failure and post-failure aspects of mechanical response of concrete structures … ·...

Failure and post-failure aspects of mechanical response of concrete structures to compression and tension

By

HOSSEIN BINESHIAN

BSc, MSc

This thesis is presented for the degree of Master of Philosphy

of The University of Western Australia

School of Civil, Environmental and Mining Engineering

2014

DECLARATION

This thesis does not contain work that I have published, nor work under review for

publication.

Hossein Bineshian

2

DEDICATION

To my love,

Leila jaan

3

ABASTRACT

The mechanical response of concrete to tension and compression including pre- and

post-peak portions are examined in this thesis. Pre-peak normally consists of three parts;

plastic, elastic, and brittle that can be obtained by any load-control testing machine.

Failure is occurred when a network of major cracks is formed at peak stress while

failure criterion describes failure behaviour of concrete under triaxial loading. Post-peak

is a mixture of material and structural responses of concrete after failure and unlike the

pre-peak is significantly influenced by geometry of specimen and stiffness of loading

frame. Stress–deformation curve is used in displacement-control mode. Strain softening

is a post-peak highly localized phenomenon caused by coalescence of micro-cracks that

forms a damage zone after peak stress while load-carrying capacity is diminished.

In total 102 uniaxial/triaxial compression/tension tests are conducted on 102 concrete

cylindrical 150 × 300 mm specimens in accord with ASTM standards. Major fracture

pattern of cone/shear and cone for concrete specimens under uniaxial and triaxial

compression tests are recognized respectively. An empirical classification; Concrete

Quality Designation (CQD), is introduced in this research that classifies concrete to six

classes, based on the UCS. It can be applicable in practical purposes.

A new strength criterion is developed for concrete based on the mathematical function

of Bineshian criterion (2000) having three parameters; however, new constant values

are determined for different CQDs. It can provide linear and nonlinear envelopes

capable of describing triaxial actual data for both compressive and tensile quadrants and

provides credible estimations of these values similar to those determined in laboratory

with correlation and accordance coefficients adjacent to 1 and 0 respectively.

Concrete Damage Plasticity (CDP) constitutive model is used for a CQD-G class plain

concrete in Abaqus. Suitable constitutive parameters for the behavioural model are

determined using experimental data, analytical methods, and other sources while proper

results obtained from numerical simulations using FEA. Findings in this sector of the

research are applicable in numerical modelling and simulations of failure and post-

failure responses of concrete and especially in design of high strength concrete (CQD-G

class) structures and concrete-like brittle materials like rocks.

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TABLE OF CONTENTS

DECLARATION 2

DEDICATION 3

ABSTRACT 4

LIST OF FIGURES 7

LIST OF STMBOLS 10

LIST OF TABLES 14

ACKNOWLEDGEMENTS 17

CHAPTER 1:

INTRODUCTION 1.1. Preface…….……………………..………………………………………….18 1.2. Objectives…………………………………………………………………...20 1.3. Methodology…….…………..……………………………………………....22 1.4. Thesis Structure………………………………………………………….…23

CHAPTER 2:

MECHANICAL BEHAVIOUR OF CONCRETE 2.1. Introduction...……………………………………………………………....25 2.2. Pre-peak Response and Failure…..……………………………………….27 2.3. Post-peak Response………………………………………………………...30

2.3.1. Stiffness……………………………………………………....30

2.3.2. Geometry and Boundary Restraint…………………………...32

2.3.3. Strain Localization…………………………………………...35

2.3.4. Damage Plasticity……..……………………………………...36

2.4. Conclusions………………………………………………………………….41

CHAPTER 3:

EXPERIMENTAL AND EMPIRICAL WORKS 3.1. Introduction.………………………………………………………………..43 3.2. Plain Concrete Preparation………………………………………………..44

3.2.1. Mix Proportion and Design………………………………….44

3.2.2. Casting and Capping………………………………………...45

3.3. Strain Measurements……………………………………………………....46 3.4. Testing Plan………………………………………………………………....47 3.5. Test Results………………………………………………………………....49 3.6. Concrete Quality Designation……………………………………………...52 3.7. Conclusions………………………………………………………………….54

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CHAPTER 4:

NEW STRENGTH CRITERION FOR CONCRETE 4.1. Introduction.………………………………………………………………..56 4.2. Bineshian’s Strength Criterion (2000) for Intact Rocks and Coals….….57 4.3. Development of Bineshian Strength Criterion for Concrete…………….65

4.3.1. Introduction…………………………………………………..65

4.3.2. Performance………………………………………………….67

4.3.3. Parameters of the Criterion…...……………………………...79

4.4. Conclusions………………………………………………………………….81

CHAPTER 5:

CONCRETE DAMAGE PLASTICITY MODEL 5.1. Introduction.………………………………………………………………..83 5.2. CDP Theory………………………………………………………………...84

5.2.1. Mechanical Response………………………………………...84

5.2.2. Uniaxial Cyclic Response…………………………………....87

5.2.3. Multiaxial Response………………………………………….90

5.2.4. Post-failure Stress-Strain Relation…………………………...90

5.2.5. Fracture Energy Cracking Criterion………………………….92

5.2.6. Response to Compressive Load……………………..…….....94

5.2.7. Yield Surface……………………………………………..…..95

5.2.8. Plastic Flow Potential……………….………………..……....96

5.3. Visualization of Crack Directions………………………………..………..97 5.4. Output…………………………………………………………….……..…..97 5.5. Identification of Constitutive Parameters……………………………..….98 5.6. 3D FEA using CDP………………………………………………………..103 5.7. Conclusions………………………………………………………………...113

CHAPTER 6:

CONCLUSIONS 6.1. Observations……………………………………………………………….114 6.2. Future Research…………………………………………………………...116

REFERENCES 118

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LIST OF FIGURES

Figure 2.1. Complete stress-strain diagram for concrete consisting pre-peak and

post-peak responses.

Figure 2.2. Load-control and displacement-control modes in compression testing.

Complete 𝜎𝜎 − 𝜀𝜀 curve only in displacement-control mode is obtained.

Figure 2.3. Failure process on concrete under uniaxial compression including flaw

initiation and finally coalescence of network of cracks causing failure.

Figure 2.4. Failure envelops for actual triaxial data (Johnston, 1985).

Figure 2.5. Stability of machine-specimen system considering stiffness of machine,

and geometry of specimen (Hudson et al., 1972).

Figure 2.6. Typical force–displacement diagrams of large/small structure (Jansen

and Shah, 1997; Nemecek and Bittnar, 2004; Gamino et al., 2004).

Figure 2.7. Softening curves for specimens with different height (van Mier, 1986).

Figure 2.8. Stress–strain curves obtained from cylinder tests on high strength

concrete using high friction steel platens (van Mier et al., 1997).

Figure 2.9. 𝜎𝜎 − 𝜀𝜀 curves for normal strength concrete specimens of different

geometry loaded between rigid high friction steel platens (Gobbi and

Ferrara, 1995; van Mier et al., 1997).

Figure 2.10. Influence of end friction for concrete specimens under uniaxial

compression (van Vliet and van Mier, 1996).

Figure 2.11. Bazant’s model fitting on 𝜎𝜎 − 𝜀𝜀 data of different slenderness ratio

(Bazant, 1989 based on van Mier’s (1984) test results).

Figure 2.12. Mechanical response of concrete to uniaxial loading in compression.

Figure 2.13. Mechanical response of concrete to uniaxial loading in tension.

Figure 2.14. Definition of the cracking strain 𝜀𝜀𝑡𝑡~𝑐𝑐𝑐𝑐 used for the definition of tension

stiffening data for CDP model.

Figure 3.1. Cylindrical concrete mold used in this research.

Figure 3.2. 𝜎𝜎 - 𝜀𝜀 curve for concrete specimens under uniaxial compression.

Figure 3.3. Sketches of types of concrete fracture under uniaxial compression

(redrawing from ASTM C39/C39M-04).

Figure 3.4. Baldwin loading frame used for uniaxial/triaxial compression testing.

Figure 4.1. Fitting the proposed strength criterion to the data of Johnston (1985) on

Westerly Granite.

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Figure 4.2. Accordance coefficient for strength criteria with pairs of tensile data.

Figure 4.3. Observed value against predicted value by proposed strength criterion for

concrete specimens with CQD-VP and UCS130 MPa.


𝜀𝜀𝑐𝑐~𝑝𝑝𝑝𝑝 is the equivalent compressive plastic strain, 𝑑𝑑𝑐𝑐 is the compressive

damage variable, 𝐸𝐸0 is the initial elastic stiffness, 𝜎𝜎𝑐𝑐0 is the value of

initial yield in compression, and 𝜎𝜎𝑐𝑐𝑐𝑐 is the ultimate uniaxial stress.

Figure 5.2. Mechanical response of concrete to uniaxial loading in tension. 𝜀𝜀𝑡𝑡~𝑝𝑝𝑝𝑝 is

the equivalent tensile plastic strain, 𝑑𝑑𝑡𝑡 is the tensile damage variable,

𝐸𝐸0 is the initial elastic stiffness, and 𝜎𝜎𝑡𝑡0 is value of initial yield in tension.

Figure 5.3. Effect of the compression stiffness recovery.

Figure 5.4. Definition of the cracking strain 𝜀𝜀𝑡𝑡~𝑐𝑐𝑐𝑐 used for the definition of tension

stiffening data for CDP model. 𝜀𝜀𝑡𝑡~𝑝𝑝𝑝𝑝 is the equivalent tensile plastic

strain, 𝑑𝑑𝑡𝑡 is the tensile damage variable, 𝐸𝐸0 is the initial elastic stiffness,

and 𝜎𝜎𝑡𝑡0 is the value of initial yield in tension.

Figure 5.5. Post-failure stress-displacement curve.

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Figure 5.6. Post-failure stress-fracture energy curve.

