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
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.
ii
Table of contents
Table of contents ........................................................................................................................ iii
Acknowledgements ...................................................................................................................... v
Abstract ...................................................................................................................................... v
Chapter summary ....................................................................................................................... vi
List of tables and figures ........................................................................................................... vii
Nomenclature ............................................................................................................................. xi
Chapter 1 Introduction .............................................................................................................. 1
1.1.
General ......................................................................................................................... 1
1.2.
Objectives ..................................................................................................................... 2
Chapter 2 Literature review ........................................................................................................ 3
2.1.
General ......................................................................................................................... 3
2.2.
Shrinkage ...................................................................................................................... 3
2.3.
Compressive creep ........................................................................................................ 7
2.4.
Tensile creep ................................................................................................................. 9
2.5.
Tensile strength .......................................................................................................... 10
2.6.
Modelling time dependent behaviour .......................................................................... 10
Chapter 3 Time dependent behaviour in concrete ..................................................................... 12
3.1.
Time dependent properties ......................................................................................... 12
3.2.
Time dependent modelling – step by step method ..................................................... 17
3.3.
SSM assumptions ........................................................................................................ 20
Chapter 4 Cross sectional analysis ............................................................................................ 21
4.1.
Background ................................................................................................................. 21
4.2.
Uncracked formulation ............................................................................................... 21
4.3.
Cracked formulation ................................................................................................... 22
4.4.
Uncracked example – layered approach ...................................................................... 24
4.5.
Cracked example – layered approach.......................................................................... 26
iii
4.6.
Comparison of results ................................................................................................. 27
Chapter 5 Finite element method.............................................................................................. 28
5.1.
Assumptions and comments ....................................................................................... 28
5.2.
Formulation ................................................................................................................ 29
5.3.
Degrees of freedom and consistency ............................................................................ 29
5.4.
Time dependency ........................................................................................................ 30
5.5.
Transformation from local to global axes ................................................................... 31
5.6.
Shrinkage .................................................................................................................... 31
5.7.
Cracking ..................................................................................................................... 32
5.8.
Gaussian quadrature ................................................................................................... 33
5.9.
Programming .............................................................................................................. 34
5.10.
Cracked example ..................................................................................................... 35
5.11.
Uncracked validation .............................................................................................. 36
5.12.
Cracked validation .................................................................................................. 39
5.13.
AS3600-2009 comparison......................................................................................... 45
Chapter 6 Measurement of shrinkage profiles............................................................................ 47
6.1.
Previous techniques .................................................................................................... 47
6.2.
Development of new sensors ....................................................................................... 48
Chapter 7 Conclusion ................................................................................................................ 50
Appendix A Comparison of cross sectional methods to analyse time dependent behaviour ...... 51
A.1.
Constant Deformation ................................................................................................ 51
A.2.
Constant load ............................................................................................................. 56
Appendix B Step by step cross sectional analysis formulation .................................................. 59
Appendix C Finite beam element formulation .......................................................................... 62
C.1.
Displacement field ...................................................................................................... 62
C.2.
Weak Formulation ...................................................................................................... 62
Appendix D Matlab finite element program.............................................................................. 65
References.................................................................................................................................. 67
iv
Acknowledgements
Thank you to my supervisor, Gianluca Ranzi, who was passionate, generous with his time and
gave me the freedom to explore avenues of interest in this thesis. Thank you to my parents for
always supporting me. Finally thank you to my wife, Carly, whose support and patience for my
obsession allowed me this far.
Abstract
The main objective of this thesis is to predict the long term behaviour of reinforced concrete.
To this end, a method of cross sectional analysis (based on the step by step method) is
developed using a layered approach to model time dependent behaviour, including cracking, in
beams under axial and/or bending loads. Calculated strains from this model are shown to agree
with results from the literature.
The cross sectional method is extended to a finite element framework, and a formulation for
beam elements incorporating time dependent effects is presented. This formulation is
implemented in Matlab, and calculated deflections are shown to agree with cracked and
uncracked experiments on beams.
The assumed shrinkage profile used to predict time dependent behaviour is explored, and nonuniform, curved shrinkage profiles are shown to significantly change calculated deflection by
inducing cracked behaviour.
As a result of the above finding, the measurement of shrinkage profiles is explored, and a
humidity sensor developed.
Finally, a comparison is made between results calculated by the refined FEM method, and the
simplified method provided in AS3600-2009. It is suggested that improved accuracy in refined
methods must be weighed up against complexity and additional time requirements.
v
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 some
modelling 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.
vi
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 f’c(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 plane
19
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
vii
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 calculation
38
Figure 5.11: Comparison of mid-span strains as measured by experiment and by FEM
calculation 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 validation
test.
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 prism
43
Figure 5.17: Comparison of deflection s as calculated by the FEM model and as
measured by experiment for the beam
43
Figure 5.18: Comparison of deflections as calculated by the FEM model and as
measured by experiment for the slab
44
Figure 5.19: Comparison of cracking for the one way slab as calculated and as recorded
by experiment at 400 days
44
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
viii
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
shrinkage
52
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
ix
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
x
Nomenclature
, , ௖ , ௖ , ௖
௦ , ௦ , ௦
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
௦௧ , ௦௖
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 ௦
Elastic modulus of steel
௖௥,௝
Creep loading vector at time ௝
௦௛,௝
Shrinkage loading vector at time ௝
௘,௝,௜
Creep factor at time ௝ for stresses applied at ௜
′௖
Characteristic compressive strength of concrete
′௖௧.௙
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
௖௦
Long-term to short term deflection factor
௘
Element stiffness matrix
ଷ
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 ௝
ࢋ
Vector of external actions
xi
࢏
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
௦௛
Shrinkage strain
∗
௖௦
Final design shrinkage strain
Curvature
௪ , ௖௪
Tensile web reinforcement ratio and compressive web reinforcement ratio
௖,௝
Stress in the concrete at time step j
௖௦
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 xii
Chapter 1
Introduction
1.1. General
Reinforced 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 first
applied. 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.
1
1.2. Objectives
This 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 behaviour
including cracking.
2. To develop a finite element program that will evaluate deflections in cracked and
uncracked 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 AS36002009 with experimental and finite element results.
5. To develop a method to measure the humidity profile through a cross section which can be
used as a proxy for the shrinkage profile.
2
Chapter 2
Literature review
2.1. General
Time 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. Shrinkage
Shrinkage 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
Shrinkag
normal strength concrete are shown in figure 2.1.
Drying shrinkage
Autegenous shrinkage
Age of concrete
Figure 2.1: Relative magnitudes of drying and autogenous shrinkage.
2.2.1. Plastic shrinkage
When 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.
3
Figure 2.2: Meniscus that forms as a
result of evaporation of bleed water
Figure 2.3: Forces acting on the meniscus.
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).
4
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 shrinkage
Drying 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
5
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 shrinkage
Hydration 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).
6
2.2.4. Thermal shrinkage
Thermal 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 creep
Creep 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.
Drying
Strain
Basic creep
Strain
Basic creep
Shrinkage
Elastic
Time
Elastic
Time
Figure 2.5: Comparison of creep in a sealed specimen (left)
and creep in a drying specimen (right)
7
2.3.1. Factors influencing creep
There are a number of factors that influence the rate of creep in general:
•
It is heavily influenced by relative humidity, with lower relative humidities causing
increased 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 the
potential 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 f’c, the relationship between stress and creep is
linear, 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 creep
occurs (Bazant, Cusatis & Cedolin 2004).
•
There is a size effect where an increase in surface area to volume ratio leads to a
decrease in creep.
2.3.2. Mechanisms behind creep
Many 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 and
part 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 viscoelastic 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).
