Service Behaviour of Reinforced Concrete Members (Thesis)

7/27/2019 Service Behaviour of Reinforced Concrete Members (Thesis)

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Service Behaviour of Reinforced Concrete Members

A thesis

submitted in partial fulfillment of the requirements

for the degree of

Bachelor of Engineering

in

Civil Engineering (Structures)

John van Rooyen

309243947

Supervisor: Associate Professor Gianluca Ranzi

School of Civil Engineering

University of Sydney, NSW 2006

Australia

November 2012


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Disclaimers

Student Disclaimer

The work comprising this thesis is substantially my own, and to the extent that any part of

this work is not my own I have indicated that it is not my own by acknowledging the source of

that part or those parts of the work. I have read and understood the University of Sydney

Student Plagiarism: Coursework Policy and Procedure. I understand that failure to comply

with the University of Sydney Student Plagiarism: Coursework Policy and Procedure can lead

to the University commencing proceedings against me for potential student misconduct under

chapter 8 of the University of Sydney By-Law 1999 (as amended).

Departmental Disclaimer

This thesis was prepared for the School of Civil Engineering at the University of Sydney,

Australia, and describes the time dependent behaviour of reinforced concrete. The opinions,

conclusions and recommendations presented herein are those of the author and do not

necessarily reflect those of the University of Sydney or any of the sponsoring parties to this

project.


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Table of contents

Table of contents ........................................................................................................................ iiiAcknowledgements ...................................................................................................................... vAbstract ...................................................................................................................................... vChapter summary ....................................................................................................................... viList of tables and figures ........................................................................................................... viiNomenclature ............................................................................................................................. xiChapter 1 Introduction .............................................................................................................. 1

1.1. General ......................................................................................................................... 11.2. Objectives ..................................................................................................................... 2

Chapter 2 Literature review ........................................................................................................ 32.1. General ......................................................................................................................... 32.2. Shrinkage ...................................................................................................................... 32.3. Compressive creep ........................................................................................................ 72.4. Tensile creep ................................................................................................................. 92.5. Tensile strength .......................................................................................................... 102.6. Modelling time dependent behaviour .......................................................................... 10

Chapter 3 Time dependent behaviour in concrete ..................................................................... 123.1. Time dependent properties ......................................................................................... 123.2. Time dependent modelling step by step method ..................................................... 173.3. SSM assumptions ........................................................................................................ 20

Chapter 4 Cross sectional analysis ............................................................................................ 214.1. Background ................................................................................................................. 214.2. Uncracked formulation ............................................................................................... 214.3. Cracked formulation ................................................................................................... 224.4. Uncracked example layered approach ...................................................................... 244.5. Cracked example layered approach.......................................................................... 26


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4.6. Comparison of results ................................................................................................. 27Chapter 5 Finite element method .............................................................................................. 28

5.1. Assumptions and comments ....................................................................................... 285.2. Formulation ................................................................................................................ 295.3. Degrees of freedom and consistency ............................................................................ 295.4. Time dependency ........................................................................................................ 305.5. Transformation from local to global axes ................................................................... 315.6. Shrinkage .................................................................................................................... 315.7. Cracking ..................................................................................................................... 325.8.

Gaussian quadrature ................................................................................................... 33

5.9. Programming .............................................................................................................. 345.10. Cracked example ..................................................................................................... 355.11. Uncracked validation .............................................................................................. 365.12. Cracked validation .................................................................................................. 395.13. AS3600-2009 comparison ......................................................................................... 45

Chapter 6 Measurement of shrinkage profiles............................................................................ 476.1. Previous techniques .................................................................................................... 476.2. Development of new sensors ....................................................................................... 48

Chapter 7 Conclusion ................................................................................................................ 50Appendix A Comparison of cross sectional methods to analyse time dependent behaviour ...... 51

A.1. Constant Deformation ................................................................................................ 51A.2. Constant load ............................................................................................................. 56

Appendix B Step by step cross sectional analysis formulation .................................................. 59Appendix C Finite beam element formulation .......................................................................... 62

C.1. Displacement field ...................................................................................................... 62C.2. Weak Formulation ...................................................................................................... 62

Appendix D Matlab finite element program .............................................................................. 65References .................................................................................................................................. 67


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Chapter summary

Chapter 1 provides a rationale behind the thesis and outlines its key objectives

Chapter 2 describes current knowledge behind key time dependent properties including creep

and shrinkage, compares conflicting research and identifies some gaps. It also describes somemodelling considerations raised in the literature.

Chapter 3 outlines the key time dependent properties in concrete, and introduces the step by

step method for modelling time dependent behaviour.

Chapter 4 shows the development of a cross sectional method of analysis based on the step by

step method that considers axial loading and bending in cracked and uncracked sections. The

method is validated against results in the literature.

Chapter 5 presents a finite element formulation based on the step by step method outlined in

chapter 4. Implementation of the formulation in Matlab is described. Experimental results are

compared to results from the model, using uniform and non-uniform shrinkage profiles.

Chapter 6 describes the development of a sensor to measure humidity in concrete as a means

to identify the shrinkage profile.

Chapter 7 outlines the conclusions of this thesis.


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List of tables and figures

Figure 2.1: Relative magnitudes of drying and autogenous shrinkage 3

Figure 2.2: Meniscus that forms as a result of evaporation of bleed water 4

Figure 2.3: Forces acting on the meniscus 4

Figure 2.4: Electron microscope images of the formation of plastic shrinkage cracking 4

Figure 2.5: Comparison of creep in a sealed specimen (left) and creep in a drying

specimen (right)7

Figure 3.1: Development of shrinkage with time 12

Figure 3.2: Shrinkage without restraint 13

Figure 3.3: Free shrinkage strains 13

Figure 3.4: Concrete under creep with no shrinkage 14

Figure 3.5: Creep coefficient vs time 15

Figure 3.6: Generalised concrete compressive strength development over time for

normal strength concrete relative to fc(28)16

Figure 3.7: Stress strain curve for concrete 16

Figure 3.8: Stress strain curves for varying strengths of concrete 16

Figure 3.9: Discreet stress intervals used in the SSM 17

Figure 3.10: Strain in a beam under axial and bending loads where plane sections

remain plane19

Figure 4.1: Cross section divided into layers 23

Figure 4.2: Cross Section for example (all units in mm) 24

Figure 5.1: Assumed axis system 28

Figure 5.2: 7 degree of freedom beam element 29

Figure 5.3: Relationship between local element axis and global axis 31

Figure 5.4: Sampling points and weights for a cubic 33

Figure 5.5: Results from FEM for first time step 35

Figure 5.6 Cross sections for Slabs A and B used for long term deflection tests 36

Figure 5.7: Support and loading conditions for long term deflection tests 36

Figure 5.8: Shrinkage results from test cylinders 37


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Figure 5.9: Creep coefficients calculated from cylinder tests 37

Figure 5.10: Comparison of mid-span deflection as measured by experiment and by

FEM calculation38

Figure 5.11: Comparison of mid-span strains as measured by experiment and by FEMcalculation for Slabs A and B

39

Figure 5.12: Slab cross section, and support and loading conditions for long term

cracked tests (dimensions in mm)39

Figure 5.13: Beam cross section and support and loading conditions for long term

cracked tests (dimensions in mm) 40

Figure 5.14: Progression shrinkage strain profiles over time assumed for the validationtest. 42

Figure 5.15: Comparison of free shrinkage stresses as a result of non-uniform shrinkage

strains. The left hand side shows those measured by experiment and the right hand

side, those calculated by the FEM program based on an assumed shrinkage profiles.

