreinforced
struts. One value for all compressive struts in concrete.
CHAPTER 7
89
Another possible reason for the big difference between the results is due to the high stiffness of
the model. The concrete model in ATENA utilizes smeared approach which is known for stiff
behavior. In addition, the effect of aggregate size was included in the concrete mix which also
increased the stiffness of the model. The appropriate measures taken to reduce the over stiff
behavior of the model were; a) modelling the piles and column with non-linear concrete
material instead of linear elastic, b) introducing an interface element at the contacts between
column-pile cap and pile-pile cap.
Concerning the stress in reinforcement, based on strut and tie method, the structure is divided
into a series of load paths representing struts and ties. Therefore, separate failure checks for ties
and struts were performed i.e. strengthening one part does not affect the strength in the other.
Consequently, the failure load for model C with lower reinforcement ratio and model A with
higher reinforcement are the same (crushing of concrete strut). Whereas in numerical analysis,
it was proven that an increase in reinforcement ratio, increased the ultimate failure load by 13
%. Based on sectional approach, however, the combined effect of both reinforcement and
concrete are considered. Therefore, an increase in longitudinal reinforcement ratio increased
the shear resistance of the concrete both in numerical analysis and analytical calculations.
Concerning design based on sectional approach, the safety margin is very small, though using
the strut and method for pile cap is highly questionable due to failure mode that occurred in the
pile cap (splitting due to transverse tensile stresses along the strut).
DISCUSSION, CONCLUSION AND FURTHER RESEARCH
90
Conclusion
The results of the numerical models using non-linear finite element show:
1) The failure mode for the pile cap models with reinforcement placed at the top of piles
is the crushing of concrete strut at the superior node (node below column) and failure
mode for pile cap model with reinforcement placed at the bottom of the slab is
splitting of the pile cap due to transverse tensile stresses perpendicular to the inclined
compressive strut.
2) The ultimate bearing capacity increased by 23 % when the reinforcement was placed
at the top position in models where reinforcement does not reach yielding. And
increased by 18 % in the models where the reinforcement yielded prior to failure.
3) The models with bottom reinforcement position had a better crack control at
serviceability limit state loads, i.e, when comparing the crack widths in models prior
to appearance of shear cracks, the models with bottom reinforcement position had
significantly smaller crack widths.
4) The increase in the longitudinal reinforcement barsโ diameter from ร 14mm to ร
18mm increased the bearing capacity in models with top reinforcement position by 13
%. And in models with bottom reinforcement position by 8%. This phenomenon is not
captured in strut and tie method.
5) The tensile and compressive stress flow in concrete, the stress and strain in
reinforcement, and the failure mode occurred in the model approve the strut and tie
method as the better method for the calculation of pile caps.
6) Modifying the tensile strength, fracture energy and compressive strength values had
considerable effects on the load carrying capacity, number of cracks and crack width.
7) The results are also dependent on the stiffness of individual pile. Therefore, the
assumption in the numerical model and strut and tie method that each pile carries the
same amount of load is questionable.
CHAPTER 7
91
Further research:
Though it was demonstrated that placing the reinforcement at the top of the piles, recommended
by strut and tie analogy, yields better results compared to the bottom position, there are still so
many interesting topics that can be considered in the future research:
1) Understanding the multi-axial state of stresses in the compressive strut and how it
affects the strength of the strut. Accordingly, breakdown of the general equation for
the strength of compressive strut in Eurocode 2 for the specific cases of the multiaxial
stresses.
2) Study of the pile cap behavior with the transverse reinforcement in the models.
3) A study about effect of lateral stiffness and vertical stiffness of piles on pile cap
behavior.
4) Mechanism of stress distribution between concrete and reinforcement in a pile cap.
5) Effect of pile cap height on the bearing capacity of pile cap.
DISCUSSION, CONCLUSION AND FURTHER RESEARCH
92
BIBLIOGRAPHY
93
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Appendix A
95
A
Design of pile cap based on strut and
tie method
Appendix A
Hand calculations
96
Appendix A
97
Pile cap dimensions:
Column cross section
Specifications
Spacing between piles
Breadth of pile cap
Height of pile cap
Input data
Pile foundation Column
Concrete material properties:
Class for pile caps: C30/37
Mean Strength Characteristic Strength Design Strength Modulus of Elasticity
Exposure class:XC2
Design life: 100 years XC2
Hand calculations
98
Reinforcement material properties:
Angles between struts and ties:
Force distribution between model A and B
Total area of column cross section
Area of reinforcement steel
Area of concrete
Converting factor
Equavalent column cross section area
Applied load
Stress in concrete
Appendix A
99
Stress in reinforcement
Load carried by concrete part- Model A
Load carried by reinforcement part- Model B
Calculation of forces in struts and ties
MODEL-A MODEL-B
Lever arm
Angle S & T
Load
Reaction
Force in tie
Force in strut, x
Force in strut, y
Total strut force
Hand calculations
100
Ultimate Model
Load
Reaction
Force in tie
Force in strut, x
Force in strut, x
Total strut force
Angle S & T
Lever arm
Depth of lever arm
Reinforcement provision and verification
XX direction YY direction
Appendix A
101
Stress in Nodes:
Node 1 (CCTT) properties:
Stress in Node 1:
Plane: Force: Stress:
Pile connection
Strut
Check of stresses in node 1:
Hand calculations
102
Node 2- (CCCC) properties:
Stress in Node 2:
Plane Force Stress
Column conn.
Vertical (Tx)
Vertical (Ty)
Strut
Appendix A
103
Check node 2:
Verification of strut forces:
Hand calculations
104
Check of struts:
Appendix B
105
B
Design of pile cap based on sectional
approach
Appendix
Hand calculations
106
Input Data
For rectangular stress block the cross section factors, when
Factor for effective strength
Factor for Effective height of compressive zone
Bending Analysis of the section
Effective depth layer 1:
Effective depth layer 2:
Lever arm level 1
Appendix B
107
Lever arm level 2
Moment resistance in y direction
Moment resistance in x direction
Height of concrete compressive zone
Check of reinforcement
Applied moment at the face of pile cap according to EC 2-1-1/clause 5.3.2.2
Hand calculations
108
Check of moment resistance
Calculation of shear force at critical section
Applied shear force at the face of the support:
Force in one column
Maximum Shear force in the face of the pile cap
Shear resistance:
Considering the small effective depth:
Appendix B
109
Reduction of shear force:
According to EC 2-1-1/clause 6.2.2 (6)
The shear force is reduced by :
(Flexural shear span including 20% of pile section diameter)
Applied reduced shear force
Check of shear resistance
Appendix B
111