8
2. A mechanism behind drying creep was proposed by Wittman, who suggested that tensile
stresses 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 creep
Research 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).
9
2.5. Tensile strength
Tensile 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 attract
high 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 behaviour
Modelling 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
10
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 0.4f’c (Kawando & Warner 1996).
For ease of application to experimental data, and because computational efficiency is not
critical, the integral-type approach is used in this thesis.
11
Chapter 3
Time dependent behaviour in concrete
3.1. Time dependent properties
Deformations 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. Shrinkage
After 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
Shrinkage
limit as shown in figure 3.1.
εsh
Time
Figure 3.1: Development of shrinkage with time
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 of
−200 × 10 to −1100 × 10 (Wight & Macgregor 2012). The majority of this strain is reached
within 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 is
not made between shrinkage types, and shrinkage is given simply as .
12
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.
Free shrinkage – no
induced stresses
Restraint ‘pulls’ the concrete
specimen into tension from
its free shrinkage state.
Figure 3.2: Shrinkage without restraint
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.
ߝ௦௛
(a) Slab subject to free
shrinkage
Δߝ௦௛
(b) Shrinkage
strain profile
ߝ௦௛
(c) Resulting strain =
elastic strains +
shrinkage strains
Figure 3.3: Free shrinkage strains
13
3.1.2. Creep
Creep 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.
Load
Load
Load
Load
Elastic
Creep
recovery
recovery
Load removed
Strain
Load applied
Elastic
Creep strain
recovery
Creep recovery
Elastic or
0
instantaneous
Permanent
strain
deformation
t0
t
Time
T→∞
Figure 3.4: Concrete under creep with no shrinkage
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.
14
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 elastic
deformation at time , and , is known as the creep coefficient. The creep coefficient can be
measured or calculated. AS 3600-2009 Concrete Structures provides 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.
߮(‫ݐ‬, ߬)
߮(‫ݐ‬, ߬଴ )
߮(‫ݐ‬, ߬ଵ )
߬଴
߬ଵ
Time
Figure 3.5: Creep coefficient vs time (Ranzi & Gilbert 2011)
3.1.3. Ageing
Over 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 concrete
must exceed this strength. Though this thesis is not concerned with ultimate strength, the
15
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.
Ratio fc(T)/fc(28)
1.4
1.0
0.6
0.2
1
3
7
90
28
365
Time (days)
Figure 3.6: Generalised concrete compressive strength development over
time for normal strength concrete relative to f’c(28) (Wight & Macgregor
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.
f’c
Stress
Strength (MPa)
80
Secant modulus
60
40
20
(elastic modulus)
Strain
0
10
20
Strain x10
Figure 3.7: Stress strain curve for concrete
30
40
-6
Figure 3.8: Stress strain curves for varying
strengths of concrete (Neville 1995)
16
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 method
To 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.
ߪ(߬଴ )
Δߪ௜
ߪ௖ (‫)ݐ‬
Stress
߬଴
߬ଵ ߬ଶ
߬ଷ
߬௞
Time
Figure 3.9: Discreet stress intervals used in the SSM
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:
+ =
Δ Δ +
(, )
This can be expressed more compactly as:
17
+ = , Δ ( )
(3.4)
Where , is known as the creep or compliance function. It represents the combined elastic
and 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
= + + ()
= , +
బ
, + ()
(3.6)
Equation 3.6 can be approximated by:
= , + , Δ + ()
(3.7)
This can be re-written in short hand as follows:
, − , = ,
,
+ , Δ
,
(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
18
, − ,
, − , =
+
,
,
,
,
=
, (,
− , ) + ,,
where ,, =
,
(3.10)
,
ೕ,೔ ೕ,೔శభ
(3.10a)
ೕ,ೕ
And from equation 3.5, , = , since , = 0 (there is no creep because no time has
೎,ೕ
elapsed).
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)
d_ref
ߢ
x
ߝ௥
y
Cross section A-A
Strain
Figure 3.10: Strain in a beam under axial and bending loads where plane sections remain plane
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.
19
3.3. SSM assumptions
There 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 f’c, 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.
20
Chapter 4
Cross sectional analysis
4.1. Background
A 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 formulation
At 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)
21
After some manipulation (full derivation can be found in Appendix B), the resulting
equilibrium equation is found to be:
, = + , − ,
(4.9)
Where
,
,
=
,
,
And !,, = !,
=
!,
,
+ ,
!,
!,,
,
, = " # , , = ,, , , = !,
,
!,, ,
,
!,, = $
,
+ $
and !,, = %
,
+ %
(4.10a-e)
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 time
step 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 subsequent
time steps as necessary.
4.3. Cracked formulation
The 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.
22
d_ref
yl
w
∆h
Layer ‘l’
Figure 4.1: Cross section divided into layers
For the layer shown in figure 4.1, = ' × Δℎ , $ = , % = .
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:
≈ ,,
≈ Where
,,
,,
(4.13)
(4.14)
is the stress in concrete layer h, at time tj.
23
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 layer’s 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 approach
Strains 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.
Table 4.1: Loading, elastic modulus, shrinkage
parameters and times steps for example
τj
Ec,j
(days)
(MPa)
28
100
30,000
Table 4.2: Creep coefficients for example
εsh(t)
25,000
28,000
30,000
0
-300E-06
-600E-06
ϕ(τj,τi)
28
100
30,000
28
100
30,000
0.0
1.5
2.5
0.0
2.0
0.0
b = 300
dst(1) = 50
Ast(1) = 620
dref = 200
D = 600
‫ݔ‬
dst(2) = 550
ys(1) = -150
ys(2) = +350
Ast(2) = 1800
‫ݕ‬
Figure 4.2: Cross Section for example (all units in mm)
24
4.4.1. Analysis at ࣎૙ = ૛ૡ days
The 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 × 10
557.4 × 10 #
226.9 × 10
"−30 × 10 # = " −42 ×10 #
50 × 10
324 × 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 calculated
from = 28 days by equations 4.13 and 4.14. The results are summed giving:
= "−44.4 × 10 # − 1.8 = " 79.9 × 10 #
−39.3 × 10
−70.8 × 10 ((
In order to calculate , !,,and !,, are calculated using
#
= " 5.04 × 10 # − 300 × 10 = " −1.51 × 10
504.0 × 10
−151.2 × 10 ((
= 28,000 giving:
Thus strains may be calculated using equation 4.11 as:
,
" # = " 4.98 × 10 557.4 × 10
557.4 × 10 #
226.9 × 10
,
" # = " −385 ×10 #
825 × 10 ((
)"−30 × 10 # − " 79.9 × 10 # + " −1.51 × 10 #*
50 × 10
−70.8 × 10
−151.2 × 10
25
4.4.3. Analysis at ࣎૛ = ૜૙, ૙૙૙ days
!, = 5.88 × 10 (( , !, = 647.4 × 10 (( , !, = 262.9 × 10 ((
#, = " −3.24 × 10 #
= " −106.1 × 10
−155.4 × 10 ((
−324 × 10 ((
,
" # = " −669 ×10 #
1.20 × 10 ((
4.5. Cracked example – layered approach
The same cross section (and parameters) is analysed assuming the concrete can take no tensile
stress, so that f’ct.f = 0 MPa.
