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International Journal of Pavement Engineering

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Modelling of creep effect on moisture warping andstress developments in concrete pavement slabs

Siming Liang & Ya Wei

To cite this article: Siming Liang & Ya Wei (2018) Modelling of creep effect on moisture warpingand stress developments in concrete pavement slabs, International Journal of PavementEngineering, 19:5, 429-438, DOI: 10.1080/10298436.2017.1402595

To link to this article: https://doi.org/10.1080/10298436.2017.1402595

Published online: 16 Mar 2018.

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InternatIonal Journal of Pavement engIneerIng, 2018vol. 19, no. 5, 429–438https://doi.org/10.1080/10298436.2017.1402595

Modelling of creep effect on moisture warping and stress developments in concrete pavement slabs

Siming Lianga and Ya Weia,b

aDepartment of Civil engineering, tsinghua university, Beijing, China; bKey laboratory of Civil engineering Safety and Durability of China, education ministry, Department of Civil engineering, tsinghua university, Beijing, China

ABSTRACTOne of the major reasons for the fatigue failure in concrete pavements attributes to the curling and warping deformations and the traffic loads, while creep effect can reduce such deformation and consequently the stress generated in slabs. The literature on the influence of creep effect on slab warping and stress generation is found rare. In this study, a test set-up was designed to measure the flexural creep of concrete beams exposed to sealed and drying conditions. The measured creep properties were then incorporated in finite element analysis to model the creep effect on warping deformation and stress generated in slabs under the conditions of bounded and unbounded with base. It is found that creep effect is significant in slab bonded with base, it reduces warping deformation and stress by 36 and 45%, respectively. The total stress is reduced by 34%. Therefore, it is of importance to take into account creep effect when conducting deformation and stress analysis in concrete pavement slabs.

1. Introduction

Portland cement concrete (PCC) pavements exposed to the envi-ronment normally undergo volume changes due to temperature and moisture variations (Ytterberg 1987, Springenschmid 1998). When the distribution of temperature or moisture along the slab depth is differential, the pavement will curl or warp (Hansen et al. 2008, Hajibabaee and Ley 2015). Since the curling and warping are always restrained to a certain degree by self-weight of the pavement, base layer and dowel bar, etc., stress will generate in the pavement slab. Generally, the stress induced by the envi-ronmental effects alone is not great enough to cause damage. However, when combined with the traffic loadings, the tensile stress may become large and cause fatigue failure.

Research on moisture warping of concrete pavements has drawn significant interest due to its severe consequence on pave-ment performance (Fang 2001, Yu et al. 2004, Chang et al. 2008, Jeong et al. 2012). According to these studies, warping occurs quite early in concrete slabs, more than 70% of the maximum warping deformation was reached after 28 days in the plain con-crete slabs (Rao and Roesler 2005). The equivalent temperature gradient caused by moisture warping could reach −107 °C/m under some severe drying condition (Wei et al. 2013), which is comparable to the daily temperature gradient. Thus, it’s necessary to consider moisture warping to ensure well performing and long-lasting concrete pavements.

Although significant amount of investigations have been conducted on the moisture warping of concrete pavement,

few of them considered the creep effect (Yeon et al. 2012, Lim et al. 2014). It is well acknowledged that creep can contribute remarkably to the stress relaxation of concrete structures (Zhou et al. 2014), it will also affect the magnitude of moisture warp-ing and stress in slabs (Jeong and Zollinger 2004, Asbahan and Vandenbossche 2011). Yeon et al. (2012) and Lim et al. (2014) have investigated the creep effect on warping stress in early-age PCC pavements. It was found that 45–60% warping stress is reduced due to creep. However, neither of these researches have conducted tests to measure the creep directly, the creep property was back-calculated from the measured strain history in the field (Yeon et al. 2012) or the restrained ring test (Lim et al. 2014), a more direct and accurate investigation on the creep property of concrete slab needs to be performed. Moreover, only warping stress was evaluated. As mentioned previously, the fatigue failure of concrete pavement depends on the total stress caused by simul-taneous traffic and environmental loadings. Since the magnitude of moisture warping could impact the contact status between slab and base, which would lead to changes in total stress. Thus, it is also essential to investigate the creep effect on moisture warping and consequently the total stress combined with traffic loading.

