Constitutive relationship of plain concrete under uniaxial load

Table 4.5 Mix Details and Basic Information for Ahmad Test 77Table 4.7 Summary of The Experiments Selected For Simulation 78 Table 4.11 Results of Simulation – Logarithmic λ Expression,

CONSTITUTIVE RELATIONSHIP OF PLAIN CONCRETE UNDER RAPID UNIAXIAL LOAD

LIU DING

NATIONAL UNIVERSITY OF SINGAPORE

2005

CONSTITUTIVE RELATIONSHIP OF PLAIN CONCRETE UNDER RAPID UNIAXIAL LOAD

LIU DING (M.Eng.)

A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF CIVIL ENGINEERING

NATIONAL UNIVERSITY OF SINGAPORE

2005

ACKNOWLEDGEMENT

The author wishes to express his most sincere appreciation and deep gratitude to his academic supervisors, Assoc Professor W.A.M Alwis and Assoc Professor Quek Ser Tong for their invaluable guidance, patient encouragement and comments throughout the course of this study

In addition, special appreciation to National University of Singapore for the support granted for this period of study

Finally, but never the least, extremely thanks are presented to his family members for their endless love and understanding

TABLE OF CONTENTS

ACKNOWLEDGEMENT ……… i

TABLE OF CONTENT ……… ii

LIST OF TABLES……… v

LIST OF FIGURES……… viii

NOMENCLATURE……… xvii

SUMMARY……… xx

CHAPTER 1

INTRODUCTION 1

1.1 General 1

1.2 Literature Review 3

1.2.1 Constitutive Modeling on Deformational Behaviour of Concrete 3

1.2.1.1 Introduction 4

1.2.1.2 Viscoelastic Models 5

1.2.1.3 Elastic / Viscoplastic Models 7

1.2.1.4 Endochronic Models 9

1.2.1.5 Damage Models 9

1.2.1.6 Discrete-Element Method (DEM) 10

1.2.2 Empirical Relationships 10

1.3 Aims and Objectives 13

1.3.1 Scope of Work 13

1.3.2 Thesis Layout 14

CHAPTER 2

LOADING RATE SENSITIVE BEHAVOUR OF CONCRETE AND CREEP MODELS 20

2.1 General 20

2.2 Comparison Between Creep and Dynamic Response of Concrete 20

2.2.1 Phenomenological Similarities 20

2.2.2 Similarities in Mechanism 22

2.2.2.1 The Microcracking Theory 22

2.2.2.2 Similarities 23

2.3 Creep Models 24

2.4 B3 Model 30

2.5 CEB 1990 Model 33

2.6 The Sakata Model 34

2.7 Elastic Deformation 35

2.7.1 Initial Modulus 35

2.7.2 Determination of Asymptotic Modulus 37

CHAPTER 3

APPLICATION OF CREEP MODELS FOR RAPID AXIAL COMPRESSION 42

3.1 General 42

3.2 Creep Strain Due To Varying Stress History 43

3.2.1 Effective Modulus Method (EM Method) 43

3.2.2 Method of Superposition 44

3.2.3 Rate of Creep Method (RC Method) 45

3.2.4 Rate of Creep Compliance Method (For the B3 Model Only) 47

3.3 Rapid Axial Compression Modelling 49

3.3.1 Computational Model 49

3.3.2 Time-Marching Calculation Algorithm 49

3.3.3 Convergence of the Simulation Results 53

CHAPTER 4

SIMULATION AND DISCUSSION 59

4.1 General 59

4.2 Variation of Secant Modulus 59

4.2.1 General 59

4.2.2 Stress-Time Paths 60

4.2.3 Characteristic Variation of Secant Modulus 61

4.2.4 Effects of Strains 61

4.2.5 Effect of Stress Path 62

4.2.6 Discussion 62

4.3 Rapid Axial Compression Experiments 63

4.3.1 Experimental Data 63

4.3.1.1 Deformational Characteristics 64

4.3.1.2 Strength Characteristics 64

4.4 Nonlinearity Of Stress-Strain Curves 65

4.4.1 Nonlinearity of Concrete Creep 65

