A mechanistic approach to predict built-in temperature concrete slabs

Cite this article
Research Article
Sok T and Lee SW (2021)
A mechanistic approach to predict built-in temperature in concrete pavements.
Proceedings of the Institution of Civil Engineers – Transport 174(4): 227–238,
https://doi.org/10.1680/jtran.18.00065
Paper 1800065
Received 15/04/2018;
Accepted 26/07/2018;
Published online 06/09/2018
Keywords: pavement design/roads &
highways/thermal effects
ICE Publishing: All rights reserved
Transport
A mechanistic approach to predict built-in
temperature in concrete pavements
Tetsya Sok MSc
Seung Woo Lee PhD
PhD candidate, Department of Civil Engineering,
Gangneung-Wonju National University, Gangneung-si, Gangwon-do,
South Korea
Professor, Department of Civil Engineering, Gangneung-Wonju
National University, Gangneung-si, Gangwon-do, South Korea
(corresponding author: [email protected])
The built-in temperature difference (BITD), defined as the temperature difference between the top and bottom of a
slab at the final setting time, is an important parameter for analysing curling stress and slab deformation in concrete
pavements. However, the available methods for estimating this parameter are very limited. A method to predict the
BITD in a concrete pavement was therefore developed. To do this, a numerical model was developed to predict the
temperature distribution in a concrete slab at early age using a transient one-dimensional finite-difference method.
A mathematical equation for predicting concrete final setting time presented in the literature was used and
incorporated in the numerical model to predict the BITD. The results of the numerical model showed good agreement
with field data. Using the proposed model, the effects of climatic conditions, placement time, pavement thickness and
concrete mix proportion on the BITD were also evaluated. The results showed that climatic conditions, placement time
and concrete mix proportion have substantial effects on the BITD, whereas pavement thickness has only a slight
effect. The proposed model can be used to predict the BITD in concrete slabs for given concrete mix design, placement
time, pavement thickness and environmental conditions.
Notation
Cc
cc
cp
E
FB
Fcloud
Hu
hconv
If
i
k
kc
kp
pcem
pC2S
pC3A
pC3S
pC4AF
pFA
pFA-CaO
pFreeCaO
pMgO
pslag
pSO3
q
qabs
3
cementitious materials content (kg/m )
heat capacity of concrete layer (J/kg.°C)
specific heat of pavement material (J/kg.°C)
activation energy (J/mol)
Blaine fineness (m2/kg)
cloud cover factor
total heat of hydration of cementitious materials (J/kg)
heat convection transfer coefficient (W/m2/°C)
intensity factor
node
time step
thermal conductivity of concrete layer (W/m.°C)
thermal conductivity of pavement material (W/m.°C)
weight ratio of cement to total cementitious content
weight ratio of dicalcium silicate to total
cement content
weight ratio of tricalcium aluminate to total
cement content
weight ratio of tricalcium silicate to total
cement content
weight ratio of tetracalcium aluminoferrite to total
cement content
fly ash mass ratio to total cementitious content
calcium oxide mass ratio to total fly ash content
weight ratio of free calcium oxide to total cement
content
weight ratio of magnesium oxide to total cement
content
slag mass ratio to total cementitious content
weight ratio of sulfate to total cement content
rate of heat liberation during cement hydration (W/m3)
solar absorption of concrete (W/m2)
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qconv
qir
qsolar
N
R
T
Ta
Tactual
Tagg
Tc
Tcem
Tconcrete
Tdp
Tr
Ts
Tset
Tsky
Tw
t
te
vwind
Wa
Wc
Ww
Wwa
x
α
αcr
αu
β
Δt
Δx
Δxc
heat convection transfer (W/m2)
thermal irradiation from the surface (W/m2)
peak solar radiation during daytime period (W/m2)
cloud cover
universal gas constant (8·3144 J/mol/K)
pavement temperature (°C)
air temperature (°C)
actual concrete temperature (°C)
temperature of aggregate (°C)
nodal concrete slab temperature (°C)
temperature of cement (°C)
concrete placement temperature (°C)
local dew point temperature (°C)
reference concrete slab temperature (21·1°C)
surface pavement temperature (°C)
concrete temperature at final set (°C)
effective sky temperature (K )
temperature of water (°C)
time (s)
equivalent age of concrete (h)
wind speed (m/s)
weight of dry aggregate (kg/m3)
weight of cement (kg/m3)
weight of water (kg/m3)
weight of wet aggregate (kg/m3)
depth below pavement surface (m)
degree of hydration
critical degree of hydration
ultimate degree of hydration
hydration shape model
time increment (s)
space increment of pavement layers (m)
space increment of concrete layer (m)
227
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Volume 174 Issue 4
δsurface
εsky
εp
γabs
ρ
ρc
σ
τ
1.