42

Figure 5.16: Progression of the relative humidity profile over the first 7 days of curing

for a concrete prism43

Figure 5.17: Comparison of deflection s as calculated by the FEM model and as

measured by experiment for the beam43

Figure 5.18: Comparison of deflections as calculated by the FEM model and as

measured by experiment for the slab44

Figure 5.19: Comparison of cracking for the one way slab as calculated and as recorded

by experiment at 400 days44

Figure 6.1: Measuring RH by measuring electrical conductivity 47

Figure 6.2: Measuring RH in sealed cavities by direct measurement 47

Figure 6.3: Measuring RH in sealed cavities by direct measurement Invalid source

specified.48

Figure 6.4: Circuit board design for the sensor 48

Figure 6.5: Data logger used to connect to humidity sensors Invalid source specified. 49

Figure 6.6: Finished humidity sensor 49


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Figure A.1: Concrete beam subject to shrinkage and creep 51

Figure A.2: Concrete stresses over time as a result of immediate and sustained

shrinkage52

Figure A.3: Concrete stresses over time calculated using EMM 53

Figure A.4: Concrete stresses over time calculated using AEMM 54

Figure A.5: Creep assumptions in the RCM 54

Figure A.6: Concrete stress over time as a result of instant and constant shrinkage 55

Figure A.7: Strain over time as a result of instant and constant shrinkage 55

Figure A.8: concrete column subject to constant load and creep 56

Figure A.9: Axial strain over time as calculated by each cross sectional method 57

Figure A.10: Concrete stress over time as calculated by each cross sectional method 57

Figure A.11: Steel stress over time as calculated by each cross sectional method 57

Figure C.1: Admissible displacement field under the Euler-Bernoulli beam assumptions 62

Figure C.2: Generalised beam loading 63

Figure D.1: Main GUI input for FEM 65

Figure D.2: Properties GUI input for FEM 66

Table 4.1: Loading, elastic modulus, shrinkage parameters and times steps for example 24

Table 4.2: Creep coefficients for example 24

Table 4.3: Comparison of strains for cracked and uncracked methods 27

Table 6.1: Gauss-Legendre sampling points and weights 33

Table 5.2: Results from FEM 35

Table 5.3: Cross sectional results from FEM, matching those of a cross sectional

analysis.36

Table 5.4: Creep coefficients and shrinkage strain values 40

Table 5.5: Tensile strength and modulus of elasticity values 40


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Table A.1: Creep and shrinkage values calculated as per AS 3600-2009 51

Table A.2: Elastic modulus and shrinkage values used in constant loading example 56


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Nomenclature

, ,Area, first moment of area, second moment of area, respectfully,

calculated about the reference axis

, , Area, first moment of area, second moment of area, respectfully, for

concrete calculated about the reference axis

, , Area, first moment of area, second moment of area, respectfully, for steel

calculated about the reference axis

, Area of tension steel, area of compressive steel.

Matrix of cross sectional (geometric) properties

d_ref Distance from top of section to reference axis

( ) Elastic modulus of concrete at time

Elastic modulus of steel

, Creep loading vector at time

, Shrinkage loading vector at time

,, Creep factor at time for stresses applied at

Characteristic compressive strength of concrete

. Characteric flexural tensile strength of concrete

, Cracked and uncracked second moments of area

Effective second moment of area after cracking

( , ) Creep function representing elastic and creep strain per unit of stress

Long-term to short term deflection factor

Element stiffness matrix

Factor in AS3600-2009 adjusting creep for age at loading

Length of span

Moment

Cracking moment

Design service moment

, Internal axial force and bending moment resisted by the concrete

, External force and moment applied to the cross section

, Internal force and moment in the cross section

,, ,, , Cross sectional rigidities at time

Vector of external actions


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Vector of internal actions

t Time

Vector of nodal displacements

Deflection

Uniformly distributed load

Distance from the reference axis to the neutral axes

Z Section modulus

Concrete strain

Creep strain

Elastic strain

Strain at the reference axis

Shrinkage strain Final design shrinkage strain

Curvature

, Tensile web reinforcement ratio and compressive web reinforcement ratio

, Stress in the concrete at time step j

Shrinkage induced tensile stress in the concrete

( , ) Creep coefficient at time t, for loads applied at time

( , ) Ageing coefficient at time t, for loads applied at time


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Chapter 1

Introduction

1.1.GeneralReinforced concrete is widely used in the construction of high rises, bridges, floor slabs, pipes

and other structures. Compared to other methods of construction, it is low cost, durable, and

widely available.

In the design of reinforced concrete structures, self-weight can be a significant component of

total load. Sustained long term loads, such as self-weight, lead to deformation in concrete

which occurs gradually over time, in addition to that which occurs when the load is firstapplied. These time-based deformations are not insignificant. For example, it is not uncommon

for deflection in a simply supported beam to double over a period of one year. Over 30 years,

deflections can be 2.5 times those occurring instantaneously.

In addition to loading and material considerations such as those above, deflection also depends

on span and cross section. Trends in building design have required increased spans and thinner

cross sections, a result of a combination of developers wanting to maximise building floor space

and minimise storey heights, and architects pushing the limits of concrete design. Deflections

are therefore often critical in concrete design. That is, the design will be governed by

serviceability rather than strength. Rigorous methods to calculate deflection, however, are not

well understood or widely used by practicing engineers (Ranzi & Gilbert 2011).

The basis for any rigorous method to predict deflection is the interaction of creep and

shrinkage, both time dependent properties of concrete, and the inclusion of cracking which

considerably reduces the stiffness of a member and increases deflection in flexural members.


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1.2.ObjectivesThis thesis seeks to predict the behaviour of reinforced concrete members over time under

service loading using numerical models. It is the intention this work will provide some insight

into the effect the shrinkage profile has on long term behaviour.

Specific objectives are as follows:

1. To develop a cross sectional method of analysis that incorporates time dependent behaviourincluding cracking.

2. To develop a finite element program that will evaluate deflections in cracked anduncracked beams.

3. To assess the impact of the assumed shrinkage profile on calculated deflection.4. To compare the simplified method for calculating long term deflection given by AS3600-

2009 with experimental and finite element results.

5. To develop a method to measure the humidity profile through a cross section which can beused as a proxy for the shrinkage profile.


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Chapter 2

Literature review

2.1. GeneralTime dependent behaviour in concrete is a result of the interaction of creep, shrinkage,

elasticity, and tensile strength. These properties change over time, and of particular interest is

their development at early ages as this has significant bearing on cracking. This literature

review will focus on shrinkage, compressive creep, tensile creep and early age behaviour, in that

order.

2.2.ShrinkageShrinkage can be divided into four categories: plastic, drying, autogenous, carbonation and

thermal shrinkage. At early ages, the concrete goes through three phases particulate

suspension, skeleton formation and initial hardening (Nehdi & Soliman 2011). While the

concrete is wet and acts as a fluid (with particles in suspension) it may be subject to plastic

shrinkage, but as soon as the skeleton is formed drying, autogenous and thermal shrinkage

occurs. At a glance, drying shrinkage is the result of a loss of water, autogenous shrinkage a

result of chemical reactions taking place, and thermal shrinkage a consequence of temperature

changes that come about from the exothermic reactions taking place. Relative magnitudes for

normal strength concrete are shown in figure 2.1.

2.2.1.Plastic shrinkageWhen the concrete is first cast, particles settle and excess water rises to the top forming a thin

layer in a process known as bleeding. If the rate at which bleed water rises is less than the rate

at which it evaporates the water level will eventually drop below that of the surface particles

forming a meniscus as shown in figure 2.2.

Shrinka

A e of concrete

Drying shrinkage

Aute enous shrinka e

Figure 2.1: Relative magnitudes of drying and autogenous shrinkage.


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The surface tension in the meniscus of the water acts upwards as shown in figure 2.3. To

achieve equilibrium the water pressure must decrease to balance the external air pressure.

Because the concrete is wet and the particles are mobile, this pressure differential induces

shrinkage (Slowik, Schmidt & Fritzsch 2008). The mechanism is known as the capillary effect.

As evaporation continues, capillaries become smaller and the meniscus radii sharper, inducing a

greater pressure difference. Eventually the forces required by the menisci are too big, and the

pressure reaches what is known as the air entry value (Slowik, Schmidt & Fritzsch 2008), at

which point air breaks through the meniscus. This creates high localised stresses, with particles

subject to relatively large tensile forces by menisci on one side and negligible forces on the side

where air is entrained. These localised stresses can lead to what is known as plastic shrinkage

cracking (Slowik, Schmidt & Fritzsch 2008). As this shrinkage occurs while the concrete is wet,

bonds have not yet formed between concrete and reinforcing steel. The result is two-fold. On

one hand cracking is not restrained by the steel, and cracks may carry across the entire section

(Slowik, Schmidt & Fritzsch 2008). On the other hand, there are no internal restraints creating

tension in the concrete.

