lin yang early-age cracking of concrete and calorimetry
TRANSCRIPT
-
8/17/2019 Lin Yang Early-age cracking of concrete and calorimetry
1/13
Thermal analysis and adiabatic calorimetry for early-age concretemembers
Part 2. Evaluation of thermally induced stresses
Yun Lin1 • Hung-Liang Chen1
Received: 20 July 2015 / Accepted: 24 October 2015 / Published online: 18 November 2015 Akadémiai Kiadó, Budapest, Hungary 2015
Abstract In this study, a finite element model was
developed to perform the stress analysis on early-ageconcrete members to predict the thermally induced stresses
and the associated cracking risk. FORTRAN subroutines
were created for ABAQUS finite element program to
enable solution-dependent material properties in the ther-
mal stress calculation. Young’s modulus development,
strength development, tensile creep, and compressive creep
behaviors at early age were experimentally obtained, and
these material properties were incorporated in the subrou-
tines. Two 1.2-m concrete cubes were constructed with
embedded temperature sensors and vibrating wire strain
gages to verify the simulation results. Results showed that
the calculated temperature and strain values correlated wellwith the measured field data. Additionally, visual cracks
were confirmed at the predicted locations on the concrete
cube. It is concluded that the method developed in this
study is capable of determining the thermally induced
stresses of early-age concrete members.
Keywords Mass concrete Heat of hydration Early-ageconcrete Thermal stress ABAQUS
Introduction
The heat generation from cement hydration leads to a
temperature rise, especially at the core of a large concrete
member. At concrete surfaces, the temperatures are rela-
tively lower due to surface heat loss from external ambient
cooling. The created temperature differential may lead to
high tensile stresses at the concrete surfaces and produce
surface cracking. The most common thermal control
practice is to limit the temperature differential between the
center and the surface of the concrete structures. However,
temperature differential is not conclusive enough to
determine the cracking risks due to thermal stresses. Nagy
and Thelandersson [1] pointed out that the development of concrete Young’s modulus is very important in thermal
stress modeling. Gutsch and Rosatasy [2] suggested the
importance of tensile strength development and the tensile
creep behavior in terms of cracking potentials.
Lawrence et al. [3] reported that temperature differential
alone was not sufficient to determine thermal stresses.
Instead, a thermal stress analysis considering the changes
of concrete material properties, such as thermal expansion
coefficient, Young’s modulus and viscoelasticity should be
used.
During early age, the nonuniform temperature profile
distribution causes disproportionate thermal expansions
within the concrete body. The surface of concrete in lower
temperatures can be under high tensile stresses due to
relative thermal expansions from internal concrete. The
heating effect due to hydration and the cooling effect due
to surface heat loss occur simultaneously. Therefore, the
surface of concrete is under tension once concrete is set
until the hydration heat is fully dissipated to the environ-
ment. The reversal of stress may occur beneath the surface
of concrete when the concrete passes from the heating
& Yun [email protected]
Hung-Liang [email protected]
1 Department of Civil and Environmental Engineering, WestVirginia University, P.O. Box 6103, Morgantown,WV 26506-6103, USA
1 3
J Therm Anal Calorim (2016) 124:227–239
DOI 10.1007/s10973-015-5131-x
http://crossmark.crossref.org/dialog/?doi=10.1007/s10973-015-5131-x&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1007/s10973-015-5131-x&domain=pdfhttp://orcid.org/0000-0002-2450-222X
-
8/17/2019 Lin Yang Early-age cracking of concrete and calorimetry
2/13
phase to the cooling phase. Whether the high surface ten-
sile stresses can cause cracking is depending on the stress-
to-strength ratio at the critical locations. During the
hydration process of the early-age concrete, both the
thermally induced stresses and the concrete strength are
being developed but at different rates. Cracks are most
likely to occur at the critical locations where tensile stress
exceeds the tensile strength. Figure 1 originally presentedby Tia et al. [4] depicts an example of thermal stress and
concrete tensile strength development. The cracking zone
in the figure refers to the time when tensile stress exceeds
tensile strength. In practice, this cracking time zone is most
likely to occur within 1–2 days after concrete placement,
depending on the member geometry, size, boundary
restraint and the ambient temperature variations.
The development of thermally induced stresses is a
complicated phenomenon which includes the variability of
temperature distribution, concrete thermal and mechanical
properties, and the viscoelastic behavior of early-age con-
crete. In recent years, finite element models have been usedto predict the thermally induced stresses of early-age
concrete members. Waller et al. [5] presented a model
using CESAR-LCPC which included two modules, TEXO
and MEXO, to perform the thermal analysis and stress
analysis on concrete structures. Wu et al. [6] described the
procedures calculating thermally induced stresses for a
wall element using ANSYS. Tia et al. [7] evaluated bridge
footing elements with wooden formwork using TNO Diana
software. Their research findings are very helpful to this
topic; however, the modeling procedure of the viscoelastic
behaviors due to tensile or compressive stresses was not
detailed enough for replication purposes.Researchers have emphasized the importance of con-
crete’s viscoelasticity, which is crucial in calculating
thermal stresses. Bažant’s B3 model [8], which was
designed for long-term creep behaviors, has been widely
adopted recently to describe creep behaviors of early-age
concrete. Østergaard et al. [9] improved the B3 model on
the early-age creep behavior by adjusting the ‘‘aging’’ term,
while Wei and Hansen [10] made an adjustment on the
later ages by modifying the ‘‘flow’’ term of B3 model.
However, the temperature, although often ignored in the
creep calculation, has a significant effect on early-ageconcrete creep behavior. Using the equivalent age to con-
sider the temperature effect in creep was suggested by
Bažant and Baweja [11]. Atrushi [12] also showed the
usage of equivalent age on the modified double power law
(DPL) with some experimental proof. Luzio and Cusatis
[13] validated the solidification–microprestress–mi-
croplane (SMM) model considering moisture variation and
moisture diffusion associated with environmental exposure
and internal water consumption. This paper described the
thermal stress calculation of early-age concrete using a
modified B3 model considering the aging and temperature
effect in a variable loading and temperature environment.A thermal stress calculation algorithm with experimental
verification is presented in this paper.
