DOI:10.14311/APP.2021.30.0024 Acta Polytechnica CTU Proceedings 30:24–29, 2021 © Czech Technical University in Prague, 2021 available online at http://ojs.cvut.cz/ojs/index.php/app MODELING TIME-DEPENDENT DEFORMATIONS OF CONCRETE MEMBERS SUBJECTED TO SYMMETRIC AND ASYMMETRIC DRYING Pavel Horák∗, Petr Havlásek Czech Technical University in Prague, Faculty of Civil Engineering, Department of Mechanics, Thákurova 7, 166 29 Prague 6, Czech Republic ∗ corresponding author: pavel.horak@fsv.cvut.cz Abstract. The time-dependent behavior of concrete members with uniform cross-section subjected to symmetric drying can be predicted with one of many models anchored in the design codes. Since these models use the cross-sectional approach, only the average quantities can be obtained. More com- plex phenomena can be investigated by means of a coupled hygro-mechanical finite element analysis. Owing to insufficient or missing experimental data, the input parameters for the moisture transport model are often tuned to match the development of axial shrinkage or moisture loss measured on small-scale laboratory specimens. In this paper it is assessed whether this approach leads to a unique set of parameters of the Bažant-Najjar model for moisture diffusion. Additionally, for the selected sets of parameters the analysis is repeated with modified size and/or boundary conditions leading to asymmetric drying and the results are discussed. Keywords: Concrete, drying shrinkage, finite elements, moisture transport, parameter identification. 1. Introduction and motivation Creep and shrinkage are complex time-dependent phenomena in concrete whose magnitude and evolu- tion is influenced by many factors. The distinction between creep and shrinkage, which are both related to delayed strains, is that creep is caused by loading. At constant temperature the creep deformation can be additively decomposed into basic creep and dry- ing creep. Basic creep is the deformation of hygrally sealed concrete while drying creep is an additional deformation caused by drying. In a similar fashion, autogenous and drying shrinkage refer to the volume change (typically contraction) of sealed and drying concrete, respectively. In normal-strength structural concretes with w/c ≥ 0.4, the autogenous shrinkage is typically negligible (≈ 40 × 10−6) in comparison to the magnitude of drying shrinkage (≈ 780 × 10−6 at 40% relative humidity according to ACI 209). This time-dependent behavior of typical concrete members can be estimated using the prediction mod- els (e.g. B3, B4, fib Model Code) or the formulae from the design codes (e.g. ACI 209, Eurocode 2). How- ever, since these models operate on the cross-sectional level, only the average quantities can be obtained. In order to describe more complex phenomena, e.g. tensile cracking produced by stress gradients caused by nonuniform drying or the structural response to non-symmetric drying, a more versatile approach is a necessity. One of the options is a coupled hygro-mechanical finite element analysis. The bottleneck of this ap- proach resides in often insufficient number of exper- iments which are necessary for proper calibration of the complex constitutive models. To overcome this obstacle, certain parameters are set to their recom- mended values or the parameters are adjusted to re- produce the observed macroscopic behavior. To illus- trate this, in the recent paper [1], this approach was used by several research groups to calibrate the trans- port model based on the axial shrinkage and moisture loss of a standard laboratory specimen without any information on the distribution of relative humidity. The primary objective of this paper is to assess whether such approach yields a unique set of material parameters and what macroscopic measurements can narrow the range of the identified values. 2. Methodology All results presented further in this paper were ob- tained using a one-way coupled hygro-mechanical simulations in open-source finite element package OOFEM [2]. The constitutive models described in more detail in the following sections were calibrated on creep and shrinkage data reported by Bryant [3]. The parameters of the structural material model are then fixed while the parameters in the model for mois- ture transport are subject to optimization under var- ious assumptions. The identified parameters are con- sequently utilized for predicting the curvature of an infinite slab subjected to asymmetric drying. 