Images of plastic shrinkage crack formation are shown in figure 2.4.

Figure 2.4: Electron microscope images of the formation of plastic shrinkage cracking (Slowik,

Schmidt & Fritzsch 2008).

Figure 2.2: Meniscus that forms as a

result of evaporation of bleed waterFigure 2.3: Forces acting on the meniscus.


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Plastic shrinkage is determined by the rate at which evaporation and bleeding occurs. It is also

highly dependent on the rigidity of the concrete mix (Neville 1995). Plastic shrinkage increases

for increasing cement content and decreasing water content (Neville 1995).

As the concrete starts to set and a solid skeleton forms, the forces exerted by the capillary

pressures have less effect and the importance of capillary action reduces dramatically

(Wittmann 1976).

2.2.2.Drying shrinkageDrying shrinking occurs once the concrete skeleton is formed and is the result of a loss of

moisture. There are four mechanisms which are suggested to cause drying shrinkage: capillary

action, disjoining pressure, surface free energy and loss of interlayer water (RILEM 1988).

Capillary action is discussed in section 2.2.1 and also applies to drying shrinkage but with

reduced effect compared to plastic shrinkage, due to the restraint provided by the rigid

skeleton.

Disjoining pressure is the pressure that separates two parallel surfaces attracted to each other

by the Gibbs energy of the two surfaces (International Union of Pure and Applied Chemistry

2012). In concrete these surfaces are the cement particles. Surrounding each particle is a film of

adsorbed water which separates it from a layered neighbouring particle. The particles are

attracted to each other, but repelled by the film of adsorbed water resulting in a disjoining

pressure in the film of water. The water films, or small gaps between the cement particles, are

known as gel pores (Neville 1995) or micropores (RILEM 1988). There is also a network of

larger pores that are longer but not completely continuous through the concrete, called

capillary pores. If the relative humidity of the environment is lower than that of the capillary

pores, water is drawn from the micropores to maintain equilibrium. The movement of water

from the micropores reduces the thickness of the water films separating the particles and

results in shrinkage (RILEM 1988). This process is reversible so that concrete may expand if

subject to environments with higher relative humidities.

Surface free energy is related to the surface tension of solids. Atoms at the surface of a solid are

in a higher state of energy than those inside. This is because there is an imbalance of forces

with greater attractive forces between atoms of the solid, than those with the external

environment. This creates a net hydrostatic compressive force on the solid. Water adsorption,

however, reduces this imbalance, and decreases the surface free energy of the solid cement

particle. As a result the hydrostatic compression is also reduced. Thus increased water in

micropores leads to decreased surface energy and therefore expansion, while decreased water


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leads to shrinkage (RILEM 1988). The water in micropores is governed by the relative

humidity of the external environment and therefore so is shrinkage through changes in surface

free energy.

Loss of Interlayer water is governed by the loss of water between sheets of Calcium Silicate

Hydrates (CSH). CSH are one of the products from the cement reaction, also known as

hydration. (The other is tricalcium aluminate hydrate. Together these are (and have been)

referred to as the cement particles.) There is not a clear distinction between the layers of

water between CSH particles and the micropores referred to previously, however it is suggested

that a small amount of water lost in these regions can lead to a large bulk shrinkage strains

(RILEM 1988). The effect is greatest below 11% relative humidity and the shrinkage induced is

found to be partially reversible. Little information is available in the literature on the extent to

which each of these mechanisms affects drying shrinkage and in which conditions.

A final consideration regarding drying shrinkage is the development of the drying front, or

shrinkage profile within a cross section. Of the limited research that has been done in this area,

none relates to the effect on time dependent behaviour.

2.2.3.Autogenous shrinkageHydration of cement requires water which is drawn from capillary pores. This process is known

as self-desiccation. The volume of the hydrated cement (the product) is less than the reacting

constituents. This volume change is known as chemical shrinkage. It is also known as the

internal volume change. Autogenous shrinkage is caused by chemical shrinkage but is measured

as the external volume change (Nehdi & Soliman 2011). This means during the plastic stage

when the cement is still wet and the particles are mobile, chemical shrinkage is identical to

autogenous shrinkage (these effects are minor however, and not usually considered in plastic

shrinkage because hydration is minimal in the first two hours (Wittmann 1976)). As the

concrete sets, the skeleton provides some resistance to the chemical shrinkage, and autogenous

shrinkage drops below chemical shrinkage with no external supply of water (if external water is

available concrete can expand from continued hydration and water absorption (Neville 1995)).

As hardening progresses, autogenous shrinkage becomes increasingly restrained (Nehdi &

Soliman 2011). Autogenous shrinkage is typically minor for normal water to cement ratios, but

can be as large as drying shrinkage for very low water to cement ratios as in high performance

concretes (Faria, Azenha & Figueiras 2006). It is less affected by size and shape of the member

and relative humidity than drying shrinkage (Ranzi & Gilbert 2011).


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2.2.4.Thermal shrinkageThermal shrinkage (or expansion) is governed by the coefficient of thermal expansion (CTE).

However, the behaviour of concrete subject to a change in temperature depends on the

temperature gradient within the section, which can create internal stresses. This is determined

by thermal conductivity. The CTE for concrete is a result of the mixed coefficients for

aggregate and hydrated cement paste.

2.3.Compressive creepCreep can be broadly defined as deformation as a result of constant load (Neville 1995)

(excluding shrinkage deformations).

It can be divided into recoverable and non-recoverable parts. Recoverable creep, termed

delayed elastic strain, occurs immediately after loading, and constitutes approximately 10

20% of total creep (40-50% of elastic strain) (Ranzi & Gilbert 2011). It is determined by

subtracting the instantaneous elastic recovery from the total strain recovered when a load is

removed (Neville 1995). Bazant, however, points out that the separation of elastic and creep

recovery strains is often ambiguous (RILEM 1988). Generally, it is found that creep recovery is

independent of the factors that govern the magnitude of irrecoverable creep (Neville 1970).

Creep may also be defined according to whether the concrete is subject to drying or not.

Consider two identical specimens: one where drying is prevented and which is subject to

sustained compression, and one which is unsealed and subject to drying conditions, but not

loaded. Intuition would suggest the combined response of these two specimens would be the

same as a specimen subject to drying and loading. However, it is found that deformation

exceeds the sum of the strains of the two independent tests. The difference is known as drying

creep. Creep which occurs in the absence of shrinkage is known as basic creep. Figure 2.5

expresses this graphically.

Basic creep

Elastic

Time

Strain

Shrinkage

Elastic

Time

Strain

Basic creep

Drying

Figure 2.5: Comparison of creep in a sealed specimen (left)

and creep in a drying specimen (right)


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2.3.1.Factors influencing creepThere are a number of factors that influence the rate of creep in general:

It is heavily influenced by relative humidity, with lower relative humidities causingincreased creep strains (Neville 1995).

As the age at which loading occurs increases, creep decreases. It is found to be non-linearly related to the volumetric contents of cement paste and

aggregate (Neville 1995).

It is also influenced by the stiffness of the aggregate which provides restraint to thepotential creep of the paste alone (Neville 1995).

The porosity of the aggregate appears to influence creep rates (Neville 1995).

Up to a stress to strength ratio of 0.5 fc, the relationship between stress and creep islinear, but becomes non-linear after this as micro cracking occurs (Neville 1995).

It is found as temperature increases or decreases an increase the transient rate of creepoccurs (Bazant, Cusatis & Cedolin 2004).

There is a size effect where an increase in surface area to volume ratio leads to adecrease in creep.

2.3.2.Mechanisms behind creepMany theories have been proposed to explain creep behaviour including the factors that

influence it. The most relevant developments are outlined below:

1. Basic creep is explained by the cement paste being a visco-elastic material: part elastic andpart viscous (Neville 1995). Under load, the viscous phase of the cement paste flows. The

flow is a result of bonds breaking and reforming. Alone, however, this theory does not

explain drying creep, ageing or any of the influencing factors outlined above. Bazant

suggested an improvement, known as Solidification theory which still assumes a visco-elastic material, but explains the ageing of concrete by an increase in the volume fraction

of load bearing concrete. This increase is brought about by continuing cement hydration

over time. Thus, the load bearing volume fraction of the concrete increases, along with

stiffness. This explains short term ageing of concrete and importantly from a modelling

point of view also allows visco-elastic parameters to remain time independent. It was found

however, long term ageing could not be explained by solidification because volume growth

of hydrated cement is too short lived (Bazant et al. 1997).