4.5.1. Analysis at ࣎૙ = ૛ૡ days
The first calculation for the instantaneous analysis is the same as for the uncracked example
with resulting strains:
,
" # = " −42 ×10 #
324 × 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 ×10 #
324 × 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 are
subject to tensile stresses. and are found to be:
= "
80 × 10 #, = "−1.51 × 10 #
−71 × 10 ((
−151 × 10
,
" # = " −385 ×10 #
825 × 10 ((
26
4.5.3. Analysis at ࣎૛ = ૜૙, ૙૙૙ days
!, = 1.97 × 10 , !, = −67 × 10 , !, = 70.8 × 10
= "
260 × 10 #, = "−894 × 10 #
−33.5 × 10 ((
105 × 10
,
" # = " −526 ×10 #
2.16 × 10 ((
4.6. Comparison of results
Results 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).
Table 4.3: Comparison of strains for cracked and uncracked methods
εr,j
κj
Analytical uncracked
Layered uncracked
−42.7 × 10
−42.3 × 10
−42.2 × 10
−386 × 10
−385 × 10
−385 × 10
331 × 10
324 × 10
324 × 10
−670 × 10
−669 × 10
Layered cracked
−526 × 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.
27
Chapter 5
Finite element method
5.1. Assumptions and comments
In 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 used
assumption 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 this
assumption holds true (Bazant 1988)
•
The cross section is symmetric about the y axis as shown in figure 5.1 so that no
torsional 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.
z
x
x
y
y
Figure 5.1: Assumed axis system
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.
28
5.2. Formulation
Using 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
̂
−0,′′
2
. = " #, the axial and transverse distributed loads
(
/,
+, = " #, the virtual displacements
0,
=3
!
0
0
−
మ
! మ
4 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 consistency
The 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
29
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"
(5.2)
/
Where + = "0 #, 5#" = [6 6 6 7 7 7 7 ] and,
1−
= 3
$
+
0
$ మ
మ
$
−
$ మ
0
$
− +
మ
$ మ
మ
0
0
1−
$ మ
మ
+
$ య
య
-−
0
$ మ
0
+
$య
మ
$ మ
మ
−
0
$ య
య
−
$మ
+
$య
4
5.4. Time dependency
To 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:
8 - =
+ − . 5
8 .. 5
(5.4)
which can be rearranged to form:
8 - =
# + − . 5
8 # .. 5
(5.5)
using the matrix property 9:. ;< = ;# 9:. <, equation 5.2 and equation C.8, equation 5.5 can be
expressed as:
#
( )5 - =
(# . − # + # ) -
(5.6)
This is equivalent to . = = 5 , 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.
30
5.5. Transformation from local to global axes
The 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 @ = ?A
(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:
EFGH
D−GI2H
C
C 0
?=C 0
C 0
C 0
B 0
GI2H
EFGH
0
0
0
0
0
0
0
0
0
1
0
0 EFGH
0
0
0
0
0
0
0
0
0
0
EFGH
−GI2H
0
0
0
0
0
GI2H
EFGH
0
0
0L
K
0K
0K
0K
0K
1J
(5.8)
where H is the angle between the local and global x axis, taken counter clockwise from the
global x axis as shown in figure 5.3.
Global Y
ߠ
Global X
Figure 5.3: Relationship between local element axis and global axis
Equation 5.7 is combined with equation 5.6 to give:
A = ? # = ?
So that the stiffness for the element in global terms becomes =% = ? # = ?
(5.9)
5.6. Shrinkage
A shrinkage profile in a concrete member is often assumed to be uniform across its crosssection. 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.
31
The approximating polynomial is given as:
= M + M + M … + M& &
(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 D L D1 C K CC
C K = C1 C ⋮ K C⋮ ⋮
B& J B1 &
or = N:
⋮
&
⋯
⋯
⋯
⋯
& M
L
& K D M L
C K
& K CM K
KC ⋮ K
⋮ K
&& J BM& J
(5.11)
Solving for a, gives:
: = N (5.12)
Where is the vector of known shrinkage values, at points , , … & . Shrinkage strain at
any point along the y axis can now be given by:
= O1 ⋯
&P :
(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. Cracking
A 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 with
stress 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 using
,
stored stress values at a cross section in each of the layers, and integrated along the element.
32
5.8. Gaussian quadrature
Integration 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
Table 6.1: Gauss-Legendre sampling points and weights
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, 5/9
The sampling locations in table 6.1 are based on the roots for Legendre functions of order ‘n’,
while the weightings are given by R =
'! మ ()*೙ᇲ '!೔ (+
మ
(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.
3.0
1.53
2.0
f(x)
1.14
ࢌሺ࢞ሻ = ࢞૜ + ࢞૛ + ૚
1.0
0.0
x
-1.0
-0.5
0.0
0.5
1.0
Figure 5.4: Sampling points and weights for a cubic
To convert from the normalised domain to a domain of a-b, the following operation is used:
SS =
T−M
2
.
U
T−M
M+T
S+
V S
2
2
(5.14)
33
5.9. Programming
Matlab 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 assuming
layers are uncracked
If time period is greater than 1, calculate creep vector
based 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, and
identify uncracked layers
Calculate and store stresses in each layer using equation
3.10 for current time period
Calculate geometric properties based on uncracked
layers
Loop
Loop
Loop
Remove stresses from previous time periods for layers that are now cracked
Loop
34
5.10. Cracked example
The 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.
Rx = -30 kN
Dx = -0.211 mm
Ry = 1.09e-14kN
Ry = -1.09e-14kN
Dzz = 0.00811
Dzz = -0.00811
Figure 5.5: Results from FEM for first time step
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.
Table 5.2: Results from FEM
Time (Days)
N1x (mm)
N1y (mm)
N1zz (rad)
N2x (mm)
N3x (mm)
N3y (mm)
N3zz (rad)
28
100
30000
0
0
0
0
0
ି଺
811 × 10
0
ି଺
2100 × 10
5400 × 10ି଺
−106 × 10ିଷ
−962 × 10ିଷ
−1300 × 10ିଷ
−211 × 10ିଷ
−1900 × 10ିଷ
−2600 × 10ିଷ
0
0
0
811 × 10ି଺
2100 ×
10ି଺
5400 × 10ି଺
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
ࢿ = ‫ܰܣ = ࢋܣ‬௘ ࢊࢋ
(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.
35
Table 5.3: Cross sectional results from FEM, matching those of
a cross sectional analysis.
Time (days)
e
28
100
30000
−42.3 × 10ି଺
−385 × 10ି଺
−526 × 10ି଺
324 × 10ିଽ
825 × 10ିଽ
2,160 × 10ିଽ
K (mm-1)
5.11. Uncracked validation
Tests 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-weight
with 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.
900
900
Slab A
Slab B
180
5 x N16 reinforcing bars
165
180
162
62
10 x N16 reinforcing bars
Figure 5.6 Cross sections for Slabs A and B used for long term deflection tests
180
Bearing
Roller support
150
3,000
150
Figure 5.7: Support and loading conditions for long term deflection tests
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.
36
Shrinkage vs time
Shrinkage
Creep coefficient vs time
1.8
Creep coefficient ϕ(15,t)
400E-6
300E-6
200E-6
100E-6
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0
0
50
100
Time (days)
150
Figure 5.8: Shrinkage results from test cylinders
0
50
100
150
Time (days)
Figure 5.9: Creep coefficients calculated from
cylinder tests
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)
37
Thus for any other time periods where the age at loading is not 15 days, the creep coefficient is
adjusted and is given by:
U
2.7
V
1 + log
, ′ =
(15, ′)
2.7
U
V
1 + log15
(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.
Young’s modulus was found to be 19,000 MPa at 15 days. The increase in Young’s modulus
over time was estimated using a modified relationship from Eurocode 90, with s = 0.38 for
normal strength concrete (Comité Euro-International du Béton 1993):
= WX
1
/0 2
Y
.1
(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.