4.4.2 Nonlinear Amplification of Creep Strain 66

4.4.3 Material Constatant λ 68

4.5 Simulation Results 70

4.5.1 Performance of Creep Models 71

4.5.2 Performance of Strategies for Creep Calculation 72

4.5.3 Performance of λ Expression 73

4.5.4 Comparison with CEB Emprical Formula 73

CHAPTER 5

APPLICATION OF CREEP MODELS FOR CYCLIC LOADING 99

5.1 General 99

5.2 Simulation On Cyclic Loading Test 99

5.2.1 Variation of Stress and Strain 99

5.2.2 Tentative Analysis 100

5.2.3 Simulations On Experiments 100

5.3 Discussions 101

5.3.1 Creep Model Performances 101

5.3.2 Creep Calculation Strategies Performances 103

CHAPTER 6

CONCLUSION AND RECOMMENDATION 113

6.1 General 113

6.2 Conclusions 113

6.3 Recommendations 114

REFERENCES 116

PUBLICATION 130

APPENDIX 131

Table 4.5 Mix Details and Basic Information for Ahmad Test 77

Table 4.7 Summary of The Experiments Selected For Simulation 78

Table 4.11 Results of Simulation – Logarithmic λ Expression, B3 Model, Rate of

Creep Compliance Method

Table 4.14 CEB Empirical Formula Results and Creep Model Simulation

(B3Model, Rate of Creep Compliance Method)

85

Table 4.15 CEB Empirical Formula Results and Creep Model Simulation

(CEB Model, Rate of Creep Method)

86

Table 4.16 CEB Empirical Formula Results and Creep Model Simulation

(Sakata Model, Superposition Rate of Creep Method)

87

Tables listed in Appendix

App-10 Results of Simulation – Linear λ Expression, Sakata Model, Rate of Creep

Figure 2.1 Strain Development of Concrete under Creep Phenomena 39

Figure 2.3 Components of Concrete Deformation Under Rapid Load 40Figure 2.4 General form of Strain-time Curve for Concrete Subjected to Creep 40

Figure 2.6 Flow Chart For Determination of Asymptotic Elastic Modulus 41Figure 3.1 Stress History Effect On Effective Modulus Method 56

Figure 3.4 Discretized Representation of the Specimen 58

Figure 4.10 Comparison with Test and Simulation Data (Mix B) 93

Figure 4.11 Comparison with Test and Simulation Data (Mix C) 93

Figure 4.12 Comparison with Test and Simulation Data (Mix II) 94

Figure 4.13 Comparison with Test and Simulation Data (Mix IV) 94

Figure 4.15 Curves of ν versusε According to Relationshipν =(1000ε)λ 95Figure 4.16 Relation Between λ and Compressive Strength for B3 Model

Figure 4.19 Results of Simulation - Logarithmic λ Expression, B3 Model, Rate of

Creep Method, Test 1

97

Figure 4.20 Results of Simulation - Logarithmic λ Expression, B3 Model, 98

Superposition Method, Test 1 Figure 4.21 Results of Simulation - Logarithmic λ Expression, CEB Model, Rate of

Creep Method, Test 1

98

Figure 4.22 Results of Simulation - Logarithmic λ Expression, CEB Model,

Superposition Method, Test 1

99

Figure 5.2 Simulated Stress-Strain Curves for Strain-Controlled Cyclic Loading 106Figure 5.3 Simulated Stress-Strain Curves for Stress-Controlled Cyclic Loading 106Figure 5.4 Simulated Stress-Strain Curves for Repeat Loading Case C-R4, B3

Model, Rate of Creep Compliance Method

107

Figure 5.5 Simulated Stress-Strain Curves for Repeat Loading Case C-R7, B3 Model,

Rate of Creep Compliance Method

107

Figure 5.6 Simulated Stress-Strain Curves for Repeat Loading Case C-R4, B3 Model,