Figure 5.7. Definition of the compressive inelastic (crushing) strain 𝜀𝜀𝑐𝑐~𝑖𝑖𝑖𝑖 used for the

definition of compression hardening data. 𝜀𝜀𝑐𝑐~𝑝𝑝𝑝𝑝 is the equivalent

compressive plastic strain, 𝑑𝑑𝑐𝑐 is the compressive damage variable, 𝐸𝐸0 is

the initial elastic stiffness, 𝜎𝜎𝑐𝑐0 is the value of initial yield in compression,

and 𝜎𝜎𝑐𝑐𝑐𝑐 is the ultimate uniaxial stress.

Figure 5.8. Yield surface in plane stress corresponding to different values of the ratio

of the second stress invariant (𝐾𝐾𝑐𝑐) on the tensile meridian. CM and TM

indicate compressive and tensile meridians respectively.

Figure 5.9. Uniaxial tensile test conducted on concrete class CQD-G.

Figure 5.10. Uniaxial compressive test conducted on concrete class CQD-G.

Figure 5.11. Yield surface in plane stress. The values of 𝜎𝜎𝑏𝑏0 and 𝜎𝜎𝑐𝑐0 can be extracted

from this curve. 𝛼𝛼 is a material constant, 𝜎𝜎𝑐𝑐0 is the uniaxial compressive

strength, 𝜎𝜎𝑡𝑡0 is the uniaxial tensile strength, �̅�𝑝 is the effective hydrostatic

pressure, and 𝑞𝑞� is the Mises equivalent effective stress.

Figure 5.12. Concrete specimen instance modeled in Abaqus.

Figure 5.13. Assembled parts as an instance (the whole concrete specimen) in Abaqus.

Figure 5.14. Loading system’s amplitudes applied on the model in Abaqus.

Figure 5.15. Loads configuration for the concrete specimen model in Abaqus.

Figure 5.16. Mesh generated for the concrete specimen model in Abaqus.

Figure 5.17. Strain energy density.

Figure 5.18. Scalar stiffness degradation.

Figure 5.19. Mises stress component.

Figure 5.20. Damage.

Figure 5.21. Max principal strain component.

Figure 5.22. Logarithmic max principal strain component.

Figure 5.23. Equivalent plastic strain.

Figure 5.24. Damage dissipation energy density.

Figure 5.25. Plastic dissipation energy density.

Figure 5.26. Displacement.

Figure 5.27. Magnitude of reaction forces.

Figure 5.28. Fracture pattern obtained from Abaqus FEA using constitutive

parameters identified in this research for CDP model for CQD-G

concrete specimen in comparison to actual fracture pattern obtained from

test for concrete specimens categorized in CQD-G.

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LIST OF SYMBOLS AND ABBREVIATIONS

𝛽𝛽 A factor in Bineshian criterion that defines RUCS in terms of AUCS

𝜓𝜓2 Accordance coefficient

ACI American Concrete Institute

ASTM American Society for Testing and Materials

AUCS Apparent Unconfined Compressive Strength in Bineshian criterion

𝐶𝐶 Apparent Unconfined Compressive Strength in Bineshian criterion

bp Bedding planes

𝐹𝐹 CDP yield function

𝑅𝑅2 Coefficients of determination

CAE Complete Abaqus Environment; Computer Aided Engineering

𝑑𝑑𝑐𝑐 Compressive damage variable

DAMAGEC Compressive damage variable, 𝑑𝑑𝑐𝑐

CDZ Compressive Damage Zone

PEEQ Compressive equivalent plastic strain, 𝜀𝜀𝑐𝑐~𝑝𝑝𝑝𝑝

CM Compressive meridian

𝑤𝑤𝑐𝑐 Compressive weight factor

CDP Concrete Damaged Plasticity

𝜆𝜆 Concrete/rock/coal constant in Bineshian strength criterion

𝜁𝜁 Concrete/rock/coal constant in Bineshian strength criterion

CQD Concrete Quality Designation

𝜀𝜀𝑡𝑡~𝑐𝑐𝑐𝑐 Cracking strain

𝛼𝛼 Dimensionless material constant

𝛾𝛾 Dimensionless material constant

U Displacement

𝜎𝜎�𝑐𝑐 Effective compressive cohesion stress

�̅�𝑝 Effective hydrostatic pressure

𝜎𝜎�𝑡𝑡 Effective tensile cohesion stress

𝜎𝜎𝑐𝑐′ Effective unconfined compressive strength

ALLDMD Energy dissipated in the whole or partial model by damage

DMENER Energy dissipated per unit volume by damage

EDMDDEN Energy dissipated per unit volume in the element by damage

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εc~pl Equivalent compressive plastic strain

𝜀𝜀𝑐𝑐~̇𝑝𝑝𝑝𝑝 Equivalent compressive plastic strain rates

𝜀𝜀𝑡𝑡~𝑝𝑝𝑝𝑝 Equivalent tensile plastic strain

𝜀𝜀𝑡𝑡~̇𝑝𝑝𝑝𝑝 Equivalent tensile plastic strain rate

E Excellent quality concrete class in CQD classification

F Fair quality concrete class in CQD classification

FEA Finite Element Analysis

𝐺𝐺𝑓𝑓 Fracture energy

𝜑𝜑 Friction angle in degree

𝑠𝑠𝑐𝑐 Function of the compressive stress state

𝑠𝑠𝑡𝑡 Function of the tensile stress state

GF Gauge Factor

GB General-purpose Blended cement type in accordance with AS 3972

GP General-purpose Portland cement type in accordance with AS 3972

G Good quality concrete class in CQD classification

HE High Early strength cement type in accordance with AS 3972

𝜀𝜀𝑐𝑐~𝑖𝑖𝑖𝑖 Inelastic or crushing strain

𝐸𝐸0 Initial or undamaged elastic stiffness of concrete

𝐷𝐷0𝑒𝑒𝑝𝑝 Initial or undamaged elasticity matrix

𝑆𝑆 Internal cohesive strength

ISRM International Society for Rock Mechanics

LE Logarithmic maximum principal strain component

〈. 〉 Macauley bracket

RF Magnitude of reaction forces

mc Main cleats

𝜎𝜎1′ Major effective principal stress

𝜎𝜎1 Major principal stress at failure

𝜎𝜎��𝑚𝑚𝑚𝑚𝑚𝑚 Maximum principal effective stress

PE Maximum principal strain component

𝜎𝜎3′ Minor effective principal stress

𝜎𝜎3 Minor principal stress at failure

𝑞𝑞� Mises equivalent effective stress

S Mises stress component

𝑟𝑟(𝜎𝜎�) Multiaxial stress weight factor

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𝜎𝜎𝑖𝑖 Normal stress at failure

𝜎𝜎1𝑒𝑒𝑚𝑚𝑝𝑝 Observed value of 𝜎𝜎1

𝑓𝑓𝑐𝑐 Peak stress

PENER Plastic dissipation energy density

PLT Point Load Test

P Poor quality concrete class in CQD classification

𝑓𝑓𝑖𝑖 Predefined field variables

𝜎𝜎𝚤𝚤� Principal stress components

𝐾𝐾𝑐𝑐 Ratio of the second stress invariant on the tensile meridian

RUCS Regulated Unconfined Compressive Strength in Bineshian criterion

𝜎𝜎𝑐𝑐 Regulated Unconfined Compressive Strength in Bineshian criterion

𝑑𝑑 Scalar stiffness degradation variable

𝜏𝜏 Shear stress at failure

𝐵𝐵 Slope of Mohr-Coulomb linear envelope

𝜆𝜆 Slope of the complete force–displacement curve

𝑙𝑙0 Specimen length

SDEG Stiffness degradation variable, 𝑑𝑑

𝐾𝐾 Stiffness of loading machine

𝜀𝜀 Strain

𝜎𝜎1𝑐𝑐𝑐𝑐𝑖𝑖 Strength value calculated by failure criterion

𝜎𝜎 Stress

𝜃𝜃 Temperature

DAMAGET Tensile damage variable, 𝑑𝑑𝑡𝑡

𝑑𝑑𝑡𝑡 Tensile damage variable; post-peak tensile softening

PEEQT Tensile equivalent plastic strain, 𝜀𝜀𝑡𝑡~𝑝𝑝𝑝𝑝

TM Tensile meridian

𝑤𝑤𝑡𝑡 Tensile weight factor

ELSE The recoverable part of the energy in the element

ALLSE The recoverable part of the energy in the whole (partial) model

SENER The recoverable part of the energy per unit volume

ESEDEN The recoverable part of the energy per unit volume in the element

TML Tokyo Sokki Kenkyujo

ELDMD Total energy dissipated in the element by damage

ALLIE Total strain energy

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𝜎𝜎𝑡𝑡𝑡𝑡 Ultimate tensile stress

UCS Uniaxial Compressive Strength

𝑓𝑓𝑐𝑐′ Uniaxial compressive strength of concrete at 28 days

𝐶𝐶 Uniaxial compressive strength in Mohr-Coulomb criterion

𝜎𝜎𝑡𝑡 Uniaxial tensile strength

UTS Uniaxial tensile strength

𝑟𝑟∗(𝜎𝜎11) Unit step function

𝜎𝜎𝑐𝑐0 Value of initial yield

𝜎𝜎𝑡𝑡0 Value of tensile failure stress

VG Very Good quality concrete class in CQD classification

VP Very Poor quality concrete class in CQD classification

W/C Water-cement ratio

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LIST OF TABLES

Table 2.1. Failure criteria.

Table 3.1. Concrete mix proportion and design.

Table 3.2. Strain gauges characteristics.

Table 3.3. Triaxial testing results for concrete specimens – Batch # 1.










Table 3.13. Legend for Concrete Quality Designation introduced in this research.

Table 4.1. Evaluation of correlation between criteria and triaxial data for intact

rocks and coals.

Table 4.2. Accordance of strength criteria with actual triaxial data for different

types of limestone.

Table 4.3. Comparison of uniaxial tensile strengths calculated by the criteria with

uniaxial tensile strengths obtained from Brazilian testing on coal by

Hobbs (Sheorey, 1997; Bineshian, 2000).

Figure 4.2. Accordance coefficient for strength criteria with pairs of tensile data.

