1 7 Hardening Concrete 7.1 Introduction For quite a number of reasons, the behaviour of hardening concrete has gained more and more interest in recent years. The most important reasons are: • Automation of production and building processes. Successful introduction of automation in the (concrete) building industry requires an accurate understanding of the hardening process. Criteria, with which production and building processes are controlled, are strength development and control or prevention of cracking. • More stringent requirements with relation to quality and durability More stringent quality and durability requirements are formulated, either in terms of the maximum temperature and temperature differences in the hardening stage, or in terms of the acceptability of the stresses or the magnitude of deformations. This represents micro- and/or macro cracking. • Functional requirements with relation to concrete structures Walls of liquid retaining structures, such as tunnels and basements are desired (preferred) to remain crack free. • The development of high performance types of concrete In order to benefit from the specific properties of high performance concretes, one has to be convinced that these properties are obtained. This asks for an accurate control of the hardening process. • The availability of high performance pc-computers The introduction of (rapid) personal computers made it possible to describe and calculate the complex behaviour of hardening concrete more accurately. This fact generates a need for more interest in the hardening stage and a move towards increasingly strictly formulated requirements. For all these cases, emphasis is on the hardening process and the associated phenomena such as temperature, strength and stress development and finally, cracking. Besides the presence of thermal stresses, the use of low water-cement ratio concretes substantially contributes to the development of stresses due to autogenous shrinkage (see section 7.6.3.). 7.2 Stress development and cracking in hardening concrete The mechanism responsible for the development of stresses and the formation of cracks in hardening concrete is known qualitatively. Also in the field of the qualification of the behaviour of hardening concrete, much is achieved. Nevertheless, there still exist a number of problems, especially when it comes to the practical applications of the achieved understandings. It is a highly complex problem, where we have to deal with the highly non-linear behaviour of the
Transcript
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7 Hardening Concrete
7.1 Introduction For quite a number of reasons, the behaviour of hardening concrete has gained more and more interest in recent years. The most important reasons are: • Automation of production and building processes.
Successful introduction of automation in the (concrete) building industry requires an accurate understanding of the hardening process. Criteria, with which production and building processes are controlled, are strength development and control or prevention of cracking.
• More stringent requirements with relation to quality and durability
More stringent quality and durability requirements are formulated, either in terms of the maximum temperature and temperature differences in the hardening stage, or in terms of the acceptability of the stresses or the magnitude of deformations. This represents micro- and/or macro cracking.
• Functional requirements with relation to concrete structures
Walls of liquid retaining structures, such as tunnels and basements are desired (preferred) to remain crack free.
• The development of high performance types of concrete In order to benefit from the specific properties of high performance concretes, one has to be convinced that these properties are obtained. This asks for an accurate control of the hardening process.
• The availability of high performance pc-computers The introduction of (rapid) personal computers made it possible to describe and calculate the complex behaviour of hardening concrete more accurately. This fact generates a need for more interest in the hardening stage and a move towards increasingly strictly formulated requirements.
For all these cases, emphasis is on the hardening process and the associated phenomena such as temperature, strength and stress development and finally, cracking. Besides the presence of thermal stresses, the use of low water-cement ratio concretes substantially contributes to the development of stresses due to autogenous shrinkage (see section 7.6.3.).
7.2 Stress development and cracking in hardening concrete The mechanism responsible for the development of stresses and the formation of cracks in hardening concrete is known qualitatively. Also in the field of the qualification of the behaviour of hardening concrete, much is achieved. Nevertheless, there still exist a number of problems, especially when it comes to the practical applications of the achieved understandings. It is a highly complex problem, where we have to deal with the highly non-linear behaviour of the
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material. Without the help of adequate numerical tools, quantification of this behaviour is impossible. Factors that play an important role for the quantification of the stress development and the probability of cracking in early-age concrete are: • The temperature development T(t) • The development of the thermal properties
- heat conduction coefficient - specific heat - expansion coefficient
• The hardening shrinkage • The development of mechanical properties
- compressive and tensile strength - elastic modulus
• Rheology properties - creep - relaxation
• Mechanical boundary conditions (degree of restraint and deformations) The factors mentioned, i.e. properties, show a time dependency. It is better to say that there exists a relationship with the development (progress) of the hydration or hardening process. An accurate description of the hydration process is therefore indispensable for achieving a consistent description of the behaviour of hardening concrete. The hardening process of concrete is the result of a chemical-physical reaction of cement and water. This is an exothermic reaction, which is a reaction during which heat is liberated. Due to this liberated heat, the temperature of the concrete rises (Fig. 7.1a) and the concrete expands. The strain increment Δε(ΔT(τ)) developed is:
)())(( τατε TT c ∆⋅=∆∆ (7.1)
In which: αc = expansion coefficient of concrete ΔT(τ) = temperature increment at time t = τ Besides the temperature induced strains, deformations due to swelling and/or hardening shrinkage might also occur. This issue is discussed in more detail in section 7.6. The hardening shrinkage, which is almost equal to the autogenous shrinkage εaush (see section 7.6.4) plays an important role for low water cement ratio concretes. For the total deformation increment of concrete in the hardening stage at time t = τ it holds: aushaushΔT )( εταε ∆+∆⋅=∆ + Tc (7.2)
If this deformation is restrained, stresses develop according to (Fig. 2.1.d): ( , ) ( ( ) ( )) ( ) ( )c aush ct T E rσ τ α τ ε τ τ τ∆ = ⋅ ∆ + ∆ ⋅ ⋅ (7.3)
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where: Δσ(t,τ) = stress increment at “loading” time t = τ Ec(τ) = elastic modulus of concrete at “loading” time t = τ r(τ) = degree of restraint of the deformation at “loading” time τ (r = 0,0 .. 1,0) Strain increment at time Δt(τ)
)())(( τατε TT c ∆⋅=∆∆
temp. (°C)T(t)
tτ
∆ τT ( )
To
a. temperature distribution T (t) tτ
shrinkage
∆εaush
b. autogenous shrinkage (t) ( hardening shrinkage) function of degree of hydratation
d. stress increment ( )∆σ τ Probability of cracking = P{σ(t) ≥ fctm(t)}
tens
ion
com
pres
sion
t ime
stress = Σ∆σ τ( )
normal distribution
tensile strength = f (t)ctm
e. stress versus tensile strength Fig. 7.1 The mechanism that leads to stress development and cracking in hardening concrete.
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Due to the influence of relaxation, stresses are reduced in accordance with (Fig. 7.1d): [ ]( , ) ( ( )) ( ) ( , ) ( ) ( )aush ct T t E rσ τ ε τ ε τ ψ τ τ τ∆ = ∆ ∆ + ∆ ⋅ ⋅ ⋅ (7.4)
where: Δσ(t,τ) = resulting stress increment at time t > τ ψ(t,τ) = relaxation factor Stress reduction due to relaxation is an important issue in hardening concrete. Relaxation factors traditionally applied for the description of the stress development in hardened concrete are not applicable. The relaxation behaviour of hardening concrete, therefore, requires due attention (see section 7.8.2) Summation of the stress increments Δσ(t,τ) results in the overall (tensile) stress σct(t) according to: ∑∆=∆
i
ict ttτ
τ
τσσ0
),()( (7.5)
Cracking occurs whenever the tensile stresses σct(t) exceeds the actual tensile strength (Fig. 7.1e). Expressed as a formula: )}(f)({)}({ ttPtFP ctctcr <= σ (7.6)
where: P{Fcr}(t) = probability of cracking at time t
σct(t) = tensile stress at time t (summation of the stress increments Δσ(t,τ))
fct(t) = tensile strength at time t
As mentioned before, the description of the hydration process is the basis for the description of the behaviour of hardening concrete. The quantification of the hydration process is discussed in more detail in the following sections.
7.3 The hydration process
7.3.1 Processes and mechanisms One can imagine the hardening process very schematically as follows. As soon as cement and water come into contact, a thin layer of reaction products forms at the surface of the cement particles (Fig. 7.2). This is called a phase-boundary reaction. Initially, the reaction process is retarded quite significantly due to the precipitation of the reaction products at the surface of the cement particles. The period during which hardly any observable reaction activity is exhibited, is called the dormant stage. The duration of this dormant stage is influenced substantially by means of addition of reaction accelerators or retarders. The duration of the dormant stage ranges from several hours up to more than twenty hours. At the end of the dormant stage, the reaction process changes towards the acceleration period. In a relatively short time, a substantial part of the cement reacts with water. Within this process, the
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layer of hydration products around the cement particles becomes thicker and particles are in contact with each other. After this stage, the process becomes diffusion controlled. A microstructure develops, existing of unhydrated cement, hydration products (“gel”) and a (capillary) pore system, partly filled with water. The gel consists of a solid material and water filled gel pores. This gel water is physically bound to the surface of the gel particles. After a certain period, the acceleration stage changes towards the “ceasing stage”.
acceleration period
de-celeration perioddormant stage
hydratationcurve (t)αh
time (t)
Q*max
Qmax
Qmax
Q*max
heat
-dev
elop
men
t Q (
)τ Q ( )τ
Q ( )τ
degr
ee o
f hyd
ratio
n (
)α
τh
degree of hydration ( ):α τh
α τh ( ) =
α τr ( ) =
degree of reaction ( )α τr
0.5
1.0
0
Fig. 7.2 Schematic representation of different characteristic stages during hydration.