This work investigated creep effect on moisture warping and stress developments in concrete pavements by incorporating flex-ural creep into finite element modelling. A test set-up was devel-oped to measure the flexural creep of concrete beams exposed to sealed and drying conditions. The creep effect is assessed for concrete slabs bonded and unbonded with the base. This study is

© 2018 Informa uK limited, trading as taylor & francis group

KEYWORDSCement concrete pavement; flexural creep; moisture warping; total stress; bonded and unbonded pavement

ARTICLE HISTORYreceived 26 october 2017 accepted 3 november 2017

CONTACT Ya Wei yawei@tsinghua.edu.cn

430 S. LIANG AND Y. WEI

tensile stress of about 1.4 MPa (according to Equation 1), which is below 40% of the tensile strength:

where σmax is the maximum tensile stress caused by exter-nal load P (98 N) and self-weight G (60 N/m); a is the dis-tance between support and loads (200 mm); l is the distance between supports (1150  mm); e is the cantilever length (35 mm); MR is the moment of resistance of the beam section (2.08 × 104 mm3).

Both creep properties of concrete beams under sealed and drying condition were investigated. For the sealed case, the beams were sealed by three layers of self-adhesive aluminium foil in all faces, while for the drying case, the top and bottom surfaces were exposed to the environment with RH = 50% and temperature of 23 °C. All concrete flexural creep tests started at the age of 7 days and the duration was about 30 days.

2.3. Direct tensile strength and elastic modulus test

Direct tensile modulus and strength tests were conducted on cylindrical specimens with size of Φ100 mm × 400 mm at the ages of 1, 3, 7 and 28 days. Three specimens were tested at each age, and the average values were reported. Much more detailed test procedures for direct tensile modulus and tensile strength can be found in the published paper (Yao and Wei 2014). The measured direct tensile modulus and tensile strength were sum-marised in Table 1. An exponential function was used to fit the measured results utilising least square method by Equations (2) and (3):

The developments of direct tensile modulus and tensile strength are well captured by Equations (2) and (3). The predicted direct tensile modulus and strength over ages will be used in the finite element analysis.

(1)�max =Pa + 1

8Gl2 − 1

2Ge2

MR

(2)E(t�) = 32.4 ×(1 − 0.3219e−0.2602t

�)

(3)ft(t�) = 3.95 ×

(1 − 0.572e−0.2025t

�)

expected to provide a more realistic and accurate methodology to the current performance evaluation of concrete pavement.

2. Experimental

2.1. Materials and proportions

Ordinary Portland cement was employed in concrete mixes. The water/cement ratio used in this study was 0.30, the water content was 243 kg/m3 and the mass ratio of fine aggregate to coarse aggregate was 0.67. The average compressive strength measured by 100 × 100 × 100 mm cubes at the age of 28 days was 63.1 MPa. Crushed limestone with a maximum aggregate size of 12.5 mm was used as coarse aggregate. The fine aggregate was natural sand with a fineness modulus of 2.56. Naphthalene-based high-performance super plasticiser was used to adjust the workability. All specimens were cured for 1  day before demoulding, and then sealed by three layers of self-adhesive aluminium foil and stored in a room with relative humidity (RH) and temperature controlled at 50% and 23 °C until the age of testing.

