Acta Polytechnica CTU Proceedings https://doi.org/10.14311/APP.2023.40.0104 Acta Polytechnica CTU Proceedings 40:104–110, 2023 © 2023 The Author(s). Licensed under a CC-BY 4.0 licence Published by the Czech Technical University in Prague HYGRO-MECHANICAL MODEL FOR CONCRETE PAVEMENT WITH LONG-TERM DRYING ANALYSIS Jakub Veselý∗, Vít Šmilauer Czech Technical University in Prague, Faculty of Civil Engineering, Department of Mechanics, Thákurova 7, 166 29 Prague 6, Czech Republic ∗ corresponding author: jakub.vesely.2@fsv.cvut.cz Abstract. Concrete pavements are subjected to the combination of moisture transport, heat transport and traffic loading. A hygro-mechanical 3D finite element model was created in OOFEM software in order to analyse the stress field and deformed shape from a long-term non-uniform drying. The model uses a staggered approach, solving moisture transfer weakly coupled with MPS viscoelastic model for ageing concrete creep and shrinkage. Moisture transport and mechanical sub-models are calibrated with lab experiments, long-term monitoring on D1 highway and data from 40 year old highway pavement. The slab geometry is 3.5 × 5.0 × 0.29 m, resting on elastic Winkler-Pasternak foundation. The validation covers autogenous and drying strain on the slab. The models predict drying-induced tensile stress up to 3.3 MPa, inducing additional loading on the slab, uncaptured by current design methods. Keywords: Concrete pavement, hygro-mechanical analysis, creep, shrinkage. 1. Introduction Concrete pavements present a proven solution for high- ways and airports due to its ability to withstand high mechanical loads with long service life compared to bituminous alternatives [1]. Effects of wheel load, temperature and moisture generally need to be con- sidered, as they influence stress, strain, deformations and service life [2]. For a detailed pavement analysis, finite element simulations with various factors have been carried out [3–7]. Hydration and moisture transport generally induces autogeneous shrinkage strain, drying shrinkage strain with an impact on creep. Early microcracks may occur due to internal or external restraints [8]. In ad- dition, moisture gradients generate differential shrink- age between the top and the bottom, leading to warp- ing [9, 10]. Stress relaxation plays a role on the warp- ing effect [11]. There are several approaches for creating a thermo- hygro-mechanical model for concrete. The most sim- ple one uses a superposition principle, solving mois- ture, thermal, traffic load and ASR effects sepa- rately [12]. Due to a coupled nature of the problem, a staggered solution strategy offers more accurate analysis [13], solving moisture and heat transport and passing the obtained fields to a mechanical model. Fully coupled problems with chemical submodels were proposed as well [14–16]. This paper presents a new hygro-mechanical 3D model, implemented in an open-source software OOFEM [17]. A staggered solution strategy uses solving of the weakly-coupled sub-models in the dis- cretized time steps. The geometry uses 3D repre- sentation of a single rectangular concrete slab rest- ing on Winkler-Pasternak foundation without dowel bars. The aim is to inspect overall performance of the concrete pavement under long-term drying loading scenario. A similar model has been published with limited calibration and interpretation [18]. 2. Formulation of a hygro-mechanical model Hygro-mechanical model solves transport of moisture and mechanical behavior of the slab. Staggered solu- tion strategy is adopted in the multiphysical model, partially separating transport transient problem from the mechanical one and passing transient moisture field to the mechanical problem. Such an approach allows to define equivalent time in creep models or to calibrate constitutive laws independently. The models were implemented in OOFEM, an open-source and object-oriented software for finite element method [17]. 2.1. Moisture transport A nonlinear moisture transport model describes con- crete as a single-fluid medium with the governing equation: k(h) ∂h ∂t = ∇ · [c(h)∇h] , (1) where h is the dimensionless pore relative humidity, k(h) [kg/m3] is the humidity-dependent moisture ca- pacity (k(h) = ∂w ∂h which is derivative of the moisture content w(h) [kg/m3] with respect to the relative hu- midity), c(h) [kg/m/s] is the moisture permeability in the Bažant-Najjar’s form: c(h) = c1  α0 + 1 − α0 1 + ( 1−h 1−hc )n   . (2) Newton (convection) boundary condition is used in the form qn = β(h − h∞) with the moisture transfer coefficient β. 104 https://doi.org/10.14311/APP.2023.40.0104 https://creativecommons.org/licenses/by/4.0/ https://www.cvut.cz/en vol. 40/2023 Hygro-mechanical model for concrete pavement E0 E1 E2 EM η1 η2 ηM solidification CS S σ σ S εe εv εf εsh εT ε ksh αT Figure 1. Rheological scheme of MPS model. 