Lin Yang Early-age cracking of concrete and calorimetry

8/17/2019 Lin Yang Early-age cracking of concrete and calorimetry

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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


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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

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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

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(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]


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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


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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


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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


1 3


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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


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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


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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


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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


0

50

100

Calculated

Experiment

Calculated

Experiment

Calculated

Experiment

Calculated

Experiment

150

200

250


0

50

100

150

200

250


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


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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.

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Lin Yang Early-age cracking of concrete and calorimetry

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