Reinforcement Layout in Concrete Pile Foundations: A study based on non - linear finite element analysis

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Master of Science Thesis

Reinforcement Layout in Concrete Pile

Foundations

A study based on non-linear finite element analysis

MOHAMMAD MUSTAFA ANGAR

TRITA-ABE-MBT 2020:20418 ISBN: 978-91-7873-598-3 kth royal institute of technology M OHA M M AD M US TA FA A NG AR Re inf or ce m en t L ay ou t in C onc re te P ile Fo und at ion s K TH

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Concrete Pile Foundation

s

A study based on non-linear finite element analysis

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TRITA-ABE-MBT- 20418, Master Thesis ISBN: 978-91-7873-598-3

KTH School of ABE SE-100 44 Stockholm Sweden

Department of Civil and Architectural Engineering Division of Concrete Structures

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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

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Sammanfattning

Det huvudsakliga ämnet för den här examensarbetet handlar om beteendet hos pålfundament som stöds av fyra pålar med två olika positioner av längsgående armering. Positionerna inkluderar placering över pålarna och placering i botten av fundamentet, dvs under pålavskärningsplanet. För detta ändamål skapades icke-linjära finita elementmodeller av en pålfundamentet med mjukvaran ATENA 3D. Målet var att observera vilket armeringsläge som ger den högre bärkapaciteten och att identifiera brottmekanismen i modellerna.

För att uppnå ovanstående mål utfördes en kort genomgång av den teoretisk bakgrunden rörande skjuvningsfenomenet. Detta för att förbättra kunskapen om skjuvspänningar, skjuvöverföringsmekanism, faktorer som påverkar skjuvkapacitet, skjuvbrott och att relatera dem till beteendet av ett pålfundament. Beräkningar av skjuvkapaciteten baserad på Eurocode2 med hjälp av Srut and tie-metod och sektionsmetod presenteras.

Den numeriska analysen började med att skapa fyra pålfundament i ATENA 3D. Skillnaden mellan modellerna är positionen och innehållet av den längsgående armeringen. Syftet med två armeringsinnehåll var att observera beteendet av pålfundamentet i två fall: a) när brott inträffar innan armering plasticeras; b) när brott inträffar medan armeringen plasticeras. Modellerna analyserades sedan med hjälp av programvaran ATENA Studio.

Resultaten visade att placering av armeringen ovanpå pålarna i fall a) ökade modellens kapacitet med 23,5% och i fall (b) ökade kapaciteten med 18,5%. Detta på grund av att dragspänningarna visade sig vara koncentrerade på toppen av pålarna snarare än på botten av pålfundamentet. Det slutliga brottet i modellen med armering över pålarna var krossning av den lutande trycksträvan vid noden under pelaren. I modellen med armering i botten av fundamentet spräcktes trycksträvan på grund av dragspänningar vinkelrätt mot den lutande strävan. The potentiella fördelen med placering av armoring I botten av pålfundamentet är ren bättre sprickkontroll och en något segare brottmod i jämförelse med placering av armering över pålarna.

En parametrisk studie genomfördes också med modellen för att observera effekterna av olika parametrar på de erhållna resultaten. Det visade sig att brottenergi hade den mest signifikanta effekten på de erhållna resultaten.

Slutligen genomfördes en jämförelse mellan resultaten från numerisk analys och analytiska dimensioneringsmetoder baserade på fackverksmetoden och tvärsnittsmetoden. Jämförelsen avslöjar att de kapaciteter som erhölls med fackverksmetoden var mycket konservativa. I synnerhet var ekvationen för kapaciteten hos det lutande trycksträvorna baserad på Eurocode 2

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Preface

In the name of God, the compassionate, the merciful

This report presents a master’s thesis which was initiated in a joint effort between the division of concrete structures at the KTH Royal Institute of Technology and Tyréns AB in Stockholm, Sweden. The completion of this thesis is the requirement for the last semester of a two-year master’s 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 Tyréns 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 Tyréns, 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 master’s program. Finally, I want to thank Swedish institute, a government agency in Sweden for providing me a fully funded scholarship to complete a master’s degree program in Sweden.

Stockholm, May 2020

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Symbols

Greek Symbols

ϵ1 Principle tensile strain

ϵ2 Principle compressive strain

∆P Small load increment

α Modulus of elasticity ratio

Shear reduction factor

β* Angle between strut and tie in plan

η Factor for effective strength

θ Rotation angle

θ* Angle between strut and tie in elevation

λ Factor for effective height of compressive zone

μ Friction coefficient

Longitudinal reinforcement ratio

ρlx Reinforcement ratio

σc Stress in concrete

σRd.max Maximum resistance of strut

σw Stress surrounding a crack

Compressive stress in concrete

τxy 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

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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

k’ Safety factor

k1 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

σw 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

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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

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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 60’s and the reinforcement was positioned further down at the bottom of the slab. The engineers at Tyréns 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 Tyréns 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.