2.2. Flexural creep test

It is common to conduct compressive or tensile creep test to investigate the concrete creep behaviour, and many experimental results have indicated that there exists an obvious difference between the compressive and tensile creep (Hilaire et al. 2014). To represent concrete slab’s real loading condition, a flexural creep test is considered the most appropriate (Bissonnette et al. 2007). The test set-up using a four-point bending configuration shown in Figure 1 was designed. Concrete beams with dimension of 50 mm in height, 50 mm in width and 1220 mm in length, under the conditions of loaded and unloaded, were measured by their deflection using linear variable differential transformers (LVDTs) with the precision of 1 μm. The minimum size of the beam cross section (50 mm) is four times of the maximum size of the aggregate (12.5  mm), such geometric dimension can ensure the homogeneity of the concrete without the effect of aggregate size (China Academy of Building Research 2011). The measuring locations were at the mid-span and the 350 mm from the mid-span. During the test, the loaded beam was subject to two loads of 10 kg symmetrically which were 200 mm away from the beam supports. This load is expected to produce a maximum

Figure 1. flexural creep test set-up: (a) layout of the set-up and (b) side view of the set-up.

INTERNATIONAL JOURNAL OF PAVEMENT ENGINEERING 431

3. Methodology

3.1. Determining flexural creep

As mentioned above, the stress level in the loaded beam was lower than the elastic limit of strength (i.e. usually less than 40% of the strength), thus the superposition principle (ACI Committee 209 2008) of different strains can be applied to evaluate the creep strain of the loaded beam.

According to the classical beam theory (Bauchau and Craig 2009), the instantaneous deflection caused by external load and self-weight for the loaded beam can be calculated from Equations (4) and (5), respectively:

where fins-L(t′) is the instantaneous deflection caused by external load P at loading age of t′; fins-U(t′) is the instantaneous deflection caused by self-weight G at loading age of t′; b is the distance between support and the nearest measuring location (225 mm); I is the moment of inertia(5.21 × 105 mm4); Ec(t′) is the elastic modulus of concrete at age t′.

In this study, the flexural creep coefficient φ(t,t′) was defined as the ratio of the time-dependent deflection to the instantaneous deflection caused by simultaneous self-weight and external load in the loaded beam, which can be easily expressed as Equation (6).

(4)fins-L(t�) =

Pa ⋅ (3l2 + 12b2 − 12lb)

24Ec(t�)I

(5)fins-U

(t�) =G

24Ec(t�)I

⋅

(5

16l4 − b4 + 2lb3 − 6e2b2 − l3b

+ 6e2lb −3l2e2

2

)

(6)�(t, t�) =fm-L(t, t

�) − fins-L(t�)

fins-L(�) + fins-U(t

�)

where fm-L (t,t′) is the measured deflection of the loaded beam.The specific creep C(t,t′), defined as the induced creep strain

at age t per unit stress since loading age t′, can be calculated from Equation (7).

3.2. Predicting moisture gradient and equivalent temperature difference

Before calculating the creep effect on the moisture warping and stresses developments, moisture diffusion was analysed to obtain the RH profile in concrete slab based on the finite difference method. It has been reported that the combined effect from top surface drying and bottom surface capillary absorption of con-crete slab plays a major role in causing extensive slab warping (Wei et al. 2013). On the other hand, self-desiccation, which involves in early-age concrete, also has a non-negligible effect on RH development. Thus, the moisture transport equation con-sidering these three processes can be expressed as Equation (8).

where Hd is concrete RH due to external drying; Hs is concrete RH due to self-desiccation; Hw is concrete RH due to water absorption.

For the case of top surface drying, the moisture diffusion coef-ficient D (H) can be expressed as Equation (9) (Wei et al. 2013), which is a function of concrete RH.

For the case of bottom surface absorption, the hydraulic diffu-sivity D (θ) is expressed as Equation (10), which is a function of volumetric water content θ. An equation based on extended Brunauer–Emmett–Teller model can be adopted for transform-ing water content θ to H (Nielsen 2007, Wei et al. 2013).

Thus, the governing equation (Equation 8) can be used for char-acterising both drying and wetting. The initial RH of the concrete pavements at the age of 7 days was set as 88% based on the previ-ous experimental results (Wei et al. 2013). The top surface of con-crete slab was assumed to be exposed to environment with RH values of 50% and bottom surface was exposed to environment with RH = 100%. The algorithm was programmed in MATLAB, and the analysis period was about 40 days.