7.3.2 The degree of hydration The ratio of the amount of cement converted into reaction products and the originally available amount of cement is called the degree of hydration αh (Fig. 7.3): ce ce
hce
amount of hydrated cement (V (0) V ( )) at time toriginally available amount of cement V (0)
− τ = τα = (7.7)
When it is assumed that the amount of liberated heat is proportional to the amount of cement that has reacted, the degree of hydration can also be presented with the ratio:
max
)()(QQ
hττα = (7.8)
where: Q(τ) = the amount of liberated heat at time t = τ
Qmax = the amount of liberated heat at complete hydration of all available cement.
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V (0)ce V (0)ce
V (0) -ce Vce
Vce
Vg
Vcw
V
αh =
Vl = air volume Vcw = volume of capillary water Vg = gel volume Vce = original unhydrated cement volume Vce(0) = original total cement volume
Fig. 7.3 Schematic representation of the degree of hydration αh(τ). The amount of heat developed at complete hydration depends on the type of cement. For Portland cement in particular, the value of Qmax is between 350 and 500 J/g cement. A relatively high C3A1) and/or C3S2) content results in a high value of Qmax. A high amount of C2S3) provides a lower and slower heat development. Depending on the type of cement, a water-cement ratio (wcr) of about 0,4 is theoretically required to obtain complete hydration of all the cement present. In practice however, a wcr of 0,4 does not result in complete hydration. Fig. 7.4 provides an impression of the maximum degree of hydration αh,max expected in practice as a function of the wcr. For finer cements, higher values for αh,max can be achieved and for more coarse cements lower values for αh,max will be reached than the results presented in the figure suggest.
00
0.2
0.4
0.6
0.8
1.0
0.2 0.4 0.6 0.8 1.0wcr
degree of hydration
Fig. 7.4 Practical values for maximum degree of hydration in concrete as a function of the wcr [1].
7.3.3 Degree of reaction In the literature, the term “degree of reaction αr” is sometimes used. The degree of reaction is defined as the ratiot between the amount of heat liberated Q(τ) and the (practical) maximum liberated amount of heat Q*max (see also Fig. 7.2):
max*
)()(QQ
rττα = (7.9)
For Q*max, the amount of heat liberated after 28 days of hardening can be used or, instead, the value determined in an adiabatic hydration test after e.g. 7 days of hardening. After this period, the hydration process almost ceases. As a result of this, the sum of the heat produced hardly changes. For a certain mixture, i.e. a mixture with a certain wcr, an unambiguous linear relationship between the degree of reaction and the degree of hydration exists: )()(
max*max τατα hr Q
Q= (7.10)
Based on Fig. 7.4, it is seen that, for most commonly used concrete mixtures in practice with wcr’s ranging between 0,40 and 0,55, a degree of reaction of 100% corresponds with a degree of hydration between 0,7 and 0,8. For low water-cement ratio mixtures, like high strength concrete (wcr ≈ 0,3), the degree of hydration is limited to about 0,5 – 0,6. The high strength of these concretes is therefore not a result of the high degree of hydration, but is the result of a high packing density of the particles.
7.4 Temperature calculation in hardening concrete
7.4.1 Adiabatic temperature rise In case all heat liberated is used for heating the concrete itself, it is called an adiabatic process. The temperature follows a so-called adiabatic temperature rise. This adiabatic temperature rise is calculated from:
cc
ha c
QCttT⋅
⋅⋅=∆
ρα max)()( (7.11)
where: ΔT(t) = adiabatic temperature rise at time t [°C] C = cement content of the concrete [kg/m3] ρc = specific mass of the concrete [kg/m3] cc = specific heat of the concrete [kJ/kg.°C] To illustrate this, Table 7.1 provides an example of an adiabatic temperature rise in a normal strength concrete mixture.
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temperature [°C]
timeT0
degr
ee o
f hyd
ratio
n α h
0.10.20.30.40.50.60.70.80.91.0
αh (adi)
Tadi
Fig. 7.5 Adiabatic hydration process and temperature rise – schematic view. The heat capacity ρc cc [kJ/m3 °C] of hardening concrete is, strictly speaking, not a constant value, but is a function of the degree of hydration. If, to avoid complexities, the heat capacity were a constant (which, for most practical applications, is acceptable) a linear relation exists between the adiabatic temperature rise and the degree of hydration. This implies that both have the same course in time (Fig. 7.5). For this particular case, the adiabatic temperature rise is often used to determine the degree of reaction, using a so-called adiabatic calorimeter. Table 7.1 Example of adiabatic temperature rise calculation Input parameters: Heat capacity: ρc.cc = 2500 kJ/m3°C Max. degree of hydration: αh = 0,8 Max heat production: Qmax = 450 kJ/kg Cement content: C = 350 kg/m3 Question: Adiabatic temperature rise ΔTa The adiabatic temperature rise is (formula (7.11)): ΔTa = 0,8 . 350 . 450 / 2500 = 50,4 °C The heat transfer to the surroundings prevented, the temperature of a concrete cast at 15 °C would rise up to about 65 °C!
7.4.2 Shape of the adiabatic curve The adiabatic hydration curve, or simply speaking the ‘adiabatic curve’, is an important input parameter for computer software to calculate the temperature development in hardening concrete structures. More and more (besides the traditional mix characteristics as slump and flow), a concrete producer might also be asked to provide an adiabatic curve of a concrete mixture. One should be aware that the shape of the experimentally determined adiabatic curve always depends on the initial (casting) temperature of the mixture!
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The shape of the adiabatic curve also depends on the following set of factors: Chemical composition of the cement Particle size distribution of the cement (Blaine) Water-cement ratio Cement content Temperature Availability of agents (accelerators, retarders) In Fig. 7.6, for three different concrete mixtures, the adiabatic heat development is presented as a function of time. It concerns a normal strength concrete (C = 350 kg/m3 Portland cement), a high strength concrete (C = 475 kg/m3) and a Granulate concrete (C = 150 kg/m3 blast furnace slag cement CEM III/B). The latter two adiabatic curves provide a clear impression of the range of adiabatic temperature rises to be expected. Fig. 7.7 provides three adiabatic curves for Portland fly ash cement for three different casting temperatures. What is striking is that a lower casting temperature initially shows a lower heat development, but in the longer run shows an increase of the heat production. The cause of this phenomenon is the influence of the hardening temperature on the structure of the hydration products formed [2].
200 24 48 72 96 120 144 168
40
60
80temperature [°C]
b
a
c
time [hours]
concrete characteristics
a. C25/C35
b. C85
c. granulate concrete
Fig. 7.6 Adiabatic temperature rise for three concrete mixtures [19].
10
0
0
10
20
30
40
50
1 2 3 4 5 12 24 36 48 72 96 120 168time [hours]
adiabatic temperature rise T [K]∆
TA 51
TA 52
TA 53
T =300 ° 20° 10°
Fig. 7.7 Adiabatic temperature rise at different casting temperatures.
7.4.3 The semi-adiabatic hardening process In case of practical circumstances (depending on the dimensions of a structure and/or the presence of cooling pipes), heat is transferred to the surroundings with elapse of time. The hydration process then no longer is adiabatic. The temperature of the concrete decreases and the hydration process will retard. A hardening process that initially is adiabatic and shows indications of retardation is a semi-adiabatic process. In Fig. 7.8, both the development of an adiabatic and a semi-adiabatic hydration process are schematically presented. The semi-adiabatic process takes place at the indicated semi-adiabatic temperature development Tp(t).
0
0.5
1.0
degr
ee o
f hyd
rata
tion
(t)
α h
time
temperature ( °C)
adiabatic curve
αh (t)
Tsemi adi
Fig. 7.8 Semi-adiabatic hardening process – schematically. The course of a semi-adiabatic hydration process, also called the process curve, is determined by the temperature Tp(t) at which the reaction process takes place. Oppositely, the temperature on its turn depends, among other things, on the amount of heat produced per unit of time. Thus, the temperature development and the heat development are coupled quantities.
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The mutual dependency between the rate of hydration and the concrete temperature implies that the calculations of both the temperature development and the hardening process are to be carried out using a step-wise calculation procedure in which the effect of the actual temperature and the rate of reaction are modelled explicitly.