Concrete cube construction
To study the development of thermally induced stresses,
two 1.2-m concrete cubes were constructed (Fig. 2). For
both cubes, temperature measurements were taken at the
center of the cube and 5 cm from the side surface and the
top surface. The details about the temperature predictions
and measurements were presented in Lin and Chen [14].Additionally, Geokon vibrating wire strain gages (Model
4200, gage length 15.25 cm) were installed to measure the
strain changes within a concrete member during the early
Liquid Solid
Cracking zone
T e n s i l e s t r e s s a
n d s t r e n g t h
Time
Tensile strength
Tensile stress
Fig. 1 Thermal stress and tensile strength development with crack initiation Fig. 2 Pictures of internal sensor installations before cube 2 casting
228 Y. Lin, H.-L. Chen
1 3
-
8/17/2019 Lin Yang Early-age cracking of concrete and calorimetry
3/13
ages after concrete placement. Strain measurements were
taken at location A for cube 1 and locations A, B, and C for
cube 2. The locations of these sensors are illustrated in
Fig. 2; locations A and B are 10 and 5 cm from the center
of the side surface, while location C is 2.5 cm from the
center of the top surface. The mix design and cement
chemical compositions used for the two cubes are shown in
Table 1 and Table 2. The cement chemical composition
shown in Table 2 was analyzed by the material testing
laboratory of West Virginia Department of Transportation
(WVDOT). Figure 3 shows the pictures of the vibratingwire strain ages and the temperature loggers used and the
1.2-m concrete cubes.
Mechanical properties of early-age concrete
In order to calculate the thermally induced stresses, accu-
rate estimations of the concrete tensile strength and the
modulus of elasticity development are crucial. The degree
of hydration (a) calculated using Eq. (3.1), used to estimate
the concrete strength and modulus at any given time, is a
function of the equivalent age, t e. The equivalent age can
be calculated using the Arrhenius equation, Eq. (3.3) [15],
which is depending on the concrete temperature history and
the activation energy, E a. The activation energy of this
particular mix design from Table 1 was determined to be
41,800 J mol-1 by Yikici and Chen [16] following ASTM
C 1074-10 procedures. The ultimate degree of hydration,au, can be calculated using Eq. (3.2) [17]. The hydration
parameters, s and b, were two constants depending on the
mix design. s = 14.0 and b = 0.94 were determined from
the adiabatic temperature rise tests by Lin and Chen [14].
a t eð Þ ¼ au exp st e
b ! ð3:1Þ
au ¼ 1:031 w=c0:194 þ w=c ð3:2Þ
t e ¼
r t
0
exp E a
R
1
273 þ T r 1
273 þ T c t ð Þ dt ð3:3Þwhere s, b hydration parameters, R universal gas constant,
T c(t ) concrete temperature at time t , T r reference temper-
ature, 23 C, E a activation energy/J mole-1.
Concrete strength testing
The compressive strength development curves obtained
from the compressive cylinder test results using 0.15 by
0.3 m cylinders cured at a constant temperature of 23 C
Table 1 Concrete mix design/kg m-3
Material Cement Water CA FA AE/Lm-3 WR/Lm-3
Quantity 335 139 969 844 0.067 1.0
CA coarse aggregates, FA fine aggregates, AE air entraining agent,WR High-range water reducer
Table 2 Cement chemical composition/%
Components CaO SiO2 Al2O3 Fe2O3 MgO SO3 Na2O K 2O
Percentages 62.3 20.22 4.8 3.1 2.51 3.0 0.034 0.76
Fig. 3 Pictures of vibratingwire strain gage andtemperature logger and cube 1and cube 2
Thermal analysis and adiabatic calorimetry for early-age concrete members 229
1 3
-
8/17/2019 Lin Yang Early-age cracking of concrete and calorimetry
4/13
(Fig. 4a). Schutter [18] reported that concrete compressive
strength and degree of hydration (a) had a linear relation-
ship. Results from the current mix design (Table 1) also
showed a strong linear correlation between the degree of
hydration and the compressive strength (Fig. 4b). This
linear relationship, Eq. (3.1.1), will be used to describe the
concrete strength at any given degree of hydration.
f 0c ¼ 45:53a 1:71 a 0:04; f 0c 0 ð3:1:1Þ
Splitting tensile strength development was determined
according to ASTM C496 using 0.15 by 0.3 m cylinders.
Wight and MacGregor [19] presented Eq. (3.1.2) obtained
from the mean split cylinder strength ( f ct) from a massive
database. The curve described by Eq. (3.1.2) has a high
correlation with the current splitting tensile experiment
results; the comparison of the test results and the predicted
values using Eq. (3.1.2) are shown in Fig. 5. For modeling
purposes, the splitting tensile strength development can
also be expressed in terms of degree of hydration shown in
Eq. (3.1.3) by inserting Eqs. (3.1.1)–(3.1.2).
f ct ¼ 0:53 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
f 0c inMPað Þq
ð3:1:2Þ
f ct ¼ 0:53 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
45:53a 1:71p
a 0:04; f ct 0ð Þ ð3:1:3Þ
Young’s modulus development
In this study, accurately assigning elastic modulus for the
concrete under tension is essential for thermal stress calcu-
lations. The tensile modulus development curve was experi-
mentally obtained. The specimens used in tensile modulus
test were 0.9-m-long dog-bone specimens, each with a
0.1 m 9 0.1 m mid-cross section (Fig. 6a). One vibrating
wire strain gage was embedded in the middle of the concrete
specimen. A steel hook was placed at each end in order to
applytension. The specimen was loaded in direct tension non-
destructively (Fig. 6b) using a force between 600 and 2400
Newton (approximately 10 % stress-to-strength ratio)
depending on theconcrete maturity. The strain due to external
tensile stress was measured by the vibrating wire strain gage.
The specimen was loaded four times for each data point to
ensure the accuracy. Each loading lasted approximately 10 s
to minimize any creep effect. The tensile modulus values
shown in Fig. 6c was obtained based on the measured stress-
to-strain ratios. The relationship between compressive
strength and Young’s modulus for this particular mix design
can be determined using curve fitting method. A best-fit
exponential function as shown in Eq. (3.2.1) is used to
describe the development of the Young’s modulus. In
Eq. (3.2.1), the compressive strength f 0c can be expressedwith degree of hydration (Eq. 3.1.1). The elastic modulus can
also be expressed in terms of degree of hydration as in
Eq. (3.2.2). The Young’s modulus was assumed identical in
both tensile and compressive directions.