The calculated RH profiles in concrete slab with thickness of 25 cm at different ages are shown in Figure 2. It can be seen that the depths of the drying front and the absorption front both pro-gress towards the middle region of the slab with increasing age. The RH distribution profile is also affected by self-desiccation, as the RH in the middle region that is not affected by top drying and bottom absorption drops (Wei et al. 2013).

To quantify the magnitude of warping resulted from moisture gradient along the slab depth, an equivalent temperature differ-ence ΔTe was introduced. Based on the assumption that the ΔTe causes the same magnitude of slab uplift as the moisture gradient does (Wei et al. 2011), the ΔTe can be expressed as Equation (11),

(7)C(t, t�) = �(t, t�)∕Ec(t�)

(8)�H

�t=

�Hd

�t+

�Hs

�t+

�Hw

�t= ∇ ⋅ [D(H) ⋅ ∇H]

(9)

D(H) = 2.16 × 10−6 ×

{0.018 +

0.982

1 + [(1 −H)∕0.02]1.3

}(m2∕d)

(10)D(�) = 6.48 × 10−6 × exp(6�) (m2∕d)

Table 1. measured tensile modulus and strength of concrete at different ages.

Age (days) 1 3 7 28Direct tensile modulus (gPa) 24.3 28.0 30.4 32.2Direct tensile strength (mPa) 2.15 2.63 3.48 3.93

Figure 2.  rH development along the concrete slab depth over ages when top surface exposed to drying (environment with rH values of 50%) and bottom surface exposed to wetting at the age of 7 days.

432 S. LIANG AND Y. WEI

location in slabs when combined with a negative temperature difference. The contact area of each wheel was 250 × 167 mm. The total time of ABAQUS job was about 33 days, and the maxi-mum and minimum time increment was set as 0.001 and 0.5 day, respectively. The total computation time of each case was about 230 seconds on 2.3 GHz CPU and 4 G of computer memory.

User-defined material subroutines (UMAT) incorporating the creep behaviour as shown in Section 3.1 were developed in ABAQUS. Equation (12) was adopted to fit the specific creep, because these simple expressions could capture the long-term creep behaviour well and were convenient for programming (Zhu 1983):

where ai, bi, ci and m are constants, which can be obtained by fitting the flexural creep results. In Equation (12), the first two terms on the right-hand side represent the recoverable creep of concrete, and the irrecoverable creep can be characterised by the last term.

For the sake of step-by-step finite element analysis, an incremental stress and strain relationship need to be formu-lated. Assume that the stress increases linearly in [tn, tn+1], Δtn+1  =  tn+1  −  tn is the generic time step, Δσn+1 and Δεn+1 are associated to the stress increment and the total strain incre-ment, respectively. Then the elastic strain increment Δεe

n+1 and the creep strain increment Δεc

n+1 can be calculated according to Equations (14) and (15):

where �in+1 = �i

ne−aiΔtn + Δ�nΨi(tn−0.5)e

−0.5aiΔtn; �i0 = 0,

tn+0.5 = tn + 0.5Δtn+1.When neglecting the thermal strain, the total strain increment

is the sum of elastic, creep, and shrinkage strain increment, as expressed in Equation (16):

Combining Equations (14)–(16), the relationship of stress incre-ment and strain increment in one-dimensional is expressed as Equation (17):

where Δεsn+1 is shrinkage strain increment, and it can be calcu-

lated by Δεsn+1 = α[ΔTe(n+1) − ΔTe(n)]; E(tn+0.5) is elastic modulus at

the ages of tn+0.5; En+0.5 =E(tn+0.5)

1+E(tn+0.5)C(tn+1,tn+0.5) is effective viscoelastic

modulus.