7.4.4 Hydration and temperature development in a hardening concrete structure
For determining the development of the hydration process (i.e. the degree of hydration) and the temperature development in a hardening concrete structure, use is made of the differential equation of Fourier including a heat source. For a one-dimensional heat flow, it holds: ),(
.1
ccc
2
2
c txqcx
TatT
⋅+∂∂
⋅=∂∂
ρ (7.12)
where ac is the temperature levelling coefficient according to:
cc
cc . c
aρ
λ= (7.13)
where: ac = temperature levelling coefficient [m2/h]
λc = heat conduction coefficient [W/m.K] cc = specific heat of the concrete [kJ/kg.K] ρc = specific mass of the concrete [kg/m3] T(x,t) = temperature of the concrete [K] qc(x,t) = heat source [kJ/m3.h] x = coordinate [m] t = time [h] With qc(x,t) = ∂Qc(x,t) / ∂t and the transition towards a finite difference approach, the differential equation is expressed as:
ttxQ
cxTa
tT
∆∆
⋅+∆∆
⋅=∆∆ ),(
.1
)(c
cc2
2
c ρ (7.14)
where ΔQc(x,t) [kJ/m3] is the heat development within a time step Δt. The temperature at time t+Δt is found by adding the temperature rise caused by the heat liberation ΔQc(x,t) to the temperature T(t). This temperature rise is calculated by dividing the amount of heat ΔQc(x,t) liberated within a time step Δt by the heat capacity ρc.cc. For this particular case, the heat increment ΔQp(x,t), in which the index p implies the “process”, can be determined according to (see Fig. 7.9): ),(f jp;jh;1jp;1ja; TQQ α⋅∆=∆ ++ (7.15)
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where f(αh;j,Tp;j) is the Arrhenius-based relation:
jp;ja;
jp;ja;jp; )(
jp;jh; ),(f TTTT
RTE A
eT ⋅
−⋅
=α (7.16)
where: EA(Tp) = (apparent) activation energy [J/mol] R = universal gas constant [8,31 J/mol.K] ΔQa;j+1 = heat development in time step Δtj+1 in the adiabatic process ΔQp;j+1 = heat development in time step Δtj+1 in the semi-adiabatic process Ta;j = temperature in K at an adiabatic process at time tj Tp;j = temperature in K where the process actually happens at time tj The quantity f(αh;j,Tp;j) relates the rate at which the hydration process takes place to the rate at which the process would take place if the conditions were adiabatical (see Fig. 7.9).
∆Qp;j+1∆Qa;j+1
Qa;jTa;jQp;jTp;j
∆tj+1
∆tj+1
tj+1tj
1
3
2
time0
QmaxQ (t)
T0
tem
pera
ture
(°C
)
Fig. 7.9 Procedure to determine the rate of hydration in a non-adiabatic process – schematic.
7.4.5 Apparent activation energy The reliability of the calculated temperature development and rate of reaction strongly depend on the activation energy EA adopted. This quantity accounts for the temperature sensitivity of the hydration process. A high activation energy implies a high temperature sensitivity. The activation energy is not a constant, but depends on the chemical composition of the cement, the degree of hydration and the temperature range considered [2]. An indication for the range of the activation energy is in Fig. 7.10. For many practical cases, the following values for the activation energy can be used [3]: For T ≥ 20 °C: EA = 33.5 kJ/mol For T < 20 °C: EA = 33.5 + 1.47·(20-T) kJ/mol
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0
1
0.5
2
3
4
5Krel
0 10
273 293 313 333
20 30 40 50 60 70 °C
Ktemperature
E =50 kJ/molA
E =35 kJ/molA
E =30 kJ/molA
b
a
b
E =45 kJ/molA
a. Danish cementb. Swedish cements
Fig. 7.10 Temperature function Krel = f(αh,T) for different values of the apparent activation energy EA.
A quantitatively comparable result as achieved with formula (7.16) is obtained with the relation: 10
ph
pa
),(fTT
gT−
=α (7.17)
where “g” is a temperature sensitivity factor. This expression is comparable to the one used for the activation energy EA. The “g”-value depends on the temperature, the degree of hydration and the chemical composition of the cement. With g-values between 1,6 and 2,2, in general good results are obtained. To solve the differential equation of Fourier with a heat source, solution procedures described in chapter 2 are used. The temperature rise due to heat liberation in a certain time step and at a certain location is accounted for in each time step. The result of a temperature calculation in hardening concrete is given in table 7.2.
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Table 7.2 Summary of temperature calculation in hardening concrete. The output of a temperature calculation as described previously is twofold. Calculated are: □ The temperature T(x,t) □ The degree of hydration αh(x,t)
T(x,t+1)αh(x,t+1)
αh(x,t)
T(x,t)
x x
temperature T(x,t) degree of hydratation (x,t)αh The degree of hydration is the most important parameter for the description of the material
properties. These concern the thermal, physical and mechanical properties. The accuracy of the temperature calculation depends, at given thermal boundary conditions,
on the accuracy when describing the temperature sensitivity of the reaction process and the development of the thermal properties.
7.4.6 Effect of thermal boundary conditions on temperature development
7.4.6.1 Influence on the heat transfer coefficient With the presence of for example a formwork or an insulating layer with a thickness di and a heat-conduction coefficient λi, the resulting heat transfer coefficient αres is found by: ∑+=
i d1 i
i
mres
11λαα
(7.18)
where αm is the heat-conduction coefficient of, e.g., the formwork material and the surroundings, i.e. air.
7.4.6.2 Wind effect during hardening When wind blows along the hardened concrete surface, additional cooling takes place. An indication of the effect of the transfer coefficient, i.e. the wind speed, on the temperature development in a hardening concrete element can be deduced from Fig. 7.11a-b. A 1 m thick concrete plate is cast on a soil foundation. The plate’s topside is assumed to be exposed to the surrounding air. The plate is constructed from high strength concrete, 475 kg/m3 cement (50% Portland cement and 50% Blast furnace slag cement), 25 kg/m3 silica fume and cast at a temperature of 20 °C.
15
20
30
40
50
60
70temperature ( °C)
temperature ( °C)0 24 48 72 96 120 144 168
time (hours)
0
0.2
0.4
0.6
0.8
1.0floor thicknes (m) top side
20 30 40 50 60 70
7 days 3 days 1 day
α(W/m K)2
51525
α(W/m K)2
51525
middle
top side
a. Temperature development in time b. Temperature development versus height
Fig. 7.11 Effect of the heat transfer coefficient on the temperature development in a 1 m thick
hardening concrete plate made of high strength concrete. The temperature development is calculated for transfer coefficients of about 5, 15 and 25 W/m2K, associated with wind speeds of 0, 2 and 5 m/s. Fig. 7.11a gives the temperature development for different transfer coefficients in time, for both the centre and the top side of the concrete plate. Fig. 7.11b provides the temperature development over the thickness of the concrete plate at three different ages, i.e. 1, 3 and 7 days. In order to show the effect of the thermal boundary conditions most explicitly, a high cement content mixture is investigated. It is quite clear that the presence of a surface wind results in a significant increase of the temperature differences in a concrete cross-section. As a result of the temperature decrease at the surface, the hardening process evolves more slowly. The elastic modulus also develops more slowly. This is a positive by-effect since it results in a more moderate development of the thermal stresses in this surface zone. Thanks to the moderate development of the stresses, stress relaxation is more effective. Quantification of the different influences on the stress development requires the use of advanced numerical programs.
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with solar radiation, no formwork
no solar radiation, no formwork
with solar radiation, removal offormwork after 48 hours
no solar radiation, removal offormwork after 48 hours
0 20 40 60 80 100 120time (hours)
0
20
40
60
80surface temperature ( °C)
Fig. 7.12 Effect of solar radiation on the temperature development in a hardening concrete wall
cast in high strength concrete [27].
7.4.6.3 Solar radiation During the first stage of hardening, also when formwork is present, solar radiation might influence the hydration process of concrete significantly. The additional input of heat namely stimulates the reaction process and leads to an additional temperature increase of 10 °C or even more. Fig. 7.12 provides an example of the influence of solar radiation of the maximum temperature in a 0,16 m thick concrete wall cast in high strength concrete. Due to the effect of solar radiation, the temperature rise is about 10°C higher than without the influence of solar radiation. As a result of this, the temperature drop will also be 10 °C larger. This might result in a substantial increase of the probability of cracking.
7.5 Thermal properties of hardening concrete In the differential equation of Fourier, the temperature leveling coefficient ac is determined according to (formula (7.13)):
cc
cc . c
aρ
λ=
It represents the rate at which the heat transport takes place. During hardening, the structure of the concrete will change, and, as a result, also the heat conduction coefficient λc and the specific heat cc. In the literature, contradictory data is given on the change of the heat conduction coefficient λc with elapse of the hydration process. It seems that with an ongoing degree of hydration, a slight increase of about 10 % looks reasonable. For the specific heat cc at increasing degree of hydration, a slight decrease is observed.
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A slight increase of the heat conduction coefficient and a reduction of the specific heat results in an almost constant value of the temperature leveling coefficient ac. For practical applications it is, therefore, often allowed to adopt a constant value for the thermal properties of concrete. For fundamental research, however, the changes in the heat diffusion coefficient during progress of the hydration process should be accounted for.
7.6 Deformations in hardening concrete The hardening process of concrete goes along with deformational changes. These changes can be attributed to different causes such as: Thermal changes Plastic shrinkage Chemical shrinkage Autogenous shrinkage Expansion of reaction products Expansion due to moisture adsorption (submerged hardening) Restraining the expansion caused by the expanding reaction products in the early age of hardening and the moisture adsorption, results in relatively low compressive stresses. Since the stiffness of the concrete is still very small at that time, the compressive stresses will also be quite low. In this report, these stresses will not be considered. Special types of cement have been developed of which swelling is a main characteristic. An example is the American type K cement. Swell cements, as well as the phenomenon of swelling, as holds for normal cements, will not be considered.
7.6.1 Linear expansion coefficient αc As for hardened concrete, the expansion coefficient of hardening concrete is a function of the expansion coefficients of the mix constituents. Since the cubic expansion coefficient of water is much larger than the coefficient for concrete (about 5 times), the deformations in the early age of hardening mainly depend on the presence of water. In this stage of hardening, the stiffness of the concrete is still very low. The resulting stresses will therefore also be very low. In addition to this, there is also a substantial stress relaxation. It seems therefore allowed for practical purposes, to adopt a constant value for the expansion coefficient. Other indicative values of the linear expansion coefficient are in section 2.5.