E c ¼ 5407 f 0c0:492 ð3:2:1ÞE c ¼ 5407 45:53a 1:71ð Þ0:492 a 0:04; E c 0ð Þ
ð3:2:2Þ
Thermal expansion coefficient
After performing the tensile modulus testing, the dog-
bone sample with embedded vibrating wire strain gage
0 2 4 6 8 00
5
10
15
20
25
30
35
0
5
10
15
20
25
30
35
0.2
y = 45.53x – 1.71
R 2 = 0.9969
0.4
Degree of hydrationEquivalent age/day
C o m p r e s
s i v e s t r e n g t h / M P a
C o m p r e s
s i v e s t r e n g t h / M P a
0.6 0.8
(a) (b)Fig. 4 Relationship betweencompressive strength anddegree of hydration
00
0.5
1
1.5
2
2.5
3
2 4
Test result
Wight & Macgregor
Eqvalent age/day
T e n s i l e s t r e n g t h / M P a
6 8
Fig. 5 Splitting tensile strength test results in comparison with Wightand Macgregor [19]
230 Y. Lin, H.-L. Chen
1 3
-
8/17/2019 Lin Yang Early-age cracking of concrete and calorimetry
5/13
was reused to test the coefficient of thermal expansion
(CTE). The dog-bone specimen was submerged into a
temperature controlled water tank and placed on fric-
tionless base provided by ball bearings. The specimen
was able to freely expand and contract due to temperature
changes. The strain data of the dog-bone specimen was
recorded, while the water temperature was controlled to
slowly rise and drop. The CTE test was repeated three
times, and the test results correlated closely with an
average thermal expansion coefficient of 8.53 micros-trains per C at 28 days of age. The thermal expansion
coefficient is assumed to be a constant for simplicity. The
variation of CTE of concrete is difficult to measure
because of the temperature influence of the concrete
maturity especially at early age; it was shown by
McCullough and Rasmussen [20] that concrete CTE
variation after 24 h of age could be assumed negligible
where the CTE before 24 h decreased noticeably. It is
noted that CTE is also depending on the moisture level
inside concrete, and it was assumed that the moisture
level in the current concrete cube is close to 98–100 %
[14].
Basic creep of early-age concrete under constant
load
The viscoelastic behavior of early-age concrete plays an
important role in calculating thermal stresses. The tensile
creep behavior of early-age concrete is complicated.
Tensile creep tests performed by Gutch and Rostasy [2]
showed pronounced viscoelasticity when load was applied
at early age. Umehara and Uehara [21] and Atrushi [12]
demonstrated the influence of temperature on early-age
tensile creep. Østergaard et al. [9] and Atrushi [12] showed
the strong loading age dependency in the early ages.
Bažant and Baweja [11] presented a mathematical
expression of structural creep law (B3 model) shown in
Eq. (3.4.1). With experimentally determined empirical
constants (q1–q4), B3 model was often found accurate interms of correlating with the experimental data.
J t ; t 0ð Þ ¼ e t ð Þr
¼ q1 þ q2Q t ; t 0ð Þ þ q3 ln 1 þ t t 0ð Þ0:1h i
þ q4ln t t 0
ð3:4:1Þwhere
Q t ; t 0ð Þ ¼ Qf t 0ð Þ 1 þ Qf Z t ; t 0ð Þ r 1=r
Qf t 0ð Þ ¼ 0:086 t 0ð Þ29þ1:21 t 0ð Þ49h i1 Z t ; t 0ð Þ ¼ t 0ð Þ1=2ln 1 t t 0ð Þ0:1
h ir ¼ 1:7 t 0ð Þ0:12þ8:0t current age in days (t = 0 is when water is added to the
mixture), t 0 loading age in days, q1, q2, q3, and q4 empiricalconstants
(a)
(b) (c)
LoadedConcreteSpecimen
Vibratingwire gage
Weights
RigidFrame
010000
14000
18000
22000
26000
30000
2 4
Test results
Eq. (3.2.2)
Equivalent age/day
T e n s i l e m o d u l u s / M P a
6 8
Fig. 6 Tensile modulus testingsetup and results
Thermal analysis and adiabatic calorimetry for early-age concrete members 231
1 3
-
8/17/2019 Lin Yang Early-age cracking of concrete and calorimetry
6/13
Østergaard et al. [9] found that for early-age tensile
creep, B3 model may underestimate the specific creep. The
early-age concrete exhibits much greater viscoelasticity.
They modified the q2 constant using Eq. (3.4.2) to amplify
the age dependency for early-age concrete, where q5 is
always less than the physical loading age (t 0). In theirresearch, with a very early loading age at 16 h, the best-fit
value of q5 was found to be 14 h.
q02 ¼ q2t 0
t 0 q5 ð3:4:2Þ
Temperature is also an important factor, which has two
different effects on the creep behavior of early-age con-
crete. From the maturity concept, higher curing tempera-
ture will accelerate the hydration process and increase the
concrete maturity at the time of loading and therefore
decrease the specific creep. However, the creep deforma-
tion at early age increases significantly as the temperature
increases. Atrushi [12] stated that the increasing effect is
much greater than the decreasing effect. In Atrushi’sexperimental results, a significant increase in tensile creep
was found due to the effect of temperature increase. In
order to consider the effect of the temperature, the equiv-
alent age concept was used by Bažant and Baweja [11] and
Atrushi [12]; the equivalent age was used to replace the
regular age in the places of the loading age and the loading
duration, where they found better agreements between the
theoretical and experimental results. Hence, the modified
B3 model can be expressed as shown in Eq. (3.4.3).