Rate of Creep Method

108

Figure 5.7 Simulated Stress-Strain Curves for Repeat Loading Case C-R7, B3 Model,

Rate of Creep Method

Figure 5.10 Simulated Stress-Strain Curves for Repeat Loading Case C-R4, CEB

Model, Rate of Creep Method

110

Figure 5.11 Simulated Stress-Strain Curves for Repeat Loading Case C-R7, CEB

Model, Rate of Creep Method

110

Figure 5.12 Simulated Stress-Strain Curves for Repeat Loading Case C-R4, CEB 110

Model, Superposition Method Figure 5.13 Simulated Stress-Strain Curves for Repeat Loading Case C-R7, CEB

Model, Superposition Method

111

Figure 5.14 Simulated Stress-Strain Curves for Repeat Loading Case C-R4, Sakata

Model, Rate of Creep Method

111

Figure 5.15 Simulated Stress-Strain Curves for Repeat Loading Case C-R7, Sakata

Model, Rate of Creep Method

112

Figure 5.16 Simulated Stress-Strain Curves for Repeat Loading Case C-R4, Sakata

Model, Superposition Method

112

Figure 5.17 Simulated Stress-Strain Curves for Repeat Loading Case C-R7, Sakata

Model, Superposition Method

113

Figures listed in Appendix

App-A13 Graph for Mix C under Loading Case L1-A 157

App-A38 Graph for Mix II under Loading Case LM-B 169

App-B1 Relation Between λ and Water Content for B3 Model (Logarithmic) 177App-B2 Relation Between λ and Sand Content for B3 Model (Logarithmic) 177App-B3 Relation Between λ and Density for B3 Model (Logarithmic) 178App-B4 Relation Between λ and Compressive Strength for CEB Model

(Logarithmic)

178

App-B5 Relation Between λ and Cement Content for CEB Model (Logarithmic) 179App-B6 Relation Between λ and Water Content for CEB Model (Logarithmic) 179App-B7 Relation Between λ and Sand Content for CEB Model (Logarithmic) 180App-B8 Relation Between λ and Density for CEB Model (Logarithmic) 180

App-B9 Relation Between λ and Compressive Strength for Sakata Model

(Linear)

188

App-B25 Relation Between λ and Cement Content for Sakata Model (Linear) 189App-B26 Relation Between λ and Water Content for Sakata Model (Linear) 189App-B27 Relation Between λ and Sand Content for Sakata Model (Linear) 190App-B28 Relation Between λ and Density for Sakata Model (Linear) 190

App-C1 Results of Simulation - Logarithmic λ Expression, Sakata Model, Rate of 191

Creep Method, Test 1 App-C2 Results of Simulation - Logarithmic λ Expression, Sakata Model,