Table 4.4. Parameters for Bineshian criterion applicable to intact igneous rocks.

Table 4.5. Parameters for Bineshian criterion applicable to intact sedimentary rocks.

Table 4.6. Parameters for Bineshian criterion applicable to intact metamorphic

rocks.

Table 4.7. Parameters for Bineshian criterion applicable to coals.

Table 4.8. Evaluation of correlation of the proposed criterion and triaxial data for a

variety of concrete.

Table 4.9. Accordance of the proposed criterion with actual triaxial data for

different types of concrete.

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Table 4.10. Comparison between UCS obtained by uniaxial compression test, AUCS

(𝐶𝐶) measured using Mohr-Coulomb criterion, and RUCS (𝜎𝜎𝑐𝑐) measured

using proposed criterion.

Table 4.11. Comparison between UTS obtained by splitting tensile strength test and

𝜎𝜎𝑡𝑡 measured using proposed criterion.

Table 4.12. Difference between observed values by experiments and calculated

values by proposed strength criterion for concrete with CQD-VP,

UCS130 MPa.

15

Table 4.22. Constant values for proposed criterion’s parameters.

Table 5.1. Constitutive material parameters for concrete class CQD-G, to be used in

Concrete Damaged Plasticity (CDP) model.

Table 5.2. Concrete compression hardening.

Table 5.3. Concrete compression damage.

Table 5.4. Concrete tension stiffening.

Table 5.5. Concrete tension damage.

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ACKNOWLEDGEMENT

I would like to express my gratitude to my supervisors Prof. Arcady V. Dyskin from UWA’s School of Civil and Resource Engineering and Prof. Elena Pasternak from UWA’s School of Mechanical and Chemical Engineering for the useful comments, remarks, and engagement through the learning process of this master thesis.

Furthermore I would like to thank my loved ones, who have supported me throughout entire process, both by keeping me harmonious and helping me putting pieces together. I will be grateful forever for your love.

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CHAPTER 1:

INTRODUCTION

1.1. Preface

Mechanical response of concrete for both failure and post-failure stages is the subject of

this research. Mechanical behaviour of concrete consists of two main stages: pre-peak,

and post-peak. Pre-peak stage of mechanical response of heterogeneous and

approximately isotropic materials such as concrete and some rocks in stress–strain co-

ordinates contains elastic, elastoplastic, and brittle regions until the specimen fails at

peak stress. Peak stress is the ultimate strength of concrete. Post-peak stage of

mechanical response of aforesaid materials can be seen in four different circumstances

of post-failure regime; strain hardening, ductile behaviour, strain softening, and brittle

behaviour (Cook, 1965; Hobbs, 1971; Gobbi and Ferrara, 1995; Geel, 1998; Crowder

and Coulson, 2006; Jaeger et al., 2007).

In the pre-peak stage, a very short initial non-linear behaviour is seen that relates to

closing process of existing voids and fissures; however it is not a plastic behaviour as it

is reversible. Major mechanical response in pre-peak stage is a linear part of stress–

strain curve followed by a short brittle, ductile, or elastoplastic stage until the stress

reached the ultimate strength. In triaxial loading, increasing the strength with increasing

the stress that is accompanied by plastic deformation is called work hardening, which

mechanical behaviour of the material in this state depends upon the confining pressure.

Increasing the deformation in almost constant level of stress is called ductile behaviour

that can be seen in ductile concrete under uniaxial compression. Strain softening of

concrete is the decline of stress at increasing strain. Brittle or very brittle behaviour

happens at exactly ultimate stress, which causes a violent failure that no more strain can

be measured after the peak stress (Nishihara, 1957; Rummel and Fairhurst, 1970;

Sangha, 1972; Vonk, 1992; Choi et al., 1996; van Mier et al., 1997; Torrenti, 1986;

Shang and Song, 2006; Carpinteri and Brighenti, 2010).

Study of both failure and post-failure responses of concrete by experiments can be

possible using load control testing machine for pre-peak stage and displacement control

loading machine for post-failure stage. Stress–strain curve should be used for pre-peak

18

region and stress–displacement curve should be used for post-peak region (Hudson et al.,

1972; Hustrulid and Robinson, 1972; Sangha and Dhir, 1972; Timco and Frederking,

1984; Chuan and Xiaohe, 1990; Yamaguchi and Chen, 1990; van Vliet and van Mier,

1996; van Mier and Ulfkjaer, 2000; Watanabe et al., 2004).

Factors influencing the post-peak mechanical response of concrete are very important;

however study of their effects is not the main goal of this research. These factors are the

relation between the stiffness of the loading machine and post-peak stiffness of concrete,

composition and geometry of specimen, boundary condition in loading process, loading

rate, allowable rotations of the loading platens, gauge length, and, the type of the

feedback signal of the data logger. Some of these factors affect the pre-peak behaviour;

however the effect of stiffness and geometry in pre-peak response is not significant. All

these factors should be taken into account in laboratory testing for studying concrete’s

mechanical response to compression and tension (Kesler, 1959; Symon, 1970; Bordia,

1971; Dhir et al., 1972; Gonnerman, 1975; Labuz and Biolzi, 1991; Jansen and Shah,

1997; Gamino et al., 2004; Nemecek and Bittnar, 2004; del Viso et al., 2008).

Mechanical response of concrete to compression and tension in triaxial loading

condition can be defined in terms of major and minor principal stresses at failure, which

can be described by failure/strength envelopes. Failure criteria are mathematical

functions that provide failure/strength envelopes for both compressive and tensile

quadrants in triaxial loading. These mathematical functions can be obtained by

analytical investigations – these types of criteria are called theoretical failure criteria, or

can be obtained by analyzing experimental data – these are called empirical criteria.

Failure criteria can be used to predict failure of concrete (Coulomb, 1776; Mohr, 1900;

Griffith, 1921; Balmer, 1952; Murrell, 1963; Fairhurst, 1964; Bieniawski, 1974; Hoek

and Brown, 1980; Ramamurthy et al., 1985; Bineshian, 2000).

Analytical and numerical methods are widely used to obtain post failure behaviour of

concrete and therefore some constitutive models are developed by researchers to

simulate concrete’s post-peak mechanical response to compression and tension. Most

applicable constitutive models are the Series Copping Model, Concrete Damage Zone

(CDZ) model, and Concrete Damaged Plasticity (CDP) model. Finite element analysis

can be used in simulations for abovementioned failure criteria and constitutive models

for both failure and post-failure stages of mechanical response of concrete to tensile and

19

compressive loading. Abaqus FEA is one of the most applicable software in this field

for concrete (Hillerborg et al., 1976; Hillerborg, 1988; Bazant, 1989; Lubliner et al.,

1989; Markeset, 1995; Lee and Fenves, 1998; Tang et al., 2000; Abaqus Theory Manual,

2002; Abaqus/CAE User’s Manual, 2011).

This thesis presents results obtained from the study of mechanical response of concrete

in tensile and compressive loading for both pre-peak and post-peak stages of complete

stress–strain curve of concrete. It aims to develop a failure criterion to model failure

behaviour of concrete for both tensile and compressive quadrants using analytical and

empirical procedures, simulating post-failure behaviour of concrete by numerical

models using finite elements methods to propose new constitutive parameters for CDP

model to be applicable in design of concrete structures, and suggestion of an empirical

concrete classification to be used in practical purposes.

The proposed failure criterion together with the proposed constitutive parameters for

Concrete Damaged Plasticity (CDP) model, and the suggested empirical classification

entitled “Concrete Quality Designation (CQD)” in terms of uniaxial compressive

strength are the major outcomes of the present dissertation that can be used in analytical,

numerical, and practical works in the field of concrete mechanics and technology.

1.2. Objectives

The main aims of the present research in the field of mechanical response of concrete

are:

• Developing a new comprehensive strength criterion for different types of

concrete applicable for estimating the strength for both tensile and compressive

quadrants,

• Providing practical constant values for the parameters of the proposed strength

criterion,

• Developing a new empirical concrete quality classification in terms of uniaxial

compressive strength to be applicable in practice, and finally

20

• Proposing new constitutive parameters for a Concrete Damaged Plasticity model

using numerical, analytical, and experimental procedures to simulate failure and

post-failure responses of concrete to be applicable in design of concrete

structures.

To obtain the goals, following jobs including study and laboratory experiments are

conducted:

• A comprehensive literature review of mechanical properties and behaviour of

concrete, rocks, coals, and soils, especially on the failure/strength criteria,

constitutive models, post-peak strain softening, concrete technology, and

numerical modelling and simulation of fracture and failure of concrete,

• Concrete batching, composition and mix design, and preparation in accordance

with ASTM C192/C192M-05, ASTM C143-78, and ACI 211.1,

• Concrete specimens manufacturing and ends preparation in accordance with

ASTM C617-10,

• Uniaxial compression tests on prepared concrete specimens in accordance with

ASTM C39/C39M-04,

• Splitting tensile strength tests on prepared concrete specimens in accordance

with ASTM C496/C496M-11,

• Brazilian tests on prepared concrete and rock specimens in accordance with

ASTM D3967-08,

• Triaxial compression tests on prepared concrete specimens at desired confining

pressures in accordance with ASTM C801-98,

• Mathematical study, practice and modelling of the proposed strength criterion’s

function,

21

• Regression analysis on obtained triaxial data pairs from uniaxial/triaxial

compression and splitting/Brazilian tensile tests,

• Calculations for suggested constant values for proposed strength criterion’s

parameters,

• Preparing the suggested empirical classification for different concrete types in

terms of their uniaxial compressive strength based on practical experience of

author, and

• Finite element study to find out suitable constitutive model and its parameters to

simulate prepared and tested concrete specimens’ failure and softening

behaviour.

Conventional strain measurements using foil strain gauges are used for uniaxial

compression tests.

The next section provides the research plan’s sectors briefly, which made this research

possible.