J t e; t 0e ¼
e t eð Þr
¼ q1 þ q2 t 0e
t 0e q5Q t e; t
0e
þ q3 ln 1 þ t e t 0e
0:1h i þ q4ln t et 0e
ð3:4:3Þ
To measure the basic tensile creep of early-age concrete,
surface sealing is important because tensile loading can
significantly accelerate the drying effect and lead to more
load-induced drying shrinkage. In this study, during each
creep testing, two identical 0.9-m dog-bone specimens with
a 0.1 m 9 0.1 m cross section at the mid-span region were
used. The concrete molds and sensor installation were the
same as shown in Fig. 6. Both specimens were sealed withepoxy paint plus four layers of plastic wraps immediately
after unmolding (1 h prior to the loading). Epoxy paint
creates an adhesion between the plastic wrap and the
concrete surfaces to further prevent surface drying. Both
specimens were kept in the same room with a controlled
temperature of 23 C and 50 % humidity level. One of the
specimens was loaded with a tensile stress of 0.13 MPa
(approximately 10 % stress-to-strength ratio), while the
other was kept free to deform. Although the loading
magnitude is small with respect to its tensile strength, the
specific creep was assumed to be un-affected.
Hauggaard et al. [22] reported that the specific creep
response of early-age concrete was found to be unchanged
when a stress-to-strength ratio is below 60 %. Similar
conclusion was drawn by Atrushi [12] in his tests up to
80 % stress-to-strength ratio.
The strain measurements for both specimens (loadedand free) were taken using Geokon vibrating wire strain
gages. The difference in the monitored strain between the
two specimens divided by the loading magnitude is cal-
culated to show creep compliance. The tensile creep test
for the specific mix design (Table 1) was performed three
times at three different loading ages (0.75, 1, and 10 days),
and the results are shown in Fig. 7. As shown in Fig. 7, all
of the three test results can be described by the modified B3
model (Eq. 3.4.3). The best-fit empirical constants are
shown in Table 3.
Although tensile and compressive creep models were
often assumed to be the same for simplicity, differenttensile and compressive creep behaviors were discovered
by numerous researchers [12, 23, 24]. The compressive
creep model of the current mix design was obtained from
an existing model by Atrushi [12] shown in Eq. (3.4.4).The
double power law (DPL) developed by Bažant and Osman
[25] has been widely used to model compressive creep
behavior for hardened concrete. Atrushi [12] modified the
DPL for early-age concrete by incorporating the tempera-
ture effect observed in early age. The equation includes
equivalent age at loading (t e0
), the current equivalent age
(t e), the Young’s modulus at loading (E (t e0
)), and three
additional creep parameters (/, d and p). This modified
0
0
10
20
30
40
50
60
70
80
90
100
1 2 3 4 5 6
Eq. 3.4.3 (0.75 day)
Eq. 3.4.3 (1 day)
Eq. 3.4.3 (10 day)
Experiment (0.75 day)
Experiment (1 day)Experiment (10 day)
Time/day
C r e e p c o m p l i a n c e / µ
M P a – 1
7 8 9 10 11 12 13
Fig. 7 Comparison of creep compliance between experimentalresults and Eq. (3.4.3)
Table 3 Best-fit empirical constants for Eq. (3.4.3)
Constant q1 q2 q3 q4 q5
Value 0.3 24.0 65.0 0.5 0.2
232 Y. Lin, H.-L. Chen
1 3
-
8/17/2019 Lin Yang Early-age cracking of concrete and calorimetry
7/13
double power law (M-DPL) is shown in Eq. (3.4.4).Since
the mix design used in this study (Table 1) was similar to
the ‘‘Base-0 mix’’ from Atrushi [12], same creep parameter
values were used for the current calculation. The values of
/, d , and p were 0.75, 0.2, and 0.21, respectively, obtained
from Atrushi [12]. The M-DPL with these pre-determined
creep parameters was used to account for compressive
creep of concrete in the FEM calculations.
J t e; t 0e
¼ 1E t 0e 1 þ /t 0de t e t 0e p ð3:4:4Þ
Modeling of viscoelastic behavior
Before cracking, the concrete material is usually assumed
linear elastic as in Eq. (3.5.1) for one-dimensional stress;
the elastic modulus is the ratio of the stress and the
instantaneous strain (eins). However, the early-age concrete
exhibits high viscoelastic behavior which causes a change
in effective modulus, and the analytical response is illus-trated in Fig. 8. To simplify the creep calculation, an
effective modulus (E eff ) is used in the FEM modeling. The
effective modulus (Eq. 3.5.2) represents the ratio of the
stress and the total deformation as plotted in Fig. 8a.
Figure 8b demonstrates a typical creep compliance J (t, t 0)of a concrete specimen under constant loading. J (t, t 0) isdefined as the ratio of the total strain (e(t ) = etotal =
ecr ? eins) and the stress ( J (t, t 0) = e(t ) / r). The creep
coefficient (C cr) is defined as the ratio of the creep strain
(ecr) and the instantaneous strain (eins) due to the loading
(Eq. 3.5.3). Equation (3.5.4) can be derived according to
Fig. 8b. The growth of Young’s modulus is considered inthis model for early-age concrete as shown in Fig. 8b.
E ¼ reins
ð3:5:1Þ
E eff ¼ retotal
¼ reins 1 þ C crð Þ ¼
E
1 þ C cr ð3:5:2Þ
C cr t ; t 0ð Þ ¼ ecr t ð Þ
eins t ð Þ ¼etotal t ð Þ eins t ð Þ
eins t ð Þ ð3:5:3Þ
C cr t ; t 0ð Þ ¼ J t ; t
0ð Þr eins t ð Þeins t ð Þ ¼
J t ; t 0ð Þ 1E t ð Þ
1E t
ð Þ¼ E t ð Þ J t ; t 0ð Þ 1 ð3:5:4ÞThe creep behavior becomes more complicated when
the concrete member is under variable loading such as
thermally induced stresses which would change due to the
variations of the temperature gradient and the mechanical
properties. The variable loading problem can be solved
using the superposition principle. At time n, the total load
can be decomposed to n small increments (Drt). Each
loading increment has its individual loading time (t 0 = i)and loading duration (n - t 0 = n - i) (Fig. 8c). Withoutconsidering creep, the total stress at t = n, r total(n) can be
expressed as the summation of the load increments(Eq. 3.5.5). When creep is considered, each loading
increment (Drt_cr) can be derived as in Eq. (3.5.6) and the
total load can be expressed as shown in Eqs. (3.5.7) or
(3.5.8). The actual overall stress release percentage due to
all of the load increments can be expressed as the ratio of
rtotal_cr(n) and rtotal(n). Thus, the overall creep coefficient
and effective modulus can be derived as in Eqs. (3.5.9) and
(3.5.10), respectively. In the FEM analysis, only basic
creep was considered. Basic creep refers to the strain
observed on sealed specimens due to sustained loading
[26]. In reality, all the surfaces of the two 1.2 m3 were
covered for entire 5 days after casting. The influences fromdrying effect are assumed negligible.