Superposition Method, Test 1

191

App-C3 Results of Simulation - Logarithmic λ Expression, B3 Model, Rate of

Creep Compliance Method, Test 2

192

App-C4 Results of Simulation - Logarithmic λ Expression, B3 Model, Rate of

Creep Method, Test 2

192

App-C5 Results of Simulation - Logarithmic λ Expression, B3 Model,

Superposition Method, Test 2

193

App-C6 Results of Simulation - Logarithmic λ Expression, CEB Model, Rate of

Creep Method, Test 2

193

App-C7 Results of Simulation - Logarithmic λ Expression, CEB Model,

Superposition Method, Test 2

194

App-C8 Results of Simulation - Logarithmic λ Expression, Sakata Model, Rate of

Creep Method, Test 2

194

App-C9 Results of Simulation - Logarithmic λ Expression, Sakata Model,

Superposition Method, Test 2

195

App-C10 Results of Simulation - Logarithmic λ Expression, B3 Model, Rate of

Creep Compliance Method, Test 17

195

App-C11 Results of Simulation - Logarithmic λ Expression, B3 Model, Rate of

Creep Method, Test 17

196

App-C12 Results of Simulation - Logarithmic λ Expression, B3 Model,

Superposition Method, Test 17

196

App-C13 Results of Simulation - Logarithmic λ Expression, CEB Model, Rate of

Creep Method, Test 17

197

App-C14 Results of Simulation - Logarithmic λ Expression, CEB Model,

Superposition Method, Test 17

197

App-C15 Results of Simulation - Logarithmic λ Expression, Sakata Model, Rate of

Creep Method, Test 17

198

App-C16 Results of Simulation - Logarithmic λ Expression, Sakata Model,

Superposition Method, Test 17

198

NOMENCLATURE

c

a Specific contents of aggregate of concrete

c Specific contents of cement of concrete

C Creep strain vector of the system in integration

g Degree of hydration at loading for B-3 model

M The mass matrix

u Perimeter of the member in contract with the atmosphere

V a Volume concentration of aggregate

V uc Volume concentration of un-hydrated cement

V cp Volume concentration of cement paste

S

w Specific contents of water of concrete

α Compliance of spring in Kelvin constitutive model

η Retardation time of Kelvin constitutive model

δ Acceleration of node j at step i

λ Material constant in nonlinear multiplier expression

µ Nonlinear multiplier

ν Coefficient in nonlinear multiplier

χ Material constant in nonlinear multiplier expression

SUMMARY

Creep models are usually rate-type constitutive models and in the case of concrete creep models, they had originally been developed for the prediction of long-term deformation Potential exploitation of similarities between the dynamic and creep behaviour of concrete is the focus of the present work An attempt is made here to extend the application of creep theories to modelling of concrete behaviour under rapid uniaxial loading

Three creep models, namely CEB model, B3 model and Sakata Model, were chosen for simulating the behaviour of concrete under various rates of rapid loading Two creep calculation strategies, namely the rate of creep method and superposition method, were applied for calculating creep In addition, another method developed specially for the B-3 model was implemented as well As a key contribution in the present work, a new creep multiplier term is proposed and implemented to enable modelling of strain-softening and unloading response To establish the applicability of the creep models under rapid loading conditions as well as to determine the level of accuracy rendered by the computational approach adopted, a wide range of reported experiments were simulated The experimental results are compared with simulated results obtained It is apparent from comparisons that the creep model could be used to simulate the axial stress-strain relationship of concrete under different rates of loading fairly satisfactorily

Keywords: creep, dynamic, rate of loading, compressive strength, secant

modulus, strain rate, uniaxial loading

The mechanical behaviour of most materials is known to be influenced significantly

by the rate of loading The quest for knowledge about the influence of loading rate on the behaviour of materials goes beyond the confines of scientific curiosity, driven by important practical engineering needs

Structures are often subjected to transient dynamic loads These dynamic loads may result, for instance, from impact by air blast, wind gusts, accidental collision and earthquake loading The strain rates associated with impact or other similar loading, are typically one million times higher than those usually associated with static loading (see

Figure 1.1) Most engineering materials, when subjected to such high strain rates, will

exhibit some changes in their response, indicating variations in material properties such as strength, stiffness, and ductility Generally, these changes would enhance their engineering performance, enabling structures to withstand impact loading better than one would have otherwise expected (Bischoff and Perry 1991)

In order to design structural components well, engineers need to understand the material behaviour and thereby be able to predict structural performance Given the behavioural complexity of impact response, computational models of material behaviour are preferred by engineers Material behaviour in such instances is characterised by the relationship among stress, stress rate, strain and strain rate

Concrete is a loading rate sensitive material of great engineering significance However, the understanding on dynamic behaviour of concrete is still not understood

sufficiently (Bischoff and Perry 1991) On the other hand, a great deal of knowledge on the long-term deformational behaviours of concrete, in terms of the phenomena known as creep and shrinkage, has been accumulated Many theoretical models also have been developed for long-term deformation of concrete (Mehta 1993) The primary interest of this study is to investigate the potential use of the established knowledge on the long terms behaviour of concrete for predicting behaviour of concrete over very short time periods under dynamic conditions