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2. A mechanism behind drying creep was proposed by Wittman, who suggested that tensilestresses induced by shrinkage, caused microcracks in unloaded specimens, reducing

measured shrinkage. Axially loaded specimens however, are not subject to any cracking,

and shrinkage deformations are therefore greater. However, experiments with symmetrical

members under pure bending, which shrinkage does not affect, still display drying creep,showing microcracks do not explain all of drying creep (Bazant & Xi 1994). Bazant

proposed that stress induced shrinkage may explain additional drying creep. It is based on

the notion that micro-diffusion between the micro-pores (which occurs as a result of

drying) increases the ability of bonds to break and reform and therefore increases creep

(Bazant & Chern 1985). No physical explanation behind this behaviour could be found

however.

3. Bazant solved the issues in 1 and 2, with the development of microprestress theory.Microprestress is proposed to develop in the micropores as a result of differences in the

energy of the water vapour and adsorbed water. These energy differences can be brought

about by volume or temperature changes in the micropores. Because microprestress is

transmitted through the bonds that exist between the opposing walls of micropores, this

increases the breakage of these bonds, and promotes shear slip (viscous flow). Bazant

shows this not only resolves issues in 1 and 2, explaining ageing and drying creep, but also

explains the temperature effects on creep. Together, Bazant suggests the micrprestress and

solidification theories explain almost all creep behaviour and together form a grand unified

theory.

2.4.Tensile creepResearch described for creep so far is based on compressive creep. No consensus was found in

the literature as to the magnitude of tensile creep compared to compression creep with

conflicting research suggesting it was bigger, the same, and smaller (Neville 1970). Bissonnette

found that Tensile creep was subject to drying creep as in compressive creep (Bissonnette

2007). However, research done by Illston suggests drying has no influence on the magnitude of

creep in tension, while studies done by Davis et al. on plain concrete beams showed that drying

creep on the compression face was three times greater than drying creep on the tension side

(Neville 1970). Microcracking appears to play a minor role in tensile creep according to

Bissonnette, who explains that micro cracking reduces the modulus of elasticity and is found

not to be significantly changed after loading. As for compression creep, tensile creep is found to

be proportional to applied stress, up to a limit of 50 67% of short term ultimate tensile

strength (Bissonnette 2007) (Neville 1970).


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2.5.Tensile strengthTensile strength in concrete is a function of the propensity for concrete to fracture. If the

concrete is assumed to be homogenous and flawless, theoretical tensile strengths are calculated

to be 2000 times actual. The discrepancy is explained by the presence of flaws, which attracthigh stress concentrations despite low average stresses in the medium. Bigger flaws attract

bigger stress concentrations. This can lead to microscopic failures but not necessarily entire

failure. The propensity of the entire medium failing depends on the behaviour and state of the

material surrounding the local failure. The number and size of flaws is stochastic, and means

that strength is governed by probability. Therefore larger specimens are more likely to have a

great number of bigger flaws leading to reduced tensile strength. All of this may explain why

tensile strengths based on flexural tests are greater than those based on uniaxial stress (such as

the Brazilian test). There is a size effect (less material is subject to tensile stress in a flexure

test) and also a difference in the state of the material surrounding a potential flaw. In flexure,

stresses reduce as distance from the extreme tensile fibre increases reducing the likelihood of

cracks propagating (Neville 1995).

2.6.Modelling time dependent behaviourModelling time dependent behaviour requires spatial and time discretisation. Spatial

discretisation refers to breaking down a structure into constituent elements, elements into cross

sections, and cross section into layers. For each time step, the discretised structure must be

solved by iteration, where deformations are adjusted progressively until equilibrium is reached

(Kawando & Warner 1996).

Time dependent behaviour is the result of two effects those that are stress dependent such as

creep, and those that are stress independent such as shrinkage. To separate these two effects,

stress dependent behaviour is measured as the difference in deformation between a loaded

specimen and an identically sized and aged specimen that has undergone the same

environmental conditions but unloaded (Bazant 1975).

Stress-dependent behaviour over time may be modelled in essentially two ways. Firstly by an

integral-type model, and secondly by a rate-type creep model.

The integral-type model is based on the assumption that the relationship between stress and

strain is linear. This is roughly true under serviceability conditions, where stresses are less than

40% concrete strength (Neville 1995). As a result of this linearity the stress dependent

deformation may be expressed by the compliance function , , which is defined as the strain


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at time t, caused by a unit application of constant stress applied at time . It incorporates both

creep and elastic strains. Strain is then calculated as the sum of the stress changes over time

by their respective compliance functions. A shortfall of this approach is that it does not

directly model some extrinsic state variables that affect the rate of creep. Extrinsic state

variables are factors that can change creep after casting, and are properties within the material.They include things such as temperature, degree of hydration and pore humidity (Bazant

1988). To account for this, behaviours such as drying creep, are often incorporated into the

compliance function (as is done by AS3600-2009), rather than being modelled directly. Another

disadvantage of the integral-type model is that stress increments for each time period for each

discretised element or layer must be stored, decreasing computational efficiency and increasing

memory requirements (Kawano & Warner 1996). It has been found the integral type

formulation does not model creep recovery accurately, and should not be used in scenarios

where unloading occurs (stress reduction as a result of redistribution is not problematic)

(Bazant 1988).

Under the rate-type method, concrete is modelled as a viscoelastic material. That is, it

undergoes a time dependent shearing strain under shearing stress as would a liquid (albeit

highly viscous), and also undergoes a non-time dependent elastic strain as a result of an applied

stress. It is essentially represented by dampers (dashpots) and springs combined in series and

parallel as required to produce the appropriate response. This is the same method used to

model polymers, however unlike polymers, concrete is also subject to ageing. This means time

dependent behaviour in concrete is not only a function of time lag, but of time lag and the time

of loading. A result of this is that solutions must be solved numerically, not analytically

(Bazant 1975). Bazant maintains the rate-type approach is most realistic, as it is based on the

physical processes behind the solidification-microprestress theory (see section 2.3.2) and can

incorporate the effects of ageing, varying pore humidity and temperature (Bazant 1997). The

rate-type method is particularly suited to finite element applications because creep calculations

are not dependent on stress histories and therefore do not need to be stored improving

computationally efficiency. Warner, however, shows that this method can be unstable if time

discretisation is not fine enough.

Warner shows that the two methods, integral-type and rate-type, produce similarly accurate

results for given stress histories, as long as the integral-type method is not used for unloading

scenarios or where stresses in the concrete reach more than 0.4fc (Kawando & Warner 1996).

For ease of application to experimental data, and because computational efficiency is notcritical, the integral-type approach is used in this thesis.


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Chapter 3

Time dependent behaviour in concrete

3.1.Time dependent propertiesDeformations in concrete can be classified as either instantaneous or time dependent. When

subject to load, concrete will effectively deform instantly. The extent of this deformation will

depend on the stiffness of the concrete at the time, and the magnitude of the load. After

loading and instantaneous deformation, the concrete will continue to deform over time. This is

a result of three phenomena: shrinkage, creep and ageing.

3.1.1.ShrinkageAfter concrete is poured and begins to set, it will shrink as water is lost and chemical reactions

take place. This process occurs gradually, with shrinkage approaching an asymptotic upper

limit as shown in figure 3.1.

As shrinkage depends on a range of factors as outlined in chapter 2, such as aggregate type,

mix, and drying conditions, shrinkage strains can vary, but are typically in the range of200 10 to 1100 10 (Wight & Macgregor 2012). The majority of this strain is reachedwithin 100 days and can be attributed to drying. The exception to this is high performance

concretes with very low water to cement ratios which undergo significant autogenous shrinkage,

making up as much as 50% of total shrinkage strain (Yang, Sato & Kawai 2005). In

AS3600-2009 shrinkage is given as the sum of drying and endogenous shrinkage (autogenous

and thermal shrinkage), so that = + . For the purposes of this thesis, distinction isnot made between shrinkage types, and shrinkage is given simply as .