rtotal nð Þ ¼ Dr1 þ Dr2 þ Dr3 þ . . . þ Drn2 þ Drn1þ Drn
ð3:5:5Þ
rtotal nð Þ ¼Xni¼1
Dri
E
1 1E (t ′) E (t )
E eff
σ
σ
σ
t ′ t Time
C r e e p c o m
p l i a n c e J
( t , t
′ )
Time
L o a d
0 1 2 3 n –2 n –1 n
Load duration of
σ n–2
σ n
n–1
n–2
3
2
1
ε ins ε
ε
total
ε ins(t )
ε cr(t )
(a) (b)
(c)
∆
∆
σ ∆
σ ∆
σ ∆
σ ∆
σ ∆
Fig. 8 Illustration of effective modulus, the creep compliance andvariable load decomposition
Thermal analysis and adiabatic calorimetry for early-age concrete members 233
1 3
-
8/17/2019 Lin Yang Early-age cracking of concrete and calorimetry
8/13
Drt cr ¼ Drt1 þ C cr n; t 0ð Þ ¼
Drt
E nð Þ J n; t 0ð Þ ð3:5:6Þ
rtotalcr nð Þ ¼ Dr1
E nð Þ J n; 1ð Þ þ Dr2
E nð Þ J n; 2ð Þ þ Dr3
E nð Þ J n; 3ð Þ þ . . .
þ Drn2E nð Þ J n; n 2ð Þ þ
Drn1E nð Þ J n; n 1ð Þ
þ DrnE nð Þ J n; nð Þ ð3:5:7Þ
rtotal cr nð Þ ¼Xni¼1
Dri
E nð Þ J t ¼ n; t 0 ¼ ið Þ ð3:5:8Þ
C cr overall nð Þ ¼ rtotal nð Þrtotal cr nð Þ 1 ¼
Pni¼1 DriPn
i¼1Dri
E nð Þ J n;ið Þ 1
ð3:5:9Þ
E eff nð Þ ¼ E nð Þ1 þ C cr overall nð Þ ð3:5:10Þ
It is also noted that relationship between the creepcompliances and loading could become nonlinear if the
loading stress and strength ratio becomes very high [26];
the creep deformation would be further increased due to
nonlinear creep behavior at high stress-to-strength ratio
[12]. In this study, since only linear creep behavior is
considered, when stress-to-strength ratio is beyond 80 %,
the current creep model would overestimate the thermal
stresses, and this will be discussed in Sect. 5.
Discussion of maturity method on this application
The maturity method has been used to estimate in situ
concrete strength since late 1940s. Many researchers have
discovered that high-temperature curing may have a nega-
tive effect on the long-term concrete strength gain. Carino
and Lew [27] described the ‘‘crossover’’ effect due to high-
temperature curing. They suggested that maturity method is
more reliable on predicting the relative strength rather than
absolute strength. Tepke et al. [28] concluded that high-
temperature curing affect the strength–maturity relation-
ship. Kim and Rens [29] experimented on three sets of
concrete cylinders in three different curing temperatures of
40, 50, and 60 C. Their results showed higher-temperature-
cured samples exhibited lower ultimate strength at the
equivalent age of 28 days. In this study, three sets of con-crete cylinders with the same mix design (Table 1) were
cured at 23, 40, and 50 C. The strength development curves
plotted versus equivalent age (Eq. 3.3) are shown in Fig. 9a.
Concrete cylinders cured at 23 and 40 C showed very
similar strength–maturity relationship, while the cylinders
cured at 50 C showed lower strength. It suggests that
maturity method works for concrete with curing temperature
between 23 and 40 C but may not work for 50 C or higher
temperature. For verification purposes, another batch of
concrete with the same mix design was cast and cured at 23,
30, and 40 C. As shown in Fig. 9b, maturity method
worked quite well up to 7 days of equivalent age. In massconcrete applications, the concrete temperatures are nor-
mally higher due to the relatively larger member sizes. At
the center of a mass concrete member, the temperature can
be kept higher than 50 C for an extended period. However,
since only the surface tensile stresses are critical, the tem-
perature near the surface is of particular concern and the
temperature is usually much lower due to external heat loss.
For both 1.2-m concrete cubes constructed, the surface
maximum temperatures were about 45–46 C and quickly
decreased after the maximum temperatures were reached.
To verify whether the concrete surface strength of these
two cubes can be predicted using the maturity method,another compressive strength test was performed using a
set of concrete cylinders (0.1 m 9 0.2 m) cured in a tem-
perature history similar to the surface temperatures expe-
rienced by the surfaces of the cubes (Fig. 10a). Similar to
Fig. 4a, the strength development curve from cylinders of
35
30
25
20
15
10
5
00 2 4
23 °C
40 °C
50 °C
23 °C
30 °C
40 °C
Equivalent age/day
C o m p r e s s i v e s t r e n g t h / M P a
35
30
25
20
15
10
5
0
C o m p r e s s i v e s t r e n g t h
/ M P a
6 8 0 2 4
Equivalent age/day
6 8
(a) (b)Fig. 9 Strength developmentcurves for concrete cylinderscured in three differenttemperatures
234 Y. Lin, H.-L. Chen
1 3
-
8/17/2019 Lin Yang Early-age cracking of concrete and calorimetry
9/13
identical concrete cured at 23 C was also obtained. Fig-
ure 10b shows that the compressive strength of the cylin-
ders with this variable temperature curing can be predicted
by the strength–maturity relationship. These results indi-
cate that maturity method may not accurately predict thestrength for long-duration high-temperature curing at a
constant 50 C or above, but it is applicable for the strength
prediction of the concrete experiencing short-duration
high-temperature curing, such as those experienced on the
surface of the 1.2 m3.