2.2. Mechanical model The mechanical model consists of three different com- ponents. The first one takes a concrete slab with a viscoelastic model for concrete with ageing based on microprestress solidification theory (MPS) [19]. The slab is placed on elastic, 2D subsoil, Winkler- Pasternak model. The interaction between the slab and the subsoil is controlled by interface elements. 2.2.1. Linear viscoelastic material model for creep and shrinkage A constitutive model for creep and shrinkage origi- nate from B3 model [20] extended with MPS formu- lation [19]. The rheological model in Figure 1 shows non-aging asymptotic elastic spring, solidifying Kelvin chain, aging dashpot, shrinkage strain, and thermal strain. The viscosity ηf in the flow strain εf depends on the evolution of the microprestress. All these units are connected in series, manifesting total strain decom- position into individual contributions. The governing equation for MPS theory reads [21]: η̇f + 1 µST0 ∣∣∣∣Ṫ ln h + ḣh ∣∣∣∣ (µSηf ) pp−1 = ψSq4 , (3) where µS is a parameter with the dimension of fluidity, p is a dimensionless material parameter influencing the size effect (for p = ∞, the size effect disappears), ψS is a temperature and humidity dependent factor, q4 is a material parameter, T is the current temperature, and T0 is the reference temperature. Equation (3) can be simplified into Equation (4) under the assumption of constant temperature T = T0 as: η̇f + k3 ∣∣∣∣ḣh ∣∣∣∣ ηp̃f = ψSq4 , (4) with parameters: p̃ = p p − 1 , (5) k3 = µ 1 p−1 S . (6) The drying shrinkage is computed as: ˙εsh,d = kshḣ, (7) with shrinkage ratio ksh. The autogenous shrinkage εsh,au is approximated as: εsh,au = ε∞sh,au [ 1 + (τau/te) w/c 0.38 ]Rt,au , (8) where ε∞sh,au is the ultimate value of autogenous shrink- age strain, w/c is water/cement ratio and parameters τau,Rt,au control strain evolution in time te. 2.2.2. Winkler-Pasternak subsoil model Elastic subsoil is treated as a 2D Winkler-Pasternak model, capturing normal and shear stiffness with c1 and c2 parameters [22–24]. The governing equation reads: f(z) = c1w(z) − c2 ∂2w(z) ∂z2 , (9) where f(z) is surface load and w(z) is displacement. 2.2.3. Interface elements The interface elements allow separation between the slab and subsoil, eliminating tension stress. Both meshes of slab and subsoil share coinciding nodes at the interface. The traction-separation law takes the form: tn = knδ, kn = k for compression, (10) kn = 0.01k for tension, where δ is displacement between two nodes, positive in separation. Shear stiffness is assumed zero. Interface elements lead generally to slower converge or even con- vergence loss hence it is important to use reasonable time step to induce gradual slab deformations. 3. Results and discussion The presented numerical model stems from a moni- tored highway slab. The pilot project started in 2017 as a joint activity among the Road and Motorway Di- rectorate (ŘSD ČR), a contractor Skanska, a.s. and the Czech Technical University in Prague [25]. The project involved 8 978 m of concrete pavement built on D1 highway Přerov-Lipník nad Bečvou, the Czech Republic. The pavement was cast between 07/2018 and 09/2019 with the opening to traffic Dec 12, 2019. The binder used a slag-blended, slow hardening binder composed of 75 % CEM I 42.5 R(sc) + 25 % GGBFS SMŠ 400, corresponding to CEM II/B-S 42.5 N. 105 Jakub Veselý, Vít Šmilauer Acta Polytechnica CTU Proceedings Figure 2. Vibrating wire gauges under protective covers. Detail at one assembled location. 3.1. Long-term monitoring system A long-term monitoring system was designed and in- stalled in one concrete pavement slab with dimensions of 3.5 × 5.0 × 0.29 m. The system records tempera- ture and strains at six measuring locations [25]. Each location contains three vibrating wire strain gauges located 50 mm from the surfaces and in the mid-height. All the gauges have integrated temperature sensors. In addition, one thermal gauge was placed 150 mm under the pavement in order to deliver the sub-base temper- ature. Ambient air temperature and solar radiation sensors were installed as well. Two-lift concrete technology started with a bottom layer 240 mm thick with 370 kg/m3 of the binder, fol- lowing with the top layer 50 mm thick with 420 kg/m3 of the binder. A two-step installation process was adopted, utilizing protective covers which hid the strain gauges before the casting, see Figure 2. After the first finisher had passed, the covers were removed, the gauges put in their positions and