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r.v.(%
) 100
40
concrete temperature (°C)40
35
30
25
20
10
wind
spee
d (k
m/h
)
3530
25
20
15
10
5
0
9080
7060
50
40
30
5 10 20 30 40air temperature ( °C)
0
1
2
3
4ra
te o
f eva
pora
tion
(kg/
m.h
)2
Fig. 7.13 Evaporation rate as a function of different influential factors.
stress N/mm2 stress N/mm2
time time
fct fct1 fct2
no crackingcrack development
σcapillaryσcapillary
Fig. 7.14 Development of tensile stress and tensile strength of early age concrete.
19
7.6.2 Plastic shrinkage Inside the freshly cast concrete, the heaviest parts of the mixture tend to move downwards under the influence of the gravitational forces. As a result, water moves upwards. Especially for floor slabs, a thin water layer is often present on top of the concrete surface directly after casting. This phenomenon is the more pronounced when the concrete is vibrated after casting. Depending on the surrounding conditions (wind, temperature), this water layer evaporates more or less rapid. Fig. 7.13 gives an indication of the rate of evaporation as a function of a number of relevant influential factors. In case of a high evaporation rate, the thin water layer might disappear completely and plastic shrinkage is very likely to occur. The background of this phenomenon is the presence of an under-pressure in the fresh concrete. Imagine young concrete as being a system of a high number of water-filled pores. Initially, in the water-saturated system, a hydrostatic pressure is present. When the surface water evaporates, hollow menisci develop in the pores, similar to the one in the well know experiment with Torricelli’s tube. If this test is performed by inserting a tube into a closed water-filled box, an under-pressure develops inside this box. Quite similar, an under-pressure develops in the young, still plastic, concrete. This under-pressure results in tensile stresses in the hardening mass. If these tensile stresses exceed the actual concrete tensile strength, cracking appears (Fig. 7.14). In floor slabs with a thickness of 400 mm, cracks might easily develop over full slab thickness (through cracks). Note: Plastic shrinkage is not to be confused with drying shrinkage. Preventing measures A simple and often used measure to avoid plastic cracking is keeping the concrete wet during hardening. This is achieved by supplying water on top of the concrete’s surface or by preventing moisture loss (evaporation). For this purpose, the concrete surface can be covered with a carpet, plastic cover or a foil layer, or by applying a curing compound (curing compound: a hydrophobic liquid to be sprayed on top of the concrete’s surface directly after casting to act as a “vapor-loss preventing layer” through which the water in the concrete can not escape).
7.6.3 Chemical shrinkage The volume of the hydration products (the gel) is smaller than the product of both the individual reacting components cement and water. This volume reduction is denoted as chemical shrinkage (in German: Chemisches Schrumpfen). Chemical shrinkage represents itself mainly by the formation of capillary pores in the cement stone and hardly as an outer deformational change. Therefore, negligible or no stresses are generated on a macro-scale level. Chemical shrinkage corresponds to about 25% of the volume of the chemically bound water of a hydration reaction. The chemically bound water corresponds to about 25% of the weight of the reacted cement. If no additional water comes from the surroundings, the volume reduction induced capillary spaces are filled with water vapor or air.
20
7.6.4 Autogenous shrinkage With progress of the hydration process, the water available reacts slowly. This process results in “emptying” the pore system. At first, water from the largest pores in the system reacts, followed by water from the smaller ones. One can also say that the water withdraws itself into the smaller pores. This is like a drying process, where no exchange of moisture takes place with the surroundings. In the English literature, this process is denoted “self desiccation”. For low water cement ratios (wcr < 0.5 [13]), this “self drying” results in volume reduction. In the German literature, it is called “Chemisches Schwinden”. In contrast with chemical shrinkage, from now on, we will denote it as autogenous shrinkage. The German differentiation between Chemisches Schrumpfen and Chemisches Schwinden makes clear that we deal with two different phenomena. However, the term Chemisches Schwinden suggests somehow that we deal with a chemically-based shrinkage process, whereas it should be considered much more as a process with a physical i.e. thermo-dynamical origin. On the other hand, it is a fact that autogenous shrinkage is connected with chemical shrinkage. However, on a macro-scale level, effects are often combined and denoted “hardening shrinkage”. This combination is assumed to represent autogenous shrinkage. For the sake of convenience, the external deformations from chemical shrinkage are then neglected. Table 7.3 The porosity of fully hydrated cement. Complete hydration of cement theoretically requires a water cement ratio of 0,4. At hydration of X grams of cement, 0,15 X grams of water are physically bound and 0,25 X grams are bound chemically. 0,25 X gram water has a volume of 0,25 X cm3. The volume reduction of the hydration products with respect to the volume of the individual components is: ΔV = 0,25 · 0,25 X cm3 = 0,0625 X cm3 Expressed in volumetric percentages of completely hydrated cement, in which no free water is available anymore, for the porosity P op the hardened cement it holds:
ce w
V 0,0625 XP 100% 100% 8,6%X wcf X X 0,4 X3,1 1,0
∆= ⋅ = ⋅ =
⋅+ +
ρ ρ
where ρce and ρw are the specific mass of cement and water, respectively. Autogenous shrinkage manifests itself mainly for water-cement ratios below 0,4. For a cement paste with a wcr of 0,4, Grube [13] measured an autogenous shrinkage of about 0,7·10-3 after 28 days of hardening. For normal strength concrete, autogenous shrinkage is about 1/6 of the shrinkage of cement paste. In this case, εaush = 0,12·10-3. This is about the cracking strain of concrete! Large autogenous shrinkage occurs in high strength concrete. In Fig. 7.15, values of the autogenous shrinkage of cement paste and concrete are given for different values of the wcr. In Fig. 7.16, the autogenous shrinkage is a function of the degree of hydration. There is a strong correlation between the autogenous shrinkage and the degree of hydration at low wcr’s.
Fig. 7.15 Autogenous shrinkage of cement paste (left) and concrete (right) for different wcr’s
[24]. (Note: be aware of the differences in scale).
00 0.1 0.2 0.3 0.4 0.5 0.6
-0.05
-0.10
-0.15
-0.20
-0.25
degrees of hydration
autogenous shrinkage (x10 )-3
Fig. 7.16 Autogenous shrinkage of high strength concrete as a function of the degree of hydration
[4]. Note: For any concrete mixture, the autogenous shrinkage might differ substantially.
Grube [13] remarks that autogenous shrinkage, i.e. shrinkage due to “self-drying”, also has a positive effect, in particular when the frost resistance and heat loading are concerned. In those cases, the self-drying introduces pore spaces that either act as an additional expansion capacity in case of ice formation or store the expanding water vapor.
22
7.7 Development of mechanical properties of young concrete
7.7.1 Compressive strength For the description of the compressive strength of concrete, use can be made of the following concepts: 1. The maturity concept 2. The degree of hydration concept 3. Codes and Standards (e.g. CEB/FIP MC’90) The maturity concept is often used in practice. The degree of hydration concept is a more recent development, at least as far as the application in practice is concerned. Some codes give guidelines for including the effect of temperature on the strength development. Mostly, these guidelines are conceptual. For a description of the strength development in the very early age of hardening, these guidelines are often inadequate.
7.7.1.1 The Maturity concept The principle The maturity concept gives a relation between the compressive strength fc and the maturity M. The concept maturity as such is ambiguous. One has to check for each case which specific method to use. In the original definition of Saul et al. [5], the maturity M [°C.h] is the product of the temperature and the hardening period (Fig. 7.17a): ∫ +=
tdtTtTM
0 r ))(( [°C.h] (7.19)
where T(t) is the concrete temperature at time t and Tr is the reference temperature. For the reference temperature, Saul adopted a value Tr= -10°C. For the maturity as considered, there is a linear relationship between the strength and the logarithm of the maturity (Fig. 7.17b).
∆t
∆T
T (t)
temperature ( °C)
time-10
0
n (maturity) (°C.h)
fc cm
a. Temperature as a function of time b. Strength as a function of the logarithm of maturity Fig. 7.17 Principle of the Maturity method [5].
23
The calibration curve In order to be able to work with the maturity concept in practice, the relationship between the concrete used and its strength must be determined on beforehand. To do so, the compressive strength should be determined on cubic specimens that harden at a constant temperature, i.e. the control temperature. This provides the so-called “calibration curve” (Fig. 7.17b). The control temperature is mostly 20°C. Weighted maturity Different researchers have found that the linear relationship, as proposed by Saul, does not hold for any arbitrary type of concrete. This has resulted in the introduction of several modifications to the original concept, e.g. Bergström [6], Plowman [7], Papadakis [8], Bresson [9] and Carino [10]. A number of these modified methods is known as “weighted maturity methods” (in Dutch: gewogen rijpheid). A Dutch alternative for the weighted maturity method was developed at
0 20 40time of hardening (hours)
-10
0
10
20
30
40temperature ( °C)
n- 2n- 1.5n- 1n- 0.5n 0n 0.5n 1n 1.5
Fig. 7.18 Graph with n-values used to calculate the weighted maturity Mw according to the CEMIJ-method [11].