2.0
Calculated vs Actual Mid-Span Deflection
Mid span deflection (mm)
1.8
1.6
1.4
1.2
1.0
0.8
Slab A - actual
0.6
Slab A - FEM calculated
0.4
Slab B - actual
0.2
Slab B - FEM calculated
0.0
0
20
40
60
80
100
120
Time (days)
Figure 5.10: Comparison of mid-span deflection as measured by experiment
and by FEM calculation
38
Long term strain profiles for Slab A
30 days
200
180
160
140
120
100
80
60
Theoretical
40
Actual
20
0
-600E-06
-400E-06
-200E-06
0
119 days
60 days 45 days 30 days 200
180
160
140
120
100
80
Theoretical
60
Actual
40
20
0
-600E-06
-400E-06
-200E-06
0
Strain
Strain
Figure 5.11: Comparison of mid-span strains as measured by experiment and by FEM
calculation for Slabs A and B
5.12. Cracked validation
Tests 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 5.12 and 5.13.
161
130
2.9 kN.m-1 (plus dead load)
3 x N12 reinforcing
161
Bearing
Roller
400
Distance from bottom of cross section (mm)
60 days 45 days
Distance from bottom of cross section (mm)
119 days
Long term strain profiles for Slab B
3,500
Figure 5.12: Slab cross section, and support and loading conditions for long term cracked tests
(dimensions in mm)
39
18.6 kN
18.6 kN
348
300
348
Bearing
Roller
2 x N16
3,500
Figure 5.13: Beam cross section and support and loading conditions for long term cracked tests
(dimensions in mm)
Shrinkage, creep, flexural tensile strength and modulus of elasticity values were separately
measured using cylinders and prisms according to AS 1012-2000 Methods of testing concrete,
with results shown in tables 5.4 and 5.5.
Table 5.4: Creep coefficients and shrinkage strain values
Age (days)
14
16
21
27
53
96
136
200
242
332
394
ϕcc
0
0.14
0.36
0.48
0.92
1.15
1.29
1.4
1.5
1.64
1.71
εsh
0
-14
-109
-179
-403
-591
-731
-772
-784
-816
-825
Table 5.5: Tensile strength and modulus of elasticity values
Age (days)
7
14
21
28
f'ct (MPa)
3.0
3.7
4.3
5.6
21,090
22,820
23,990
24,950
E (MPa)
As for the uncracked validation, creep coefficients and shrinkage values are fitted to a curve for
use in the model, and the family of creep coeffficients calibrated against AS3600-2009. The
values for the modulus of elasticity beyond 28 days are calculated using equation 5.19.
The tensile strength of concrete is critical to the calculation of deflection for cracked sections,
because it determines the stress at which the concrete will crack in the model as described in
chapter 4. AS3600-2009 provides a relation between concrete compressive strength and flexural
tensile strength where
′ = 0.6√
3
(5.20)
Equation 5.20 is not ideal because the relationship between the two variables is weak (Wight &
Macgregor 2012). The flexural tensile strength shown in table 5.5 is the most useful measure of
tensile strength for the model. It should be noted, it is still not ideal because in measuring the
flexural tensile strength, it is difficult to remove the effects of shrinkage. That is, the measured
40
flexural tensile strength does not take into account the additional tensile stresses near the
surface caused by non-uniform shrinkage. Thus, tensile stresses reported may be understated.
In order to calculate tensile strength beyond 28 days for use in the model, increase in tensile
strength over time is based on the increase in compressive strength. Since tensile strength is
given at 28 days, strengths beyond this point in time are calibrated by using figure 3.6 and
equation 5.20. For example, the tensile stress at 90 days is calculated as follows;
From figure 3.6, f’c at 90 days is given as:
3
90
= 1.2 3 (28)
The ratio of flexural tensile strength at 90 days to that at 28 days is then given by:
90 0.6 1.2 3
=
= 1.1
28
0.6 3
So that
3
90 = 1.1
3
28 = 1.1 × 5.6 = 6.1 MPa.
Deflections are calculated using both uniform shrinkage and an assumed shrinkage profile as
shown in figure 5.14. The assumed shrinkage profile is based on a measured relative humidity
profile from the literature (Cement Concrete and Aggregates Australia 2007), and for any given
time period is shaped such that the average of the shrinkage strains is equal to the uniform
shrinkage value for the same time period.
To validate the shrinkage profiles, the resulting free shrinkage stresses generated at 14 days in
the concrete without reinforcing steel are compared with those measured by Grasley and Lange
on a 76 mm thick mortar prism at five days allowed to dry on two opposing sides only, and
shown in figure 5.15. The maximum stresses in the samples from Grasley are limited by the
tensile strength of the concrete at approximately 2 MPa, while the limit for the FEM
calculation is 3.7MPa. This aside, the similarity between the two graphs confirms the shrinkage
profiles shown in figure 5.14 are reasonable estimates. The average of each of the non-uniform
profiles in figure 5.14 is the same as the uniform shrinkage values.
41
Assumed shrinkage profile over time
136 days
96 days
53 days
28 days
160
140
120
100
80
60
40
20
0
-900E-6 -800E-6 -700E-6 -600E-6 -500E-6 -400E-6 -300E-6 -200E-6 -100E-6
0
Cross section height (mm)
400,332,
242,200
days
Free shrinkage strain
Figure 5.14: Progression shrinkage strain profiles over time assumed for the validation test.
Free shrinkage stresses - calculated
Free shrinkage stresses - measured
0
1
2
Stress (MPa)
3
140
120
100
80
60
40
20
0
-1
0
1
2
3
Height in cross section (mm)
-1
160
Height in cross section (mm)
80
70
60
50
40
30
20
10
0
Stress (MPa)
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.
The development of the assumed shrinkage profiles over time in figure 5.14 is based on a
similar progression found experimentally over the first 7 days of curing and shown in figure
5.16.
42
Progression of relative humidity profile
Height in cross section (mm)
80
70
60
50
3 days
40
5 days
30
7 days
20
10
0
0
10
20
30
40
50
% change in relative humidity since pouring
Figure 5.16: Progression of the relative humidity profile over the first 7
days of curing for a concrete prism (Grasley & Lange 2004)
Calculated deflections for the slab and beam are shown in figures 5.16 and 5.18 respectively.
These include deflections calculated using uniform shrinkage at each time period, and those
calculated assuming the shrinkage profiles in figure 5.14.
Calculated vs Actual Mid-Span Deflection
Mid span deflection (mm)
14.0
12.0
10.0
8.0
6.0
Actual
4.0
FEM model - non-uniform shrinkage
2.0
FEM model - uniform shrinkage
0.0
0
100
200
300
400
Time (days)
Figure 5.17: Comparison of deflection s as calculated by the FEM model and as
measured by experiment for the beam
From figure 5.17, FEM calculated deflections for the beam are similar to actual deflections
regardless of shrinkage profile. In contrast to this, results for the slab in figure 5.18 show
significant variance between deflections calculated with and without a uniform shrinkage
profile.
43
Calculated vs Actual Mid-Span Deflection
Mid span deflection (mm)
30
25
20
Actual
FEM model - Non-uniform shrinkage
15
FEM model - Uniform shrinkage
10
5
0
0
50
100
150
200
250
300
350
400
Time (days)
Figure 5.18: Comparison of deflections as calculated by the FEM model and as
measured by experiment for the slab
Calculated deflections with uniform shrinkage for the slab are much lower than actual, because
the slab does not crack under these conditions. This is confirmed by comparison of the cracking
moment to the midspan moment. The cracking moment is calculated by transformed section
and found to be 6.85 kN.m, while the mid span moment is given by ଶ /8, and found to be
6.79 kN.m.