the empty space filled back with concrete using hand vibrators, see Figure 2. Finally, the top layer finalized the slab. Almost 55 000 m3 of concrete was placed in both layers, complying to required C30/37 strength class, see Figure 3. Vibrating strain gauges measure relative head dis- placements, which can be decomposed to: ε = εve + εT + εas + εds + εf + . . . , (11) where partial strains represent viscoelasticity, temper- ature effects, autogenous shrinkage, drying shrinkage, fracturing strain, etc. Figure 4 shows partial strains on the mid-plane, ze- roed at 2 hours after the end of setting for all gauges. Autogenous shrinkage plays a dominant role in the first week, reaching −70 µε in the transversal direction (W62). In the longitudinal direction, continuous cast- ing led to prestressing, which adds additional strain. A small average drying shrinkage strain from −30 to −120 µε is apparent after 3 years of drying. The measured strains allowed calculating the curva- ture of the slab, assuming a planar deformation of the cross-section. Figure 5 shows the total curvatures, cap- turing the temperature variations and demonstrating a slow positive drift due to top drying. 0 10 20 30 40 50 60 0 10 20 30 40 50 60 C u b e m e a n c o m p r. s tr e n g th ( M P a ) Concrete age (day) CEM 𝖨 42.5 R(sc) Mokrá, top layer bottom layer 75% CEM 𝖨 42.5 R(sc) + 25% slag SMŠ 400, top bottom 0 1 2 3 4 5 0 10 20 30 40 50 60 M e a n s p li t. t e n si le s tr e n g th ( M P a ) Concrete age (day) CEM 𝖨 42.5 R(sc) Mokrá, top layer bottom layer 75% CEM 𝖨 42.5 R(sc) + 25% slag SMŠ 400, top bottom Figure 3. Compressive and tensile strength evolution of top and bottom concretes. -250 -200 -150 -100 -50 0 50 20 18 /0 8/ 01 20 19 /0 8/ 01 20 20 /0 8/ 01 20 21 /0 8/ 01 20 22 /0 8/ 01 P a rt ia l st ra in , ε -ε T ( 1 0 -6 ) Location 1 Location 2 Location 3 (discarded) Location 4 Location 5 Location 6 Figure 4. Partial strains on 6 locations. -0.5 0.0 0.5 1.0 20 18 /0 8/ 01 20 19 /0 8/ 01 20 20 /0 8/ 01 20 21 /0 8/ 01 20 22 /0 8/ 01 To ta l c u rv a tu re , κ ( k m -1 ) Location 1 Location 2 Location 3 Location 4 Location 5 Location 6 Figure 5. Total curvatures at 6 locations. 106 vol. 40/2023 Hygro-mechanical model for concrete pavement Parameter Value Unit Parameter Value Unit M ec ha ni ca lm od el fc 45.0 MPa c1 70.0 MNm−3 cc cement content 370.0 kg/m 3 c2 60.0 MNm−1 w/c ratio 0.4 - α 10.0 · 10−6 °C−1 a/c ratio 4.91 - k 200.0 MNm−1 t0 7 days k3 35 - τ1 3 · 10−5 days τM 5000 days ksh 1.5 · 10−3 - E 35.0 GPa ε∞sh,au −87.0 · 10 −6 - ν 0.2 - τau 1.05 days Rt,au −4.5 - αau 1.166 - M oi st ur e tr an sf er m od el c1 6.0 · 10 −4 kg/m·s α0 1.0 - k(h) = k1 160.0 kg/m3 βtop 0.05 kg/m2 · d βbottom 0.01 kg/m2 · d Table 1. List of material input parameters. 3.2. Slab geometry The slab is meshed using 3D, quadratic, 20 node brick elements. Interaction between neighboring slabs is neglected. In moisture transfer simulation, the influence of air and soil is represented with Newton boundary conditions. The overview of used material parameters for concrete and subsoil is summarized in Table 1. Parameters ε∞sh,au,B4, τau, αau and Rt,au control the evolution of autogenous shrinkage. The basic creep in the MPS theory is influenced by the same four parameters q1–q4 as in the model B3, these are computed from fc, cc, w/c and a/c. The units of solidifying Kelvin chain are defined with the lowest τ1 and the highest retardation time τM . Time at the onset of drying is t0 = 7 days (for lab experiment t0 = 0.1 day), size effect for drying is neglected setting p = ∞. 3.3. Hygro-mechanical simulation of long-term slab drying To simulate the long-term drying, the determination of material properties of concrete and relevant boundary conditions are necessary. Material properties used in this paper are derived from drying shrinkage lab exper- iments. The kinetics of slab drying can be estimated from small lab samples 60 × 100 mm exposed to 50 % RH from the largest side. Permeability (and hygric exchange coefficient) was calibrated from moisture transport model. Boundary conditions are derived from a 40 year old concrete slab, which was removed during the demo- lition of D1 highway. The thickness of the old slab is only 240 mm compared to 290 mm used in current simulation. The moisture profile (see Figure 6) and the fully saturated state were determined. We assume that the drying process is stabilized after 40 years, which identified ambient relative humidity as 65 % at the top and 80 % at the bottom. Moisture transfer coefficients (βtop > βbottom) are set in such a way that the top drying becomes domi- nant. The linear moisture profile from Figure 6 also implies that the permeability