(differential) pavement thickness for the energy
balance (m)
sky emissivity
surface emissivity
solar absorptivity of concrete
density of pavement material (kg/m3)
density of concrete layer (kg/m3)
Stefan–Boltzmann constant (5·669 10−8 W/m−2.K−4)
hydration time parameter (h)
ΔTset > 0
Temperature difference = +ΔTset
BITD = +ΔTset
(a)
ΔTactual = 0
Introduction
In a concrete pavement, the presence of a temperature difference
between its top and bottom causes curling deformation and
stresses (Westergaard, 1926). A slab can curl upwards or downwards depending on whether the magnitude of the temperature
difference is negative (i.e. when the slab surface is cooler than
the bottom) or positive (i.e. when the slab surface is warmer
than the bottom). Since a slab is not free to curl due to restraint
from its self-weight, tensile stresses are generated in the slab
(Westergaard, 1926). In addition, a concrete slab curls upwards
when there is a moisture difference between the top surface and
the bottom of the slab. Moisture differences are caused by
drying of the top of the slab surface, but below about 50 mm
from the surface, the moisture level remains at a relatively constant high level even in very dry areas (ARA, 2002). This leads
to concrete slabs curling upwards. The amount of drying shrinkage in the upper portion of a concrete slab depends on various
influencing factors, such as the curing method and concrete
mixture properties. When a slab curls upwards due to the presence of a negative temperature gradient combined with moisture
warping, the edges of the slab do not contact the ground, thus
leaving a portion of the slab unsupported (ARA, 2002).
When a concrete pavement slab is initially placed, the temperature difference through its depth is always zero. However, if the
air temperature is warmer than the concrete temperature,
the temperature of the top part of the slab starts to increase
before the concrete hardens. In this period, the concrete is
plastic and does not have sufficient structural integrity to
deform or change shape as the temperature difference develops.
When concrete first begins to set and harden, a large positive
temperature difference can develop in a flat slab, without deformation occurring. In this situation, the concrete slab is in full
contact with the foundation and is in a zero-stress condition
even if there is a temperature or moisture gradient through the
slab because the concrete is just leaving the plastic state, as
shown in Figure 1(a). Another situation when a large positive
temperature difference would develop in a flat slab is when a
concrete pavement is placed during the morning of a hot
sunny day. This condition tends to expose the freshly paved
concrete slab to a large temperature gradient from intense
solar radiation plus the heat of hydration (ARA, 2002). When
the actual temperature gradient through the slab becomes zero
later on, the slab will attempt to curl upwards, analogous to a
slab with a negative temperature gradient (Figure 1(b)).
228
A mechanistic approach to predict built-in
temperature in concrete pavements
Sok and Lee
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Actual temperature difference = 0
Equivalent temperature difference = –ΔTset
(b)
Figure 1. Illustration of a positive built-in temperature gradient in
a concrete slab: (a) zero-stress condition; (b) zero gradient but
stressed condition
On the other hand, if a concrete pavement slab is placed
later in the afternoon or at night, the highest temperature
from the heat of hydration does not coincide with the most
intense solar radiation (ARA, 2002). This can lead to a lower
temperature difference or a potentially negative difference, as
shown in Figure 2(a). The slab will attempt to curl downwards
if the actual temperature difference is larger than the built-in
temperature difference (BITD), as shown in Figure 2(b).
Previous studies have shown that the BITD is a critical input
parameter for the performance of a concrete pavement predicted using the Mechanistic–Empirical Pavement Design Guide
(MEPDG) (Coree, 2005; Hall and Beam, 2004; Kannekanti
ΔTset > 0
Temperature difference = –ΔTset
BITD = –ΔTset
(a)
ΔTactual = 0
Actual temperature difference = 0
Equivalent temperature difference = +ΔTset
(b)
Figure 2. Illustration of a negative built-in temperature gradient in
a concrete slab: (a) zero-stress condition; (b) zero gradient but
stressed condition
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A mechanistic approach to predict built-in
temperature in concrete pavements
Sok and Lee
and Harvey, 2005). According to Yu and Khazanovich (2001),
in an analysis of temperature effects in concrete pavements, the
actual temperature difference in a concrete slab can be added
to the BITD to obtain the total effective temperature
difference.
incorporating the given initial and boundary conditions
(Narasimhan, 1999). The 1D heat-transfer equation is
Currently, there are no available methods or guidelines to
estimate the BITD in concrete pavements. According to the
MEPDG, the BITD has significant effects on the behaviour
and performance of concrete pavements. The BITD is determined, by calibration, to minimise the differences between predicted and observed distresses. This BITD can then be used as
the default value for general concrete pavements.
where kp is the thermal conductivity of the pavement material
(W/m.°C), ρ is density (kg/m3), cp is specific heat (J/kg.°C),
T is temperature (°C), t is time (s), x is the depth below the
pavement surface (m) and q is the heat generated heat from the
cement hydration process (W/m3).
Hansen et al. (2006) quantified the magnitude of the BITD by
measuring the temperature distribution in concrete pavements
together with final set time, which was determined based on
the standard test method for the setting time of concrete
mixtures by penetration resistance (ASTM, 2008) in the laboratory then converting to the final setting time of a field
concrete pavement using the maturity concept. Other
researchers have used back-calculation methods on in-service
pavement sections to determine the amount of permanent curl
or uplift (BITD combined with drying shrinkage gradient
and its associated creep effect) using a combination of the
finite-element method coupled with surface profile measurements of concrete slabs in the field (Rao et al., 2001; Wells
et al., 2006; Yu et al., 1998). According to Wells et al.