Figure 3.1: Development of shrinkage with time

Shrinkage

Time

sh


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There are some important considerations regarding the effects of restraint and shrinkage on

behaviour worth elaborating at this point. Consider a beam subject to shrinkage as shown in

figure 3.2. Without restraint, the concrete will deform without stress. If restraint is applied, the

concrete will want to shrink, but because it is restrained from doing so, will be drawn in

tension.

Restraint can be in the form of end restraints as shown in figure 3.2 or as internal restraint in

the form of reinforcing.

Shrinkage profiles need not be, and in most cases are not, linear across a section. Shrinkage

occurs more quickly the closer regions are to drying surfaces, and slower the further away they

are.

Consider a plain concrete slab drying from top and bottom only as shown in figure 3.3a. The

outer surface will shrink more than the core, as shown by the free shrinkage strains in figure

3.3b . This induces stresses that produce strains acting in the opposite direction resulting in a

uniform strain profile as shown in figure 3.3c. For design purposes it is common to assume a

uniform shrinkage profile as these effects are not usually considered to affect calculated

deflections significantly.

Figure 3.2: Shrinkage without restraint

Free shrinkage no

induced stresses

Restraint pulls the concrete

specimen into tension from

its free shrinkage state.

Figure 3.3: Free shrinkage strains

(a) Slab subject to free

shrinkage

(b) Shrinkage

strain profile

(c) Resulting strain =

elastic strains +

shrinkage strains


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3.1.2.CreepCreep describes the deformation of concrete under load over time. It is mostly irrecoverable

deformation, so that once the load is removed the concrete does not go back to its original

shape, but remains deformed. A small portion is recoverable, however the distinction between

this and instantaneous elastic deformation is not easily made.

Consider a specimen under constant load, disregarding shrinkage for the moment, as shown in

figure 3.4. Load is applied over a certain time period, and then removed. The strains that occur

as a result are shown in the strain vs time diagram, and shown schematically in the specimens

above the graph.

The magnitude of creep depends on the strength of the concrete, the age of the concrete when

loaded, the composition of the concrete, dimensions of the specimen and humidity (Wight &

Macgregor 2012). If the specimen is unsealed and allowed to dry, creep will increase, through a

process termed drying creep, discussed in chapter 2. Typical values for creep are of the order of

2.5 times instantaneous deformation.

Figure 3.4: Concrete under creep with no shrinkage

Creep strain

Elastic

recovery

Creep

recovery

Load

Load

Load

Load

Strain

Load applied

Load removed

Elastic

Creep recovery

Permanent

deformation

T Timett00

Elastic or

instantaneous

strain


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From a material modelling perspective, creep strain can be expressed as a proportion of initial

elastic strain;

, = , (3.1)Equation 3.1 may be also expressed as a function of stress:

, = , (3.2)In equations 3.1 and 3.2, , is the creep strain at some time t past the initial elasticdeformation at time , and , is known as the creep coefficient. The creep coefficient can bemeasured or calculated. AS 3600-2009 Concrete Structuresprovides a method to calculate the

creep coefficient based on empirical studies, allowing for concrete strength, humidity, exposed

concrete and concrete maturity. Accuracy of the resulting coefficient is in the order of 30%

(Standards Australia 2009). A typical curve showing the creep coefficient versus time is shown

in figure 3.5.

3.1.3.AgeingOver time, concrete strength (compressive and tensile) and stiffness gradually increase due to

the continued hydration of the cement paste and other reasons outlined in chapter 2. Creep

deformations also depend on age. For a given load, creep strains are smaller the later the load

is applied. Collectively, these effects are termed ageing. Each of these will be discussed briefly.

Compressive strength in concrete is usually specified as the lower characteristic cylinder

strength at 28 days, denoted . Standards dictate 95% of cylinder tests of the same concretemust exceed this strength. Though this thesis is not concerned with ultimate strength, the

( , )

( , )

(

,

)

Time

Figure 3.5: Creep coefficient vs time (Ranzi & Gilbert 2011)


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development of compressive strength with time, shown in figure 3.6, is important as it reveals

an aging process also associated with tensile strength and stiffness.

Compared to compressive strength, tensile strength develops over at a slower rate. As a result

the relationship between the two is not linear. AS3600-2009 gives this relationship as

. = 0.6 for tensile strength in flexure, and = 0.36 for uniaxial tensile strength.The elastic modulus is measured as the slope of the secant for the linear portion of the stress

strain curve as shown in figure 3.7. It is a measure of material stiffness, and is a result of the

combined stiffness of the cement and aggregate.

Stress

Strain

Secant modulus

(elastic modulus)

fc

Figure 3.7: Stress strain curve for concrete

80

60

40

20

0 10 20 30 40

Strength(MPa)

Strain x10-6

Figure 3.8: Stress strain curves for varying

strengths of concrete (Neville 1995)

Ratiofc(T)/

c(28)

1.4

1.0

0.6

0.2

1 3 7 28 90 365

Time (days)

Figure 3.6: Generalised concrete compressive strength development overtime for normal strength concrete relative to fc(28) (Wight & Macgregor


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From figure 3.8 it can be seen that the greater the concrete strength, the greater the elastic

modulus. It follows from this the elastic modulus must increase with time if compressive

strength does. This development of elastic modulus with time is reflected in various codes

including AS3600 and Eurocode.

3.2.Time dependent modelling step by step methodTo model time dependency, creep and shrinkage components must be included in the

expression for strain in the concrete, thus:

= + + ( ) (3.3)Where ( ) is the elastic strain, ( ) creep strain, and ( ) shrinkage strain.

There are many methods that can be used as a basis for calculating creep strain of which the

step by step method (SSM) is the most accurate and general. For brevity, explanation and

comparison of the other methods is relegated to appendix A.

The SSM is based on a stepwise approach, where gradual changes in stress are broken down

into discreet intervals as shown in figure 3.9.

For any given stress change in the concrete there will be both an elastic strain and creep strain

component, which using equation 3.2, can be given by:

+ = + ( , )

This can be expressed more compactly as:

( )

Stress

Time

( )

Figure 3.9: Discreet stress intervals used in the SSM


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+ = , ( ) (3.4)Where , is known as the creep or compliance function. It represents the combined elasticand creep strain for the time period ( ) resulting from the application of one unit of stress,and is given by:

, = 1 + , (3.5)Total elastic and creep strain in the concrete can then be calculated by summing the elastic

and creep strains for each of the changes in stress. From equation 3.3

= + + ( ) = , + , + ( ) (3.6)

Equation 3.6 can be approximated by:

= , + , + ( ) (3.7)This can be re-written in short hand as follows:

, , = , , + , ,

(3.8)

where j represents the current time step t = tj.

Equation 3.8 can be re-arranged as follows:

, , = , , + ,( , ,) + ,( , ,)

, , = , , + , , , , + , ,

, ,

,

,=

, ,+

, ,

, ,+

, ,

, ,

, , = , , + , , , , + , , + , ,

, ,

, ,

, , = , , , + , , + , , , +( ,

,) , , , = , , +( ,

,) , (3.9)

Equation 3.9 can now be solved for the stress in the concrete at time tj


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, = , ,, + , ,,

,

, = ,( , ,) + ,,

, (3.10)

where , , = ,,, (3.10a)And from equation 3.5, , = ,, since , = 0 (there is no creep because no time haselapsed).

Stress in the steel is given as

, = , (3.11)

To maintain compatibility, the strain in the concrete must match the strain in the steel at a

given position in the cross section. Thus;

, = , = (3.12)

represents the strain at any point in the cross section as shown by figure 3.10 and is given

by:

= + (3.13)

In non-time dependent analyses, the x-axis is normally set to the position of the neutral axis of

the cross section, where the first moment of area about the x-axis is zero. However, in an

analysis involving cracking over time, the position of the neutral axis changes, making it more

practical to refer to the x-axis by an arbitrary reference distance, d_ref, as shown in

figure 3.10.

x

Cross section A-A

d_ref

Strain

Figure 3.10: Strain in a beam under axial and bending loads where plane sections remain plane

y


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3.3.SSM assumptionsThere are a number of assumptions behind the SSM.