2.1. Material models Concrete drying is in the present study described by a widely recognized model proposed by Bažant and Najjar [4]. The governing equation for the diffusion of water vapor was derived under the assumption of 24 http://dx.doi.org/10.14311/APP.2021.30.0024 http://ojs.cvut.cz/ojs/index.php/app vol. 30/2021 Modeling of Concrete Subjected to Symmetric and Asymmetric Drying constant moisture capacity k (slope of the desorption isotherm). When the moisture sink is neglected the equation reads ∂h ∂t = ∇ · (C(h)∇h) (1) where ∇h is the gradient of relative humidity and C(h) is the humidity-dependent diffusivity. This de- pendence is for the cementitious materials highly non- linear and can be approximated as C(h) = C1 ! "α0 + 1 − α0 1 + # 1−h 1−hc $n % & (2) where C1 is the maximum diffusivity at h = 1, α0 determines the ratio between minimum diffusivity at h = 0 and C1, and parameters hc and n de- scribe the relative humidity and the rate of the tran- sition. This model was embedded into the fib Model Code 2010 [5] which recommends the following values α0 = 0.05, hc = 0.8, n = 15, while C1 is to be deter- mined from the compressive strength. In the simula- tions, the initial value of relative humidity is set to h = 0.98 which is the typical value for the sealed con- ditions and normal-strength concrete with negligible self-desiccation. The ambient relative humidity is ide- alized as constant and is prescribed using the mixed boundary condition which uses the surface factor f to relates the humidity flux Jh with the difference be- tween the relative humidity at the boundary and the environment Jh = f · (henv − h) (3) In each time step the moisture transport is followed by the structural sub-problem which utilizes the com- puted field of relative humidity. The constitutive vicoelastic model is the modified [6] Microprestress- solidification (MPS) theory [7]. Under constant tem- perature and hygrally sealed conditions the mate- rial model reduces to the B3 model [8]. Changes in relative humidity give rise to additional compli- ance, referred to as drying creep or the Pickett effect, and to volume changes–shrinkage or swelling. The model has been modified to minimize the size-effect on drying creep which is controlled by parameter k3. Shrinkage strain and relative humidity are linearly linked via their rates, ε̇sh = kshḣ (4) where ksh is a material parameter usually treated as humidity- and age-independent constant. The model can be enhanced by tensile softening but this feature is not activated in the present study. 2.2. Modeling strategy Despite the unidirectional nature of the transport problem, the analysis was performed in two dimen- sions to provide the subsequent structural analysis with the distribution of relative humidity. The me- chanical problem was carried out under the assump- tion of plane stress which allowed to run the com- putations very efficiently and repeatedly as described later. The assumption of plane stress is very realistic in the case of the experimental data. The parametric study is done on an idealized plane section cut out from an “infinite slab” drying from one or two sur- faces. Yet, the model which is hereafter referred to as “slab”, neglects the stresses perpendicular to the section and its behavior corresponds to a beam with prismatic cross-section, sealed lateral sides and dry- ing from top and/or bottom surfaces. The effect of self weight is omitted in the study. The finite ele- ment meshes of the two sub-problems were identical and were composed of 60 quadrilateral elements with linear approximation of humidity and displacement. The meshes were refined near the drying surfaces. All problems were solved in 284 time steps with increas- ing length. As mentioned earlier, the material parameters were calibrated on experimental data by Bryant and Va- hanavikkit [3]. Their study was focused on creep and shrinkage of structural concrete (fc = 50.1 MPa, w/c = 0.47) measured on prismatic members with cross-section ranging from 100×100 to 400×400 mm. The specimens were either entirely sealed to investi- gate basic creep or partially sealed to study drying creep and shrinkage. Drying at 60% relative humidity began at the age of 8 days. The majority of the creep tests imposing 7 MPa compression started at the age of 14 days. Additional tests were done to identify aging. All loaded specimens had a companion with- out external loading. The constant room temperature was maintained throughout the experiment. Calibration began with the basic creep compliance function. The parameters q1 − q4 of the B3 model were adopted from [9] and their values were q1 = 9, q2 = 75, q3 = 28 and q4 = 6.5 all in 10−6/MPa. Drying creep and shrinkage was calibrated on exper- imental data of prism 150 × 150 mm drying from two lateral sides (i.e. 150 mm thick infinite wall). This specimen is the largest one which has reached the ultimate values and for this reason was selected as reference. With crudely calibrated parameters of the transport model the shrinkage constant was identi- fied as ksh = 1.9 × 10−3, and drying creep parameter k3 = 25. The parameters of the Bažant-Najjar model and the surface factor were identified using a brute-force method implemented in a computer program. The re- peatedly run analysis used the values with the ranges and stepping summarized in Table 1 which makes al- together 31 360 combinations. The accuracy of the fit was assessed by means of a weighted error which considered the drying period on the log scale. The reference set of parameters which gave the best fit of with experimental data is listed in Table 1 (see solid 25 Pavel Horák, Petr