Simulation process and results
The computation of thermally induced stresses for early-
age concrete contains two parts: thermal analysis and stress
analysis. Thermal analysis was first conducted using 3-di-
mensional finite element method (FEM), and the calculated
temperature compared quite well with the experimental
measurements [14]. Figure 11 shows the temperature
comparisons between the FEM predictions and the mea-
surements at the side surface (5 cm inside the surface) and
at the center of the 1.2-m concrete cube. The details of the
temperature calculations were described in Lin and Chen
[14]. The temperature profile results are used for the stress
analysis herein.
For thermal stress analysis, the complexity of variation
in material properties and viscoelastic behavior required usto develop a subroutine ‘‘USDFLD’’ to account for the
change of material properties and viscoelastic behavior at
every time increment. For each individual element, the
program first calculates the equivalent age (t e) and degree
of hydration (a) based on the calculated temperature his-
tory. The compressive elements and tensile elements are
treated independently. The difference in creep behavior
between elements in tension and compression is considered
(Sect. 3.4). The modified B3 model is used to describe the
tensile creep, and the modified double power law (M-DPL)
is used to describe the compressive creep behavior. To
simplify the analysis, it was programmed to check the
maximum principal stress in each element at every time
step to identify tensile and compressive elements.
The creep stress loading magnitude changes because of
temperature variation. Load decomposition and superpo-
sition rules were used to calculate the overall creep coef-
ficient. The effective modulus was then used to incorporate
0 1015
20
25
30
35
40
45
50
20
Cube #123 °C
23 °C (Fig. 4(a))
Match curingCube #2
Curing temperature
30 40
Time/h
50 60 70 80 00
5
10
15
20
25
30
35
30 60 90 120 150 180
Equivalent age/h
C o m p r e s s
i v e s t r e n g t h / M P a
T e m
p e r a t u r e / ° C
(a) (b)Fig. 10 a Surface temperaturehistories of the two cubes andthe variable curing temperature,b measured compressivestrength of the specimens curedin different temperaturehistories
010
20
30
40
50
60
70Cube 1
Experiment (Center)
FEM (Center)
FEM (side)
Ambient
Experiment (side)
Experiment (Center)
FEM (Center)
FEM (side)
Ambient
Experiment (side)
Cube 2
10
0
20
30
40
50
60
70
20 40 60 80
Time/h
100 120 0 20 40 60 80
Time/h
100 120
T e m p e r a t u r e / ° C
T e m p e r a t u r e / ° C
(a) (b)Fig. 11 Center and the sidetemperature predictions of cube1 and cube 2
Thermal analysis and adiabatic calorimetry for early-age concrete members 235
1 3
-
8/17/2019 Lin Yang Early-age cracking of concrete and calorimetry
10/13
both elastic deformation and viscoelastic deformation due
to the creep behavior (Sect. 3.5). At each time step, the
thermal stress was computed for each element based on the
calculated thermal gradient, current elastic modulus, and
overall creep coefficient. Finally, the equivalent age,
degree of hydration, and stresses in principle directions for
each element were stored for the next time step. Figure 12
shows the programming algorithm of the stress analysis.The analysis used a fixed time increment of 1 h. The entire
algorithm was executed for each individual element at
every time step. For simplicity, the Poisson’s ratio and
coefficient of thermal expansion (CTE) were assumed
constants. Poisson’s ratio was assumed to be 0.2. CTE was
experimentally obtained to be 8.53 microstrains per C
(Sect. 3.3). The frictional interaction between the bottom
of the concrete cube and the wood base was neglected for
simplicity.
Stress analysis was performed for both 1.2-m concrete
cubes using the ABAQUS program and the above FOR-
TRAN subroutine. The model had 15,625 nodes and13,824 elements using 3-D 8-node linear element (C3D8R)
with 5-cm element size. Results showed that due to the
temperature evolution and the thermal expansion, the inner
elements were expanding which caused the surface element
to be in tension. The calculated surface tensile stress con-
tour patterns for the two cubes are similar. As shown in
Fig. 13 (cube 1), the tensile strength of the concrete was
exceeded by maximum thermal stress at the center loca-
tions of the edges (shown as gray color) at 16 h after
concrete placement. The predicted tensile stresses and the
estimated tensile strength at the critical locations for both
cubes are compared in Fig. 14. The estimated tensile
strength history was calculated using Eq. (3.1.3) for the
concrete at that location; concrete strength was temperaturehistory dependent, and hence, location dependent. The
FEM result showed that cube 1 was likely to crack at these
locations because the maximum thermal stress exceeded
the tensile strength. From the experimental observation,
four large cracks with an approximate cracking length of
0.6 m were found at the center of the top edges (0.3 m on
the top surface and 0.3 m downward to the side surfaces).
Figure 13 shows the calculated stress distribution and the
actual crack locations for cube 1. No other crack was found
at side or bottom edges of cube 1. The reason can be the
nonuniform strength distribution due to vibration com-
paction; the top surface is typically found to be the weakestpart of the concrete cube [16]. For cube 2, the predicted
maximum stress was shown meeting the estimated tensile
strength at the critical locations (Fig. 14); however, no
thermal crack could be visually identified on any surface of
cube 2. The reason for cube 1 to have larger stress mag-
nitudes in comparison with cube 2 was because of a larger
Start (time = n)
Return to start
S u b r o u t i n e : U S D F L D
Assemble global stiffness matrix and calculate element stressbased on the temperature profile and material properties.
Store history values of calculated element equivalent age (t e ) and stress (σ).Time step Advance: n = n+1.
Input Material properties:Modulus: E eff (n) Thermal expansion coefficient: 8.53 µ °C–1
(Sect. 3.3)Poisson’s ratio; 0.2 (constant)
Yes
Calculate tensile modulus,E t (n) [Eq.(3.2.2)]. Calculate compressive modulus,E c (n) [Eq.(3.2.2)].
Calculate each individual compressive creepcompliance, J(t,t ′ ) (t = 1,2,...,n ) using M-DPLmodel [Eq. (3.4.4)].Calculate the overall creep coefficient, C cr_overall[Eq. (3.5.9)]Calculate compressive effective modulus, E eff(n)
using Calculated E c (n) and Eq. (3.5.10).