1.3. Methodology

Research plan for present study contains four main sectors that have been conducted

during Feb 2010 to July 2011:

• Pure study sector including literature review,

• Analytical and statistical investigations,

• Experimental sector including laboratory studies and field experience, and

• Numerical modelling and simulations using FEA software.

22

1.4. Thesis Structure

The thesis is organized as follows:

• Chapter 1 (the current chapter) is an introduction to the whole thesis containing

objectives and the methodology applied in the research plan.

• Chapter 2 contains literature review conducted in this research including two

sections; pre-peak response and post-peak response of concrete to tension and

compression. An overview of failure criteria covering both tensile and

compressive quadrants with linear and non-linear envelopes is presented. Effects

of stiffness of specimen and loading system, and influence of geometry

including size, shape, and slenderness ratios and boundary restraint of loading

platens and specimen’s ends on strain softening are presented in this chapter as

well. A study on strain localization phenomenon as a structural response is also

considered in this chapter. Chapter 2 finally presents damaged plasticity

considering three different appropriate constitutive models for damage

behaviour of concrete.

• Main parts of experimental and empirical investigation conducted in this

research are presented in Chapter 3. It includes concrete batch’s composition and

mix design, specimen manufacturing and ends preparation, strain measurements,

test plan, and test results. All specimens are prepared and tested in this research

in accordance with American Society for Testing and Material standards.

Concrete Quality Designation in abbreviation form of CQD as an applicable

empirical classification for concrete in terms of uniaxial compressive strength is

introduced in Chapter 3.

• New strength criterion for concrete is introduced in Chapter 4. Its performance

in prediction of tensile and compressive strength using statistical methods is

verified in this chapter. Parameters of proposed strength criterion for different

types of concrete from very poor to excellent quality concrete are calculated and

suggested in this chapter as a guide to its constant values to be used in practical

purposes.

23

• Chapter 5 presents the numerical study conducted in this research to simulate

failure and post-peak response of concrete by Concrete Damaged Plasticity

(CDP) constitutive model using Abaqus FEA. The suggested constitutive

parameters for CDP behavioral model and the methodology followed by

interpretation of results of numerical simulation conducted using Abaqus are

provided in this chapter too.

• Conclusions from all sectors of this research including experimental, empirical,

analytical, and numerical works conducted in this research are presented in

Chapter 6. This chapter also contains a section entitled “Future Research”,

which includes recommendations for future study and complementary research

on subjects similar to present dissertation. This section mainly contains

recommendations on improvements to the empirical classification called

Concrete Quality Designation that is introduced in this research by practical

experience, further experimental works to obtain more precise constant values

for the proposed strength criterion, further investigations on transition limit from

brittle to ductile behaviour of concrete types, numerical crack propagation study

of failure and strain softening of concrete, and finally calculation of Concrete

Damaged Plasticity modeling parameters for all classes of concrete that can be

done using Abaqus FEA.

24

CHAPTER 2:

MECHANICAL BEHAVIOUR OF CONCRETE

2.1. Introduction

Relationship between stress and strain for concrete under compression or tension is

called mechanical behaviour of concrete (Kiendle and Maldari, 1938). A complete

stress–strain curve of concrete under uniaxial compression (Figure 2.1) can be split to

two main portions; pre-peak region and post-peak region. The first complete

compressive stress–strain curve is obtained by Kiendle and Maldari (1938) and Brace

(1964) for concrete and rocks respectively.

Figure 2.1. Complete stress–strain diagram for concrete consisting pre-peak and post-peak responses.

Pre-peak region of complete stress–strain curve of concrete is related to the mechanical

response of concrete before failure (ultimate/peak stress), which normally consists of

three parts; plastic, elastic, and brittle. This part of curve is also called ascending

branch, strength zone, and unfailing regime. Typically, the initial short length non-linear

part does not occur and only the elastic and brittle parts can be observed. The initial

linear part of pre-peak region of complete 𝜎𝜎 − 𝜀𝜀 response of concrete is seen until the

stress level is reached to about 35% of ultimate uniaxial stress. Afterward until the peak

loading, non-linear behaviour is observed. The reason for the non-linear behaviour is

growth and possible coalescence of micro-cracks at the boundary of matrix-aggregates

of concrete (see Section 2.2). Failure occurs when a network of major cracks is formed.

𝜎𝜎

𝜀𝜀

Brittle Behaviour

Ductile Behaviour

Pre-peak Response

Post-peak Response

25

At this state the peak stress is reached while micro-cracks are merged and a critical flaw

in the matrix is propagated. The pre-peak region can be obtained by any load-control

testing machine (Figure 2.2). In load-control machines, a constant rate of load is applied

to the specimen and if the load exceeds the peak level, catastrophic failure occurs such

that its violent sound can be heard.

Figure 2.2. Load-control and displacement-control modes in compression testing. Complete 𝜎𝜎 − 𝜀𝜀 curve

is obtained only in displacement-control mode.

The post-peak stage is the mechanical response of concrete after failure. In the complete

stress–strain curve it can be seen after the peak stress and is called descending branch,

damage zone, and failing regime. The post-peak behaviour of concrete is directly

affected by the factors such as loading frame stiffness in relation to the concrete

specimen post-peak stiffness, the strain rate applied by the loading frame, and the

specimen geometry.

The descending branch of 𝜎𝜎 − 𝜀𝜀 curve often manifests itself in strain localization. Post-

peak region can be obtained by a displacement-control testing machine, which applies

small increments of displacement to concrete specimen (Figure 2.2). Strain softening

occurs when micro-cracks, which begin forming during the pre-peak portion of the

stress–strain curve, coalesce to form a zone of damage (fracture, fault, shear band or

rupture zone) that weakening the concrete so its load-carrying capacity is diminished

and therefore additional deformation of the zone of damage weakens it further and

continued softening occurs (Jansen and Shah, 1997).

Forc

e

Displacement

Load-Control

Displacement-Control

26

2.2. Pre-peak Response and Failure

Solid line in Figure 2.3 represents the pre-peak response for a concrete specimen under

uniaxial compressive loading. A load-control mode could be used to record the pre-peak

response of concrete; however, a displacement-control loading machine should be used

to obtain a complete 𝜎𝜎 − 𝜀𝜀 curve. As can be seen in Figure 2.3 the micro-cracks and

fissures are coalesced at the boundary of the aggregates and cement. It happens at 35%

of peak stress. When the load is increased to 50% of peak stress, cracks are developed

and propagate toward the matrix. By increasing the load level to about 75% of peak

stress, cracks are merged and major flaws are occurred. At the peak stress a network of

major flaws are occurred that cause failure. In this state, the specimen is not able to

carry load anymore and in displacement-control mode its strain softening can be

recorded.

Intact

Concrete

35%

Peak Stress

50%

Peak Stress

75%

Peak Stress

100%

Peak Stress

Figure 2.3. Failure process on concrete under uniaxial compression including flaw initiation and finally

coalescence of network of cracks causing failure (Bineshian, 2000).

𝜎𝜎

𝜀𝜀

%50 Ultimate Stress

%35 Ultimate Stress

%75 Ultimate Stress

Ultimate Stress (Failure happens at this level of stress)

27

When the specimen is under triaxial compression a similar mechanical behaviour can be

seen in 𝜎𝜎 − 𝜀𝜀 co-ordinates; however, study of failure behaviour in major and minor

principal stresses co-ordinates 𝜎𝜎1 − 𝜎𝜎3, or in shear strength and normal stress co-

ordinates 𝜏𝜏 − 𝜎𝜎𝑛𝑛, are more applicable.

Under triaxial loading the peak stress is replaced with failure criteria, which are surfaces

in 𝜎𝜎1 and 𝜎𝜎3 or 𝜏𝜏 and 𝜎𝜎𝑛𝑛 spaces. The first failure criterion was developed by Coulomb in

1776 (Jaeger et al., 2007). It was based on Coulomb’s (1776) friction hypothesis. After

that many functional dependences were tried to find the most accurate model to estimate

the failure behaviour. Figure 2.4 shows an example of failure envelope obtained by two

failure criterion using actual triaxial data.

Figure 2.4. Failure envelops for actual triaxial data (Source of triaxial data is Johnston, 1985).

Table 2.1 shows a brief history of failure/strength criteria. Most of them are developed

for rocks, coals, and soils, but can be used for concrete as well. Some of these criteria

provide linear strength envelope and some of them provide non-linear envelope for

triaxial test data. Some of them are applicable for compression only and some of them

can be applied for both types of stress (compression and tension). It also should be

noted that some of these criteria have limitations that restrict their comprehensiveness.

0

70

140

210

-15 -10 -5 0 5 10 15 20

σ 1 (

MPa

)

σ3 (MPa)

Bineshian Criterion R^2 = 0.996

Mohr-Coulomb R^2 = 0.893

Compressive Quadrant

Tensile Quadrant

Linear Envelope

Non-linear Envelope

Bineshian Criterion (2000)

𝝈𝝈𝟏𝟏=𝝈𝝈𝟑𝟑+𝝈𝝈𝒄𝒄 (𝝈𝝈𝒄𝒄+𝝀𝝀𝝈𝝈𝟑𝟑𝝈𝝈𝒄𝒄+𝜻𝜻𝝈𝝈𝟑𝟑

)

28

Table 2.1. Failure criteria.