Firstly it assumes a linear relationship between stress and strain. This holds true for stresses up

to 0.4 fc, and under service loading this is a valid assumption (Bazant 1988).

Secondly, the SSM assumes the principal of superposition for creep. Creep strain (at a given

point in time) is calculated as the sum of creep strains from loads, regardless of when the loads

were placed. This has been found to agree with experimental observations when the stress

history is increasing. However, when the stress history is decreasing, creep strains are found to

be overestimated by super position (Kawando & Warner 1996).

Finally, all methods, including the SSM, are limited by the accuracy of available inputs.Tensile creep, as mentioned in chapter 2, is not well researched compared to compressive creep.

That which has been done shows conflicting results. For lack of a better alternative, this model

will assume tensile creep is the same as compressive creep.


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Chapter 4

Cross sectional analysis

4.1.BackgroundA cross sectional analysis may be used to calculate strain and curvature at a cross section

based on the moment and axial force at that point. Using the SSM as a basis, formulations are

developed for cracked and cracked sections and examples given.

4.2.Uncracked formulationAt a cross section, at any time, the internal forces will be equal to the external applied forces.

The external forces and moments at a cross section refer to the external moments and axial

forces that would need to be applied to maintain the position of the beam if the beam was cut

at that point. Thus:

= (4.1) = (4.2)

Ne and Me are the externally applied axial loads and moments, and Ni and Mi are the equal and

opposite internal resisting forces, a portion of which comes from the steel, and a portion of

which comes from the concrete. Thus;

= + (4.3) = + (4.4)

The forces and moments in the concrete are given by equations 4.5 and 4.6 respectively.

=

,

(4.5)

= , (4.6)Combining equations 3.10, 3.13 and 4.5 and 4.6 yields:

= ,( , + ,) + ,,

, (4.7) = ,( , + ,) + ,,

,

(4.8)


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After some manipulation (full derivation can be found in Appendix B), the resulting

equilibrium equation is found to be:

, = + , , (4.9)Where

, = ,, , = , ,, , , = , , , = ,, ,,

, , = ,,,, , (4.10a-e)

And ,, = , + , ,, = , + and ,, = , +

Where subscript c denotes concrete, and s steel. Equation 4.9 is then solved for strain as

follows:

= , , + , (4.11)The first time step will have no creep history so that , = . The solution to the first timestep will then be passed into equations 4.7 and 4.8. Values calculated for N0 and M0 will then

be used in the calculation of , for the next time step. The process is repeated for subsequenttime steps as necessary.

4.3.Cracked formulationThe previous section described the solution for an uncracked section, where geometrical

properties do not change with time. When cracked sections are taken into account, the concrete

cross section must be analysed in layers. The greater the number of layers in the cross section

the greater the accuracy of the solution. The appropriate number of layers can be determined

by ensuring the second moment of area, as calculated by equation 4.12c, is within 1% of the

analytical value .

To begin the analysis, it is assumed the layers are uncracked, and the cross section properties

Ac, Bc and Ic are calculated assuming each layer is a rectangle with width calculated at the

centre of the layer, and height as shown in figure 4.1. For clarity, the number of layers in the

diagram has been limited. Typical calculations could involve 500 layers.


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For the layer shown in figure 4.1, = , = , = .Ac, Bc and Ic are then given as:

,

,

(4.12a-c)

Where m is the number of layers in the cross section.

Equation 4.11 is then called, and strains calculated. These strains are used to calculate stress in

each layer with equation 3.10. Any layer with a stress greater than the tensile strength of the

concrete are ignored for the recalculation of Ac, Bc and Ic using equations 4.12a-c.

Equation 4.11 is again called, and strains, and concrete stresses recalculated and compared

with tensile strength. New concrete geometric properties are calculated. This process is

repeated until the value for strain converges to an acceptable limit.

Once this limit is reached, and strains for the first (instantaneous) time period have been

determined, it is necessary to include the values for Nc,0 and Mc,0 in the next time step. For this

purpose, the integrals in equations 4.5 and 4.6 are approximated by the stresses calculated in

the previous time step, so that:

,,

(4.13)

,,

(4.14)

Where ,, is the stress in concrete layer h, at time tj.

Layer l

d_ref

yl

h w

Figure 4.1: Cross section divided into layers


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The converging process outlined in the first time period is called again to solve for strains in

the next time period. The process continues for as many time periods as required.

It may be noted from this formulation that stresses in each layer must be stored for each time

period so that solutions for subsequent time periods can be found.

There are additional rules regarding stresses and stress histories which should be mentioned.

Firstly, the stress history in a given layer is completely removed when that layer is cracked

(when stress is greater than tensile strength). Secondly, once a layer is cracked it may only

take compressive stresses from that point in time onwards. If it does take subsequent

compressive stresses, these should become part of the layers new stored stress history.

The limiting tensile strength may also be set to 0, rather than the tensile strength, if a

conservative answer is required. It may also be set so that the concrete cannot crack, so the

resulting solution will closely match that of the uncracked solution, enabling a check of the

layered procedure.

4.4. Uncracked example layered approachStrains for the cross section in figure 4.2 are to be calculated with the parameters in tables 4.1

and 4.2, assuming a constant load and moment of -30 kN and 50 kNm respectively, dividing

the section into 500 layers.

j Ec,j sh(t)

(days) (MPa)

28 25,000 0

100 28,000 -300E-06

30,000 30,000 -600E-06

(j,i) 28 100 30,000

28 0.0

100 1.5 0.0

30,000 2.5 2.0 0.0

dst(1) = 50

dref= 200

b = 300

Ast(1) = 620

dst(2) = 550D = 600

Ast(2) = 1800

ys(1) = -150

ys(2) = +350

Figure 4.2: Cross Section for example (all units in mm)

Table 4.1: Loading, elastic modulus, shrinkage

parameters and times steps for exampleTable 4.2: Creep coefficients for example


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4.4.1.Analysis at = daysThe concrete section is divided into 500 layers. Each layer is of width 300mm, and height

600/500 = 1.2mm.

,, , and , are calculated with E= 25,000 MPa, and assuming no cracking., = 4.50 10 + 484.0 10 = 4.98 10 , = 450.0 10 + 107.4 10 = 557.4 10 , = 180.0 10 + 46.9 10 = 226.9 10

Shrinkage is zero during the first time period, as is creep, so equation 4.11 reduces to:

=

,

, = 4.98 10 557.4 10557.4 10 226.9 10 30 1050 10 = 42 10324 10 4.4.2.Analysis at = days,, , and , are calculated in a similar fashion to the previous time step, however = 28,000 MPa is now used, giving the following values:

, = 5.524 10 , , = 611.4 10 , , = 248.5 10 and must now be calculated in order to determine strains.

In order to calculate , Nc,0 and Mc,0 are calculated for each layer using the strains calculatedfrom = 28 days by equations 4.13 and 4.14. The results are summed giving:

= 44.4 1039.3 10 1.8 = 79.9 1070.8 10 In order to calculate

,

,

,

and

,

,

are calculated using

= 28,000 giving:

= 5.04 10504.0 10 300 10 = 1.51 10151.2 10 Thus strains may be calculated using equation 4.11 as:

, = 4.98 10 557.4 10557.4 10 226.9 10 30 1050 10 79.9 1070.8 10 + 1.51 10151.2 10 , = 385 10825 10


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4.4.3.Analysis at = , days, = 5.88 10 , , = 647.4 10 , , = 262.9 10 =

106.1 10

155.4 10

, =

3.24 10

324 10

, = 669 101.20 10 4.5.Cracked example layered approachThe same cross section (and parameters) is analysed assuming the concrete can take no tensile

stress, so that fct.f = 0 MPa.