The loading rate sensitive behaviour of concrete and its constituents has been under investigation for a long time Initially, most researchers studying high strain-rate effects of concrete have been concerned with the increase of strength Many experiments have been performed to observe the variation of strength under rapid loading (Bischoff and Perry 1991) Some theories and empirical expressions for the increase of concrete strength with the rate of loading have been suggested in some of these studies With this emphasis mainly on observed strength increases, little or no attention had been paid to the changes in deformational behaviour under rapid loading

From the viewpoint of engineering mechanics, the elastic stiffness of material is a key parameter that determines the deformational response It controls the natural frequency spectrum of a given structure and therefore determining the overall response of the structure under the action of any impact load An assessment of such response would be important especially in cases where the relevant serviceability requirements are stringent

In extreme cases of long span or slim structures, P−∆effects caused by large deformation could even lead to catastrophic consequence

It is generally accepted that the elastic modulus and strength of concrete will increase

with increasing of strain rate (see Figure 1.2) Nevertheless, a form of an explicit

relationship between strain / stress rate and the elastic modulus / strength has not yet been established, a primary reason being the broad scatter of available experimental results

Empirical relationships between elastic modulus / strength and strain rates tend to be

logarithmic (Ahmad 1981, Konig and Dargel 1982, Parviz et al 1986, CEB 1988)

However, each of such relationships has been found to relate to a limited scope of experimental results

Some theoretical models that could explain the enhancement of concrete stiffness and strength have been proposed Examples are the visco-elastic theory (Flugge 1975),

plastic-fracture theory (Bazant and Kim 1979) and the endochronic theory (Bazant and

Bhat 1976) Certain fracture theories attribute this property of concrete to the cracking at the mortar-coarse aggregate interface According to these explanations, the decrease of internal micro-cracking with increase of strain rate is the main reason for the increase of dynamic elastic modulus Some endochronic theories regard the visco-elastic property of concrete as the source of the increase of dynamic elastic modulus The higher the strain rate, the lesser the development of viscous deformation concrete develops Such variations in deformation are manifested as variations in elastic modulus Elastic / viscoplastic theories have also been applied in explaining the dynamic elastic modulus In terms of applicability, the disadvantages of such theories are the limitation of their applicable ranges or the high degree of complexities

The prediction of the dynamic elastic modulus and strength of concrete subjected to rapid loading has been addressed in some reported tests Some theoretical constitutive models and empirical formulas have also been proposed Several cases are discussed in the following

1.2.1.1 Introduction

The elastic theory, associated with one of the oldest and most developed constitutive models, is regularly applied for concrete for various analytical purposes The main disadvantage of the elastic theory when applied to concrete is derivation due to the significant non-linearity of concrete deformation Equivalent elastic moduli have been introduced depending on the application and situation in order to overcome this difficulty From the viewpoint of rheology, the rate dependence of concrete deformation can be explained by viscosity Viscosity admits that deformation of concrete is decided not only

by the magnitude of the force applied, but also the rate of force application The higher the load rate, the lesser time allowed for viscous deformation to develop for a given load magnitude at the instant of its application This manifests as the rate dependence of concrete deformation Nevertheless, viscosity alone cannot explain the behaviour of concrete well, especially under dynamic loading conditions

With inclusion of many assumptions and postulates, the plastic theory, which was initially developed for metals, has also been tried out for problems of concrete These attempts have been largely unsuccessful from the viewpoint of deformation estimation Although the success of applications of elastic, viscous and plastic constitutive models in isolation had been somewhat limited, the combined models appear to capture some essential features of concrete reasonably well In rheology, the idealized features of elasticity, viscosity, and plasticity are represented by spring, dashpot, and friction elements respectively The relevant individual bodies with these properties are referred to as Hookean solid, Newtonian liquid, and St Venant body respectively These rheological models do not denote the microscopic deformation mechanism of the materials They are adopted to represent the macroscopic point of view only

A perfect elastic body is one that exhibits completely reversible deformation ability

If the load-deformation relationship is linear, it can be represented by a linear spring using the equation:

where v is the viscosity coefficient of the dashpot and x is the displacement of the piston