Failure Criteria Mathematical Function Failure Envelope Stress Quadrants Linear Non-linear Tensile Compressive

Coulomb (1776) 𝜎𝜎1 − 𝜎𝜎3

2= �𝐶𝐶 cot𝜑𝜑 +

𝜎𝜎1 + 𝜎𝜎32

� sin𝜑𝜑

Mohr-Coulomb (1900) 𝜎𝜎1 = 𝐶𝐶 + 𝐵𝐵𝜎𝜎3

Griffith (1921) (𝜎𝜎1 − 𝜎𝜎3)2 = 8𝜎𝜎𝑡𝑡(𝜎𝜎1 + 𝜎𝜎3)

Balmer (1952) 𝜎𝜎1 = 𝜎𝜎𝑐𝑐(1 +𝜎𝜎3𝜎𝜎𝑡𝑡

)𝑏𝑏

Price (1960) 𝜎𝜎1 =1𝐴𝐴

(𝜎𝜎3 − 𝐶𝐶)1𝐵𝐵

Fairhurst (1964) (𝜎𝜎1 − 𝜎𝜎3)2 = 𝑎𝑎 + 𝑏𝑏(𝜎𝜎1 + 𝜎𝜎3)

Hobbs (1964) 𝜎𝜎1 = 𝜎𝜎𝑐𝑐 + 𝜎𝜎3 + 𝐵𝐵𝜎𝜎3𝛼𝛼

Murrell (1965) 𝜎𝜎1 = 𝜎𝜎𝑐𝑐 + 𝐵𝐵𝜎𝜎3𝛼𝛼

Mogi (1966) 𝜎𝜎1 − 𝜎𝜎3 = 𝜎𝜎𝑐𝑐 + 𝐵𝐵𝜎𝜎3𝛼𝛼

Hoek (1968) 𝜏𝜏 − 𝐶𝐶𝜎𝜎𝑐𝑐

= 𝐷𝐷(𝜎𝜎𝑛𝑛𝜎𝜎𝑐𝑐

)𝛼𝛼

Hobbs (1970) 𝜏𝜏 = 𝐶𝐶 + 𝑘𝑘𝜎𝜎𝑛𝑛𝛼𝛼

Hobbs (1971) 𝜎𝜎1 = 𝜎𝜎𝑐𝑐 + 𝑐𝑐𝜎𝜎3𝑑𝑑

Franklin (1971) 𝜎𝜎1 − 𝜎𝜎3 = 𝜎𝜎𝑐𝑐1−𝐵𝐵(𝜎𝜎1 − 𝜎𝜎3)2

Ohnaka (1973) 𝜎𝜎1 − 𝜎𝜎3𝜎𝜎𝑐𝑐

= 1 + 𝑏𝑏(𝜎𝜎3𝜎𝜎𝑐𝑐

)𝛼𝛼

Bieniawski (1974) 𝜎𝜎1𝜎𝜎𝑐𝑐

= 1 + 𝐵𝐵 �𝜎𝜎3𝜎𝜎𝑐𝑐�𝛼𝛼

Panek (1979) 𝜎𝜎1 − 𝜎𝜎3

2= 𝑎𝑎 + 𝑑𝑑(

𝜎𝜎1 + 𝜎𝜎32

)

Brook (1979) 𝜏𝜏 𝜎𝜎𝑐𝑐⁄ = 𝐵𝐵(𝜎𝜎𝑛𝑛 𝜎𝜎𝑐𝑐⁄ )𝛼𝛼

Hoek-Brown (1980) 𝜎𝜎1′

𝜎𝜎𝑐𝑐′=𝜎𝜎3′

𝜎𝜎𝑐𝑐′+ �𝑚𝑚

𝜎𝜎3′

𝜎𝜎𝑐𝑐′+ 𝑆𝑆�

0.5

Johnston-Chiu (1984) 𝜎𝜎1′

𝜎𝜎𝑐𝑐′= �1 +

𝑀𝑀𝐵𝐵𝜎𝜎3′

𝜎𝜎𝑐𝑐′�𝐵𝐵

Ramamurthy et al. (1985) 𝜎𝜎1 − 𝜎𝜎3𝜎𝜎3

= 𝐵𝐵 �𝜎𝜎𝑐𝑐𝜎𝜎3�𝛼𝛼

Kwasniewski (1987) 𝜎𝜎1 = 𝜎𝜎𝑐𝑐 + 𝑑𝑑𝜎𝜎3

Sheorey et al (1989) 𝜎𝜎1 = 𝜎𝜎𝑐𝑐�1 − (𝜎𝜎3 𝜎𝜎𝑡𝑡⁄ )�𝑏𝑏

Yoshida et al. (1990) 𝜎𝜎1 = 𝜎𝜎3 + 𝐴𝐴𝜎𝜎𝑐𝑐�𝑆𝑆 + (𝜎𝜎3 𝜎𝜎𝑐𝑐⁄ )�𝐵𝐵

Carter et al. (1991) σ1 = σ3 + �mσcσ3 + σc2

1 + w(σ3 σc⁄ )�0.5

Bineshian (2000) 𝜎𝜎1 = 𝜎𝜎3 + 𝜎𝜎𝑐𝑐 �𝜎𝜎𝑐𝑐 + 𝜆𝜆𝜎𝜎3𝜎𝜎𝑐𝑐 + 𝜁𝜁𝜎𝜎3

�

All other parameters presented in this table are criteria’s parameters that can be determined by regression analysis. 𝜎𝜎1 and 𝜎𝜎3 are the major and minor principal stresses at failure and 𝜎𝜎1′ and 𝜎𝜎3′ are the major and minor effective principal stresses. 𝜏𝜏 and 𝜎𝜎𝑛𝑛 are shear and normal stresses at failure. Applicable Not applicable

29

2.3. Post-peak Response

Post-peak region of complete 𝜎𝜎 − 𝜀𝜀 curve represents the softening behaviour (Crowder

and Coulson, 2006), which is a mixture of material and structural behaviour. The main

parameters affecting the compressive softening of concrete are (van Mier et al., 1997):

• Stiffness of the testing machine,

• Stiffness and the composition of concrete,

• Geometry of the specimen,

• Boundary condition between the loading platens and the specimen,

• Loading rate,

• Allowable rotations of the loading platens,

• The strain gauge length, and

• Type of the feedback signal.

2.3.1. Stiffness

Stiffness of the loading machine has a significant effect on the stress–strain curve of

concrete specimen. By increasing the stiffness of testing machine, reducing the violence

of fracture of the specimen is possible. Behaviour of concrete-like material in the post-

failure region depends upon the relative stiffness of the loading system and specimen. If

the stiffness of the loading system is lower than that of the specimen, the specimen will

fail violently, instead if the stiffness of the loading system is greater than that of the

specimen, failure will occur in a quasi-stable manner with the decreasing of the

specimen strength in the damage regime more or less linearly with increasing

displacement. A stiff loading machine can be a hydraulic, hybrid, or a very fast response

servo controlled loading machine. The condition for controlled fracture is determined by

the ability of the hydraulic loading system to unload rapidly. Failure during the

compression test with flexible loading frame is violent but fracture occurs during the

test with the stiff loading frame is almost non-violent. Almost all the energy stored by

the stiff machine was absorbed in non-elastic deformation and fracturing of specimen

(Cook, 1965; Rummel and Fairhurst, 1970; Salamon, 1970; Hustrulid and Robinson,

1972).

30

Stability of a compression test of a brittle specimen depends on the relation between the

stiffness of the loading machine (𝐾𝐾) and the slope of the complete force–displacement

curve of the specimen (𝜆𝜆; It is positive in unfailed regime and negative in the failing

regime). If 𝐾𝐾 + 𝜆𝜆 > 0 therefore complete 𝜎𝜎 − 𝜀𝜀 curve of the specimen in compression

test can be recorded (Salamon, 1970).

In other words, the complete process of specimen collapse can be controlled if the

longitudinal stiffness of the testing machine including load frame, actuator, load cell,

compression platens etc. is always greater than the absolute value of the slope of the

complete force–axial displacement curve for the specimen during the failure process.

The presence of any elastic member in a testing system reduces the effective stiffness of

the system, which is always less than the stiffness of any single component, Figure 2.5

(Hudson et al., 1972; Timco and Frederking, 1984; Schubert and Blumel, 2006).

Figure 2.5. Stability of machine–specimen system considering stiffness of machine, and geometry of

specimen (Hudson et al., 1972).

A testing machine can also be stiffened by reducing the height of load column

component and increasing the modulus of elements and platens area (Chuan and

Xiaohe, 1990; Abdullah, 2006).

Forc

e

Displacement

Uncontrolled Failure

Forc

e

Displacement

Controlled Failure

31

2.3.2. Geometry and Boundary Restraint

While there is no influence of the specimen shape on the ascending branch, considerable

effect on the descending branch is seen (Hustrulid and Robinson, 1972).

Strain softening behaviour is highly affected by the specimen geometry including size,

shape and slenderness ratio and boundary condition. Higher strength will obtain by

increasing the end friction and decreasing the slenderness ratio of the specimen.

Slenderness ratio of the specimen has a significant effect (increasing the brittle

behaviour with increasing the slenderness ratio) on the descending branch of complete

𝜎𝜎 − 𝜀𝜀 curve (van Mier and Vonk, 1991; Vonk, 1992; van Vliet and van Mier, 1996;

Jansen and Shah, 1997; van Mier, 1998; Gamino et al., 2004; del Viso et al., 2008).

Nemecek and Bittnar (2004) stated that there is no significant effect of specimen size on

ascending branch of stress–strain curve and nominal strength of materials like concrete,

while other researchers believe that a strong increase of uniaxial compressive strength is

observed when the specimen’s size decreases, or in other words at a constant diameter,

the more slender the specimen is, the lower the strength value is. The difference is

straightforward; decreasing the specimen height results in an increase of ductility, i.e. a

decreasing slope in descending branch of the stress–strain curve; however, in pre-peak,

the curves are almost identical, independent of specimen size (Hudson et al., 1972;

Sangha and Dhir, 1972; van Mier, 1986; van Mier and Vonk, 1991; Gobbi and Ferrara,

1995; Jansen and Shah, 1997; Pellegrino et al., 1997; van Mier et al., 1997; Gamino et

al., 2004).

It also should be noted that slenderness ratio 2 commonly used for the testing of

concrete cylinders. As the maximum size of the aggregate increases, lower ratios of

specimen diameter/maximum size of aggregate are necessary in order to obtain

maximum strength for a given concrete mix (Nishihara, 1957; Kesler, 1959; Symons,

1970; Dhir et al., 1972; Sangha, 1972; Dhir and Sangha, 1973; Gonnerman, 1975).