(2006) and Nassiri and Vandenbossche (2012), the built-in
curling temperature is defined at the time corresponding to
when expansion/contraction is measured with changes in slab
temperature. Although methods to determine the BITD have
been proposed, they are very costly and require experimental
data collected in the field. A practical and simple method to
estimate the realistic BITD in concrete pavements is thus still
required.
The major objective of this study was to develop a mechanistic
approach to predict the built-in temperature distribution in
concrete pavements. To do this, a one-dimensional (1D) heattransfer model to simulate concrete pavement temperature
using the finite-difference method was developed. Existing
mathematical equations for computing the rate of heat
liberation from cement hydration and the final setting time of
concrete were integrated into the developed model to establish
the BITD. Using the proposed model, the effects of climatic
conditions, concrete placement time, concrete mix proportion
and pavement thickness on the BITD in concrete pavement
slabs were also evaluated.
1:
kp
@2T
@T
þ q ¼ ρcp
@x2
@t
The heat of hydration q, which is influenced by a number of
factors including cement chemical composition, cement fineness and concrete mixture proportions, can be defined using
the regression model developed by Schindler and Folliard
(2005). Equation 1 accounts for the temperature fluctuation in
the environment, underlying layer properties and the properties
of the pavement slab. For a concrete slab placed under field
conditions, heat will be transferred to and from its surroundings. This heat transfer can take place in four different ways
(conduction, convection, irradiation and solar absorption), as
shown in Figure 3.
2.2
Heat of hydration
The cement hydration reaction is an exothermic process and is
a critical factor for field concrete temperature predictions. The
rate of heat generation is the numerical differentiation of the
heat evolution curve with respect to time. The regression
model developed by Schindler and Folliard (2005) was used in
this study, which is written as
2:
qðtÞ ¼ Hu Cc
β τ
β
E
1
1
αðte Þ exp
te
te
R 273 þ Tr 273 þ Tc
Solar radiation
Wind
Heat convection
Incoming and
outgoing
radiation
Concrete slab
+X
Sub-base
Heat conduction
Subgrade
2.
Mechanistic modelling of BITD
2.1
Heat-transfer model of pavement structure
In a concrete pavement, the temperature distribution can
be calculated by solving the 1D heat-transfer equation by
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Figure 3. Heat-transfer model of a pavement and its surrounding
environment
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A mechanistic approach to predict built-in
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Sok and Lee
where Cc is the cementitious materials content (kg/m3) and the
other parameters are defined in the notation list. Hu is the
total heat of hydration of cementitious materials at 100%
hydration (J/kg), which can be calculated by
where qabs is solar absorption of concrete (W/m2), qir is the
thermal irradiation from the surface (W/m2), qconv is the heat
convection transfer (W/m2), q is the heat generation rate of
cement hydration (determined by Equation 2), δsurface is the
(differential) pavement thickness for the energy balance and
Ts is the surface pavement temperature (°C).
3:
Hu ¼ Hcem pcem þ 461pslag þ 1800pFACaO pFA
Hcem ¼ 500pC3 S þ 260pC2 S þ 866pC3 A þ 420pC4 AF þ 624pSO3
4:
þ 1186pFreeCaO þ 850pMgO
The degree of hydration α(te) and the ultimate degree of
hydration αu can be defined based on the mathematical model
developed by Schindler and Folliard (2005), given by
5:
6:
β !
τ
αðte Þ ¼ αu exp te
αu ¼
1031 w=c
þ 05pFA þ 03pslag 1
0194 þ w=c
where
7:
8:
0401 0804 0758
τ ¼ 66 78p0154
FB
pSO3
C3 A pC3 S
0227 0535 0558
pSO3 expð0647pslag Þ
β ¼ 1814p0146
C3 A pC3 S FB
te ðTr Þ ¼
t
X
0
E
1
1
exp
R 273 þ Tr 273 þ Tc
All the parameters in these equations are defined in the notation list.
2.3
Boundary conditions
The thermal balance equation at the pavement surface contains the foregoing heat flux through convection, radiation,
absorption and conduction within the pavement surface and
the pavement layers beneath, and the energy stored at the pavement surface. The thermal balance equation used in this study
is expressed by (Gui et al., 2007)
10:
230
11:
qconv ¼ hconv ðTa Ts Þ
where Ta is the air temperature (°C), Ts is the surface temperature (°C) and hconv is the heat convection transfer coefficient
(W/m2/°C), which can be determined as (Ashrae, 1993; TTG,
2009)
12:
hconv ¼ 3727C½09ðTs þ Ta Þ þ 320181
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðTs Ta Þ0266 1 þ 2857vwind
where vwind is the wind speed (m/s) and C is a constant correction factor depending on the heat flow condition. C is chosen
as 1·79 when the pavement surface is warmer than the air and
0·89 when the pavement surface is cooler than the air.
expð2187pslag þ 950pFA pFACaO Þ
and the equivalent age te is given by
9:
Convection is heat transfer by the mass of a gas or fluid
motion, such as air or water, when the heated gas or fluid is
made to move away from the source of heat, carrying energy
with it. Generally, Newton’s law of cooling is used to express
heat convection at the surface (Ashrae, 1993)
qabs þ qir þ qconv þ qδsurface þ kc
@T
@x
@Ts
¼ δsurface ρc cc
@t
surface
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Solar radiation is the most important source of heat to pavement structures. Solar radiation propagates energy as a short
wave arriving at the ground surface. The amount of energy
absorbed by a pavement’s surface depends on the medium’s
thermal properties and the colour of the pavement surface.