When non-uniform shrinkage strains are introduced the slab cracks due to additional shrinkage
induced tensile stresses at and near the surfaces of the slab. The resulting cracked beam profile
and stresses are compared to actual beam cracks at 400 days in figure 5.19, and show good
agreement.
Calculated cracking and stresses
Actual beam cracks
160 mm
3500 mm
Figure 5.19: Comparison of cracking for the one way slab as calculated and as
recorded by experiment at 400 days
44
It is noted, however, that autogenous shrinkage is not measured, nor included in the model, so
that at
= 14 days, ௦௛ = 0. This may introduce some error. However, in light of the
variability of the other inputs such as tensile strength and elastic modulus, a more refined
approach may not be any more meaningful.
The results in this section show the model predicts the cracked behaviour of beams and one
way slabs well. They also highlight the sensitive nature of deflection to cracking and the
critical role shrinkage can play in the onset of cracking.
5.13. AS3600-2009 comparison
AS3600-2009 provides a simplified approach to calculating long-term deflection in section
8.5.3.2, by multiplying short term deflection by a multiplier kcs, given as follows:
= = O2 − 1.2 / P ≥ 0.8
(5.21)
Where is the area of steel in the compressive zone (if any) and is the area of steel in the
tensile zone, both taken at midspan in a simply supported beam. For the uncracked beams
analysed in section 5.11, short term (immediate) deflection is given by:
0=
'Z
384 %
(5.22)
For these beams, the transformed section properties (all concrete) are; % = 484.4 × 10 mm4 and
% = 498.0 × 10 mm4 for slabs A and B respectively, giving deflections of 0.44 mm at 28 days
for both. This closely matches the short term values calculated by the FEM of 0.43mm. The
= factor is calculated to be 2 for slab A and 0.8 for slab B. For slab A, this gives a long term
deflection of 0.89 mm compared to actual long term deflection of 1.97 mm, and for slab B, 0.35
mm compared to an actual of 1.34 mm. FEM on the other hand provided closer results of 1.74
mm (vs 1.97 mm actual) and 1.21 mm (vs 1.34 mm actual).
For the cracked beam analysed in section 5.12, deflection can be calculated using equation 5.22
however I is now calculated according to clause 8.5.3.1 to incorporate cracking:
% = % + % + % U
where:
V ≤ 0.6%
(5.23)
% is the cracked transformed section found to be 28.7 × 10 ((
% is the uncracked transformed section found to be 144.3 × 10 ((
is the service design moment found to be 6.79 kNm based on a UDL of 4.43 kN.m-1.
is the cracking moment given by:
45
= [ \
3
.
−
where:
′
.
+
]
^ + ]X ≥ 0
%
(5.24)
is calculated to be 3 MPa using equation 7.XXX based on ′ = 25 MPa.
P is a prestressing/post-tensioning force not applicable to this example.
is the maximum shrinkage-induced tensile stress given by:
=
2.5_4 − 0.8_4
1 + 50_4
∗
(5.25)
where:
_4 is the ratio of tensile web reinforcement and found to be 0.0053
_4 is the ratio of compressive web reinforcement which is 0.
∗
is the final design shrinkage strain of concrete. From table 3.1.7.2 in AS3600-2009, for
25 MPa concrete, and interior environments this is found to be 789 × 10 (interpolated value).
This gives calculated values of
= 1.65 ]M , = 2.48 =( and % = 34.3 × 10 (( .
Short term deflection is calculated to be 11.3 mm, and long term deflection 22.6mm. This
compares to an actual value of 25.2mm, and 24.1 mm as calculated by FEM.
These results show that in design, the increased accuracy that comes about from a refined
approach comes at the cost of time and increased complexity, and this trade off must be
evaluated in light of the design problem. In addition, consideration must be given to the
variability of the properties that must be estimated for the calculation of time dependent
behaviour, and whether a refined and accurate model would improve results significantly.
46
Chapter 6
Measurement of shrinkage profiles
6.1. Previous techniques
As previously outlined, most shrinkage can be attributed to drying shrinkage. Drying
shrinkage, as discussed in chapter 2, occurs as a result of a loss of moisture by evaporation at
the surface. Thus the shrinkage profile will closely follow that of the moisture profile.
Previous experimental methods used to measure moisture content include overall weight
change, destructive sectioning and internal humidity/water content measurements (Weiss
1999). Weight change and destructive sectioning methods either do not provide adequate
resolution or are not practical. Humidity measurements have previously been taken by
measuring electrical conductivity of a porous medium, such as porous siltstone, inside the
concrete (Rajabipour, Sant & Weiss 2007). An example of this device is shown in figure 6.1.
Figure 6.1: Measuring RH by measuring
electrical conductivity (Rajabipour,
Sant & Weiss 2007)
Figure 6.2: Measuring RH in sealed cavities by direct
measurement (Grasley, Lange & D'Ambrosia 2006)
One disadvantage is that the porous medium itself absorbs some of the water thereby reducing
the relative humidity and electrical conductivity. This is known as hysteresis. Hysteresis can be
accounted for and the effect removed, but this requires calibration measurements. Accuracy of
this method is not confirmed.
Another method used to measure relative humidity is by measuring relative humidity directly
in sealed cavities in the concrete as shown in figure 6.2.
47
The drawback of this procedure is that the cavities run the depth of the sensor, and the air in
this large cavity must come into equilibrium with the concrete which may affect readings.
6.2. Development of new sensors
The approach taken in this thesis is to measure relative humidity by burying small humidity
sensors in the concrete, and sealing them with a breathable and waterproof cap. It is expected
impact on concrete performance would be minimised. Their small size would also suit real time
measurement if an application arises for measuring humidity in concrete in real buildings.
Sensors used are Sensirion SHT25s, with an accuracy of ±1.8% RH, an operating range of
0-100% RH, and size 3 x 3 x 1.1 mm. An image of the sensor is shown in figure 6.3.
3 mm
3 mm
Figure 6.3: Measuring RH in sealed cavities by direct measurement Invalid
source specified.
These sensors must be integrated into an electronic system. They cannot be used off the shelf,
and a circuit board is required to measured data. Design of a printed circuit board was sourced
from a local design company in Sydney. Size was minimised by printing on both sides of the
board. Final designs are shown in figure 6.4.
(b)
(a)
Figure 6.4: Circuit board design for the sensor
48
The white headerboard (4 pin connector) shown in figure 6.4a was not used, and wires directly
soldered to the PCB to reduce size. The grey cap shown in figure 6.4b is a PTFE membrane,
similar to Gore-tex commonly used in rain jackets, that is waterproof but also breathable.
The remainder of the PCB was sealed in an epoxy resin to ensure moisture would not interfere
with components. A 3m flat RJ11 (telephone) cable was used to connect to the PCB (by
soldering). The other end was connected to a male RJ45 connector, to connect to a data
logger, as shown in figure 6.5.
Figure 6.5: Data logger used to connect to humidity sensors Invalid source
specified.
Manufacture and assembly of the circuit boards and sensors was completed by a company in
South Australia. The finished product is shown in figure 6.6 and has been tested successfully in
wet environments.
Figure 6.6: Finished humidity sensor
49
Chapter 7
Conclusion
The objective of this thesis was to predict the behaviour of reinforced concrete members over
time and under service conditions.
In this vein, a cross sectional method of time dependent analysis was formulated for cracked
sections using a layered approach, and based on the step by step method. This was validated
against examples in the literature.
A finite element program was developed in Matlab, extending application of the step by step
method to frames and non-uniform loading. A comparison of experimental results from the
literature with the program showed agreement in both cracked and uncracked beams.