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INTRODUCTION

Therefore, understanding the overall behavior of pile cap in top and bottom reinforcement positions is the question of this masters’ 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 master’s 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 – covers the theory regarding non-linear finite element method, material models

used in ATENA 3D and assumptions made when building the model.

Chapter 4 – presents the results of the numerical analysis performed in ATENA Studio. Chapter 5 – presents the results of hand calculations based on strut and tie method and

numerical analysis

Chapter 6 – presents the parametric studies regarding tensile strength, fracture energy,

compressive strength, and stiffness of piles.

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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 (Skåne, Öland 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 α 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:

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THEORETICAL BACKGROUND

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 master’s 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,

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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):

• For infrastructure: VVFS 2004:43 (with changes in TRVFS 2011:12) provided by Trafikverket (the Swedish Transport Administration).

• 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 master’s 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.

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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.

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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-Venant’s 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).

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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: Ritter’s truss model recreated from (Brown, 2005)

In 1902, Ritter’s truss model was refined by Mörch who suggested that the discrete diagonal forces used in Ritter’s truss should be replaced with a continuous field of diagonal compression. (Brown, 2005) (Figure 2.6).

Figure 2.8: Ritter’s truss refined by Mörch 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

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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)

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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:

𝜎_Rd.max = f_cd 2.1

Where;

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Figure 2.11: Recreated form (Eurocode2, 2004)

B) Strut with lower strength where the transverse tensile stresses are present:

𝜎_Rd.max = k′ ∙ v′ ∙ f_cd 2.2 𝑘′ = 0.6 2.3 v′ = 1 − fck 250 2.4 Where;

𝜎_Rd.max - design strength

k'- is a safety coefficient to cover for worst case condition for multiaxial stress in strut. v′ - 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).

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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).

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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 𝑓_𝑦𝑑 of the steel (Hendy & Smith, 2007):

𝑇 ≤ 𝑓𝑦𝑑 ∙ 𝐴𝑠 2.5

Where:

T- Tension force in the model

𝑓𝑦𝑑- Yield strength of steel

𝐴_𝑠 – 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).

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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).

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THEORETICAL BACKGROUND T T C C T C C C C

CCC-node CCT-node CTT-node

Figure 2.15: Types of singular nodes

Based on (Eurocode2, 2004), the maximum allowable stress 𝜎_{𝑅𝑑.𝑚𝑎𝑥} in nodes are determined using below equations:

a) All members in a node are in compression (CCC-node)

𝜎_{𝑅𝑑.𝑚𝑎𝑥} = 𝑘₁ 𝑣′ 𝑓_𝑐𝑑 2.6

where;

k1 – nationally determined parameter which’s value is 1.0

v’ – nationally determined parameter which’s value is recommended to be: 𝑣′ = 1 − 𝑓𝑐𝑘

250

b) One member in a node in tension, others in compression (CCT-node)

𝜎_{𝑅𝑑.𝑚𝑎𝑥} = 𝑘₂ 𝑣′ 𝑓_𝑐𝑑 2.7

where;

k2 – nationally determined parameter which’s recommended value is 0.85

c) Two members in a node in Tension formed by bent bar, others in compression (CTT-node)

𝜎_{𝑅𝑑.𝑚𝑎𝑥} = 𝑘₃ 𝑣′ 𝑓_𝑐𝑑 2.8

𝜎_{𝑅𝑑.𝑚𝑎𝑥} = 𝑘₃ 𝑣′ 𝑓_𝑐𝑑 where;

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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).

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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 τxy at distance y from the

neutral axis, can be found using:

𝜏_𝑥𝑦=𝑉𝑥∙ 𝑆𝑦 𝑏 ∙ 𝐼_𝑦

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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 θ which is found using Mohr’s circle.

Figure 2.19: a) Strains in an arbitrary point in a beam recreated from (Chantelot & Mathern,

2010) b) Direction of strains based on Mohr’s circle recreated from (Chantelot & Mathern, 2010)

In Figure (2.19);

ϵ

1 - principle tensile strain.

ϵ

2 - principle compressive strain.

θ – is the angle determining direction of principle compression Figure 2.18; represents the Mohr’s circle.

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appears in locations where cracking moment Mcr is reached first. These cracks are vertical with

θ 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 θ (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 (Engström, 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

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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).