needs to stay constant, otherwise non-linear profile is obtained. The explana- tion for this phenomenon is likely a sorption isotherm and full resaturation with rain water. It was showed on prisms, that rewetting is orders of magnitude faster process than drying, thus occasional rain will lead to linear moisture profile [26]. This leads to setting α0 = 1 and c(h) = c1 according to Equation (2). Figure 7 shows shrinkage validation of numerical model both for lab experiment and for concrete pave- ment (strain gauge W52). The strain from gauge W52 was shifted by 40 µε to eliminate longitudinal prestressing. The predictions show that the slab will be drying for approximately 30 years. It should be mentioned that other gauges show smaller mid-plane shrinkage values, attributed likely to other partial strains and restraints. The initial drying takes place predominantly in top and bottom 100 mm of the slab, see Figure 6. This leads to stress induction in these areas and subsequent redistribution in time. The early age stress field (first 2 years) shows tensile stress in the top part of the slab (around 3.3 MPa after 60 days and 2.2 MPa after 1 year, see Figure 6). This will likely cause cracking in the slab when combined with temperature and traffic load. Concrete slab should be able to cope with the resulting fracture process zone since the slab is contin- uously supported and controlled by displacement to a large extent. Similar stresses were found on the anal- ysis of restrained slab with thermo-hygro-mechanical model; exposing 150 mm thick slab to 60 % RH led to tensile stresses up to 3 MPa after 100 days [27]. 107 Jakub Veselý, Vít Šmilauer Acta Polytechnica CTU Proceedings 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.5 0.6 0.7 0.8 0.9 1.0 D ep th ( m ) Relative humidity (-) After 60 days After 1 year After 5 years After 15 years After 30 years Real slab, after 40 years 0.00 0.05 0.10 0.15 0.20 0.25 0.30 -3 -2 -1 0 1 2 3 σx (MPa) Figure 6. Comparison of relative humidity profile (left) from FE simulation with real slab data after 40 years and σx (right) induced by long term drying. -800 -600 -400 -200 0 100 101 102 103 104 Autogeneous shrinkageAutogeneous shrinkageAutogeneous shrinkage A u to g en ou s + d ry in g s tr ai n ( 1 0 -6 ) Time of hydration (days) Prism 100×60 mm, t0=0.1 d, RH=50% Experiment at BUT Slab 290 mm, t0=7 d, updated Monitoring, W52+40 με Figure 7. Shrinkage validation of concrete slab with monitoring. After 2 years, the compression occurs within the top layer and the tensile stress occurs in the middle of the slab. The tensile stress reaches values up to 1.0 MPa, these are values lower than tensile strength of the concrete. The compression of the top layer of concrete is likely to help close the cracks developed in the earlier phase. The deformed shape induced by long-term drying is known as warping. The simulation predicts that the corners will raise up up to 1.7 mm when subjected to drying and self-weight (see Figure 8). 4. Conclusions This paper describes behavior of road concrete slab when subjected to the long-term drying. Individual material models were calibrated based on lab and field experimental data, the main source being a long-term monitoring system on D1 highway. Shrinkage was validated both for lab experiment and for the concrete pavement with moisture transport model. Boundary conditions were calibrated from moisture profile of 40 year old pavement. Drying induces high tensile stress on top surface, reaching 3.3 MPa after 60 days and 2.2 MPa after 1 year of drying. As the drying front advances, the stress profile changes and the surface becomes slightly compressed. Slab warping is predicted as 1.7 mm corner uplift at the maximum. 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Engineering Structures 212:110493, 2020. https://doi.org/10.1016/j.engstruct.2020.110493 110 http://www.oofem.org https://doi.org/10.3390/ma13235530 https://doi.org/10.1016/j.cemconres.2014.03.008 https://doi.org/10.1007/BF02485965 https://doi.org/10.1016/j.cemconres.2016.04.001 https://doi.org/10.1016/j.trgeo.2021.100696 https://doi.org/10.1007/s11071-019-04977-9 https://doi.org/10.14311/APP.2022.34.0001 https://doi.org/10.1016/j.engstruct.2020.110493 Acta Polytechnica CTU Proceedings 40:104–110, 2023 1 Introduction 2 Formulation of a hygro-mechanical model 2.1 Moisture transport 2.2 Mechanical model 2.2.1 Linear viscoelastic material model for creep and shrinkage 2.2.2 Winkler-Pasternak subsoil model 2.2.3 Interface elements 3 Results and discussion 3.1 Long-term monitoring system 3.2 Slab geometry 3.3 Hygro-mechanical simulation of long-term slab drying 4 Conclusions Acknowledgements References