An ideal friction element is a block resting on a flat surface Under an applied force beyond the static friction limit, the movement takes place while the friction force is independent of the displacement or its rate The friction limit simulates the yield limit

These basic constitutive model units can be combined to build up rheological models

of varying complexity, to predict the behaviour of concrete more accurately The schematic

graphs of these mechanical units are shown in Figure 1.3

1.2.1.2 Viscoelastic Models

Combination of the elastic and viscous constitutive model units, that is viscoelastic models, has been attempted in order to model concrete behaviour

When an elastic unit and a viscous unit are connected in series with each unit taking

the same load, the model generally known as Maxwell model is obtained (see Figure 1.4) Let the elongation of the spring be x s and the extension at the dashpot be x d Then the relation between the deformation rates of the dashpot and the applied force is

P dt

P dt

dP

=+

The deformation response of the Maxwell model is shown in Figure 1.5

When the spring and dashpot units are connected in parallel, that is, both undergo

same displacement, what is known as Kelvin model (see Figure 1.6)

When a load is suddenly introduced and maintained on a Kelvin model, the element exhibits no instantaneous deformation, as the dashpot would take the entire load Thereafter, the deformation would increase with time exponentially accompanied by increasing load sharing by the spring Finally, the spring would carry the entire load The rheological equation of the model is

dt

dx v x

P= ⋅ + ⋅α

1

(1.7) where α demotes the compliance of spring and v is the viscosity of the dashpot

Introducing x = 0 when t = 0 for a constant load P, the solution of the eq.(1.7) is

Using these two basic models, more complex models can be built to simulate more complex load deformation response One of the most interesting compound models is the Burgers model, which is a series combination of Kelvin and Maxwell models The

schematic graph and deformation response of Burgers Model are shown in Figure 1.8 and

Figure 1.9 The governing equation of the Burgers model is:

2

K M K K K

M M

K

M K

dx v dt

x d

αα

αα

in which x is the overall extension and the subscript K and M stand for Kelvin and

Maxwell components respectively

More complicated viscoelastic models can be built up using the basic mechanical units The general form of their differential equations can be written as follow:

0 + 1 + 2 22 + ⋅⋅ ⋅⋅ ⋅⋅= 0 + 1 + 2 22 + ⋅⋅ ⋅⋅ ⋅⋅

dt

P d D dt

dP D P D dt

x d C dt

dx C x

1.2.1.3 Elastic / Viscoplastic Models

The classical theory of plasticity is well founded on a physical and a mathematical basis with a long history of successful applications for metals The wealth of knowledge and familiarity had led to attempts to apply the theory to frictional materials such as concrete, rocks and soils Schematic stress-strain behaviour of concrete as illustrated in

Figure1.10, in which the quasi-ductile stage can be modelled by the theory of plasticity

The use of plasticity models to describe concrete behaviour has a major advantage: it accounts, in principle, for the stress-history-dependent behaviour, which is also related to the increase of dynamic modulus The stress-history-dependent behaviour of plasticity allows for unloading and reloading, while accommodating for residual strain and making the modelling of cyclic loading viable However, some features of dynamic behaviour of concrete cannot be reproduced by the plasticity model A key feature among them is the rate dependence of failure strength and elastic modulus in both tension and compression, the subject of interest in this thesis

It has been recognised that some modifications to the classical plasticity theory have

to be made in order to realistically model the dynamic behaviour of concrete In this regard,

an elastic/viscoplastic theory has been suggested This theory possesses merits of both plastic theory and viscous theory Viscous properties introduce time dependence to the state

of stress and strain, and plastic properties introduce a path dependence of these states

The elastic/viscoplasticity theory, as formulated by Perzyna (1966), postulates that the viscous properties are not essential in the elastic region and that they become manifest only after the plastic region has been reached This permits to admit that the initial yield condition can be taken to be the same as in plasticity theory Accordingly, the total strain is expressed as the sum of elastic and inelastic components, which are calculated separately