Table 7.4 C-values for Dutch cements (method CEMIJ) (e.g. [11]). Cement type Notation Strength Characteristics C-value Old cement
the Cement plant IJmuiden (CEMIJ) and is known as the method “De Vree” (see [11]). According to the CEMIJ-method, the weighted maturity Mw is directly determined using an expression that replaces the calculation time intensive integral from equation (7.19) (see Fig. 7.18):
24
nnw .COM = [°C.h] (7.20)
where On is an integral surface increment [°C.h] to which a value n is linked. The n is the power of the C-value in the calculation of the weighted maturity. In table 7.4, different indicative C-values are given. By using these so-called weight values in the calculation, there is a linear relationship between the (weighted) maturity and the strength. Application in practice The maturity method (CEMIJ-method) is described very accurately in CUR-recommendation 9 [12]. The potential of the method comes from its easy application. After having determined the calibration curve, the temperature is measured, followed by a calculation of the (weighted) maturity and, finally, the strength. The strength follows from the calibration curve. For a more extensive description of the maturity method, reference is made to the literature [11,12,14]. Remarks: 1. The term “maturity” is sometimes used to indicate the “degree of hydration” [15]. By doing so, things become
obscured. The maturity is a parameter with dimension [°C.h], whereas the degree of hydration is a dimensionless quantity that ranges from 0 to 1,0, or 0 to 100% respectively.
2. Instead of using the measured temperature to determine the maturity, the calculated temperature can also be used. In this way, one can simulate all kinds of boundary conditions in the design stage of a project. The results might be a basis for the choice of a concrete mixture, casting procedure or time of removal of the formwork.
7.7.1.2 Temperature effect on strength development CEB/FIB MC’90 In the CEB/FIP MC’90 [31] (the basis for EN 1992-1-1), the strength development is written as: cmcccm )( ftf ⋅= β (7.21)
where:
−
=
5,0
1/281s
cc )(tt
etβ (7.22)
where: fcm(t) = mean compressive cylinder strength after t days [N/mm2] fcm = mean compressive cylinder strength after 28 days [N/mm2] βcc(t) = coefficient, depending on the age t
t = age of concrete [d] (if necessary corrected for the hardening temperature acc. to formula (7.23))
t1 = 1 (with regard to dimensions) [d] s = cement depending coefficient 0,20 : for rapid hardening cement 0,25 : for normal hardening cement 0,38 : for slowly hardening cement These formulae are valid within the temperature range -20°C to + 40°C. For temperatures within the range from 0°C to 80°C, a maturity method is given. This method proposes an equivalent time (in Dutch: equivalente tijd) according to:
25
∑
=
−
∆+−
⋅∆=n
i
TtTi
ietTt1
65,13/)(273
4000
0)( (7.23)
where: t(T) = corrected age of concrete [d] T(Δti) = temperature within time increment Δti [°C] Δti = time increment [d] T0 = 1 (to have consistency of dimensions) [°C]
7.7.1.3 Degree of hydration concept 7.7.1.3a Compressive strength as a function of the degree of hydration – material science
view At increasing degree of hydration, cement particles are gradually more intense and firmly connected to one another. Therefore, the degree of hydration is an important parameter with respect to the strength development of concrete. In a large number of examinations on this topic, a clear relationship existed between the degree of hydration and the strength development of concrete. Fig. 7.19 shows some results of Fagerlund [16]. The figure illustrates that, after having reached the critical degree of hydration α0, the strength increases almost linearly with the progress of the degree of hydration. The critical degree of hydration α0 turns out to be a function of the wcr. This was to be expected. After all, for a higher wcr, the degree of hydration must be higher before the hydrating cement particles are able to bridge the inter-particle distance and are in contact with each other, which results in strength and stiffness development.
0
20
40
60
80
100concrete compressive strength (f )cm
0 0.2 0.4 0.6 0.8 1.0degree of hydration
wcr = 0.3
wcr = 0.4
wcr = 0.5
wcr = 0.6
Fig. 7.19 Compressive strength versus degree of hydration for different wcr’s ([16]). The linear relationship as found by Fagerlund is reconfirmed by a large number of research projects. Strictly speaking, the relationship is modeled even more accurately when using a 3rd or 4th degree polynomial that crosses the origin and crosses the horizontal axis at 90 degrees. For practical purposes, the linear relationship, where the strength development commences after having passed a so-called critical degree of hydration, is sufficiently accurate.
26
7.7.3.1b Practical applications – The UCON-system Application of the degree of hydration concept to determine the strength development, pre-supposes that the degree of hydration as well as the relationship between the degree of hydration and the compressive strength are known. The degree of hydration is relatively easily determined with the help of special purpose software. The input required for a calculation consists of: 1. The adiabatic hydration curve of the concrete mixture; 2. The concrete temperature during hardening. That the degree of hydration can be determined using these two input data can easily be seen if the algorithm, which has been discussed earlier in section 7.4.4 for the determination of the temperature development, is considered. The calculation algorithm is explained with the help of Fig. 7.9. In Fig. 7.20, figure 7.9 is shown again, almost without any changes. When explaining Fig. 7.9, it was stated that the heat increment ΔQp;j+1 within time step Δtj+1, can be calculated if the calculated temperature from the previous time step Tp;j is known. Instead of this calculated temperature, the measured concrete temperature can be used too. Based on this measured temperature and the adiabatic hydration curve, exactly the same calculation can be conduced as described in section 7.4.4. The result of the calculation is the degree of hydration as it develops at the prevailing temperature conditions within the concrete!
∆Qp;j+1∆Qa;j+1
Qa;jTa;jQp;jTp;j
∆tj+1
∆tj+1
tj+1tj
1
3
2
time0
QmaxQ (t)
T0
tem
pera
ture
(°C
)
1. Adiabatic hydration curve
(input: from adiabatic test) 2. Measured temperature on the
construction site 3. Calculated degree of hydration
Fig. 7.20 Calculation procedure for the determination of the degree of hydration αh(t) from the
adiabatic hydration curve and the measured concrete temperature Tp(t). The software for calculating the degree of hydration from the adiabatic hydration curve and the measured temperature is part of the UCON-system (Universal Concrete Curing CONtrol system) developed at the TU Delft [25,29]. Fig. 7.21 presents the UCON-system schematically. The adiabatic hydration curve is determined experimentally (or numerically [28]) and is imported in the computer. At pre-assigned locations within the structure, temperature measurements are performed. Their results are stored in the computer. Based on the imported data, the degree of hydration is determined for all these locations. For a known and a priori data base stored relationship between the degree of hydration and the strength, the strength in these particular locations is determined. The relationship between the compressive strength and the degree of hydration is determined quite easily, using results from compressive strength tests on concrete cubes hardened in
27
compliance with a known predefined curing temperature. With assistance of the UCON-system, the degree of hydration that corresponds with the temperature profile is determined. By measuring the compressive strength at different times, i.e. different degrees of hydration, the relation between the degree of hydration and the compressive strength is determined for a specific concrete. If the relation between the degree of hydration and the strength is not determined in advance, use is made of a graph as shown in Fig. 7.19 or by consulting other relations for similar concrete mixtures from the literature. It is noted that the relation between the compressive strength and the degree of hydration for high strength concrete as proposed by Fagerlund, shows and underestimation of the compressive strength [17]. The reason for this is most likely the effect of silica fume in the concrete. This aspect was at that time not considered by Fagerlund.
thermocouple
thermocouples
concretestructure
adiabatic bath
adiabatic curve
concrete
informationtemperature
computer
strength
24 72 168 672
digitalthermometer
controller
Fig. 7.21 Schematic representation of the UCON-system [25,29]. • Relation strength/degree of hydration (fc/αh) acquired priory (from database). • Calculating degree of hydration from adiabatic curve and measured temperature. • Strength obtained from the mixture’s fc/αh-relation.
measured temperaturein center of cube 150x150x150 mm
0
0.10
0.20
0.30
0.40
0.50
0.60degree of hydration ( )αh
Fig. 7.22 Calculated degree of hydration of a concrete cube at almost isothermal hardening
conditions. Calculation conducted with the UCON-system [29]. Input: measured adiabatic curve and temperature in cube during hardening.
Fig. 7.22 shows the result of a degree of hydration calculation with the UCON-system. It concerns a high strength concrete project. The concrete has hardened under isothermal conditions (process temperature Tp(t)). The adiabatic hydration curve is determined experimentally.
7.7.2 Tensile strength Just as the compressive strength, the tensile strength is a function of the degree of hydration. Also for this, a (bi-)linear relationship is found. Fig. 7.23 shows an example. In practice, similar as to hardened concrete, the tensile strength of hardening concrete is calculated using its compressive strength. In many cases, the relation used for hardened concrete turns out to be sufficiently accurate for hardening concrete.
+++++
+
++
0 20 40 60 80degree of hydration (%)
0
1
2
3
4tensile splitting strength (N/mm )2
+wcr0.4
0.50.6
Fig. 7.23 Tensile strength as a function of the degree of hydration for different wcr’s. Blast furnace slag cement CEM II. Cement content: 350 kg/m3 [19].