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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 _{(𝑎 𝑑}⁄ ≥ 3), showed almost the same shear failure load. Beams with (𝑎 𝑑⁄ = 7 𝑜𝑟 8) showed a flexural failure due to high bending moment. And beams with (𝑎 𝑑⁄ ≥ 2.5) showed a higher resistance in shear with decreasing 𝑎 𝑑⁄ (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 (𝑎 𝑑⁄ ≥ 1 𝑜𝑟 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

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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:

𝑓_𝑣 _{= 𝑉 𝑏 ∙ 𝑑}⁄ 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 don’t 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.

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𝐶_{𝑅𝑑,𝑐} = 0.18

𝛾𝑐 is a coefficient dependent on loading case

𝑘 = 1 + √200 𝑑⁄ ≤ 2.0 (k) is the size factor (d) is the effective depth in (mm)

d- is the effective depth of slab in (mm)

𝜌_𝑙 = 𝐴𝑠𝑙

𝑏_𝑤∙𝑑 ≤ 0.02 is the longitudinal reinforcement ratio

𝐴_𝑠𝑙 is the area of tensile reinforcement extending ≥ 𝑙_𝑏𝑑+ 𝑑

𝑏_𝑤 is the smallest width of cross-section in (mm).

𝑓_𝑐𝑘 concrete characteristic compressive strength (cylinder) in (MPa).

𝑘₁= 0.15 set as per (BFS, 2011)

𝜎_𝑐𝑝 is the compressive stress in concrete from axial load in (MPa).

𝑣_𝑚𝑖𝑛 = 0.035 ∙ 𝑘3⁄2_{∙ 𝑓}_𝑐𝑘1⁄2_{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:

𝑉_{𝑅𝑑,𝑐} = 𝑣_𝑚𝑖𝑛∙ 𝑏_𝑤∙ 𝑑 2.12

For calculating the shear stresses in pile cap, the smallest shear resistance width 𝑏_𝑤should be carefully selected. For loads close to support, a reduction factor β is considered:

𝛽 = { 𝑎_𝑣 2𝑑, 𝑓𝑜𝑟 0.5 ∙ 𝑑 ≤ 𝑎𝑣 ≤ 2 ∙ 𝑑 0.5 ∙ 𝑑 2 ∙ 𝑑 , 𝑓𝑜𝑟 0.5 ∙ 𝑑 } 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: 𝑉𝑅𝑑,𝑐 = 𝐼 ∙ 𝑏_𝑤 𝑆 ∙ √𝑓𝑐𝑡𝑑 2 + 𝜎𝑐𝑝∙ 𝑓𝑐𝑡𝑑 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;

𝑉_𝐸𝑑 ≤ 0.5 ∙ 𝑣₁ ∙ 𝑓_𝑐𝑑∙ 𝑏_𝑤 ∙ 𝑑 2.15

where v1 is a factor of reduction due to cracks in concrete;

𝑣₁ = 0.6 (1 − 𝑓𝑐𝑘 250)

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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).

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FINITE ELEMENT ANALYSIS

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 master’s 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.

[𝐾]{𝑑} = {𝐹}

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.

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3.2.1 Iterative procedure

In a linear analysis, the load {𝐹} and displacement {𝑑} are linearly related through a constant stiffness matrix [𝐾]. Therefore, it is possible obtain {𝑑} 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 ∆P. 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 toua. 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

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3.2.2 Crack opening laws and f racture 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 ≥ 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 (σw) is zero at the transition point between macro crack and

fracture process zone and maximum (σw,max= ft) at the crack tip (Malm, 2016).

Figure 3.3: Stress distribution in crack tip for failure mode I, recreated from (Hillerborg et.

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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 master’s 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).

𝜀𝑡𝑜𝑡𝑎𝑙 = 𝜀𝑒𝑙𝑎𝑠𝑡𝑖𝑐 + 𝜀𝑐𝑟𝑎𝑐𝑘.

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).

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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 σ-ε (stress-strain) whereas the non-linear (descending) part is defined with σ-w (stress-crack opening displacement) (Malm, 2016).

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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:

𝐺_𝑓 = 73 ∙ 𝑓_𝑐𝑚0.18 _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

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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:

𝑤_𝑐 = 5.14 ∙𝐺𝑓 𝑓_𝑡

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: 𝑤_𝑐 = 2 ∙𝐺𝑓 𝑓_𝑡 3.3 For bilinear: 𝑤_𝑐 = 3.6 ∙𝐺𝑓 𝑓_𝑡 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.

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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

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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)

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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 which’s 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).

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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

Reinforcement Layout in Concrete Pile Foundations: A study based on non - linear finite element analysis