In the analysis of cracked beams, the incorporation of a curved shrinkage profile in the finite
element model was found to closely predict measured deflections, while a uniform shrinkage
profile predicted only half those measured. This highlights a critical role the shrinkage profile
can play in the calculation of cracking and deflection.
A comparison between the simplified method for calculating deflection provided by
AS3600-2009, and a more refined approach such as the finite element program, showed
improved accuracy for the refined model. It also highlighted the need to assess the trade-off
between accuracy and complexity, and whether a refined model would significantly improve
predictions in light of the variability in estimating time dependent properties.
50
Appendix A
Comparison of cross sectional methods to analyse time
dependent behaviour
Four methods to model time dependent behaviour are compared.: The effective modulus
method (EMM), the age-adjusted effective modulus method (AEMM), the rate of creep method
(RCM) and the step by step method (SSM).
A.1. Constant Deformation
To compare methods used to analyse time dependent behaviour, consider a symmetrically
reinforced concrete beam unrestrained and subject to shrinkage of -600µε that is constant over
time (while not realistic, this will facilitate explanation of the methods), as shown in figure A.1.
100mm
100mm
Where;
f’c = 40 MPa
As = 200 mm2
Figure A.1: Concrete beam subject to shrinkage and
creep
Values for the creep coefficient and the elastic modulus are given in table A.1 (calculated using
AS 3600-2009 Concrete Structures). Shrinkage values are not realistic.
Table A.1: Creep and shrinkage values calculated as per AS 3600-2009
Time
(days)
14
50
100
200
400
1000
2000
5000
10000
ϕ
0.00
2.22
2.62
2.85
2.98
3.08
3.12
3.15
3.16
εsh(t)
-600E-6
-600E-6
-600E-6
-600E-6
-600E-6
-600E-6
-600E-6
-600E-6
-600E-6
E(τ)
26.8E+3
29.3E+3
30.1E+3
30.8E+3
31.2E+3
31.6E+3
31.8E+3
32.0E+3
32.1E+3
From an intuitive perspective, one may reasonably expect the unrestrained specimen to
contract immediately as a result of the shrinkage, with compressive stresses in the restraining
steel and tensile stresses in the concrete. Creep in concrete will reduce these stresses over time
(effectively allowing it to stretch), as shown in figure A.2.
51
Intial tensile stress as a
result of shrinkage
Δߪ
stepwise reductions in tensile stress as a
Δߪ
result of creep from the initial tensile stress
Stress
Δߪ
Δߪ
Time
Figure A.2: Concrete stresses over time as a result of immediate and sustained shrinkage
The gradual reduction in tensile stress over time may be modelled as step wise drops in tensile
stress. Each drop in tensile stress is effectively the same as an application of compressive stress.
Each compressive stress is also subject to creep, as the original tensile stress is.
A.1.1. Effective modulus method (EMM)
The EMM adjusts the modulus of elasticity to include the effect of creep. Instead of using a
value of E based on instantaneous stiffness, it is reduced to take into consideration the creep
that has occurred over time. Consider equation 3.3, now in terms of stress;
=
=
=
=
Where
+
, + ()
1 + , + ()
`
1 + , , (, )
+ + () ≈
`
1 + , + (A.1)
= () ⁄ (1 + (, )) and is known as the effective modulus.
The EMM assumes the concrete stress applied at the end of the time history, is applied
constantly from the beginning to the end of the time history. In the example, this means creep
will be based on the final tensile concrete stress as shown in figure A.3, which is less than the
52
tensile stress at the beginning of the time history, thus creep will be underestimated, stresses
over estimated, and total contraction of the beam over-estimated as shown in figure A.3.
2.5
Concrete stress vs time
2.0
This region of stress not accounted for under
Concrete Stress (Mpa)
EMM in calculation of creep and final strain
1.5
Creep based on
1.0
final stress value
0.5
0.0
10
100
1000
10000
100000
Time (days)
Figure A.3: Concrete stresses over time calculated using EMM
A.1.2. Age-adjusted effective modulus method (AEMM)
The AEMM is similar to the EMM, however creep is calculated from two components. The
first is based on the initial stress at t=0 with the effective modulus as per the EMM. The
second is based on the total change in stress over the time period with an adjusted effective
modulus. The adjustment is required because the change in stress is gradual and the resulting
creep is reduced. The adjusted effective modulus is given as:
a , =
1 + , b(, )
(A.2)
Where , is the ageing coefficient and is less than one. Determination of the ageing
coefficient requires knowledge of the step by step method and is not readily available. However,
for load durations greater than 100 days, the following values provide reasonably accurate
results (Ranzi & Gilbert 2011);
Constant load b, = 0.65
Constant deformation b, = 0.80
The results in figure A.4 agree with those of the step by step method closely.
53
Creep based partly on initial stress
Concrete stress vs time
2.5
value using effective modulus as
Concrete Stress (MPa)
defined in EMM
2.0
Creep also based on the
total change in stress
1.5
using the adjusted
effective modulus
1.0
0.5
0.0
10
100
1000
10000
100000
Time (days)
Figure A.4: Concrete stresses over time calculated using AEMM
A.1.3. Rate of creep method (RCM)
The RCM assumes that the rate of creep that occurs at a given point in time is the same
regardless of when the load is first applied. This means that the creep curves are parallel as
shown in figure A.5.
߮ሶ (‫ݐ‬, ߬଴ )
߮(‫ݐ‬, ߬)
߮(‫ݐ‬, ߬଴ )
߮ሺ‫ݐ‬, ߬ଵ ሻ
(Actual)
߮ሶ (‫ݐ‬, ߬ଵ ) = ߮ሶ (‫ݐ‬, ߬଴ )
߮ሺ‫ݐ‬, ߬ଵ ሻ
(RCM Assumed)
߬଴
߬ଵ
‫ݐ‬
Time
Figure A.5: Creep assumptions in the RCM
54
Because the assumed rate of creep for loads applied after the first is greater than the rate of
creep that is actually occurring (refer to curve for , ଵ in figure A.5), creep strains are over
estimated for increasing load histories, and under estimated for decreasing load histories. In the
current example, the compressive stresses in the concrete are increasing, so that compressive
creep strains are over estimated. This results in lower stresses, and under estimation of
contraction in the beam.
A.1.4. Step by step method (SSM)
This method is outlined in section 3.2
A.1.5. Stress and strain results
Resulting concrete stresses over time and total beam contraction (strain) are shown in figure
A.6 and A.7 respectively.
Concrete stress vs time
2.2
Concrete Stress (MPa)
2.1
EMM
2.0
AEMM
SSM
RCM
1.9
1.8
1.7
1.6
1.5
1.4
1.3
10
100
1000
10000
100000
Time (days)
Figure A.6: Concrete stress over time as a result of instant and constant shrinkage
Total strain vs time
10
100
-300E-6
1000
10000
100000
Time (Days)
Strain
-350E-6
-400E-6
-450E-6
-500E-6
EMM
AEMM
SSM
RCM
-550E-6
Figure A.7: Strain over time as a result of instant and constant shrinkage
55
A.2.Constant load
Methods for analysing time dependent properties may also be compared by looking at resulting
stresses and strains in a short column under constant load, as shown in figure A.8. In this
example, the column is free to deform, and shrinkage is ignored.
50 kN
100mm
Where:
100mm
f’c = 40 MPa
As = 200 mm2
50 kN
Figure A.8: concrete column subject to constant load and creep
The time steps, elastic modulus, and shrinkage values used in the calculations are shown in
table A.2
Table A.2: Elastic modulus and shrinkage values used in constant loading example
Time
(days)
14
50
100
200
400
1000
2000
5000
10000
ϕ
0.00
2.22
2.62
2.85
2.98
3.08
3.12
3.15
3.16
εsh(t)
0
0
0
0
0
0
0
0
0
E(τ)
26.8E+3
29.3E+3
30.1E+3
30.8E+3
31.2E+3
31.6E+3
31.8E+3
32.0E+3
32.1E+3
Resulting total strain and concrete and steel stresses calculated by the four methods are shown
in figure A.9 to A.11.