29
7.7.3 Elastic modulus In the CEB-FIP MC’90 [31] the development of the elastic modulus in time is presented as: cEc )()( EttE β= (7.24)
where: [ ] 5,0
ccE )()( tt ββ = (7.25)
where: Ec(t) = elastic modulus after t days of hardening Ec = elastic modulus after 28 days of hardening βE(t) = coefficient, depending on the age t
(age of the concrete, eventually corrected for the hardening temperature according to formula (7.23))
βcc(t) = coefficient according to formula (7.22) Quite often, the elastic modulus is determined using existing relations between the elastic modulus and the compressive strength, e.g. VBC (NEN 6720), CEB-FIP Model Code 1990 and EN 1992-1-1. In the Model Code, it holds: [ ] 3/1
ckc 8)(f9500)( +⋅= ttE (7.26)
where fck(t) is the characteristic concrete compressive strength (prism strength) at time t. For a more advanced discussion on this issue, use can be made of the theory in which concrete is a multi-phase material existing of aggregates and cement paste. In this case, the cement paste is a two-phase system containing un-hydrated cement and hydration products. The resulting elastic modulus is expressed as a function of the degree of hydration [19].
+ +++
+
wcr0.4
0.50.6
0 20 40 60 800
10000
20000
30000
40000
50000elastic modulus (N/mm )2
degree of hydration (%) Fig. 7.24 Course of the elastic modulus as a function of the degree of hydration for the
different wcr’s. Cement content 350 kg/m3, Blast furnace slag cement CEM II [19].
30
For the relation between the elastic modulus of concrete Ec and the degree of hydration αh, Gutsch proposes [18]:
0
0h0hc 1
)(αααα
−−
= EE (7.27)
where: E0 = (fictitious) elastic modulus for αh = 1,0 α0 = critical degree of hydration Fig. 7.24 provides an example of the calculated course of the elastic modulus as a function of the degree of hydration for different wcr’s.
7.8 Stresses and probability of cracking in hardening concrete
7.8.1 Stress calculation – superposition method If temperature induced deformations are restrained fully or partly, stresses arise. For the calculation of these stresses, the superposition principle is assumed to hold. The superposition principle is shown schematically in Fig. 7.25. The stress at time ti is found by summarizing the resulting stress increments Δσ(τj,ti) (j = 1...i). For the stress increments induced by an imposed thermal deformation, it holds (see also section 7.2): ( , ) ( , ) ( ) ( ) ( )i j i j c h j c jt r t T Eσ τ ψ τ α α τ τ∆ = ⋅ ⋅ ⋅ ∆ ⋅ (7.28)
with: Δσ(ti,τj,) = stress increment at time ti τj = time of application of the stress increment r = degree of restraint: r = 0 .. 1,0
ψ(ti,τj,) = relaxation factor αc(αh) = expansion coefficient of concrete, possibly as a function of the degree
of hydration ΔT(τj) = temperature increment during time step Δτj Ec(τj) = elastic modulus (average value during time step Δ τj)
31
age
relaxation ψ
∆σE
tens
ion
com
pres
sion
∆ ∆σ σ(t) = Eψ
Fig. 7.25 Schematic representation of the superposition principle [32]. The resulting stress at time t is found by adding the stress increments as discussed earlier in section 7.2.
7.8.2 Relaxation Relaxation has a considerable influence on the stress development. Exact information about this issue is however very scarce. Some programs take into account the effect of relaxation by reducing the elastic modulus by 50% during a predefined period. Due to the lack of information, relaxation is sometimes completely neglected. A relaxation formula that gives acceptably accurate results for the stresses in hardening concrete holds [26]: 1,65( ) ( )1 1,34 ( ) ( )
( ) ( )( , )d nh h
h h
t twcr t
t eα α
τ τα τ α τψ τ
− − − + ⋅ ⋅ ⋅ − ⋅
= (7.29)
with: τ = time of applying the stress increment d = constant, depending on the type of cement slow hardening cement: d = 0,3 rapid hardening cement: d = 0,4 n = factor. For compression: n = 0,3; for tension: n = 0,6
(Note: recent research shows better results with n = 0,3 for both compression and tension)
αh = degree of hydration wcr = water cement ratio With the quotient αc/αh, the effect of ongoing hydration during the period (t-τ), is taken into account in the relaxation formula. If hydration stops (which is when αc/αh = 1), formula (7.29) reduces to:
1,651,34 ( ) ( )( , )d nwcr tt e τ τψ τ
− − ⋅ ⋅ ⋅ − = (7.30)
32
or:
( )( , )d na tt e τ τψ τ
−− ⋅ ⋅ −= (7.31)
or:
( , )( , ) tt e ϕ τψ τ −= (7.32)
where: φ(t,τ) = the creep factor (“basic creep”) a = the ageing factor Fig. 7.26 gives an impression of the relaxation factor ψ(t,τ) as a function of time. In this figure, also the course of the degree of hydration is presented. It can be observed clearly that stresses that develop during the early acceleration stage of the hydration process almost completely vanish. A disadvantage of the superposition method is the fact that is requires substantial memory allocation and calculation time. For a one-dimensional problem, the current pc’s work out well, but for the calculation of three-dimensional stress fields, different relaxation formulas
0
0.2
0.4
0.6
0.8
1.0relaxation factor ψ τ(t, )
0 40 80 120 160 200time (hours)
48 h
relaxation curves calculatedwith formula (7.29)
34 h
24 h
degree of hydration (t)αh
19 h
Fig. 7.26 Relaxation curves ψ(t,τ) for young concrete, calculated with formula (7.29). should be adopted (“rate type”). By doing so, substantial computation time reductions can be reached at the cost of the availability of the “stress history” that the hardening concrete experiences. Many multi-dimensional practical problems can be reduced to a one-dimensional problem and can then be solved using the superposition method.
33
7.8.3 Degree of restraint
7.8.3.1 Some basic principles An important parameter associated with the calculation of stresses is the degree of restraint of the deformations r, simply called the degree of restraint. The magnitude is determined from
-5-4-3-2-1012345678
stress (N/mm )2
0 20 40 60 80 100 120 140 160time (hours)
increasing degreeof restrained
r = 0
r = 1.0
r = 0.25
r = 0.5
surface centre
d
Fig. 7.27 Calculated stresses in the centre of a hardening concrete wall. Semi-adiabatic
temperature conditions. Degree of restraint: 0, 25, 50 and 100% [20]. the mechanical boundary conditions. According to the definition, it holds: full restraint of the deformation: r = 1,0 free deformations: r = 0,0 In Fig. 7.27, the course of the numerically determined stresses in a hardening concrete wall is shown. The developed thermal length changes were restrained for 0, 25, 50 and 100% respectively. The stresses increase with increasing degree of restraint. Although the definition of the degree of restraint is very transparent, wrong conclusions are derived from it quite easily. The impression might be that a high degree of restraint always results in high stresses anywhere in the structure. The geometrical and mechanical boundary conditions of a structure can, however, be such that at some locations in the structure stresses might be larger or even change sign the more the structure can deform. This situation is illustrated by means of a classical case, namely the wall cast on an already hardened floor slab. The dimensions of the wall and the floor slab are in Fig. 7.28. The wall is from a normal type of concrete. Two cases are investigated: Case A: Both longitudinal and bending deformations (curvature) of the wall are not restrained
(rl = 0,0; rφ = 0,0) Case B: Longitudinal deformations are not restrained (rl = 0,0), but the bending deformations
of the structure are restrained completely (rφ = 1,0)
34
0.70m
0.80m
5.00
m
4.10m0
1.0
2.0
3.0
4.0
5.0
6.0wall height (m)
-2 -1 0 1 2stress (MPa)
r = 0.0r1Ø = 0.0
case A: r = 0.0; r1 Ø = 0.0
case B: r = 0.0; r1 Ø = 1.0
r = 0.0r1Ø = 1.0
+
+
-
-
-
a. Deformations of the structure b. Stresses over the height of the structure Fig. 7.28 Calculated stresses (after 5 days; cooling stage) over the height of wall-slab structure,
for different values of the degree of restraint for the length (rl) and bending (rφ)deformations.
The result of the stress calculations is in Fig. 7.28b. There, the stresses are over the height of the wall after 5 days of hardening. It becomes clear that for case A, where the wall is free to deform (rl = 0,0), i.e. not restrained in any way, the stresses just above the connection with the floor slab are larger than in case B, where the wall is fully restrained (rφ = 1,0). At the top of the wall, case A shows compression whereas for case B, the wall as a whole is in tension. What happens here is that the course of the deformation of an individual fiber (micro-level) is completely different from the course expected when considering the assumed degree of restraint. The degree of restraint is a parameter that defines the deformational capabilities of a structure as a whole, not the course of deformation of any individual fiber. The deformation of an individual fiber is the net result of the imposed (thermal) strain at fiber level plus the strain that comes from the deformation of the structure as a whole! Another complication when regarding the issue of the degree of restraint is that the level of restraint of the deformations changes during the progress of the hydration process. The deformation of a wall, cast on a rigid foundation slab, is initially restrained completely at the connection with the slab (the stiffness of the wall is still negligible with respect to the slab: r = 1,0). After a certain time, depending on the stiffness ratio between the wall and the foundation slab, a part of the imposed deformations results in a deformation of the structure as a whole. Only this part results in stresses. Finally, the magnitude of the effective imposed strain at fiber level is determined by the equilibrium conditions and the deformation compatibility conditions in a cross-section!
case B: rl = 0 ; rφ = 1,0
case A: rl = 0 ; rφ = 0
35
7.8.3.2 Regulations for practice A wall, cast on an already hardened concrete floor slab that does not bend, will, at the interface with the slab, experience a degree of restraint of 90 to 100%. More to the top of the wall, this degree of restraint decreases. If the wall height is more than half the wall length, the degree of restraint at the top is almost 0%. Fig. 7.29 shows such a wall schematically.