Calculate each individual tensile creepcompliance, J(t,t ′ ) (t = 1,2,...,n) usingmodified B3 model [Eq. (3.4.3)].Calculate the overall creep coefficient,Ccr_overall [Eq. (3.5.9)]Calculate tensile effective modulus, E eff(n) using Calculate E t (n) and Eq. (3.5.10).
NoS(n–1) > 0
Load element stress history (S ) in principle direction.
Load element temperature history and calculate its equivalent
age (t e ) [Eq. (3.3)].and degree of hydration (α) [Eq. (3.2)].
Fig. 12 Algorithm of the stressanalysis
236 Y. Lin, H.-L. Chen
1 3
-
8/17/2019 Lin Yang Early-age cracking of concrete and calorimetry
11/13
s/MPa2.972.392.352.101.951.801.651.511.371.221.080.930.790.640.500.350.210.06
Fig. 13 Predicted stress field of cube 1 at 16 h
00
0.5
1
1.5
2
2.5
3
3.5
0
0.5
1
1.5
2
2.5
3
3.5
20
Cube 1 Cube 2
Tensile strength
Thermal stress
Tensile strength
Thermal stress
40 60
Time/h
T e n s i l e s t r e s s / s t r e n g t h / M P a
T e n s i l e s t r e s s / s t r e n g t h / M P a
80 100 120 0 20 40 60
Time/h
80 100 120
(a) (b)Fig. 14 Comparison of calculated thermal stress and
estimated tensile strength
0 10 20 30
Time/h
M i c r o - S t r a i n
40 50
0 10 20 30
Time/h
40 50 0 10 20 30
Time/h
40 50
00
50
100
150
200
250
M i c r o - S t r a i n
0
50
100
Calculated
Experiment
Calculated
Experiment
Calculated
Experiment
Calculated
Experiment
150
200
250
M i c r o - S t r a i n
0
50
100
150
200
250
M i c r o - S t r a i n
0
50
100
150
200
250
300
10 20 30
Time/h
40 50
(a) (b)
(c) (d)
Fig. 15 Comparison of calculated and measured strainchanges at the locations near theconcrete surfaces a cube 1—location A, b cube 2—locationA, c cube 2—location B andd cube 2—location C
Thermal analysis and adiabatic calorimetry for early-age concrete members 237
1 3
-
8/17/2019 Lin Yang Early-age cracking of concrete and calorimetry
12/13
ambient temperature drop at the night right after the cube 1
was constructed (see Fig. 11); an 18 C (Fig. 11) drop in
ambient temperature at the first night after cube 1 con-
struction caused a significant increase in thermal stresses.
It is noted that the current FEM model assumes a creep
model that is linear to the applied stresses. The nonlinear
creep behavior due to high stress-to-strength ratio is not
considered in the current model. It was observed by Atrushi[12] that when applied tensile stress/strength ratio was
about 80 %, the creep coefficient became nonlinear with
respect to the applied stress; more creep strain was
observed at higher stress-to-strength ratio. Therefore, it is
assumed that the current creep model is only able to esti-
mate the allowable thermal stress up to 80 % of the tensile
strength. Because of the linear assumption, the estimation
of the stress at the level higher than 80 % is considered to
be conservative (the estimation is higher than the actual
stress value) using the current model. Although the gray
region of Fig. 13 shows thermal stress exceeded the tensile
strength, it can only be used as a qualitative indication of high cracking probability. The current modified B3 model
is assumed to be only valid to obtain the thermal stress up
to 80 % of the tensile strength. The nonlinear creep
behavior due to high stress-to-strength ratio needs further
investigation.
The finite element calculated strains were verified with
the measurements. As mentioned in Sect. 2, concrete strain
histories were measured at several locations using vibrating
wire gages. For cube 1, the strain history was measured at
location A. For cube 2, strain values were measured at
locations A, B, and C. The locations are marked in Fig. 2.
The calculated strain histories at these locations were foundto be reasonably close to the measurements as shown in
Fig. 15. The best match was shown at location C of cube 2
(2.5 cm in depth at the center of the top surface) which was
far from any steel reinforcing bar (shown in Fig. 15d). The
calculated strain histories (Fig. 15a) showed some devia-
tions from the experimental measurements, which were
possibly due to the restrains of concrete movement pro-
vided by the parallel steel reinforcing bars close to the
sensor during cube 1 testing. On the other hand, during the
cube 2 testing (shown in Fig. 15b, identical location), there
was no parallel reinforcing bar attached to the sensor. It is
noted that the current FEM calculation neglected the effects
of autogenous shrinkage and external restraint; therefore, it
was not included in the predicted strains in Fig. 15. The
autogenous shrinkage of this particular concrete mixture
was measured to be approximately 10 microstrains in sealed
condition after the first 7 days. Hence, the influence of the
autogenous shrinkage was neglected in the calculation. In
general, mass concrete structures are constructed using
concrete with low cement contents that would typically
produce low autogenous shrinkage.
Conclusions
This paper describes a method to perform thermal stress
analysis using ABAQUS program with the aid of user
subroutines. The developed subroutine program uses the
degree of hydration to estimate the variable elastic mod-
ulus and strength developments. Concrete tensile creep
and compressive creep behavior were included using astep-by-step incremental calculation algorithm. The
influences from loading age and temperature effects were
considered in each time increment of the creep models. It
was assumed that the current creep model is able to cal-
culate the thermal stress up to 80 % of the concrete
strength. The finite element simulations were verified by
the experimental data from two 1.2-m concrete cubes
testing. Strain deformations at the locations near the
concrete cube surfaces were measured and correlated
reasonably well with the calculated results.
The concrete cubes have high tensile stresses at the
surfaces, especially at the center of the edges. The tensilestrength development of the concrete at surface locations
can be estimated using the maturity method, and the
cracking risk could be assessed using the stress-to-strength
ratio obtained at the critical locations. Four visible cracks
were found perpendicular to the top four edges on cube #1
as predicted, due to a relatively high ambient temperature
drop at the first night after construction. The method
developed can be used to estimate the thermally induced
stress of concrete members so that precautions can be
implemented prior to concrete casting in order to prevent
unexpected cracking.