Pre-peak portion of 𝜎𝜎 − 𝜀𝜀 curves is independent of specimen slenderness ratio when

low friction loading platens were used meanwhile elastic energy stored during pre-peak

phase of loading increases proportionally to the specimen length (van Mier and Vonk,

32

1991; Jansen and Shah, 1997; van Mier et al., 1997; Nemecek and Bittnar, 2004; del

Viso et al., 2008).

Softening curves become less steep as the slenderness ratio decreased (Hustrulid and

Robinson, 1972; van Mier, 1984; Torrenti, 1986; Hillerborg, 1988; Yamaguchi and

Chen, 1990; van Mier and Vonk, 1991; Rokugo and Koyanagi, 1992; Vonk, 1992; Choi

et al., 1996; Jansen and Shah, 1997; del Viso et al., 2008).

There is considerable size effect on the descending branch in terms of the energy

release, Figure 2.6. Meanwhile strain softening’s stress–displacement curves calculated

from displacement of loading platens are almost identical and therefore independent of

the slenderness ratio (Sangha and Dhir, 1972; van Mier, 1984; Yamaguchi and Chen,

1990; van Mier and Vonk, 1991; van Mier and Ulfkjaer, 2000; del Viso et al., 2008).

Figure 2.6. Typical force–displacement diagrams of large/small structure (Jansen and Shah, 1997;

Gamino et al., 2004; Nemecek and Bittnar, 2004).

Softening branch is a structural property, depending on both the specimen configuration

and the applied loading as the differences in post-peak response for specimens with

different axial dimensions almost disappear completely and in fact, localized failure in

uniaxial compression is demonstrated. The size effect is applicable for both uniaxial and

triaxial tests, Figure 2.7 (van Mier, 1986).

In pre-peak regime, 𝜎𝜎 − 𝜀𝜀 diagrams should be used, whereas in post-peak regime where

localization of deformation appears, stress–deformation diagram should be used in the

spirit of fracture mechanics of concrete (van Mier, 1998).

Forc

e

Displacement

Large Structure Small Structure

33

Figure 2.7. Softening curves for specimens with different height (van Mier, 1986).

Size effect can be found in geometrically similar structures of different sizes. Specimen

geometry and boundary conditions will affect the strain softening behaviour; however,

post-peak fracture energy is essentially independent of the specimen length and this

independence confirms the occurrence of localization. For all loading systems a strong

increase of post-peak ductility was found with decreasing specimen slenderness ratio,

Figure 2.8 and 2.9 (Bordia, 1971; van Mier and Vonk, 1991; Vonk, 1992; Gobbi and

Ferrara, 1995; Jansen and Shah, 1997; van Mier et al., 1997; Geel, 1998; van Mier,

1998; Tang et al., 2000; van Mier and Ulfkjaer, 2000; Nemecek and Bittnar, 2004;

Watanabe et al., 2004; Schubert and Blumel, 2006; del Viso et al., 2008).

Figure 2.8. Stress–strain curves obtained from cylinder tests on high strength concrete using high friction

steel platens (van Mier et al., 1997).

0

1

0 0.8

Stre

ss /

Peak

Stre

ss

Displacement (mm)

L = 100 mm

L = 200 mm

L = 50 mm

0

50

100

150

0 5 10 15

𝜎𝜎 (M

Pa)

𝜀𝜀 (%)

L/D = 1

L/D = 2

L/D = 0.5

34

Figure 2.9. 𝜎𝜎 − 𝜀𝜀 curves for normal strength concrete specimens of different geometry loaded between

rigid high friction steel platens (Gobbi and Ferrara, 1995; van Mier et al., 1997).

2.3.3. Strain Localization

During triaxial testing of brittle materials, numerous cracks were developed near the

specimen surface and propagated toward the middle of the specimen, then a non-

uniform micro-fracture field is generalized within the specimen and eventually leads to

the typical shear band that is generated, what is happen now is strain localization

(Yukutake, 1989; Abdulla and Kiousis, 1997). Due to this localization, fractured areas

are more deformed while un-fractured portions recover their deformations (Tang et al.,

2000).

The softening is found to be a highly localized phenomenon (van Mier, 1984; Markeset

and Hillerborg, 1995; van Mier and Ulfkjaer, 2000). This behaviour is called the strain

localization and is considered as being a structural response rather than a material

characteristic.

Because of localization, the post-peak portion of stress–strain curve is also dependent

on specimen size (as discussed in previous section). Most experimental data indicate

that localization occurs before or at maximum load, therefore strain softening of

concrete in uniaxial compression tests, appears after localization of damage (Vonk,

1992; van Vliet and van Mier, 1996; Jansen and Shah, 1997; Gamino et al., 2004), so

that the post-peak response is due to the deformation of a structural element,

consequently if the deformation has localized before or at the peak load, the post-peak

0

40

80

0 4 8 12

𝜎𝜎 (M

Pa)

𝜀𝜀 (%)

L/D = 1

L/D = 2

L/D = 0.5

35

behaviour is not an essential property of the material, but rather a typical structural

response (Labuz and Biolzi, 1991; van Mier and Ulfkjaer, 2000; Bisby and Take, 2009).

Localization occurs for all boundary conditions, but the way in which it takes place

strongly depends on the amount of friction along the specimen–platen interface (van

Mier, 1984; Labuz and Biolzi, 1991).

With increasing the slenderness ratio the localization zone decreases in size, the cons

shaped inner parts can fully develop and the size of the specimen parts that are not

affected by the horizontal confinement (near the boundaries) increases. In the post-peak

regime, the concrete cannot be regarded as a continuum anymore, Figure 2.10 (van Vliet

and van Mier, 1996).

L = 50 mm L = 50 mm L = 100 mm L = 100 mm L = 200 mm L = 200 mm

High friction end Low friction end High friction end Low friction end High friction end Low friction end

Figure 2.10. Influence of end friction for concrete specimens under uniaxial compression (van Vliet and

van Mier, 1996).

2.3.4. Damage Plasticity

Different models for compressive softening and strain localization are developed

(Hillerborg, 1979; Bazant, 1989; Lubliner et al., 1989; Hillerborg, 1990; Markeset,

1995; Lee and Fenves, 1998; Bazant and Novak, 2000; Bazant, 2002).

Bazant (1989) classified post-peak behaviour into two response paths: one is the strain

softening localizing into a certain zone within the specimen and another one is the

unloading path that the rest of the specimen undergoes. He showed the entire specimen

response as a combination of localization and unloading response paths. He stated that

localization at strain softening in the model must begin right at the peak-stress state.

Figure 2.11 shows very close data fit for the Bazant’s model.

36

Figure 2.11. Bazant’s model fitting on 𝜎𝜎 − 𝜀𝜀 data of different slenderness ratio (Bazant, 1989 based on

van Mier’s (1984) test results).

Markeset and Hillerborg (1995) proposed a model – the Compressive Damage Zone

(CDZ) – that takes both the localized shear deformation and the deformation due to

splitting cracks into account. According to the CDZ model, the steepness of the

descending branch of a formal 𝜎𝜎 − 𝜀𝜀 curve will increase with increasing specimen

length and slenderness ratio.

Another model that is widely applicable in numerical simulations for concrete failure

behaviour is the Concrete Damaged Plasticity (CDP) constitutive model. The finite

element CDP model (e.g. in Abaqus) provides a general capability for modelling

concrete in all types of structures. It uses concepts of isotropic damage elasticity in

combination with isotropic tensile and compressive plasticity to represent the inelastic

behaviour of concrete. It can be used for plain concrete that is subject of present study.

The CDP consists of a combination of non-associated multi-hardening plasticity and

scalar (isotropic) damaged elasticity to describe the irreversible damage that occurs

during the fracturing process. It can be used in conjunction with a visco-plastic

regularization of the constitutive equations in Abaqus/Standard to improve the

convergence rate in the softening regime; however, it requires that the elastic behaviour

of the material be isotropic and linear. This section is mainly extracted from Hillerborg

et al. (1976), Lubliner et al. (1989), Lee and Fenves (1998), Abaqus theory Manual

(2002), and Abaqus/CAE User’s Manual (2011).

0

1

0 0.01

Nor

mal

ized

Stre

ss

Post-Peak Overal Mean Strain

Series Coupling Model

L = 50 mm

L = 100 mm

L= 200 mm

L = 400 mm

37

The CDP model is a continuum, plasticity-based, damage model for concrete. It

assumes that the main two failure mechanisms are tensile cracking and compressive

crushing of the concrete material. The evolution of the yield or failure surface is

controlled by two hardening variables, 𝜀𝜀𝑡𝑡~𝑝𝑝𝑝𝑝 and 𝜀𝜀𝑐𝑐

~𝑝𝑝𝑝𝑝, linked to failure mechanisms

under tension and compression loading. The model assumes that the uniaxial tensile and

compressive response of concrete is characterized by damaged plasticity (Figures 2.12

and 2.13).

Under uniaxial tension the 𝜎𝜎 − 𝜀𝜀 response follows a linear elastic relationship until the

value of the failure stress, 𝜎𝜎𝑡𝑡0, is reached. The failure stress corresponds to the onset of

micro cracking in the concrete material. Beyond the failure stress, the formation of

micro-cracks is represented macroscopically with a softening 𝜎𝜎 − 𝜀𝜀 response, which

induces strain localization in the concrete structure.