McCullough and Rasmussen (1999) express the heat flux
absorbed by a pavement surface as
13:
qabs ¼ γabs If qsolar
where γabs is the solar absorptivity of the concrete, which can
be 0·1–0·35 for a pavement with a white curing compound,
qsolar is the peak solar radiation during a daytime period and
If is an intensity factor, which depends strongly on the atmospheric conditions, diurnal cycle, latitude and the incident angle
between the sun’s rays and the pavement surface. It is assumed
to follow a sinusoidal function that ranges from zero at sunrise
and sunset to a peak value at the solar noon (McCullough and
Rasmussen, 1999).
Thermal irradiation is a long-wave heat transfer between the
ground and the sky. It depends on ground cover conditions
such as vegetation cover, the pavement surface and so on. The
Stefan–Boltzmann law is commonly used for this type of heat
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Volume 174 Issue 4
A mechanistic approach to predict built-in
temperature in concrete pavements
Sok and Lee
transfer, which is defined by (McAdams, 1954)
weather station for the previous 3 d are used as input values to
calculate the temperature profiles. The temperature distribution
within the substructure at the end of 3 d is then obtained. This
temperature is adopted as the initial temperature distribution
in the underlying layer for prediction of the subsequent concrete pavement temperature.
14:
4
Ts4 Þ
qir ¼ εp σðTsky
where εp is the surface emissivity, which depends on the pavement’s surface colour (for a perfectly black surface εp = 1), σ is
the Stefan–Boltzmann constant (5·669 10−8 W/m−2.K−4)
and Tsky is the effective sky temperature (K ), which is not
equal to the ambient temperature but is a function of the dew
point temperature and cloud cover. Based on the model proposed by Walton (1983), it is expressed as
15:
Tsky ¼ ε025
sky Ta
16:
εsky ¼ 0787 þ 0764 ln
Tdp þ 273
Fcloud
273
2.5
Concrete final setting time
The final setting time of concrete denotes the time at which
fresh concrete begins to gain the mechanical properties and
thus to carry stresses. Based on ASTM C 403 (ASTM, 2008),
the final setting time of concrete is defined in terms of a penetration resistance of 27·6 MPa. According to Schindler et al.
(2002), for any particular mixture design, the setting of concrete occurs at a certain degree of hydration. Schindler et al.
(2002) proposed that the critical degree of hydration (αcr) at
the final setting time of a particular mixture can be correlated
with water/cementitious material (w/c) ratio, expressed as
19:
17:
Fcloud ¼ 10 þ 0024N 00035N 2 þ 000028N 3
where Ta is the air temperature (K), εsky is sky emissivity, N is
cloud cover with a value from 0 to 1, Tdp is the local dew
point temperature (°C) and Fcloud is the cloud cover factor.
αcr ¼ ks ðw=cÞ
where ks is a constant value, equal to 0·26. In this study, the
final setting time of a concrete slab was assumed to occur
when the calculated degree of hydration at the mid-depth of
the slab as presented in Equation 5 reaches the critical degree
of hydration (αcr) as given by Equation 19.
2.6
2.4
Initial pavement condition
The concrete placement temperature can initially be determined based on the temperature of the concrete ingredients.
The contribution of each constituent is calculated from its
temperature, specific heat and weight fraction. According to
Mindess et al. (2003), the concrete placement temperature
(Tconcrete) can be calculated by
18:
Tconcrete
022ðTagg Wa þ Tcem Wc Þ þ Tagg Wwa þ Tw Ww
¼
022ðWa þ Wc Þ þ Wwa þ Ww
where Tagg, Tcem and Tw are the temperatures of aggregate,
cement and water, respectively and Wa, Ww, Wc and Wwa are
the weights of dry aggregate, water, cement and wet aggregate,
respectively.
For this study, an assumption for the temperature at the
bottom of the subgrade layer is made. At a ground depth
greater than 5 m, the ground temperature is at a constant
value of 12°C (Williams and Gold, 1976). The temperature
model enables prediction of the initial temperature distribution
of the sub-base layer before concrete placement as follows.