Acknowledgements The authors acknowledge the support providedby the West Virginia Transportation Division of Highways andFHWA for the research project WVDOH RP#257. Special thanks areextended to our project monitors, Mike Mance, Donald Williams, andRyan Arnold of WVDOH. The authors also appreciate the assistancefrom Alper Yikici, Zhanxiao Ma and Jared Hershberger and theWVDOH concrete technicians from Material, Control, Soil andTesting Division for the construction of the 1.2-m concrete cubes.
References
1. Nagy A, Thelandersson S. Material characterization of youngconcrete to predict thermal stresses. Thermal cracking in concreteat early ages. In: Springenschmid R, editor. E&FN Spon, 2–6,ISBN: 0419187103, 1994.
2. Gutch A, Rostasy FS, Young concrete under high tensile stres-ses—creep, relaxation and cracking. In: Springenschmidt R,editor. Thermal cracking in concrete at early ages. Proceedings of the international RILEM symposium, E & FN Spon, London,1995. p. 111–118.
3. Lawrence MA, Tia M, Ferraro CC, Bergin M. Effect of early agestrength on cracking in mass concrete containing different sup-plementary cementitious materials: experiment and finite-elementinvestigation. J Mater Civ Eng. 2012;24(4):362–72.
238 Y. Lin, H.-L. Chen
1 3
-
8/17/2019 Lin Yang Early-age cracking of concrete and calorimetry
13/13
4. Tia M, Lawrence A, Ferraro C, Smith S, Ochiai E. Developmentof design parameters for mass concrete using finite elementanalysis. The Florida Department of Transportation, reportnumber: 00054863, 2010.
5. Waller V, d’Aloı̈a L, Cussign F, Lecrux S. Using the maturitymethod in concrete cracking control at early ages. Cement ConcrCompos. 2004;26:589–99.
6. Wu S, Huang D, Lin FB, Zhao H, Wang P. Estimation of cracking risk of concrete at early age based on thermal stressanalysis. J Therm Anal Calorim. 2011;105:171–86.
7. Tia M, Lawrence A, Ferraro C, Do TA, Chen Y. Pilot project formaximum heat of mass concrete. The Florida Department of Transportation, report number: 00093793, 2013.
8. Bažant ZP. Mathematical modeling of creep and shrinkage of concrete. Chichester: Wiley; 1988.
9. Østergaard L, Lange D, Altoubat SA, Stang H. Tensile basiccreep of early-age concrete under constant load. Cem Concr Res.2001;31(12):1895–9.
10. Wei Y, Hansen W. Tensile creep behavior of concrete subject toconstant restraint at very early ages. J Mater Civ Eng.2013;25(9):1277–84.
11. Bažant ZP, Baweja S. Creep and shrinkage prediction model foranalysis and design of concrete structures: model B3. ACI SP–194. Detroit: American Concrete Institute; 2000. p. 1–83.
12. Atrushi DS. Tensile and compressive creep of early age. Ph.D.dissertation, Trondheim, Norway: Department of Civil Engineer-ing, The Norwegian University of Science and Technology, 2003.
13. Luzio GD, Cusatis G. Soli Solidification–microprestress–mi-croplane (SMM) theory for concrete at early age: theory, vali-dation and application. Int J Solids Struct. 2012;50(6):957–75.
14. Lin Y, Chen HL. Thermal Analysis and Adiabatic Calorimetryfor Early-age Concrete Members. J Therm Anal Calorim.2015;122(2):937–45.
15. Freiesleben Hansen P, Pedersen EJ. Curing of concrete structures.Draft DEB—guide to durable concrete structures, Appendix 1, 1985.
16. Yikici A, Chen HL. Effect of temperature-time history on con-crete strength in mass concrete structure.TRB 92nd annual con-ference proceeding, Transportation Research Board of theNational Academies, Washington, DC, 2013.
17. Mills RH. Factors influencing cessation of hydration in water curedcement pastes. Special Report No. 90, Proceedings of the Sym-posium on the Structure of Portland Cement Paste and Concrete,Highway Research Board, Washington DC, 1966;406–424.
18. Schutter GD. Fundamental study of early age concrete behavioras a basis for durable concrete structures. Mater Struct.2002;35:15–21.
19. Wight JK, MacGregor JG. Reinforced concrete: mechanics anddesign. 5th ed. Pearson Prentice Hall: Upper Saddle River; 2009.p. 58–9.
20. McCullough BF, Rasmussen RO. Fast-track paving: concretetemperature control and traffic opening criteria for bonded con-crete overlays. US: FHWA 1998, Final Report.
21. Umehara H, Uehara T. Effect of creep in concrete at early ages onthermal stresses, thermal cracking in concrete at early ages. In:Springenschmidt R, editors. Thermal cracking in concrete at earlyages. Proceedings of the international RILEM symposium, E&FNSpon, London, 1995. p. 79–86.
22. Hauggaard AB, Darkled L, Hansen PF, Hansen JH, ChristensenSL, Nielsen A. HETEK, Control of early age cracking in con-crete, phase 3: creep in concrete. Danish Road Directorate, Lynby, 1997.
23. Kanstad T, Bj/stergaard Ø, Sellevold EJ. Tensile and compres-sive creep deformations of hardening concrete containing mineraladditives. Mater Struct. 2012;46(7):1167–82.
24. Hilaire A, Benboudjema F, Darquennes A, Berthaud Y, Nahas G.Modeling basic creep in concrete at early-age under compressiveand tensile loading. The international conference on structuralmechanics in reactor technology, New Dekhi India, 2011;269,p. 222–30.
25. Bažant ZP, Osman E. Double power law for basic creep of concrete. Mater Struct. 1976;9:49.
26. Bažant ZP, Xi Y. Continuous retardation spectrum for solidifi-cation theory of concrete creep. J Eng Mech. 1995;121(2):281–8.
27. Carino NJ, Lew HS. The maturity method: from theory toapplication. Washington: Structure Congress & Exposition; 2001.
28. Tepke DG, Tikalsky PJ, Scheetz BE. Concrete maturity fieldstudies for highway application. J Transp Res Board.2004;1893:26–36.
29. Kim T, Rens KL. Concrete maturity method using variabletemperature curing for normal-strength concrete mixes. J MaterCiv Eng. 2008;20:20.
Thermal analysis and adiabatic calorimetry for early-age concrete members 239
1 3