Under uniaxial compression the response is linear until the value of initial yield, 𝜎𝜎𝑐𝑐0. In

the plastic regime the response is typically characterized by stress hardening followed

by strain softening beyond the ultimate stress, 𝜎𝜎𝑡𝑡𝑡𝑡. This representation, although

somewhat simplified, captures the main features of the response of concrete. It is

assumed that the uniaxial 𝜎𝜎 − 𝜀𝜀 curves can be converted into stress versus plastic-strain

curves. Thus,

𝜎𝜎𝑡𝑡 = 𝜎𝜎𝑡𝑡(𝜀𝜀𝑡𝑡~𝑝𝑝𝑝𝑝, 𝜀𝜀𝑡𝑡

~̇𝑝𝑝𝑝𝑝,𝜃𝜃,𝑓𝑓𝑖𝑖) (2.1)

𝜎𝜎𝑐𝑐 = 𝜎𝜎𝑐𝑐(𝜀𝜀𝑐𝑐~𝑝𝑝𝑝𝑝, 𝜀𝜀𝑐𝑐

~̇𝑝𝑝𝑝𝑝,𝜃𝜃, 𝑓𝑓𝑖𝑖) (2.2)

where the subscripts 𝑡𝑡 and 𝑐𝑐 refer to tension and compression, respectively; 𝜀𝜀𝑡𝑡~𝑝𝑝𝑝𝑝 and

𝜀𝜀𝑐𝑐~𝑝𝑝𝑝𝑝 are the equivalent tensile and compressive plastic strains, 𝜀𝜀𝑡𝑡

~̇𝑝𝑝𝑝𝑝 and 𝜀𝜀𝑐𝑐~̇𝑝𝑝𝑝𝑝 are the

equivalent plastic strain rates, 𝜃𝜃 is the temperature, and 𝑓𝑓𝑖𝑖 is other predefined field

variables (𝑖𝑖 = 1,2, … ).

As shown in Figures 2.12 and 2.13, when the concrete specimen is unloaded from any

point on the strain softening branch of the 𝜎𝜎 − 𝜀𝜀 curves, the unloading response is

weakened: the elastic stiffness of the material appears to be damaged or degraded. The

degradation of the elastic stiffness is characterized by two damage variables, 𝑑𝑑𝑡𝑡 and 𝑑𝑑𝑐𝑐:

38

http://abaqus.civil.uwa.edu.au:2080/v6.10/books/usb/pt05ch20s06abm38.html%23concrete-uniaxialhttp://abaqus.civil.uwa.edu.au:2080/v6.10/books/usb/pt05ch20s06abm38.html%23concrete-uniaxial

𝑑𝑑𝑡𝑡 = 𝑑𝑑𝑡𝑡(𝜀𝜀𝑡𝑡~𝑝𝑝𝑝𝑝,𝜃𝜃,𝑓𝑓𝑖𝑖) (2.3)

𝑑𝑑𝑐𝑐 = 𝑑𝑑𝑐𝑐(𝜀𝜀𝑐𝑐~𝑝𝑝𝑝𝑝,𝜃𝜃,𝑓𝑓𝑖𝑖) (2.4)

where 𝑑𝑑𝑡𝑡 and 𝑑𝑑𝑐𝑐, are functions of the plastic strains, temperature, and field variables.

The damage variables can take values from zero, representing the undamaged material,

to one, which represents total loss of strength (0 ≤ 𝑑𝑑𝑡𝑡 ≤ 1 and 0 ≤ 𝑑𝑑𝑐𝑐 ≤ 1).

The 𝜎𝜎 − 𝜀𝜀 relations under uniaxial tension and compression loading are, respectively:

𝜎𝜎𝑡𝑡 = (1 − 𝑑𝑑𝑡𝑡)𝐸𝐸0�𝜀𝜀𝑡𝑡 − 𝜀𝜀𝑡𝑡~𝑝𝑝𝑝𝑝� (2.5)

𝜎𝜎𝑐𝑐 = (1 − 𝑑𝑑𝑐𝑐)𝐸𝐸0�𝜀𝜀𝑐𝑐 − 𝜀𝜀𝑐𝑐~𝑝𝑝𝑝𝑝� (2.6)

where 𝐸𝐸0 is the initial or undamaged elastic stiffness of the material. Effective tensile

and compressive cohesion stresses can be defined as below (The effective cohesion

stresses determine the size of the yield or failure surface):

𝜎𝜎�𝑡𝑡 =𝜎𝜎𝑡𝑡

(1−𝑑𝑑𝑡𝑡)= 𝐸𝐸0�𝜀𝜀𝑡𝑡 − 𝜀𝜀𝑡𝑡

~𝑝𝑝𝑝𝑝� (2.7)

𝜎𝜎�𝑐𝑐 =𝜎𝜎𝑐𝑐

(1−𝑑𝑑𝑐𝑐)= 𝐸𝐸0�𝜀𝜀𝑐𝑐 − 𝜀𝜀𝑐𝑐

~𝑝𝑝𝑝𝑝� (2.8)


𝜎𝜎 𝑐𝑐

𝜀𝜀𝑐𝑐

𝜎𝜎 𝑐𝑐 u

𝜎𝜎 𝑐𝑐

0

𝜀𝜀𝑐𝑐~𝑝𝑝𝑙𝑙 𝜀𝜀𝑐𝑐0𝑙𝑙

(1−𝑑𝑑𝑐𝑐)E0

E0

39

Figure 2.13. Mechanical response of concrete to uniaxial loading in tension.

Post-failure stress can be defined as a function of cracking strain, 𝜀𝜀𝑡𝑡~𝑐𝑐𝑐𝑐, which it is

defined as the total strain minus the elastic strain related to the undamaged material

(Figure 2.14):

𝜀𝜀𝑡𝑡~𝑐𝑐𝑐𝑐 = 𝜀𝜀𝑡𝑡 − 𝜀𝜀0𝑡𝑡𝑒𝑒𝑝𝑝 (2.9)

𝜀𝜀0𝑡𝑡𝑒𝑒𝑝𝑝 =𝜎𝜎𝑡𝑡𝐸𝐸0

(2.10)

Figure 2.14. Definition of the cracking strain 𝜀𝜀𝑡𝑡~𝑐𝑐𝑐𝑐 used for the definition of tension stiffening data for

CDP model.

𝜎𝜎 t

𝜀𝜀t

E0

(1−𝑑𝑑t)E0

𝜎𝜎 t 0

𝜀𝜀t0𝑙𝑙 𝜀𝜀t~𝑝𝑝𝑙𝑙

𝜎𝜎 t

𝜀𝜀t

E0

(1−𝑑𝑑t)E0

𝜎𝜎 t 0

𝜀𝜀t0𝑙𝑙 𝜀𝜀t~𝑝𝑝𝑙𝑙

E0

𝜀𝜀t~ck 𝜀𝜀te𝑙𝑙

40

The choice of the tension stiffening parameters is important since, generally, higher

tension stiffening makes it easier to obtain numerical solutions (Abaqus/CAE User’s

Manual, 2011). Too little tension stiffening will cause the local cracking failure in the

concrete to introduce temporarily unstable behaviour in the overall response of the

model. Few practical designs exhibit such behaviour, so that the presence of this type of

response in the analysis model usually indicates that the tension stiffening is

unreasonably low.

2.4. Conclusions

Mechanical behaviour of concrete is the relationship between 𝜎𝜎 and 𝜀𝜀 when concrete

specimen is under compression or tension, which consists of two regions; pre-peak and

post-peak. Pre-peak region can be obtained by any load-control testing machine in load-

control mode – 𝜎𝜎 − 𝜀𝜀 diagrams should be used – while post-peak region can be

recorded in displacement-control mode using displacement-control testing machine –

stress–deformation diagrams should be used.

Failure process of concrete is started by coalescence of micro-cracks that are occurred at

35% of peak stress after initial linear part of 𝜎𝜎 − 𝜀𝜀 curve. By increasing the load to

50%, cracks are developed and propagate toward the matrix. At about 75% of peak

stress, cracks are merged and major flaws are occurred. A network of major flaws is

occurred at peak stress that causes the failure. The specimen in this condition is not able

to carry load anymore. Strain softening occurs after or exactly at peak stress, which is a

highly localized phenomenon that is mixture of material and structural behaviour. Strain

localization is actually happened in the descending branch of complete 𝜎𝜎 − 𝜀𝜀 curve of

concrete.

There are some constitutive models that are used to describe post-peak softening of

concrete; Series Coupling Model, Compressive Damage Zone, and Concrete Damaged

Plasticity.

The main parameters affecting the strain softening of concrete are the relative stiffness

of the loading system and specimen, the concrete composition, the size and the

slenderness ratio of the specimen, boundary condition between the steel loading platens

41

and the specimen, loading and strain rate, allowable rotations of the loading platens, the

strain gauge length, and the type of the feedback signals that used for recording the data

using the data logger.

All aforesaid parameters are considered in the experimental sector of the study of failure

and post-failure of plain concrete in this research. Slenderness ratio considered in this

research is 2, which is based on the suggested value by researchers. Further explanation

regarding the experimental works conducted in this research is presented in Chapter 3.

Study of theoretical and empirical failure criteria for concrete clarified that the current

criteria contain deficiencies in providing strength envelope for both stress quadrants,

linear and nonlinear surfaces of failure, and specially comprehensiveness to follow the

trend of triaxial data at failure that cause improper estimation of strength for both

compression and tension. Considering these deficiencies assisted the author to develop a

comprehensive strength criterion for different types of concrete covering both linearity

and nonlinearity of failure behaviour in both tensile and compressive quadrants, which

is presented in Chapter 4.

CDP is selected for further study in this research. Its constitutive parameters are

determined in the numerical sector of this research. A concrete specimen with

slenderness ratio equals 2 using the determined constitutive parameters for CDP is

simulated by Abaqus FEA, which its results are presented in Chapter 5.

42

CHAPTER 3:

EXPERIMENTAL AND EMPIRICAL WORKS

3.1. Introduction

All experimental works planned for this research are conducted in the concrete

laboratory of the University of Western Australia. Previous concrete experience of the

author together with the current laboratory experiments is presented in this chapter. The

main experimental works on concrete conducted in this research is as follows:

• Batch proportion and mix design,

• Batch production,

• Specimens’ casting,

• Specimens’ curing,

• Specimens’ ends cutting,

• Specimens’ ends grinding,

• Specimens’ ends capping,

• Surface treatment for strain gauges,

• Pre-coating for strain gauges,

• Bonding the strain gauges to the specimens,

• Strain measurements,

• Uniaxial compression testing,

• Triaxial compression testing, and

• Splitting tensile strength testing.