(12)C(t, t�) =

3∑i=1

Ψi(t�) ×

[1 − e−ai(t−t

�)]

(13)Ψi(t�) =

{ci + di(t

�)−bi {i = 1, 2}

me−ait�

{i = 3}

(14)Δ�en+1 = ∫tn+1

tn

1

E(�)

��

��d� =

Δ�n+1

E(tn+0.5)

(15)Δ�cn+1 =

3∑i=1

(1 − e−aiΔtn+1)�in+1 + Δ�n+1C(tn+1, tn+0.5)

(16)Δ�n+1 = Δ�en+1 + Δ�cn+1 + Δ�sn+1

(17)Δ𝜎n+1 = En+0.5[Δ𝜀n+1 −

3∑i=1

(1 − e−aiΔtn+1)𝜔in+1 − Δ𝜀sn+1]

which is a function of RH, slab depth, coefficient of thermal expansion of concrete and the aggregate content.

where α is the coefficient of thermal expansion of concrete; h is the slab thickness; VA is the volume fraction of aggregates; n is a correlation parameter controlled by aggregate restraining effects and its value can be taken as 1.68; z is distance from median plane which locates at mid-depth of slab; H(z, t) is the local relative humidity at distance z from the median plane at time t.

For a slab with thickness of 25 cm, the development of ΔTe is shown in Figure 3. The value of ΔTe reaches −18.2 °C after about 30 days’ exposure.

3.3. Finite element model incorporating creep effect

ABAQUS v6.11 was employed to numerically analyse the creep effect on the moisture warping and stresses developments. The concrete pavement structure, shown in Figure 4(a), was modelled as a slab with size of 5 × 4 m supported by a Winkler foundation. The slab was discretised into 20 × 24 × 3 C3D20 elements which are shown in Figure 4(b). The foundation was represented by 21 × 25 spring elements. The model contained 7691 nodes and 1965 elements, which is fine enough to give a negligible scatter in the modelling results. Concrete slab was simulated as bonded and unbonded with the base. As shown in Figure 4(c) and (d), when the slab was unbonded with the base, the spring elements could only bear compressive load, while for the bonded case, the spring elements bear both compressive and tensile load. The two bonding cases shown in Figure 4(c) and (d) are actually ideal situations, and the actual concrete pavement structure is mostly in between the bonded and unbonded conditions. The Poisson’s ratio of concrete was 0.2 and the coefficient of thermal expansion was 10 × 10−6/°C. The modulus of sub-grade reaction was 68 MPa/m. To evaluate warping and stress occurring in the slab, the calculated ΔTe shown in Figure 3 was imposed in the slab through a subroutine UTEMP. Then a single axle load of 10 tons with dual wheel was applied at the slab corner (as shown in Figure 4(e)) at the age of 40 days, which was the most adverse

(11)

ΔTe(t) =12

�h2(1 − VA)

n × 10−6 × ∫0.5h

−0.5h

{6150 × [1 −H(z, t)]}zdz

Figure 3.  Calculated equivalent temperature difference between slab top and bottom for moisture conditions shown in figure 2.

INTERNATIONAL JOURNAL OF PAVEMENT ENGINEERING 433

where {⋅} represents a 6  ×  1 tensor; {�i

n+1

}={�i

n

}e−aiΔtn +

{Δ�n

}[Q]Ψi(tn+0.5)e

−0.5aiΔtn; .

4. Results and discussion

4.1. Measured flexural creep

Figure 5 shows the evolution of the mid-span deflection of the loaded and unloaded beams obtained from flexural creep tests. Based on the elastic–viscoelastic correspondence principle (Christensen and Freund 1982), the deflection of the unloaded and loaded beams can be expressed as Equations (19) and (20), respectively:

[Q] =

⎡⎢⎢⎢⎢⎢⎢⎢⎣

1 −� −�

−� 1 −�

−� −� 1

0 0 0

0 0 0

0 0 0

0 0 0

0 0 0

0 0 0

2(1 + �) 0 0

0 2(1 + �) 0

0 0 2(1 + �)

⎤⎥⎥⎥⎥⎥⎥⎥⎦

(19)ft−U (t, t

�) =G

24I⋅ C(t, t�)⋅

(5

16l4 − b4 + 2lb3 − 6e2b2 − l3b + 6e2lb −

3l2e2

2

)

The three-dimensional stress-strain relationship for an iso-tropic material can be expressed as Equation (18) using the material tensor Q.