Firstly, the temperature distribution in the sub-base layer is
assumed to be linear. The temperature of the upper layer of
the sub-base is assumed to be the mean monthly air temperature while the temperature of the lower layer is the ground
temperature (Ren, 2015). Then, climate data from the nearest
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Numerical implementation of the
temperature model
Incorporating known boundary and initial conditions,
Equation 1 can be solved using the 1D finite-difference
method. The unsteady unit ∂T/∂t and ∂T/∂x are applied in a
forward difference scheme while ∂ 2T/∂x 2 adopts the central
difference scheme, giving
20:
k
k
2Tik þ Ti1
@ 2 T Tiþ1
¼
2
2
@x
Δx
21:
@T Tikþ1 Tik
¼
@t
Δt
22:
k
Tik
@T Tiþ1
¼
@x
Δx
For the boundary condition at the pavement surface (x = 0)
and assuming δsurface = Δxc/2; introducing Equations 21 and 22
into Equation 10 yields
23:
T0kþ1 ¼ T0k þ 2
qk Δt
qk Δt
kc Δt k
T T0k þ 0 þ 2 st
ρc cc Δxc
ρc cc
ρc cc Δx2c 1
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A mechanistic approach to predict built-in
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Sok and Lee
where i and k are the node and time step, respectively, and cc,
kc, ρc and Δxc are the heat capacity, thermal conductivity,
density and space increment of the concrete layer, respectively.
qkst is calculated by summing qkabs, qkir and qkconv at time step k.
qk0 is the internal heat of cement hydration rate at time step k.
For fresh concrete, the thermal conductivity kc and the specific
heat of concrete change with respect to the degree of
hydration. The empirical equations to calculate these parameters with respect to the degree of hydration can be found
elsewhere (Ruiz et al., 2001; Van Breugel, 1991).
The proposed numerical solution of the heat-transfer
problem in a pavement structure can easily be implemented in
either a high-level programming language or a spreadsheet.
In this study, prediction of temperature was programmed in
Matlab.
For an interior node within the pavement layer, the following
finite-differential expression can be determined by introducing
Equations 20 and 21 into Equation 1. Tk+1
of the interior
i
nodes within the pavement layer is explicitly calculated according to the known previous temperature data
24:
Tikþ1 ¼ Tik þ
qki Δt
kc Δt k
k
2 Tiþ1 2Tik þ Ti1
þ
ρc cc
ρc cc Δxc
In the multi-layered interface between a pavement and it subbase, as well as between the sub-base and the ground, the continuity of heat must be maintained and Tk+1
of the interface
i
nodes within the pavement layer is given by
Tikþ1 ¼ Tik þ 2
25:
þ2
þ
λa
Δt
T k Tik
ðρa ca Δxa þ ρb cb Δxb Þ Δxa i1
λb
Δt
T k Tik
ðρa ca Δxa þ ρb cb Δxb Þ Δxb iþ1
ΔtΔxa
qk
ðρa ca Δxa þ ρb cb Δxb Þ a
in which ca, λa, ρa and Δxa are the heat capacity, thermal conductivity, density and space increment of the upper layer, respectively,
while cb, λb, ρb and Δxb are the heat capacity, thermal conductivity, density and space increment of the underlying layer. qka is
the heat of cement hydration at the interface of the upper and
underlying layer when the upper layer consists of hardening concrete layer, otherwise qka is equal to zero (Ren, 2015).
The temperature of the bottom node of the subgrade is
assumed to be a constant value equal to the ground temperature, as mentioned Section 2.4. Thus
26:
kþ1
TMðx5
mÞ ¼ constant
3.
Validation of the BITD
3.1
Summary of field test section
A concrete paving project constructed in the Sokcho region of
South Korea on 28 April 2016 was monitored to determine the
temperature distribution in hardening concrete and the magnitude of the BITD. The pavement, constructed using rollercompacted concrete (RCC), is 400 m long and 5 m wide, with
a slab thickness of 0·2 m. The concrete slab was placed on
a compacted granular base, also 0·2 m thick. The mixture
design and the RCC material properties are summarised in
Table 1. Air temperature data during construction were
obtained from weather centre reported results; the maximum
and minimum air temperatures were about 12°C and 7°C,
respectively. The wind speed ranged from 0 to 2 m/s, with
an average value of about 1 m/s. Concrete placement began at
10 a.m. and the monitored concrete slab section was placed
at around 2 p.m. An appropriate amount of curing compound
was immediately sprayed on the top surface of the concrete
slab. To measure the deformation and temperature of the
concrete slab, vibrating wire strain gauges (VWSGs) (model
KM-100B) and thermocouple sensors (iButtons) were installed
during concrete placement, as shown in Figure 4. The concrete
slab was instrumented with iButton sensors at different depths
of the concrete slab (50 mm, 100 mm and 150 mm below
the pavement surface). Since temperature measurements were
taken at only three points over the depth of the pavement, a
second-degree polynomial was used to find the temperature
distribution in the overall depth of the slab, along with the
temperatures at the top and bottom of the slab. The sensors
were programmed to collect temperature data at 60 min intervals from the start of construction.
3.2
Validation of final set of the field concrete slab
A comparison was made between the final set of the field concrete slab based on the mathematical formula given in
Equation 5 and the practical method to determine the final
Table 1. Mix proportions and material properties of the RCC at
the study site
Parameter
To ensure stability of the model, an important limitation of the
explicit numerical integration scheme is considered. For 1D schematisation with time integration, the following criterion holds.