Stress–strain curves of the uniaxial compression tests on concrete specimens and their

mechanical behavior in the failure regime are assessed together with their fracture

patterns. Failure envelopes in the 𝜎𝜎3 – 𝜎𝜎1 co-ordinates are prepared for the specimens

under triaxial compression loading as well. Results of more than 15 years experience of

author in classification of the concrete types based on their uniaxial compressive

strength at 28 days under the name of Concrete Quality Designation is presented too.

The standards used in this research in preparations of the specimens and in testing

methods are also presented.

43

3.2. Plain Concrete Preparation

Plain concrete is selected in this research as the modeling material to study the failure

and post-failure of concrete. Totally 102 specimens are manufactured from plain

concrete batches prepared in laboratory.

ACI 211.1 and ASTM C192/C192M-05 are used in preparing the batches and

specimens.

Procedure for concrete batch’s proportion, mix design, and required specimens

preparation is discussed in details in following sub-sections.

3.2.1. Mix Proportion and Design

Batch mixtures in this research are designed based on binder types and percentages,

aggregate type, size, and gradation, compressive strength for 28 days age, and W/C

ratio. The W/C proportioning method based on ACI 211.1 is used in this research.

Table 3.1. Concrete mix proportion and design.

Batch

(#)

UCS

(MPa)

Mix

Ratio W/C

Cement

Content

(Kg/m3)

Cement

Type

Max.

Grain

Size

(mm)

Grain

Type

Slump*

(mm)

1 < 15 1:1.5:0 > 0.80 350 GP 0.5 Fine Sand 80

2 15 – 20 1:1:0 0.70 400 GP 2.5 Garnet 60

3 20 – 40 1:1:1.5 0.45 450 GP 5 Sandstone 45

4 40 – 70 1:1.5:3 0.40 500 GB 10 Limestone 40

5 70 – 130 1:1.25:2.5 0.35 550 HE 20 Granite 35

6 > 130 1:0.75:1.5 < 0.35 650 HE 25 Granite 30

UCS is uniaxial compressive strength at 28 days (𝑓𝑓𝑐𝑐′). GP is General-purpose Portland cement type (AS 3972). GB is General-purpose Blended cement type (AS 3972). HE is High Early strength cement type (AS 3972). * Slump test is conducted in accordance with ACI ASTM C143-78.

44

3.2.2. Casting and Capping

Concrete batches in this research are casted in cylindrical 150 × 300 mm steel molds.

Specimen preparation is conducted according to ASTM C192/C192M-05. Figure 3.1

shows the mold used in this research.

Curing process of the specimens is done in a vibration-free humid room for first 48

hours and then they have been immersed and kept in water tank in 23 ± 2°C to keep

their moisture and temperature in standard condition for 28 days.

According to ASTM C617-10 the specimens’ ends are prepared based on the following

procedure:

• Removing the additional length of the specimens using vertical diamond disk

cutter.

• Grinding the ends of the specimens. An electrical spin grinding disk is used for

this purpose using an anti-abrasive high impact Mangalloy cylindrical sleeve.

• Capping the specimens’ ends using sulfur mortar, capping plates, and alignment

device.

Between the completion of capping and the time of testing all the specimens are kept in

a moist condition.

Figure 3.1. Cylindrical concrete mold used in this research.

45

3.3. Strain Measurements

Measuring the strain in uniaxial compression testing in this research is performed using

strain gauges. For this purpose single element TML Polyester Foil strain gauges with

pre-attached lead wire are used. Table 3.2 shows technical details of the strain gauges.

One strain gauge is installed on each concrete specimen in the axial direction.

Installation of the strain gauge on each specimen is done in a precise procedure

containing surface treatment of the area that the strain gauge should be placed on, pre-

coating the installation area, bonding the strain gauge on the pre-coated area, and finally

masking the strain gauge to protect against any damages. Recording the strain during

the uniaxial compression testing is conducted using a data logger with quarter bridge

circuit. Recorded data is in Volts and then converted to mm using the Gauge Factor

(GF). Finally the displacement across the gauge is divided by the gauge length to

calculate the strain. A sample of the measured strain is presented in Figure 3.2.

Table 3.2. Strain gauges characteristics.

Strain

Gauge

Resistance

(Ω)

Gauge Length

(mm)

Gauge Width

(mm)

Backing Length

(mm)

Backing

Width (mm) GF

PFL-30-11 120 30 2.30 40 7 2.14

PFL-20-11 120 20 1.20 28 6 2.14

PFL-10-11 120 10 0.90 18 6 2.14

GF is Gauge Factor.

Figure 3.2. 𝜎𝜎 – 𝜀𝜀 curve for concrete specimens under uniaxial compression.

0

25

50

75

0 0.002 0.004

𝜎𝜎 (M

Pa)

𝜀𝜀 (%)

UCS = 71 MPaUCS = 57 MPaUCS = 40 MPaUCS = 31 MPaUCS = 21 MPaUCS = 15 MPaUCS = 7 MPa

46

3.4. Testing Plan

Three testing methods are considered to be conducted on cylindrical concrete specimens

in this research; uniaxial compression, triaxial compression, and splitting tensile

strength testing in accordance with ASTM C39/C39M-04, ASTM C801-98, and ASTM

C496/C496M-11.

Uniaxial compression testing is considered to determine the UCS values and to study

stress–strain behavior of concrete specimens for ascending and descending branches of

𝜎𝜎 – 𝜀𝜀 curves in displacement control mode.

Uniaxial compression test method consists of applying a compressive axial load to cast

concrete cylinders until failure occurs as follows according to ASTM C39/C39M-04:

• Cleaning the specimens’ ends and loading platens.

• Centering the specimen on the lower platen and aligning the axis of the

specimen with the center of thrust of the spherically seated upper platen.

• Adjusting the load after bringing the upper platen on the upper end of the

specimen to obtain uniform seating of the specimen.

• Applying the load at a loading rate of 5.49 to 13.73 N/s until the concrete

specimen fails.

According to ASTM C39/C39M-04, concrete fails in uniaxial compression producing

the fracture patterns as sketched in Figure 3.3.

Cone Cone & Split Cone & Shear Shear Columnar

Figure 3.3. Sketches of types of concrete fracture under uniaxial compression (redrawing from ASTM

C39/C39M-04).

47

Triaxial compression testing is considered to cover the requirements to study

failure/strength envelopes in 𝜎𝜎3 – 𝜎𝜎1 co-ordinates and to evaluate the new strength

criterion proposed in this research for different types of concrete.

All triaxial compression testing in this research are conducted in accordance with

ASTM C801-98 standard.

Baldwin loading frame is used to run the uniaxial and triaxial compression testing

(Figure 3.4).

Figure 3.4. Baldwin loading frame used for uniaxial/triaxial compression testing.

Splitting tensile strength testing is used in this research to determine uniaxial tensile

strength (UTS) of concrete specimens. UTS data pairs would assist to study concrete’s

mechanical behavior in tension.

All splitting tensile testing in this research are conducted in accordance with ASTM

C496/C496M-11.

48

3.5. Test Results

Results of uniaxial tension/compression and triaxial compression testing are presented

as triaxial testing results for concrete Batch # 1 to 10 as ten data groups in Tables 3.3 to

3.12.

Each data group contains at least 5 data pairs of (𝜎𝜎3, 𝜎𝜎1). Data groups presented in

Tables 3.3 to 3.12 contains 67 qualified data pairs; negative signs represent the tensile

data pairs. These qualified data pairs are selected from 102 data pairs obtained from

testing after applying eight qualification criteria (see Section 4.3.2). Amongst the

qualification criteria considered in this study, Mogi’s transition limit (1966a; 1966b)

from brittle to ductile strength (Equation (3.1)) is also applied to data groups but this

condition can be a subject for further study that is not in present research plan.

𝜎𝜎1 > 4.40 × 𝜎𝜎3 (3.1)


Concrete Batch 𝜎𝜎3 (MPa) 𝜎𝜎1 (MPa)

# 1

-0.69 0.00 0.00 7.00 3.50 36.10 7.00 46.90

14.00 68.00 21.00 88.23 28.00 107.00 35.00 113.01

𝜎𝜎3 is confining pressure (minor principal stress) at failure. 𝜎𝜎1 is axial stress (major principal stress) at failure. Table 3.4. Triaxial testing results for concrete specimens – Batch # 2.


# 2

-0.70 0.00 0.00 15.00 1.50 26.00 3.00 39.00 5.00 44.00 7.00 60.02 9.00 64.35

𝜎𝜎3 is confining pressure (minor principal stress) at failure. 𝜎𝜎1 is axial stress (major principal stress) at failure.

49



# 3

-0.90 0.00 0.00 21.00 3.50 61.00 7.00 75.00

21.00 122.00 35.00 146.10



# 4

-1.23 0.00 0.00 31.00

11.00 87.62 21.00 127.00 35.00 164.10



# 5

-2.00 0.00 0.00 39.87 5.00 77.00

10.00 99.95 15.00 119.10 20.00 140.00 25.00 152.04 30.00 167.00 35.00 182.88 40.00 200.03 45.00 209.79 50.00 226.00



# 6

-4.90 0.00 0.00 57.00 1.00 79.98

11.00 131.00 21.00 180.01 24.00 210.05

𝜎𝜎3 is confining pressure (minor principal stress) at failure. 𝜎𝜎1 is axial stress (major principal stress) at failure.

50



# 7

-7.00 0.00 0.00 71.00 2.00 106.02 8.00 169.90

31.00 221.45 𝜎𝜎3 is confining pressure (minor principal stress) at failure. 𝜎𝜎1 is axial stress (major principal stress) at failure. Table 3.10. Triaxial testing results for concrete specimens – Batch # 8.


# 8

-6.12 0.00 0.00 101.14 5.00 148.32

10.00 185.42 15.00 219.05 30.00 281.18 50.00 370.00



# 9

-8.90 0.00 0.00 131.98 1.00 150.00

10.00 190.07 21.00 220.00 30.00 290.00



# 10

-9.00 0.00 0.00 190.10

35.00 429.00 70.00 460.00

110.00 471.39

failure and post-failure aspects of mechanical response of concrete structures … ·...

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