(18)

{Δ𝜎n+1

}= En+0.5[Q]

−1[{Δ𝜀n+1

}−

3∑i=1

(1 − e−aiΔtn+1){𝜔i

n+1

}−

{Δ𝜀sn+1

}]

(a) (b)

(c)

(d)

(e)

Figure 4. finite element modelling: (a) three-dimensional model of concrete slab, (b) C3D20 brick element, (c) side view of concrete slab unbonded with base, (d) side view of concrete slab bonded with base, (e) location of the traffic loading in simulation.

Figure 5.  measured flexural deflection of unloaded and loaded beams under sealed and drying conditions.

434 S. LIANG AND Y. WEI

It shows that the fitted curves are in good agreement with the experimental results. Thus, it can be concluded that the specific creep (Equation 12) captures very well the flexural creep of con-crete beam.

Figure 6 shows the fitted specific creep of concrete beams under sealed and drying conditions. As shown in Figure 6(a), the specific creep obtained under sealed and drying conditions can be characterised as two parts. The first part with fast kinetics occurs approximately during the first several days. This behav-iour may be related with the water micro-diffusion in concrete (Bernard et al. 2003, Hilaire et al. 2014). At early ages, the sus-tained load imposed on concrete will lead to slowly squeezing and removing water from the pores of the cement paste gel to the capillaries. This has been used to explain short-term creep of concrete. Once the water diffusion comes into an equilibrium state, the concrete short-term creep ceases. As the loading age of concrete increases, the curve shows a slower kinetic trend, which may be caused by the shear sliding of the calcium silicate hydrate (C-S-H) sheet (Bažant et al. 1997). The second part of the creep, also called the long-term creep, will last for a long period of time. Similar observations have been reported by Ghezal and Assaf (2015).

The flexural creep of concrete is significant under the dry-ing conditions as compared to the sealed condition, which is consistent with the reported results of compressive creep and tensile creep (Bissonnette et al. 2007, Cagnon et al. 2015). When the age of concrete reaches 40 days, the specific creep of the concrete exposed to the environment with RH of 50% is about 1.67 times of that under sealed condition. And this observation may be explained by the hypothesis that drying condition will increase both water micro-diffusion in concrete and shear sliding rate of the C-S-H sheet (Bažant et al. 2004, Jennings 2004). As shown in Figure 6(b), the creep rate under drying condition is greater than the rate of the sealed con-ditions, and thus this will lead to larger creep under drying condition. Though the creep tests only last for 30 days, it is interesting to see that the flexural creep curves under both sealed and drying conditions show a linear characteristic in semi-log coordinate after a few days’ loading. This phenomenon has also been observed in compressive creep (Vandamme and Ulm 2009, Zhang et al. 2014).

Figure 7 shows the creep coefficient obtained from the meas-ured deflections of concrete beams under sealed and drying con-ditions according to Equation (6). Using the fitted specific creep, the calculated creep coefficients based on Equation (7) are also shown in Figure 7. It can be seen that the calculated results are consistent with the measured results. The creep coefficient is larger under the drying condition as compared to the sealed condition. Figure 7 suggests that the drying creep deformation is about two times of the elastic one after 40 days of loading. Therefore, it is important to consider the creep effect when conducting deformation and stress analysis in slabs exposed to drying environment.

4.2. Creep effect on moisture warping and warping stress

To evaluate the creep effect on moisture warping deformation and warping stress in concrete pavement, finite element anal-ysis (FEA) was conducted through ABAQUS incorporating

where ft-U(t,t′) is the time-dependent deflection of the unloaded beam; ft-L(t, t′) is the measured deflection of the loaded beam.