56
Total strain vs time
600E-6
500E-6
Strain
400E-6
300E-6
200E-6
EMM
100E-6
AEMM
SSM
RCM
Time (Days)
0
10
100
1000
10000
100000
Figure A.9: Axial strain over time as calculated by each cross sectional method
4.4
Concrete stress vs time
Concrete Stress (MPa)
4.2
4.0
EMM
3.8
AEMM
SSM
RCM
3.6
3.4
3.2
3.0
2.8
Time (days)
2.6
10
100
1000
10000
100000
Figure A.10: Concrete stress over time as calculated by each cross sectional method
120.0
Steel stress vs time
Concrete Stress (MPa)
100.0
80.0
60.0
40.0
EMM
AEMM
SSM
RCM
20.0
Time (days)
0.0
10
100
1000
10000
100000
Figure A.11: Steel stress over time as calculated by each cross sectional method
57
For constant load, the EMM calculates creep at the final time step based on the final concrete
stress. Due to creep, the column has been ‘squashed’. This deformation increases stress in the
steel, and reduces the stress in the concrete. Because the creep is based on this final value of
concrete stress, it is lower than it should be, and total strain is therefore understated.
Conversely the RCM over estimates the rate of creep, and as a result the total creep and total
strain are overstated. This leads to greater stresses in the steel and reduced stresses in the
concrete than given by the more accurate SSM.
AEMM gives similar results to the SSM as expected, as the rate of creep is based both on the
initial concrete stress, and the gradual reduction in compressive stress. The reason for the slight
difference between the two methods is the value for b, which has been set at 0.8. A more
accurate value for b, (based on the SSM) would produce an exact match for the two
methods under scenarios such as this, where loads do not change over time.
58
Appendix B
Step by step cross sectional analysis formulation
At a cross section, at any time, the internal forces will be equal to the external applied forces
at a cross section. 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 5.11 and 5.12 respectively.
=
=
೎
೎
, (4.5)
, (4.6)
Combining equations 5.3, 5.5, and 5.11 and 5.12 yields:
=
೎
=
೎
, (,
+ − , ) + ,,
, (,
+ − , ) + ,,
,
,
(4.7)
(4.8)
After some manipulation, the resulting equilibrium equation is found to be:
Equation 5.13 can be rearranged as follows;
= , ,
+ $
, − 5
, ,
Where $ = ∫ , and , = ∫
,
+ ,, ,
(B.1)
Equation 5.14 can also be rearranged in a similar fashion;
59
= $
, ,
+ %
, − $
, ,
Where % = ∫ and , = ∫ + ,, ,
(B.2)
,
Equations 5.15 and 5.16 can be more conveniently stated as:
= !,, (, − , ) + !,, + ,, ,
(B.3)
= !,, (, − , ) + !,, + ,, ,
Where !,, = , ,
!,, = $
,
(B.4)
and !,, = %
,
As with concrete, the forces and moments in steel may be expressed as;
=
=
೎
೎
, (B.5)
, (B.6)
Equations 5.4, 5.5 and 5.19 and 5.20 can be combined to give:
=
=
೎
೎
(,
+ )
(B.7)
(, + )
(B.8)
Or in more compact form:
= !, , + !, (B.9)
= !, , + !, (B.10)
Where !, = , !, = >ೞ ×
= $
and !, = >ೞ ×
= %
Here ௦ is not strictly the second moment of area of the steel. The parallel axis theorem gives
the second moment of area about an axis other than its own as = ௢௪௡ + ଶ . The calculation
for ௦ in equation 5.23 ignores ௢௪௡ , however this is not significant as the area of the steel is
small relative to the concrete (usually about 2% (Standards Australia 2009)).
Combining equations 5.16, 5.17, 5.22 and 5.23 gives:
= !, , + !, − !,, , + ,, ,
(B.11)
60
= !, , + !, − !,, , + ,, ,
(B.12)
Where !, = !, + !,, , !, = !, + !,, and !, = !, + !,,
Using equations 5.7, 5.8, 5.24 and 5.25 it is possible to express equilibrium in matrix form as
follows:
, = + , − ,
(4.9)
Where
,
,
!,
=
, = ,
!,
,
!,
!,,
,
, = " # , , = ,, , , = !,
,
!,, ,
(4.10a-e)
Strains are then solved for, as shown in equation 5.27.
= , − , + , (4.11)
61
Appendix C
Finite beam element formulation
C.1.Displacement field
For any particular point ‘P’ in the beam, the generalised displacement model is shown in figure
C.1, which shows the admissible movement of that point in the orthogonal axis system y,z as
the beam deflects.
z
y
yr
uk
P
θ
y-yr
vj
θ
Where:
P
yr is distance from the centroid to
P
the reference z axis.
y is the vertical distance to the point P
θ= v’
(y-yr)cosθ
(y-yr)sinθ
For small θ
(‫ ݕ‬− ‫ݕ‬௥ )ܿ‫ ݕ( ≈ ߠݏ݋‬− ‫ݕ‬௥ )
ሺ‫ ݕ‬− ‫ݕ‬௥ ሻ‫ ≈ ߠ݊݅ݏ‬ሺ‫ ݕ‬− ‫ݕ‬௥ ሻߠ = ሺ‫ ݕ‬− ‫ݕ‬௥ ሻ‫ݒ‬′
Figure C.1: Admissible displacement field under the Euler-Bernoulli beam assumptions
From figure C.1, the displacement field is given as:
c, - = 0d + O/ − ( − )0′Pe
(C.1)
Where the only non-vanishing strain is ௭ given as:
$ =
fc. e f$
=
= /3 − − 0′′
ff-
(C.2)
C.2. Weak Formulation
The principle of virtual work is used to derive the weak formulation (the weak formulation is
also referred to as the global balance condition because it expresses the balance between
stiffness and displacement and loading for the element as a whole). Consider the generalised
beam loading shown in figure C.2, where subscript L refers to point loads on the left of the
beam and subscript R to point loads on the right. n and m are the distributed axial loads and
moments respectively.
62
p(z)
ML
ML
NL
NR
n(z)
SL
z
SR
L
Figure C.2: Generalised beam loading
For a virtual displacement of the beam, internal work is equated to external work in equation
C.3. From C.2 the only non–vanishing strain was found to be ௭ . The hat sign represents a
virtual displacement.
$ $ -
=
_0, + 2/,- + g 0, + g6 0,6 + /, + 6 /,6 + Hh + 6 Hh6
(C.3)
Combining equations C.1 and C.3 gives:
$ [/
3
− − 0′′]- =
Recalling that = >
/3 − 0 33 - =
$ _0, + 2/,- + g 0, + g6 0,6 + /, + 6 /,6 + Hh + 6 Hh6
and = >
$ (C.4)
gives:
_0, + 2/,- + g 0, + g6 0,6 + /, + 6 /,6
(C.5)
Equation C.5 can be expressed in matrix form as:
. " /,′ # =
−0,′′
0,
0,6
g
g6
2
/,
/
,
/
"_# . " # - + i j . 3 4 + i 6 j . 3 , 6 4
−0,
Hh
6 Hh6
(C.6)
Since the nodal forces on the right hand side can be incorporated in the assembly of the loading
vector when solving the system, it is ignored for now. This gives:
. " /,′ # =
−0,′′
2
/,
"_# . " # −0,
(C.7)
or more simply:
. , - =
.. +, -
(C.8)
2
where = , , = " /,′ # = "̂ #, . = "(#, and +, = "/,#.