27:
232
λ Δt
1
ρc Δx2 2
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Cement type I: kg/m3
Fine aggregate: kg/m3
Coarse aggregate: kg/m3
Water: kg/m3
28 d flexural strength: MPa
28 d elastic modulus: MPa
Value
280
1032
1072
107
4·8
24 500
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A mechanistic approach to predict built-in
temperature in concrete pavements
Sok and Lee
raw strain started to change to be opposite to the variation
in concrete temperature about 6·1 h and 6·3 h after concrete
placement for the two respective locations. The average of
these two times was 6·2 h (372 min). This point was defined as
the final setting time of the field concrete slab. The final
setting time based on this method was slightly longer than that
obtained from Equation 5. Since the final setting time based
on this method was assumed to occur when the surrounding
concrete and measured strain exhibited composite behaviour,
the delayed final setting time might be contributed by a
function of restraint of the concrete slab (e.g. friction base) as
compared with the final setting time defined in ASTM C 403
(ASTM, 2008).
5 cm
5 cm
5 cm
Concrete slab
Granular base
VWSG
iButton
(a)
The final setting time determined by the strain–temperature
relationship found in this study was slightly longer than the
final setting time calculated based Equation 5 and it is believed
that there was no significant effect on the built-in temperature
distribution in the concrete slab.
(b)
Figure 4. Installation of VWSGs and temperature sensors:
(a) positions of the sensors; (b) sensor installation
setting time of concrete suggested by Glisic and Simon (2000).
According to Equation 5, the final set of concrete occurs when
the degree of hydration in the concrete reaches the critical
degree of hydration given by Equation 19. In this study,
the concrete mixture comprised 280 kg/m3 Portland cement
type I with a w/c ratio of 0·38 (Table 1). For the Portland
cement type I, pC3S, pC2S, pC3A, pSO3 and FB were 0·55, 0·18,
0·1, 0·026 and 365 m2/kg, respectively. Fly ash and slag were
not used in the mixture. Table 2 shows the calculated values
of the hydration parameters based on Equations 5–8 and
Equation 19.
Based on the work of Glisic and Simon (2000), the final
setting time of concrete in the field was assumed to occur
when the VWSGs and the surrounding concrete exhibited
composite behaviour based on monitored raw VWSG readings.
Figure 5 shows the raw strain readings and concrete temperature developed in the concrete slab 50 mm and 150 mm below
the concrete surface. As shown in the figure, the measured
3.3
Validation of temperature prediction
The developed numerical model for predicting the temperature
throughout a concrete slab at early age (72 h after concrete
placement) was validated with field-measured temperature
data. Table 3 lists the thermal properties of the pavement
materials used in the validation (FHWA, 2006). To ensure stability of the numerical solution, the input time step in this
analysis was 180 s and the input space increments were 2 cm
for the concrete layer, 2 cm for the granular base layer and
10 cm for subgrade soil layer.
Figure 6 shows the predicted and measured temperature
difference between the top and bottom of the concrete slab.
As shown in the figure, the temperature difference predicted
using the developed model agreed well with the field-measured
data. The slight difference between the predicted and
measured results may be due to the different depths of the
thermocouple sensors in the field and the numerical input
depth and also to the inaccuracy of using a second-degree
polynomial to find the temperature at the top and bottom of
the slab. However, overall, the agreement was good, with a
maximum discrepancy of about 2°C. Figure 6 also shows that
the measured BITD (the temperature difference between the
top and bottom of the slab at the setting time) (−0·4°C) agreed
quite well with the predicted value (−0·1°C). Therefore, the
developed numerical model can be used to establish the BITD
under different conditions.
Table 2. Analysis results of degree of hydration
Parameter
Hydration time parameter, τ
Hydration shape parameter, β
Ultimate degree of hydration, αu
Critical degree of hydration, αcr
Final setting time: min
Value
16·75
0·63
0·69
0·099
330
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4.