Since all the parameters expect C (t, t′) in Equation (20) are known, the specific creep (Equation 12) can be determined by fitting the measured deflection of the loaded beam with Equation (20), and the fitted parameters are shown in Table 2. Using the fitted specific creep, the deflections of the unloaded beams can be obtained according to Equation (19). The fitted results of the unloaded beams and loaded beams according to Equations (19) and (20) are displayed in Figure 5 along with the measured data.

(20)

ft−L(t�) =

Pa ⋅ (3l2 + 12b2 − 12lb)

24I⋅ C(t, t�) + ft−U (t, t

�) + fins−L(t�)

Table 2. Parameters in equation (12) for calculating specific creep of concrete un-der sealed and drying conditions.

Parameter Sealed Drying (RHenv = 50%)a1 1.556 7.819a2 0.272 0.560a3 0.027 0.016b1 0.688 0.567b2 1.811 1.501c1 (×10−6/mPa) 1.731 4.564c2 (×10−6/mPa) 7.921 9.551d1 (×10−6/mPa) 39.897 49.896d2 (×10−6/mPa) 120.385 188.467m (×10−6/mPa) 31.407 67.246

0

10

20

30

40

50

60

70

80

0 10 20 30 40 50

C(t

, t') ×1

0-6(M

Pa-1

)

Time (d)

drying(RHenv=50%)

sealed

0

10

20

30

40

50

60

70

80

C(t

, t') ×1

0-6(M

Pa-1

)

Time (d)

1

0.61

1

0.33

drying(RHenv=50%)

sealed

0.01 0.1 1 10 100

(a)

(b)

Figure 6. Calculated flexural specific creep of concrete beams exposed to sealed and drying conditions: (a) in rectangular coordinates and (b) in semi-log coordinates.

INTERNATIONAL JOURNAL OF PAVEMENT ENGINEERING 435

Be aware that the viscoelastic analysis incorporating the basic creep behaviour may not be realistic, because there is always moisture exchange between the field concrete slab and the environment.

Drying creep reduces moisture warping in both bonded and unbonded slabs, and the effect of creep is more significant in bonded slabs. Figure 8 shows the simulated developments of moisture warping deformation (i.e. warping difference between slab corner and slab centre) at the top surface of concrete slabs. It can be seen that the moisture warping increases with time. The unbonded slab develops greater warping under all the three cases as compared to the bonded slabs. Figure 9 displays the comparison of three-dimensional warping at the age of 40 days. When the concrete slabs are bonded with the base, the moisture warping incorporating drying creep is about 64% of that in elas-tic analysis. This implies that about 36% of moisture warping is reduced because of the creep effect. While, only 25% of warping is reduced when concrete slab is unbonded with the base.

Figure 10 shows the developments of maximum warping stress in the longitudinal direction, which occurs at the centre of the slab top surface. Similar to warping deformation, creep effect is more significant in reducing warping stress in slabs bonded with base. For instance, the maximum warping stress

creep effect. Since the duration of flexural creep test was about 30 days, the duration of the simulation was set as the same value. Both elastic and viscoelastic analysis were performed in FEA. Three cases were conducted for comparison: elastic (with no creep), basic creep (no drying) and drying creep.

Figure 7.  Creep coefficient of the concrete beam under sealed and drying conditions.

Figure 8. Creep effect on moisture warping of concrete slab under the conditions of unbonded vs. bonded base.

Slab length (m) Slab width (m)

War

ping

def

orm

atio

n (m

m)

drying creep (unbonded)

no creep (unbonded)

no creep (bonded)

drying creep (bonded)

Figure 9. Comparison of 3D moisture warping deformation of concrete slab under different conditions.

Figure 10. Creep effect on warping stress at the top surface of concrete slab under the conditions of unbonded vs. bonded base.

436 S. LIANG AND Y. WEI

be noted that the creep only has significant effect on warping stress located around the slab centre where stress is pronounced. The difference between warping stresses with and without creep diminishes when it moves away from the slab centre, as shown in Figure 11.