−0,′′
̂
0,
Equation C.8 is a statement of global equilibrium.
The strain vector may be expressed as a function of displacement as:
63
f
D
fS
=C
C
B0
0 L
K "/# = +
f K 0
− J
fS
(C.9)
Equation C.9 can be combined with C.8 to give
. +, - =
.. +, -
(5.1)
where: = This is the weak formulation for a beam element.
64
Appendix D
Matlab finite element program
Figure D.1: Main GUI input for FEM
65
Figure D.2: Properties GUI input for FEM
66
References
Unsupported source type (InternetSite) for source Ste12.
Abramowitz, M & Stegun, I 1964, Handbook of Mathematical Functions, 10th edn, NY Wiley,
New York, US.
Al-Deen, S, Ranzi, G & Uy, B 2012, ' Non-uniform shrinkage in simply-supported composite
steel-concrete slabs', Steel & Composite Structures, vol under review.
Altoubat, SA 2000, Early age stresses and creep-shrinkage interaction of restrained concrete,
University of Illinois at Urbana-Champaign, Illinois, United States of America.
Bazant, Z 1975, 'Theory of Creep and Shrinkage in Concrete Structures: A Precis of Recent
Developments', Mechanics Today, vol 2, pp. 1-93.
Bazant, Z 1988, 'State of the art in mathematical modeling of creep and shrinkage of concrete',
RILEM Committee TC-69, John Wiley and Sons, Chichester and New York.
Bazant, Z 1997, 'Prediction of concrete creep and shrinkage: past, present and future', Joint
WANO/OECD-NEA Workshop on Prestress Losses in NPP (Nuclear Power Plant)
Containments, OECD Nuclear Energy Agency, Civaux, France.
Bazant, Z & Chern, J 1985, 'Concrete creeep at variable humidity: consitutive law and
mechanism', RILEM: Materials and Structures, vol 18, pp. 1-20.
Bazant, Z, Cusatis, G & Cedolin, L 2004, 'Temperature effect on concrete creep modeled by
microprestress-solidification theory', Journal of Engineering Mechanics, vol 130, no. 6, pp. 691699.
Bazant, Z, Hauggaard, A, Baweja, S & Ulm, F 1997, 'Microprestress-Solidification Theory for
concrete creep. I: Aging and drying effects', Journal of Engineering Mechanics, vol 123, no. 11,
pp. 1188-1194.
Bazant, Z & Xi, Y 1994, 'Drying creep of concrete: consitutive model and new experiments
separating its mechanisms', Materials and Structures, vol 27, pp. 3-14.
Bissonnette, B 2007, 'Tensile creep of concrete: study of its sensitivity to basic parameters',
ACI Materials Journal, vol 104, no. 4, pp. 360-368.
Bissonnette, B & Pigeon, M 1995, 'Tensile creep at early ages of ordinary, silica fume and fiber
reinforced concretes', Cement and Concrete Research, vol 25, no. 5, pp. 1075-1085.
67
Cement Concrete and Aggregates Australia 2007, 'Moisture in Concrete and Moisture-sensitive
Finishes and Coatings', Datasheet, Sydney.
Comité Euro-International du Béton 1993, CEB-FIB Model Code 1990: Design Code., Thomas
Telford, London.
Faria, R, Azenha, M & Figueiras, J 2006, 'Modelling of concrete at early ages: Application to
an externally restrained slab', Cement & Concrete Composites, vol 28, pp. 572-585.
Gilbert, I & Nejadi, S 2004, 'An experimental study of flexural cracking in reinforced concrete
members under sustained loads', in Studies from School of Civil and Environmental
Engineering, University of New South Wales, School of Civil and Environmental Engineering,
Sydney, Australia.
Grasley, Z & Lange, D 2004, 'Modeling Drying Shrinkage Stress Gradients in Concrete',
Cement, Concrete and Aggregates, vol 6, no. 2.
Grasley, Z, Lange, D & D'Ambrosia, M 2006, 'Internal relative humidity and drying stress
gradients in concrete', Materials and Structures, vol 39, pp. 901-909.
International Union of Pure and Applied Chemistry 2012, 'Gold Book'.
Kawando, A & Warner, R 1996, 'Model Formulations for Numerical Creep Calculations for
Concrete', Journal of Structural Engineering, vol 122, no. 3, pp. 284-290.
Kawano, A & Warner, F 1996, 'Model Formulations for Numerical Creep Calculations for
Concrete', Journal of Structural Engineering, vol 122, no. 3, pp. 284-290.
Kotsovos, M & Pavlovic, M 1995, Structural Concrete: Finite-element Analysis for Limit-state
Design, Telford, New York.
Nawy, E 2008, Concrete Construction Engineering Handbook, 2nd edn, CRC Press, Florida,
United States.
Nehdi, M & Soliman, A 2011, 'Early-age properties of concrete: overview of fundamental
concepts and state-of-the-art research', Construction Materials, vol 164, no. CM2, pp. 57-77.
Neville, AM 1970, Creep of Concrete: Plain, Reinforced and Prestressed, North-Holland
Publishing Company, Amsterdam, Holland.
Neville, AM 1995, Properties of Concrete, 4th edn, Pearson Education Limited, Essex, UK.
68
Østergaard, L, Lange, D, Altoubat, S & Stang, H 2001, 'Tensile basic creep of early-age
concrete under constant load', Cement and Concrete Research, vol 31, pp. 1895-1899.
Rajabipour, F, Sant, G & Weiss, J 2007, 'Development of Electrical Conductivity-Based
Sensors for Health Monitoring of Concrete Materials', Transportation Research Board,
Washington, DC.
Ranzi, G & Gilbert, RI 2011, Time-dependent behaviour of concrete structures, Spon Press,
Oxford, UK.
RILEM 1988, Mathematical Modeling of Creep and Shrinkage of Concrete, John Wiley & Sons
Ltd, Essex, UK.
Slowik, V & Ju, JW 2011, 'Discrete modeling of plastic cement paste subjected to drying',
Cement & Concrete Composites, vol 33, pp. 925-935.
Slowik, V, Schmidt, M & Fritzsch, R 2008, 'Capillary pressure in fresh cement-based materials
and identification of the air entry value', Cement & Concrete Composites, vol 30, pp. 557-565.
Standards Australia 2009, 'AS3600 Concrete Structures', Sydney.
Sutherland, J, Humm, D & Chrimes, M 2001, Historic concrete background to appraisal,
Thomas Telford Publishing, Longon.
Vandamme, M & Ulm, F-J 2009, 'Nanogranular origin of concrete creep', PNAS, vol 106, no.
26, pp. 10552-10557.
Weiss, J 1999, 'Prediction of Early-Age Shrinkage Cracking in Concrete Elements', PhD
Thesis, Northwestern University, Illinois.
Wight, J & Macgregor, J 2012, Reinforced concrete mechanics and design, 6th edn, Pearson
Education, New Jersey, US.
Wittmann , FH 1976, 'On the action of capillary pressure in fresh concrete', Cement and
Concrete Research, vol 6, pp. 49-56.
Wittmann, FH & Roelfstra, PE 1980, 'Total deformation of loaded drying concrete', Cement
and Concrete Research, vol 10, pp. 601-610.
Yang, Y, Sato, R & Kawai, K 2005, 'Autogenous shrinkage of high-strength concrete
containing silica fume under drying at early ages', Cement and Concrete Research, vol 35, pp.
449 - 456.
69