4.5.1.Analysis at =

days

The first calculation for the instantaneous analysis is the same as for the uncracked example

with resulting strains:

, = 42 10324 10 These strains are used to calculate stresses in each of the layers, those with tensile stresses are

excluded and ,, , and , are recalculated. Strains are then calculated again, and this

iterative procedure is continued until there is negligible difference between strains of successive

iterations. Final properties in the section are found to be:

, = 4.98 10, , = 557 10, , = 227 10

With strains:

, = 42 10324 10 4.5.2.Analysis at = days, = 5.52 10, , = 611 10, , = 248 10

Nc,0 and Mc,0 for , are calculated using only the active layers excluding those layers that aresubject to tensile stresses. and are found to be:

= 80 1071 10 , = 1.51 10151 10

, = 385 10825 10


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4.5.3.Analysis at = , days, = 1.97 10, , = 67 10, , = 70.8 10

=

260 10

33.5 10

, =

894 10105 10

, = 526 102.16 10 4.6.Comparison of resultsResults for the example given in section 4.5 are compared with results from the analytical

(uncracked) method and shown in table 4.3. Results from the analytical method are sourced

from Ranzi and Gilbert (Ranzi & Gilbert 2011).

Analytical uncracked Layered uncracked Layered cracked

r,j 42.7 10 42.3 10 42.2 10 386 10 385 10 385 10 670 10 669 10 526 10

j 331 10 324 10 324 10 841 10 825 10 825 10 1,220 10 1,197 10 2,160 10

Differences in strains between the analytical method, and the layered uncracked method arise

because the layered uncracked method calculates concrete properties based on the gross area of

concrete, over stating the stiffness of the beam slightly. The analytical uncracked method

calculates properties based on concrete net area, not including concrete where the steel exists.

Results from the cracked section indicate cracking only occurs in the third time step, as strain

and curvature at this time step are greater than for the uncracked sections.

Table 4.3: Comparison of strains for cracked and uncracked methods


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Chapter 5

Finite element method

5.1.Assumptions and commentsIn this section, a finite element formulation is derived for a beam governed by Euler-Bernoulli

beam theory and extended to incorporate time-dependent effects using the SSM. Assumptions

that follow are:

Plane sections remain plane, and shear deformations are ignored. This widely usedassumption has been shown to hold true for long slender beams.

A perfect bond between concrete and steel. This assumption is not completely valid,however variability in bond slip experiments suggest inclusion of such behaviour cannot

be relied on (Kotsovos & Pavlovic 1995).

Service stresses in concrete remain in the elastic region. At service loads thisassumption holds true (Bazant 1988)

The cross section is symmetric about the y axis as shown in figure 5.1 so that notorsional or out of plane bending effects are considered.

An axis system is used where the z axis is at an arbitrary height as shown in figure 5.1.

For design, where creep and shrinkage parameters are estimated with great variability, this

formulation would not be warranted and simpler methods preferred. However, this FEM model

will be used in conjunction with experimental data that is far more accurate.

z

x

y

y

x

Figure 5.1: Assumed axis system


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5.2.FormulationUsing the principal of virtual work, the weak formulation for the Euler-Bernoulli beam is given

as:

. = . (5.1)where = , the internal axial force and moment =

= , the reference axis strain and curvature= , the axial and transverse distributed loads

= , the virtual displacements= 0

0 a differential operator

and represent displacements in the longitudinal (z axis) and transverse (y axis) directions

respectively.

Equation 5.1 equates internal strain energy (LHS) with external work (RHS). A derivation for

this can be found in Appendix C.

5.3. Degrees of freedom and consistencyThe deformed shape of the beam is assumed to take the form of a polynomial. The order of the

polynomial chosen determines the number of nodal points required in the element, so that the

number of unknowns is equal to the number of equations.

A 7 node (7 degrees of freedom) beam element as shown in figure 5.2 is chosen for this study,with a cubic describing deformation in the y direction along the beam, and a quadratic

describing the deformation in the x-direction along the beam. These order of polynomial avoid

a locking problem where elements become overly stiff when the centroid moves away from the

reference z axis - an issue where cracking is involved. Locking can arise due to uneven

contribution of strain in the x and y directions. Using cubic and quadratic polynomials ensures

that strain becomes a linear function of position in both directions (Ranzi & Gilbert 2011).

Figure 5.2: 7 degree of freedom beam element


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The polynomial used to describe the deformed shape may be expressed in terms of the

coefficients of each order, or in terms of the displacements at particular points along the

deformed shape. This latter form of the polynomial is known as the shape function, and is

expressed as:

(5.2)Where = , = [ ] and,

= 1 +

+

0 0 0 0

0 0 0 1 +

+

+

5.4.Time dependencyTo incorporate the time dependent effects of creep and shrinkage, equation 4.9 and 5.1 are

combined, where ri = re, giving:

+ . = . (5.3)Substituting equation 5.2 into 5.1 gives:

+ . = . (5.4)which can be rearranged to form:

+ . = . (5.5)using the matrix property . = . , equation 5.2 and equation C.8, equation 5.5 can beexpressed as:

( ) = ( + ) (5.6)This is equivalent to = , where is the element loading vector given by = + , and is the element stiffness given by = ( ) .

This system of equations is then solved taking the inverse of ke and using matrix partitioning.


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5.5.Transformation from local to global axesThe formulation given in 6.8 applies to a single beam element. A finite element model consists

of multiple elements, and this system must be solved by creating a global stiffness matrix and

global displacement and loading vectors. To facilitate this, local axes must be converted to a

consistent global axis by the following transformation:

= and = (5.7)

Where d and q are the local displacement and loading vectors respectively, and D and Q are

the global displacement and loading vectors respectively. T is the transformation matrix and

for a 7 degree of freedom beam element is given as:

=

0 0 0 0 0 0 0 0 0 0

0 0 1 0 0 0 00 0 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 0 0 1

(5.8)

where is the angle between the local and global x axis, taken counter clockwise from the

global x axis as shown in figure 5.3.

Equation 5.7 is combined with equation 5.6 to give:

= (5.9)So that the stiffness for the element in global terms becomes =

5.6.ShrinkageA shrinkage profile in a concrete member is often assumed to be uniform across its cross-

section. This is not what is found according to relative humidity profiles (discussed in chapter

7). To account for other possible shrinkage profiles, the FEM model is adjusted so that the

shrinkage profile is approximated by a polynomial. To achieve this, a shape function similar to

that described by equation 5.2, is used.

Global X

G

lobalY

Figure 5.3: Relationship between local element axis and global axis


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The approximating polynomial is given as:

= + + + (5.10)Values for shrinkage down the cross section are then assumed to be known by experiment or

otherwise. So that at =

=

, =

=

, =

=

, and at

= = . This can be expressed in matrix form as follows:

=

1 1 1 1

(5.11)

or =

Solving for a, gives:

= (5.12)Where is the vector of known shrinkage values, at points , , . Shrinkage strain atany point along the y axis can now be given by:

= 1 (5.13)Equation 5.13 can be used to calculate shrinkage in each layer. These values are then

multiplied by the appropriate geometric values for each layer as per equations 4.12 a) and b),

and summed giving the shrinkage vector , = ,,,, ,.5.7.CrackingA layered model is used, similar to that outlined in section 4.3, to account for cracking.

For the first iteration, it is assumed the section is uncracked and the calculated values for the

matrix D are based on the summed values in each of the layers.

Recall from equation 4.10b, = , ,, , After strains are solved for, values for ,,, ,, and ,, are recalculated where layers withstress greater than the concrete tensile strength are not included. This is repeated until

resulting strains converge. In the next time step, it is necessary to calculate the creep

component , where , = ,, ,, . , and , must be calculated usingstored stress values at a cross section in each of the layers, and integrated along the element.


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5.8.Gaussian quadratureIntegration along an element where stresses and active layers change, mean that closed form

solutions are problematic. To get around this, integration is done using Gaussian quadrature,

specifically Gauss-Legendre quadrature.

The basis for quadrature is that the definite integral of any polynomial of a particular order

may be given exactly by the sum of weighted values at certain points along the curve. The

weighting applied to each of these values, and the points at which these values are evaluated

by the polynomial depend on the order of the polynomial. If the polynomial is evaluated at n

points (known as sampling locations), the integral is exactly given for any polynomial up to

order 2n-1. To generalise the rule, the domain is normalised to [-1,1]. Weightings and sampling

locations for the first 3 gauss points are shown in table 6.1

Points Sampling location (xi) Weights (Wi)

1 0 2

2 1/3,1/3 1,1

3 3/5, 0,3/5 8/9,5/9,5/9The sampling locations in table 6.1 are based on the roots for Legendre functions of order n,

while the weightings are given by = (Abramowitz & Stegun 1964).