In spite of the good approximation of behaviour of concrete that had been reported, there exist some noticeable deficiencies of this model The viscous properties are assumed

to be present in the plastic region only This assumption is not agreeable with the increase

in initial elastic modulus with strain rate, which is far away from the plastic region Accordingly, an elastic/viscoplastic model used with the purpose of solving the dynamic problems specifically is restricted to high rates of loading (from 10-3sec-1 to 1 sec-1) The elastic modulus is not considered rate dependent but constant for the specific range of rate that is being simulated For hydrostatic compressive states of stress, the model behaves

elastically Obviously, elastic/viscoplastic model is not applicable for low strain rate or static case

1.2.1.4 Endochronic Models

The endochronic theory, originally proposed by Valanis (1971) for viscoelastical materials, was first applied to concrete by Bazant and Bhat (1976) In this model, inelastic strains are not obtained from a loading surface, but from the evolution of a measure of irreversible damage, referred to as intrinsic time, which is a non-decreasing scalar variable that depends on strain increments The inelastic strains are related to the intrinsic time through a series of mapping functions depending on the current states of stress and strain Thus, the model is non-linear incrementally

It has been shown that this theory can predict many characteristics of concrete like nonlinearities, strain-hardening, strain-softening, inelastic volume dilatancy, hydrostatic pressure sensitivity, hysteretic behaviour and rate-dependency, but at the expense of complexity and a large number of coefficients, which are difficult to be calculated and meaningless physically The model agrees very well with a large number of experimental results, but involves a complicated optimal-fitting procedure Moreover, the incremental nonlinearity of the model requires iterations within each increment of loading, which would consume a great deal of time in computational efforts

1.2.1.5 Damage Models

Damage model has been used to predict the enhancement of secant modulus and

strength of concrete under rapid loading successfully Eibl et al (1999) proposed a

dynamic damage model to address the rate dependency of concrete Especially, not only current strain rate but also the full loading history are incorporated in the dynamic

constitutive model A dynamic damage variable D dyn was introduced, whose value is depended on loading history The most distinguishing characteristic of this model is that a

time delay function was implemented, which can describe the delay on the enhancement of dynamic strength during rapid loading satisfactorily

Suaris et al (1984) proposed a constitutive damage model for dynamic loading of

concrete The microcracking is modeled through a continuous damage parameter Not only the state of strain but also the rate of strain contribute to the rate of damage increase Consequently, the variation of strength of concrete is affected by these two parameters Both compressive and tension cases were addressed by this damage model

Liu et al (1997) developed a damage model to simulate the rate dependent behavior

of concrete The isotropic damage model is applied with a damage variable that is governed

by the state of strain The rate dependency is achieved by introducing the CEB empirical relationship (see Section 1.2.2) on determining the values of some damage parameters, from which the magnitude of damage variable is obtained

1.2.1.6 Discrete-Element Method (DEM)

Some scholars, like Sebastien et al (2004), Donze et al (1999), simulated the

dynamic behavior of concrete by discrete-element method (DEM) successfully The medium of discrete-element method (DEM) is naturally discontinuous and is very adapted

to dynamic problems The rate dependency property of concrete is addressed by the inertial effect and interaction force in DEM

However, it was noted that the size of one element is of the order of the biggest

heterogeneity, around 1mm in DEM analysis (Potapov et al., 1995) The simulation on the

practical structural component will involve a huge amount of elements, which makes the simulation too complicated and time consuming to carry out Although some optimizations

(Kusano et al., 1992) have been suggested with respect to this problem, the cost of

computation is still very high

1.2.2 Empirical Relationships

Some of the available empirical relationships between the dynamic modulus of

concrete, E d,dynamic strength of concrete, f d, andstrain rates,ε , are presented below They have been based on different sets of test data and significant difference can be observed among these relationships

1 Ahmad (1981) proposed that

0 6

6

78.27)(log100692748

010263076

These equations for elastic modulus and strength are based on the regression analysis

of the experimental results by the proposer of the model only

2 CEB (1988) give the recommendations as follow:

For elastic modulus:

d ( d)0.026

E E

εε

σσ

For compressive strength:

=( )1.026 for ≤30s−1f

f

d s

d s

and

= ( )1/3 for >30s−1f

f

d s

d s

in which f cu is static cubic compressive strength and E is the elastic modulus The subscript

d and s stand for the dynamic and static respectively

3 Parviz et al (1986) suggested the following relations:

For elastic modulus:

=1.241+0.111log10ε+0.127(log10ε)2

cs

cd E

464.0061

=

ts

td E

E

(1.20)

for compressive strength:

f d f [1.48 0.16(log ) 0.0127(log )2] for air dried

10 10

These equations had been derived by least-square curve fitting to the test results of

Cowell (1966), Watstein (1953), and Shah et al (1983)

4 Konig and Dargel (1982) proposed the relations below:

≤

×

×+

1

3

191.0ln

13.030.1

191.0ln

1006.910.1

s

s

εε

εε

whereγ is the increasing factor for both elastic modulus and compressive strength

5 Tadros (1970) suggested the fomular below:

f d = f0[1.14+0.03(logε)]α (1.24) where the subscripts have the same definitions as in (1.14) and (1.15)

The models and empirical relationship mentioned herein are usually imperfect in predicting the deformational behaviour of concrete under various rate of loading Some of these models are too complicated to be used in practice, such as endochronic model; some others cannot capture the strength features of concrete reasonably well, like viscoelastic model Some methods, such as DEM, are too complicated and time consuming to be used

in practical simulation On the other hand, the empirical formulas mentioned above cannot give good prediction on the response of concrete under different rates of loading Significant differences exist among those relationships Furthermore, those formulas also bear some natural weakness For example, It is known that different concretes bear different relationship between secant modulus and strain rate, which was not reflected in those empirical formulas

A new computational approach to predict the deformational response of concrete under uniaxial rapid loading more conveniently was investigated in the present study A

systematic way to apply creep theories in modelling concrete behaviour under uniaxial rapid loading was developed

Three creep models, namely CEB model, B3 model and Sakata Model, were chosen for simulating the behaviour of concrete under various rates of rapid loading Two creep calculation strategies, namely the rate of creep method and superposition method, were applied for calculating creep In addition, another method developed specially for the B-3 model was implemented as well A new creep multiplier term is proposed and implemented

to enable modelling of strain-softening and unloading response Verification was performed based on a wide range of published experiments, which proves that such new computational approach is applicable

The introduction is presented in Chapter 1 The similarities between the creep and

dynamic behaviour of concrete with reference to the rate of loading are discussed in

Chapter 2 A description on the creep models that are supposed to be used in present study

is addressed in Chapter 2 as well The method of implementing creep models for rapid uniaxial compression simulation is elaborated in Chapter 3 The application of aforementioned new approach are presented in Chapter 4, which include the study on the

variation of the secant modulus with respect to the strain rate Simulation on reported

experiments was performed in Chapter 4 as well, which incorporates the new proposed

creep multiplier that accounts for the nonlinearity of stress-strain curve of concrete The simulation on the stress-strain behaviour of concrete under cyclic loading is displayed in

Chapter 5 The conclusion and recommendation are presented in Chapter 6

Figure 1.1 Magnitude of Stain Rates with Respect to Different Loading Cases

Figure 1.2 Typical Stress-Strain Relationship of Concrete under Different Loading Rates

Strain

High strain-time rate

Low strain-time rate

10-9 10-7 10-5 10-3 10-1 10+1 10+3

Strain Rate (sec -1)

( a ) Elastic unit ( b ) Viscous unit ( c ) Plastic unit

Figure 1.3 Basic Mechanical Constitutive Model Units

Figure 1.4 Maxwell Model

P

ν αP

Figure 1.5 Deformation Response of Maxwell Model

Figure 1.6 Kelvin Model

Figure 1.7 Deformation Response of Kelvin Model

Figure 1.8 Burgers Model