⇒ ⇒
r 1.0~~
45° 45°
almost “free”tensile trajectories
crack directions
L
H
Fig. 7.29 Deformations of a wall cast on an already hardened floor slab. Indicated are the
degree of restraint, the direction of the tensile trajectories and the direction of the cracks.
Table 7.5 presents the degree of restraint of walls as a function of the length/height ratio. The values in the table are indicative values to obtain a first estimation of the degree of restraint of the deformation at the topside of the wall. At the connection interface with the slab, the degree of restraint is about 100%, also for short walls (see Fig. 7.29). In the British Standard 8007-1987: “Design of concrete structures for retaining aqueous liquids” [33], some indicative figures are presented for different types of structures. They also give values for the degree of restraint. These are represented by the figures 7.30a-d (EN 1992-3). Table 7.5 Degree of restraint as a function of the length/height ratio (L/H) of the wall (after
[21]) (Note: values for a low L/H-ratio are only valid for the degree of restraint at the top of the wall)
Length/height ratio (L/H) 2 3 4 > 8
Degree of restraint 0 0,1 0,6 1,0
36
a. Wall cast on rigid support b. Wall cast in between rigid supports
...
c. Wall fixed at two sides d. Wall fixed at three sides Fig. 7.30 Degree of restraint r for walls and slabs at different boundary conditions (after BS
8007-1987 [33] & EN 1992-3; no relaxation included). (Note: The effective degree of restraint also depends on the amount of reinforcement applied!).
7.8.4 Probability of cracking
7.8.4.1 Calculation procedure When calculating the probability of cracking, it is often assumed that the strength and stress can be described using a normal distribution function. This is schematically shown in Fig. 7.31. The calculated values of the strength and stress are considered to be the average values of the quantities concerned. For the standard deviations of the tensile strength and the tensile stress, values of 0,5 N/mm2 and 0,6 N/mm2 , respectively, can be used. These are conservative values,
37
especially with regard to the standard deviation of the tensile strength!. (see also section 7.8.4.2.). If the strength and the stress have a normal distribution function, the difference between them, i.e. z = fct – σct, is also normally distributed. The likelihood that at a certain time z is smaller than zero, which implies that cracking appears, is determined with help of basic statistics (see also the example in table 7.6).
tens
ion
com
pres
sion
time
stress = Σ∆σ τ( )
normal distributiontensile stress = f (t)ctm
Fig. 7.31 Development of the tensile strength and stress in time (schematically). Strength and
stress are considered to be normally distributed quantities. Table 7.6 Calculation of probability of cracking P{Fcr} in hardening concrete. Tensile strength (average value) fctm = 3,0 MPa Standard deviation tensile strength s(fct) = 0,3 MPa (10% of the tensile strength) Tensile stress (average value) σct = 2,6 MPa Standard deviation tensile stress s(σct) = 0,26 MPa (10% of the tensile stress) Tensile strength and tensile stress normally distributed. For the difference u(z) = fctm - σct it holds: u(z) = fctm - σct = 3,0 – 2,6 = 0,4 MPa s(z) = [s2(fct) + s2(σct)]0,5 = 0,396 MPa Ratio u(z)/s(z) = 0,4 / 0,396 = 1,007 With help of the table for standard normally distribution functions, the probability of cracking is [30]: Cracking probability p{Fcr} = 15,8% Figure 7.32 gives the result of a calculation on the probability of cracking according to the procedure as described before. The development of the probability of cracking in time is given for walls that have different thickness. They are cast on an already hardened floor slab. Deformations are restrained by the floor slab. The wall-slab structure can bend freely (rφ = 0). The cracking probability is determined in the most critical cross-section, which in this case is about 0,5 m above the wall-slab connection. Based on the imposed boundary conditions, the 1,0 m thick wall almost certainly cracks. The 0,5 m thick wall has a cracking probability of more than 50%. As a result of the bending of the wall, compressive stresses appear at the top of the wall. The reduction
38
of the probability of cracking after having reached a maximum value comes from the gradual decrease of the temperature differences in the cross-section.
0 20 40 60 80 100 120 140 160time (hours)
0
20
40
60
80
100cracking probability (%)
centre, d=0.5m
surface, d=0.5m
surface, d=1m
centre, d=0.25m
centre, d=1msymmetrical axis
surface centre
d3
m1
m
3 m
Fig. 7.32 Probability of cracking in a 0,25 m, 0,50 m and 1,0 m thick wall as a function of
time. Portland cement CEM I 52,5 R. Casting temperature 15°C. Temperature of floor slab 10°C. rφ = 0 [20].
7.8.4.2 Cracking criteria For judging the probability of cracking, different criteria are adopted: a. Stress criterion b. Strain criterion c. Temperature criterion 7.8.4.2a Stress criterion For judging the probability of cracking in young concrete, one often wants to know how a calculated cracking probability relates to the “classical” safety factor γ. Here, the safety factor is defined as the ratio between the lower-bound characteristic values of the tensile strength and the higher-bound characteristic tensile stress (γ = fctk;0,05 / σctk;0,95). For the relation between the cracking probability and the safety factor, use is made of experimental research results from TU Delft.
39
0 24 48 72 96 120time (hours)
3
2
1
0
-1
-2stress (N/mm )2
cement kg/mwcr 0.5
3
PC:0.9 fctmspl
PC-AHOC-A
HOC: 0.9 f ctmspl
Fig. 7.33 Stress at which concrete cracks under semi-adiabatic hardening conditions. Degree of restraint rl = 1,0 [32].
Experimental results Tests carried out on concrete hardened under semi-adiabatic conditions and experiencing a partly of full degree of restraint, has shown that cracking occurs at a stress level equal to (on average) 0,85 fctm
2) (see Fig. 7.33). For this tensile stress, the probability of cracking should be 50%. If both the strength and stress have a standard deviation of 10% of their average values, a cracking probability of 50% is associated with a stress level equal to 0,85 times the average axial (short time) tensile strength. This can only be the result if we take into account a reduction factor for tensile strength of 100 – 85 = 15%. This reduction of the tensile strength is the result of the fact that the stresses develop slowly and that, therefore, the long term-effect (reduction) on the tensile strength must be taken into account. Cracking probability and safety factors Given the fact that in many cases hardening concrete cracks at a stress level which is on average 15% below the average axial short term-tensile strength, the following argumentation can be followed. Assume that for a particular case the tensile strength is 3 N/mm2. In table 7.7, the probability of cracking is found for a reduction of the tensile strength of 0% and 15%, respectively. The cracking probability is calculated at values for the average tensile stress σctm of 0,5 fctm ; 0,6 fctm ; 0,7 fctm and 0,85 fctm. Standard deviations of 10% of the average tensile strength and tensile stress are assumed. For no reduction of the tensile strength, it is observed that at a stress level σctm = 0,85 fctm , the probability of cracking is 15,8%, whereas the safety factor is already below the value 1, namely γ = 0,83. In order to be sure to achieve a safety factor above 1,0, the stress should not exceed 0,7 fctm. For this particular case it holds that γ = 1,03. In reality, as has been observed in tests, a reduction of the tensile strength of 15% is to be taken into account. When assuming a tensile strength 0,85 fctm, at an average tensile stress level σctm = 0,85 fctm, the probability of cracking is 50%. The accompanying safety factor then is γ = 0,72. At a tensile stress level of 0,7 fctm, the probability of cracking is 7% and the safety factor against cracking is still below 1,0, namely 0,88. In this example, the turning point at which the safety factor reaches a value of more than 1,0 is σctm ≈ 0,6 fctm. The probability of cracking then is about 1%. If stresses over 0,5 fctm are not allowed to occur, the safety against cracking is 1,22 at a cracking probability of 0,02%.
2) fctm = average axial tensile strength. It holds that: 0,85 fctm = 0,85 * 0,9 fctmspl = 0,76 fctmspl (see also section 4.3.6.2)
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Table 7.7 Probability of cracking and corresponding safety factors for different values of the effective tensile strength.
Quantities
Values
w.r.t tensile strength
Stress at cracking
0,5 fctm 0,6 fctm 0,7 fctm 0,85 fctm
Short term tensile
strength
Average value Standard deviation Characteristic value
Prob. of cracking
Safety factor (fctk/σctk)
N/mm2 N/mm2
N/mm2
[%] ---
3,0 0,3
2,51
1,5 0,15 1,75
--
1,4
1,8 0,18 2,1
0,03%
1,2
2,1 0,21 2,44
0,7% 1,03
2,6 0,26 3,03
15,8% 0,83
Reduced short term
tensile strength*)
Average value Standard deviation Characteristic value
Prob. of cracking
Safety factor (fctk/σctk)
N/mm2 N/mm2
N/mm2
[%] ---
2,6 0,26*) 2,14
1,5 0,15 1,75
0,02% 1,22
1,8 0,18 2,1
0,6% 1,02
2,1 0,21 2,44
7% 0,88
2,6 0,26 3,03
50% 0,71
*) Tensile strength with a reduction of 15% in relation to the short-term tensile strength fctm = 3,0 N/mm2
1. Note
If the analysis concerns concretes with a very low water cement ratio (for example high strength concretes with a wcr < 0,35), not only the calculation of the tensile stresses due to temperature effects, but also the effect of hardening shrinkage should be considered (see section 7.6.4).