Sensitivity analysis of BITD
4.1
Input parameters
Many factors can affect the built-in temperature distribution
in a concrete pavement. The effect of climatic conditions,
concrete placement time and cementitious materials on the
built-in temperature were investigated using the developed
233
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Volume 174 Issue 4
A mechanistic approach to predict built-in
temperature in concrete pavements
Sok and Lee
40
–1250
35
–1290
30
–1310
25
–1330
20
–1350
15
–1370
10
Concrete temperature: °C
Estimated
setting time
–1270
Raw strain: mm/mm
Raw strain
Concrete temperature
5
–1390
0
6
12
18
24
30
36
42
48
Time after concrete placement: h
(a)
390
Estimated
setting time
Raw strain: mm/mm
380
30
370
25
360
20
350
15
340
Concrete temperature: °C
35
Raw strain
Concrete temperature
10
330
5
320
0
6
12
18
24
30
36
42
48
Time after concrete placement: h
(b)
Figure 5. Determination of concrete setting time: (a) 50 mm below surface; (b) 150 mm below surface
Table 3. Thermal properties of pavement materials
Property
Concrete
Density: kg/m3
Specific heat: J/kg.°C
Conductivity: W/m.°C
Coefficient of solar absorption
Emissivity
Density: kg/m3
Specific heat: J/kg.°C
Conductivity: W/m.°C
Density: kg/m3
Specific heat: J/kg.°C
Conductivity: W/m.°C
Granular base
Subgrade
Value
2350
1000
3
0·35
0·85
1898
1047
2·42
1850
1200
1·5
Temperature difference: °C
Pavement material
14
12
Measured
Predicted
Estimated
setting time
10
8
6
4
2
0
–2
–4
0
12
24
36
48
60
72
Time after concrete placement: h
Figure 6. Predicted and measured temperature gradients in the
concrete slab
numerical model. Table 4 summarises the climate data for
two seasons in South Korea. For the concrete pavement, the
thickness of the concrete layer was assumed to be 0·2–0·4 m,
the granular layer was assumed to be 0·2 m thick and the
234
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subgrade soil thickness was taken as 5 m. The thermal properties of these materials were as shown in Table 3. Pavement
placing times were assumed to be 10 a.m., 2 p.m. and 6 p.m.,
A mechanistic approach to predict built-in
temperature in concrete pavements
Sok and Lee
Transport
Volume 174 Issue 4
Table 4. Average climate data for two seasons in South Korea
Season
Low temperature: °C
High temperature: °C
Solar radiation: W/m2
Relative humidity: %
Wind speed: m/s
Autumn
Summer
4
22
12
31
500
700
65
75
3
1·5
which are typical times for concrete placement in South Korea.
To investigate the effects of the concrete mix proportions on
the BITD, two types of concrete mixtures were considered:
ordinary Portland cement concrete (OPCC) (330 kg/m3) and
OPCC with 20% fly ash and 20% slag.
4.2
Effect of concrete placement time
Figure 7 shows the effects of concrete placement time on the
magnitude of the BITD in the concrete slab for both autumn
and summer construction for the OPCC mix and a pavement
thickness of 30 cm. As shown in Figure 7(a), the distributions
of built-in temperature in the concrete slab exhibited nonlinearity for all concrete placement times in both seasons.
The assumption of a linear BITD during calculations of
Temperature distribution at final set: °C
5
15
25
35
45
Pavement depth: cm
0
5
10
15
20
25
30
Autumn construction (10 a.m.)
Autumn construction (2 p.m.)
Autumn construction (6 p.m.)
Summer construction (10 a.m.)
Summer construction (2 p.m.)
Summer construction (6 p.m.)
thermal stress and deformation would thus lead to inaccurate
results and therefore it is essential to consider the non-linearity
in the built-in temperature distribution along a slab when
calculating its thermal stress and deformation. Figure 7(b)
shows the magnitude of the BITD for different concrete placement times. For construction in the summer, a large positive
built-in temperature gradient was observed for concrete placed
at 10 a.m. This large positive BITD was expected for concrete
placed in the morning of a hot summer day since the top
surface of the concrete slab would be directly exposed to solar
radiation plus the additional heat of hydration. This would
lead to the slab curling upwards when the temperature difference between the top and bottom of the slab falls below this
BITD at a later age. The BITD magnitudes decreased and
were mostly changed from positive to negative when the concrete placement times were changed to afternoons (2 p.m. and
6 p.m.). A late-afternoon paving (6 p.m.) would reach the final
setting time during the night and therefore the BITD could be
negative. A concrete slab constructed during the late afternoon
of a summer day will thus curl permanently downwards if the
temperature difference between the top and bottom of the slab
at a later age is zero. For Autumn construction, a positive
BITD of relatively small magnitude was observed when the
concrete placement time was at 10 a.m. However, concrete
placed during the afternoon (2 p.m.) and late afternoon
(6 p.m.) exhibited a negative BITD since the final setting time
of the concrete slab would occur at a cooler time of the day.
Overall, concrete pavements placed in the autumn were found
to exhibit smaller BITD magnitudes than pavements constructed in the summer months.
(a)
12
Autumn construction
Summer construction
10
7·25
8
BITD: °C
6
3·31
4
2
1·16
0
–2
–1·53
–4
–2·16
–4·23
–6
10 a.m.
2 p.m.
6 p.m.
Concrete placement time
(b)
Figure 7. Effect of concrete placement time on built-in
temperature: (a) distribution; (b) magnitude
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4.3
Effect of concrete mix design
Differences in the concrete mix proportions due to the addition
of fly ash and slag changed the magnitude of the BITD.
Figure 8 shows the effects of the concrete mix proportion on
the magnitude of the BITD for a 30 cm concrete slab placed
at different times on a typical summer day. As the addition of
fly ash and slag can reduce the rate of heat hydration and
therefore the concrete temperature, the final setting time of the
concrete slab will be delayed. For the concrete pavement
placed in the afternoon (2 p.m.) and late afternoon (6 p.m.) of
a summer day, the BITDs of the concrete slab made with fly
ash and slag were lower than those of the pavement made with
only OPCC. The results showed that the final setting time of
concrete placed at 2 p.m. occurred in the early evening, and
produced a small positive BITD for the OPCC. Therefore,
adding fly ash and slag significantly delayed the final setting
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A mechanistic approach to predict built-in
temperature in concrete pavements
Sok and Lee
12
10
8
OPCC
OPCC + 20% fly ash + 20% slag
8·50
7·25
BITD: °C
6
3·31
4
1·51
2
0
–2
–4
–6
–4·23
10 a.m.