As an example, a graphical representation for a cubic function is shown in figure 5.4. The

definite integral is given by the weighted sum of the function evaluated at the sampling points

and

, giving 1.14+1.53 = 2.67 matching the analytical solution.

To convert from the normalised domain to a domain of a-b, the following operation is used:

= 2 2 + +2 (5.14)

Table 6.1: Gauss-Legendre sampling points and weights

1.141.53

0.0

1.0

2.0

3.0

-1.0 -0.5 0.0 0.5 1.0

f(x)

x

= + +

Figure 5.4: Sampling points and weights for a cubic


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5.9.ProgrammingMatlab was chosen as the programming language to implement the FEM program. It was

chosen because of its ability to work with matrices efficiently and easily, despite difficulties

with the implementation of a user interface.

The FEM program was written with the following calculation steps:

For each time step Calculate shrinkage While the difference in displacement compared to the prior loop is greater than

1e-7

For each element For each gauss point

If first loop, calculate geometric properties assuminglayers are uncracked

If time period is greater than 1, calculate creep vectorbased on geometric properties from previous loop, and

stored stress history

Calculate shrinkage vector Assemble element stiffness matrix using geometric

properties and gauss weighting

Loop Loop For each element, transform element stiffness matrix, and loading vector

from local to global coordinates

Assemble the global stiffness matrix Solve the system of equations by partitioning the matrices For each element

For each gauss point Calculate global reactions, global displacements and local

displacements

Determine strains at the gauss sampling points, andidentify uncracked layers

Calculate and store stresses in each layer using equation3.10 for current time period

Calculate geometric properties based on uncrackedlayers

Loop Loop

Loop Remove stresses from previous time periods for layers that are now cracked

Loop


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5.10.Cracked exampleThe same cross section and loading analysed in section 4.4 is analysed using the FEM model,

with two gauss points. The tensile strength of concrete is set to 3MPa, and the beam is subject

to a constant bending moment over its length of 5m. Input into the Matlab model is via a

graphical user interface shown in appendix D. Output for the first time step is shown in figure

5.5.

Results for the three time periods are shown in table 5.2.

These results can be compared to the cross sectional analysis because the loads and moments

are constant across the beam.

Time (Days) 28 100 30000

N1x (mm) 0 0 0

N1y (mm) 0 0 0

N1zz (rad) 811 10 2100 10 5400 10

N2x (mm) 106 10 962 10 1300 10

N3x (mm) 211 10 1900 10 2600 10

N3y (mm) 0 0 0

N3zz (rad) 811 10 2100 10 5400 10

To compare strains and curvature at a cross section, the displacement results at any node (in

this case the right hand side has been selected) are converted to strain and curvature using

equations 5.2 and C.8 so that

= = (5.15)

This gives results shown in table 5.3, which are the same as for the cracked cross sectional

analysis in section 4.5.

Figure 5.5: Results from FEM for first time step

Rx = -30 kN

Ry = 1.09e-14kN

Dzz = 0.00811

Dx = -0.211 mm

Ry = -1.09e-14kN

Dzz = -0.00811

Table 5.2: Results from FEM


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Time (days) 28 100 30000

e 42.3 10 385 10 526 10K (mm-1) 324 10 8 25 1 0 2,160 10

5.11.Uncracked validationTests carried out by Al-Deen and Ranzi (Al-Deen, Ranzi & Uy 2012) described below, were

used to validate the FEM model for uncracked sections.

Two slabs (one way) with cross sections shown in figure 5.6, were subject to their self-weightwith strains and deflections measured over a period of 119 days. Specimens were cast at the

same time, and propped and moist cured for 15 days. After 15 days, props were removed

leaving the beam simply supported as shown in figure 5.7. Mid-span deflections were measured

using linear variable differential transformers (LVDTs). Mid span strains were measured with

strain gauges on the top and bottom surfaces of the concrete and internally.

Shrinkage strains were measured from two concrete cylinders allowed to deform freely. These

cylinders were poured from the same batch as for the beams and subject to the same curing

and drying conditions. Measured strains are shown in figure 5.8.

Table 5.3: Cross sectional results from FEM, matching those of

a cross sectional analysis.

900

162

180

5 x N16 reinforcing bars

Slab A165

180

10 x N16 reinforcin bars

62

900

Slab B

Figure 5.6 Cross sections for Slabs A and B used for long term deflection tests

Figure 5.7: Support and loading conditions for long term deflection tests

3,000

180

150150

Bearing

Roller support


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A polynomial curve was fitted to the data (shown by the dotted line in figure 5.8), and used to

determine shrinkage strains for the FEM model. This was done to ensure a smooth shrinkage

profile, rather than using actual values which may include outliers.

Creep strains were measured from the average response of three cylinders subject to a constant

stress of 5.75 MPa. Creep coeffients were calculated based on equations 4.1 and 4.3, so that:

15, = = (5.16)where represents the total concrete strain. Creep coefficients are shown in figure 5.9.

As for shrinkage, a curve was fitted to the creep data as shown by the dotted line in figure 5.9,

to provide a more representative value of creep for modelling purposes. As the step by step

method is used, a family of creep curves must be generated based on the creep curve in figure

5.9. This is required because of the ageing effect. The creep curve from figure 5.9 is used as a

baseline, and the remaining creep curves adjusted for the time step using AS3600-2009.

The creep coefficient, according to AS3600-2009, is given by a basic creep value multiplied by

factors to account for humidity, slab thickness, time after loading and age of concrete at

loading (Standards Australia 2009). All factors, except the age of concrete at loading, are

accounted for in the experimental creep coefficient values in figure 5.9. This factor in AS3600,

known as k3, adjusts for the age of concrete at loading and is given by the following equation

(Ranzi & Gilbert 2011):

= 2.7/(1 + log( )) (5.17)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0 50 100 150

Creepcoefficient

(15,t

)

Time (days)

Creep coefficient vs time

Figure 5.9: Creep coefficients calculated fromcylinder tests

0

100E-6

200E-6

300E-6

400E-6

0 50 100 150

Shrinkage

Time (days)

Shrinkage vs time

Figure 5.8: Shrinkage results from test cylinders


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Thus for any other time periods where the age at loading is not 15 days, the creep coefficient is

adjusted and is given by:

, = 2.71 + log

2.7

1 + log15(15, ) (5.18)

Where is the time lag ( ).

The density of concrete is taken to be 2,400 kg.m-3, and steel 8,000 kg.m-3, giving distributed

loads of 3.87 kN.m-1, and 3.92 kN.m-1 for slabs A and B respectively.

Youngs modulus was found to be 19,000 MPa at 15 days. The increase in Youngs modulus

over time was estimated using a modified relationship from Eurocode 90, with s = 0.38 for

normal strength concrete (Comit Euro-International du Bton 1993):

= .

(15) (5.19)

This estimate produces a slightly higher value for E than would occur if equation 5.19 was

based on (28) but the effect on the compliance function ( , ) is small.

Calculated deflections mostly follow actual deflections closely and are shown in figure 5.10.

Strains are compared in figure 5.11. Curvatures seem to deviate more than reference strain.

Causes for this most likely relate to variability in the estimated parameters of ( , ) and .It is also possible material parameters in the slabs are not reflected in the cylinder tests but

this is unlikely.

Figure 5.10: Comparison of mid-span deflection as measured by experiment

and by FEM calculation

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

0 20 40 60 80 100 120

Midspandeflectio

n(mm)

Time (days)

Calculated vs Actual Mid-Span Deflection

Slab A - actual

Slab A - FEM calculated

Slab B - actual

Slab B - FEM calculated


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5.12.Cracked validationTests by Gilbert and Nejadi were used to validate the FEM model for cracked sections (Gilbert

& Nejadi 2004).

Deflections for a one way slab and beam were measured over a period of 400 days under

sustained load. The cross section and loading conditions are shown in figures 5

Documents

Service Behaviour of Reinforced Concrete Members (Thesis)