4.3. Creep effect on total stress when traffic load involved

The total stress (when traffic loading is considered) contours in the longitudinal direction at the slab top surface for bonded and unbonded slabs are displayed in Figure 13. For the joint loading applied, the maximum tensile stress occurs near the slab edge. This is different from the location of the maximum warping stress that is located at the centre of the slab. Also, slab-base bonding type affects the magnitude of the total stress that greater stress is found in bonded slab in both no creep and with drying creep cases.

Figure 14 shows the creep effect on the maximum total stress under bonded and unbonded conditions. The results show that creep effect is significant in slabs bonded with the base, with about 34% of the total stress being relieved from drying creep effect as compared to the condition without creep, while only about 10% of the total stress is relaxed for slab unbonded with the base. Be aware that the creep effect is more significant in reducing warping stress than in reducing the total stress, the reason might attribute to the different locations of the maximum warping stress and the stress from traffic loading.

of concrete slab bonded with base is reduced by 45% by drying creep. However, the creep effect on reducing warping stress in unbonded slab is minor with only 15% of stress being relieved.

The distribution and magnitude of warping stress in concrete slab are affected by slab-base bonding type and creep behaviour, as the warping stress shown in Figure 11. The comparison of the maximum warping stress under bonded and unbonded condi-tions was shown in Figure 12. The stronger the constraints, the larger the warping stress is. Creep effect becomes more signifi-cant when the slab is bonded with the base. However, it should

Sx,max=1.21 MPa Sx,max=2.85 MPa

Sx,max=1.09 MPa Sx,max=1.93 MPa

(a) unbonded, no creep (b) bonded, no creep

(c) unbonded, with drying creep (d) bonded, with drying creep

Figure 11.  Warping stress contour at the top surface of concrete slabs: (a) unbonded pavement with no creep, (b) bonded pavement with no creep, (c) unbonded pavement with drying creep and (d) bonded pavement with drying creep.

Figure 12. Comparison of creep effect on the maximum warping stresses (Sx) at the top surface of concrete slab.

INTERNATIONAL JOURNAL OF PAVEMENT ENGINEERING 437

times of the sealed one. The creep rate under drying condition is also greater.

Creep reduces both warping deformation and stress in slabs bonded and unbonded with base. It was found that creep has no significant effect on the reduction of warping stress and total stress in the unbonded pavement. The most significant creep effect is observed in slab bonded with base, and the moisture warping and warping stress is reduced by 36 and 45%, respec-tively, as compared to the elastic stress. Therefore, it is of signif-icance to take into account the creep effect when conducting deformation and stress analysis in slabs exposed to drying environment.

AcknowledgementsThe authors wish to thank National Natural Science Foundation of China under [grant number 51578316] for the supports.

Disclosure statementNo potential conflict of interest was reported by the authors.

FundingThis work was supported by National Natural Science Foundation of China under [grant number 51578316].

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5. Conclusions

Differential drying shrinkage in slabs can cause pronounced warping which is equivalent to a negative temperature difference of up to −18 °C after 40 days of top surface drying and bottom surface wetting. The slab-base bonding type affects the warping deformation and stress developments. Under the same equiva-lent temperature difference (ΔTe), greater warping deformation is found in unbonded pavement, while greater warping stress is found in bonded pavement.

Flexural creep of concrete is significant under the drying conditions as compared to the sealed condition. After 40 days of flexural loading, the drying creep coefficient is about 1.67

Sx,max=2.16 MPa Sx,max=3.12 MPa

Sx,max=1.94 MPa Sx,max =2.05 MPa

(a) unbonded, no creep )b( bonded, no creep

(c) unbonded, with drying creep (d) bonded, with drying creep

Figure 13. total stress contour at the top surface of concrete slabs: (a) unbonded pavement with no creep, (b) bonded pavement with no creep, (c) unbonded pavement with drying creep and (d) bonded pavement with drying creep.

Figure 14. Comparison of creep effect on the maximum total stresses (Sx) at the top surface of concrete slab when combined with traffic loading.

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