2 p.m.
–3·75
6 p.m.
Concrete placement time
Figure 8. Effect of cementitious material on BITD for summer
construction
time of concrete to the evening, thus reducing the BITD.
Similarly, for concrete placed at 6 p.m., the final setting time
occurred in the night, producing a large negative BITD for the
OPCC pavement. However, with fly ash and slag added to
the concrete mixture, the final setting time would be delayed
until the early morning, thus resulting in a smaller negative
12
Autumn construction
Summer construction
10
BITD: °C
8
7·52
7·25
6·81
6
4
1·92
2
1·16
0·51
0
20
30
40
BITD compared with that of the OPCC pavement. In contrast
to concrete placement during the afternoon (2 p.m.) and late
afternoon (6 p.m.), concrete placement in the morning of a
summer day led to a larger BITD for the concrete with fly ash
and slag. The results indicate that, for morning placement, the
final setting time of the OPCC occurred in the early afternoon,
but did not coincide with the peak solar radiation during the
day. The addition of fly ash and slag delays the final setting
time of concrete significantly, thus leading to a final setting
time coinciding with peak solar radiation and thus generating
a large positive BITD. In summary, adding fly ash and slag to
the concrete mixture has a significant effect on the magnitude
of BITD, increasing or decreasing the BITD depending on the
concrete placement time.
4.4
Effect of pavement thickness
The effects of concrete slab thickness on the BITD were
also investigated. The proposed model was used to establish
the BITD for OPCC slab thicknesses ranging from 20 cm
to 40 cm for both autumn and summer constructions.
The concrete placement time was selected to be 10 a.m.
Figure 9 shows that when the slab thickness was increased, the
BITD slightly increased for both autumn and summer
placements.
4.5
Summary
Table 5 summarises the BITDs of the concrete slab (OPCC
mix design) for different construction seasons, concrete
placement times and slab thicknesses. The slab thickness
had little effect on the BITD, with differences of only about
1·5°C. The BITD was very similar for slab thicknesses of
30 cm and 40 cm. In contrast, the concrete placement time
and construction season had significant effects on both the
magnitude and sign of the BITD. With a change in
concrete placement time from morning to afternoon or early
evening, the BITD decreased and mostly changed from a
positive to a negative value for both autumn and summer
construction.
Slab thickness: cm
Figure 9. Effect of pavement thickness on BITD for autumn and
summer construction
5.
Conclusions
A mechanistic approach to predicting the built-in temperature
gradient of a concrete pavement slab has been suggested.
Table 5. BITDs of OPCC concrete slab under different conditions
BITD: °C
Summer placement
Slab thickness: cm
20
30
40
236
Autumn placement
10 a.m.
2 p.m.
6 p.m.
10 a.m.
2 p.m.
6 p.m.
6·81
7·25
7·52
3·59
3·31
3·25
−3·09
−4·23
−4·54
0·51
1·16
1·92
−1·52
−1·53
−1·63
−1·57
−2·16
−2·58
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Transport
Volume 174 Issue 4
A mechanistic approach to predict built-in
temperature in concrete pavements
Sok and Lee
The numerical model to predict the temperature distribution
in a hardening concrete slab was developed using the
finite-difference method. A mathematical formula to predict
the final setting time based on the concrete degree of hydration
was integrated in the temperature predictive model to establish
the BITD. The BITD results from the proposed method
showed good agreement with field-measured data. Using the
developed numerical model, the effects of climatic conditions,
concrete mix design, concrete placement time and pavement
thickness on the BITD were also investigated. The modelling
results indicated that climatic conditions have sustainable
effects on the BITD. A concrete slab constructed during the
summer was found to have large positive or negative BITD,
whereas smaller BITDs (again positive or negative) were
obtained for concrete placed during the autumn. Concrete placement time during the day was also found to have a significant effect on the BITD. In hot weather conditions (summer),
changing the placement time from morning to late afternoon
changed the BITD from a large positive value to a small negative one. The addition of fly ash and slag to the concrete mix
was found to decrease or increase the magnitude of the BITD
depending on the concrete placement time during the day.
The pavement thickness was found to have little effect on
the BITD.
Glisic B and Simon N (2000) Monitoring of concrete at very early age
In summary, the model proposed in this paper can be used to
predict the BITD in concrete slabs for a given concrete mix
design, concrete placement time, pavement thickness and
environmental condition.
Acknowledgements
This study was supported by Ministry of Land, Infrastructure
and Transport (MOLIT) and the Korea Agency for
Infrastructure Technology Advancement (KAIA): grant number
[18TLRP-B146707-01], and the 2017 Academic Research
Program funded by Gangneung-Wonju National University.
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