2. For the values of the probability of cracking and the safety factors as calculated, it is assumed that the stresses are
determined at correct thermal and mechanical boundary conditions. Significant deviations of the boundary conditions, i.e. unexpected variations of the surrounding temperature, high wind speeds etc., are not included. To gain insight into the effects of these types of variations on the probability of cracking, additional sensitivity analyses must be performed.
When summarising the results, it can be stated that tensile stresses during hardening limited to about 50% to 60% of fctm, result in a safety against cracking higher than 1,0. The probability of cracking then is negligible. If tensile stresses up to 70% of fctm are accepted, cracking is very likely to occur. 7.8.4.2b Strain criterion For the prediction of cracking of hardening concrete, the literature quite often prefers a strain criterion rather than a stress criterion. The idea behind this consideration is that the fact that the deformational behaviour of hardening concrete differs quite substantially from hardened concrete: the strain capacity is much higher for young concrete than for hardened concrete. Although this is correct, it does not make the strain criterion more suitable than a stress criterion, certainly not if it concerns numerical simulations on the hardening behaviour of concrete. After all, in these types of simulations, the constitutive material behaviour (stress-strain relation) is used explicitly by an iterative calculation procedure. The way to express the results, either as stresses or as strains, makes no difference since these two quantities are the results of one calculation. Whether cracking appears has to be verified using either an actual strength or on an actual strain capacity. When considering the situation common in building practice, a stress criterion is more likely to use. Extensive research in Germany has confirmed the reliability of the results from the stress criterion [22].
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7.8.4.2c Temperature criterion With the aim to avoid thermal cracks in hardening concrete, the current building practice uses different temperature criteria. A rule of thumb gives a relation between the probability of cracking and the temperature difference. In order to avoid cracking due to eigen stresses, the temperature difference inside a cross-section should not exceed app. 20°C. Temperature differences between structural parts cast on a rigid base and the base itself should not exceed app. 15°C. For the Danish Støre Belt project, very strict temperature criteria were applied, namely 12°C and 8°C, respectively. Extensive calculations using the simulation program TEMPSPAN, were carried out on a number of practical cases. It appeared that the “temperature criterion” covers a quite substantial part of the work field. However, in a number of cases these criteria are very conservative whereas also situations occur for which they are not sufficiently strict [20,23]. For the eigen stresses (see chapter 2) in particular, the 20°C criterion is sufficient as long as
the wall thickness is not over 1,0 m [20]. For higher wall thicknesses, problems might arise. Whether cracking occurs depends on the age at which the temperature differences develop and why they develop. A cold wind passing alongside a relatively warm wall from which the formwork is removed, makes it relatively sensitive to cracking.
For a wall cast on a hardened foundation, the order of magnitude of the degree of restraint is of dominant importance with regard to cracking. The magnitude of this degree of restraint depends on the actual location in the structure (see also [20,34]).
The conclusion to be drawn from these studies is that temperatures and temperature differences in a number, but certainly not all, cases can be considered as providing a reliable cracking criterion.
7.8.4.3 Judging levels regarding the probability of cracking For a reliable judgment of the probability of cracking, the stress criterion is to be preferred over the temperature criterion. The stress criterion, however, requires a stress calculation with the help of suitable advanced software tools. This will not be profitable under all circumstances. If the execution of this type of complex calculations for achieving insight into the probability of cracking should be avoided, the temperature criterion can be applied as a fall back option. The following judging levels based on the Swedish concept are used: Level 1. Temperature criterion It is easy to handle but is the least reliable. In order to avoid disappointments, these criteria are very conservative. For projects in which concretes with a high autogenous shrinkage are used, it is advised not to use the temperature criteria. Level 2: Dimension graphics and design curves With the use of simulation models, graphs and design curves can be constructed that represent the behaviour and characteristics of common structures. In these graphs/design curves, the influence of the different parameters on the stress development is visible. These types of graphs and design curves can be an important instrument for judging the structural behaviour and the probability of cracking. They are only used by experienced and skilled engineers. In Sweden, these instruments are already available for general use.
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Level 3: Use of advanced simulation software The most reliable judgment on the probability of cracking is obtained by using the results from advanced simulation software. Thermal and mechanical boundary conditions as well as the hardening shrinkage are taken into account accurately, whereas the most advanced creep and relaxation formulas can also be applied. For complex projects with tight limits regarding the probability of cracking, the use of advanced simulation software for judging the structural behaviour during hardening has become common practice already.
7.9 References 1. Mills, R.H. 1966. “Factors influencing cessation of hydration in water-cured cement pastes”,
ACI-SP, pp. 406-424. 2. Breugel, K. van, 1991. “Simulation of hydration and formation of structure in hardening
cement-based materials”. PhD-thesis TU-Delft, 295 p. 3. Laube, M. 1991. “Werkstoffmodell zur Berechnung von Temperaturspannungen in massigen
Betonbauteilen im jungem Alter”. PhD-thesis, Braunschweig. 4. Koenders, E.A.B., e.a. 1995. “HSB-kantoor te Brade”. Stevin report, TU-Delft, 109 p. 5. Saul, A.G.A., 1951, “Principles underlying the steam curing of concrete at atmospheric
pressure”. Magazine of Concrete Research, No. 6. 6. Bergström, S.G., 1953. “Curing temperature, age and strength of concrete”. Mag. of Concr.
Res., pp. 61-66. 7. Plowman, J.M., 1956. “Maturity and strength of concrete”. Magazine of Concrete Research,
pp. 13-22. 8. Papadakis, M., Bresson, J., “Contribution à l’etude de facteur de maturité des liant
hydrauliques ; Application à l’ industrie du béton manufacturé. CERIB-publications Technique, No. 8.
9. Bresson, J., 1980. “Prevision des resistances: facteur de maturité et temps equivalent“. Annales, No. 387, pp. 106-110.
10. Carino, N.J. e.a., 1983. “Temperature effects on strength-maturity relations of mortar”. ACI-Journal, No. 80-17, pp. 177-182.
11. “Gewogen rijpheid”, 1984. Betoniek 6/20. 12. CUR-aanbeveling 9: “Bepaling van de sterkteontwikkeling van jong beton op basis van de
gewogen rijpheid”. Gouda, 1987, 7 p. 13. Grube, H. 1990 “Ursachen des Schwindens van Beton und Auswirkungen auf Betonbauteile”,
Habilitationsschrift, Darmstadt, 81 pp. 14. Dekker, I., 1995. “Research into validity-sphere of maturity-method ‚de Vree“. Ijmuiden,
pp. 404-407. 16. Fagerlund, G., 1987. “Relations between the strength and degree of hydration and porosity of
cement paste, cement mortar, and concrete”. Research report. 17. “Toepasbaarheid van beton met hoge sterkte in de 2e Stichtse Brug”. Onderzoek in opdracht
van RWS. TU-Delft, 1996. 18. Gutsch, A., Rostàsy, F.S., 1994. „Young concrete under high tensile stresses –Creep,
Relaxation and Cracking“. Proc. RILEM Symp. Thermal Cracking in Concrete at Early Ages”, Munich, pp. 111-118.
19. Lokhorst, S.J. et al. 1996. “Ontwikkeling van de mechanische eigenschappen: sterkte en stijfheid”. In: “Het grijze gebied van het jonge beton”. Artikel series in CEMENT 1995/1996.
20. Eberhardt, M., 1993. “Rissgefahr in jungem beton”, Delft/Darmstadt.
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21. „Scheurvorming door krimp en temperatuurwisselingen in wanden“. 1978. CUR-report 85, 102p.
22. Rostàsy, F.S., 1994. “Determination and modelling of mechanical properties“. Proc. RILEM Symposium “Thermal Cracking in Concrete at Early Ages”, Munich, General report, pp. 31-45.
23. Traute, M., 1994. “Reissneigung des hochfesten Betons im jungen Alter”. Delft/Darmstadt, 161 p.
24. Tazawa, E., 1992. „Autogenous shrinkage caused by self-desiccation in cementitious material. Proc. 9th Int. Congress on the Chemistry of Cement, Vol. IV, pp. 712-718.
25. Beek, A. van., 1995. “UCON-programma”. Manual. TU-Delft. 26. Breugel, K. van., 1980. “Relaxation of young concrete”, TU-Delft, Stevin report. 27. Koenders E.A.B., e.a., 1995, “HSB-Kantoor te Breda – Vergelijking meting en berekening”.
TU-Delft. 28. HYMOSTRUC – manual. Delfttech b.v. Delft. 29. Beek, A. van, 1995. “Sterktebepaling van jong beton”. MSc-thesis, TU-Delft. 95 p. 30. Soest, J. van. “Elementaire statistiek”, DUP, Delft, 1972, 134 p. 31. CEB-FIP Model Code 1990, CEB-Bulletin No. 203, July 1991. 32. Lokhorst, S.J., 2001: Deformational behaviour of concrete influenced by hydration related
changes of the microstructure. Report 25.5-99-05, TU Delft. 33. “Design of concrete structures for retaining aqueous liquids”. BD-Code of practice BS 8007,
1987, 27 p. 34. Wessels, J., 1995. “Temperatuureffecten in gefaseerd gestorte betonconstructies ten gevolge