Havlásek Acta Polytechnica CTU Proceedings 0 100 200 300 400 500 600 700 800 0.1 1 10 100 1000 10000 t0 = 8 d, henv = 60% Calibration: experimentA xi al s hr in ka ge [1 0- 6 ] Duration of drying [day] ref exp 2 3 4 5 6 7 8 9 10 Figure 1. Best fits of axial shrinkage measured by Bryant and Vadhanavikkit [3] on 150 mm thick slabs, t0 = 8 days, henv= 60%. red line in Figs. 1 and 2). This set was later used to define the reference behavior under a modified setup (i.e. different size or boundary conditions). In the parametric study the analysis starts at the age of 28 days and the ambient relative humidity is set to henv= 50% with the intention to minimize the effect of aging and to comply with the standard lab- oratory conditions. Furthermore, as demonstrated in the following section, it was attempted to nar- row the identified set of parameters by incorporating additional reference data: moisture loss (assuming k = 100 kg/m3), response to drying at different levels of relative humidity (30% and 70%) and shrinkage of a symmetrically drying slab with different thickness (75 mm and 300 mm). Moreover, a model of 75 mm thick slab drying from one surface only is used to assess the influence of the identified diffusivity func- tions on the structural behavior. This model has the same effective thickness (and thus drying kinetics) but turns out to be significantly more sensitive to the shape of the diffusivity function. C1 α0 hc n f [mm2/day] [-] [-] [-] [mm/day] min. 35 0.02 0.68 6 1.1 max. 80 0.23 0.80 20 2.3 step 5 0.03 0.02 2 0.2 ref. 60 0.05 0.74 0.06 1.5 Table 1. Parameters of the transport problem: ranges, stepping for identification and the reference set. 3. Results and discussion The solution obtained with the reference set of mate- rial parameters (Tab. 1) is in the following figures shown consistently in thick red lines. As demon- strated in Fig. 1, the axial shrinkage measured by 0 10 20 30 40 50 60 70 80 0.4 0.5 0.6 0.7 0.8 0.9 1 t0 = 8 d, henv = 60% Calibration: experimentD i� us iv ity [m m 2 / da y] Relative humidity [-] ref, f = 1.5 2, f = 1.5 3, f = 1.7 4, f = 1.7 5, f = 1.5 6, f = 1.5 7, f = 1.5 8, f = 1.3 9, f = 1.9 10, f = 1.7 henv Figure 2. Dependence of diffusivity on relative hu- midity for the 10 best fits in Fig. 1. In the legend f is the surface factor [mm/day]. -1000 0 1000 2000 3000 4000 5000 6000 0.01 0.1 1 10 100 1000 10000 t0 = 8 d, henv = 60% Calibration: experiment C ur va tu re [1 0- 6 / m ] Duration of drying [day] ref 2 3 4 5 6 7 8 9 10 Figure 3. Prediction of curvature of 75 mm thick slab exposed to drying from one side only at henv= 50% which started at the age t0 = 28 days. The line types and the material parameters correspond to Figs. 1 and 2. Bryant on prism 150 × 150 mm drying from two op- posite surfaces is with very similar accuracy captured by numerous combinations of material parameters of the diffusivity function. Figure 1 presents only the 10 best fits, the respective functions are shown in Fig. 2 and the legend lists the values of the surface factor f [mm/day]. From the fits, it can be concluded that the evolution of axial shrinkage is almost insensitive to the diffusivity at high relative humidity (h > 0.8) where the values span from 45 to 75 mm2/day, but becomes very sensitive to the diffusivity in the region just above the ambient relative humidity henvwhere the functions form a narrow band. The response of a slab with the same effective thick- ness exposed to drying from one side only at standard laboratory conditions (t0 = 28 days, henv= 50%) is for the best 10 identified combinations of diffusivity shown in Fig. 3. In contrast to symmetric drying, the computed response for asymmetric drying be- 26 vol. 30/2021 Modeling of Concrete Subjected to Symmetric and Asymmetric Drying 0 10 20 30 40 50 60 70 80 0.4 0.5 0.6 0.7 0.8 0.9 1 t0 = 28 d, henv = 50% Calibration: simulationD i� us iv ity [m m 2 / da y] Relative humidity [-] ref, f = 1.5 2, f = 1.5 3, f = 1.5 4, f = 1.5 5, f = 1.5 6, f = 1.5 7, f = 1.5 8, f = 1.5 9, f = 1.7 10, f = 1.5 henv Figure 4. Diffusivity functions giving the best agree- ment with the axial shrinkage from the simulation of symmetrically drying concrete slab (D = 150 mm, henv= 50%, t0 = 28 days) computed with the refer- ence set of material parameters. -1000 0 1000 2000 3000 4000 5000 6000 0.01 0.1 1 10 100 1000 10000 t0 = 28 d, henv = 50% Calibration: Simulation C ur va tu re [1 0- 6 / m ] Duration of drying [day] ref 2 3 4 5 6 7 8 9 10 Figure 5. Prediction of curvature of 75 mm thick slab exposed to drying from one side only at henv= 50% and at the age t0 = 28 days. The line types and the material parameters correspond to Fig. 4. comes surprisingly different. In all presented cases, the peak curvature is reached after ≈ 20 days of dry- ing, and the individual solutions differ by up to 20% of the reference solution. In several cases the cur- vature exhibits a second local maximum one order of magnitude later (≈ 200 days). The value at the first peak is tightly connected with the diffusivity at high relative humidity, the higher this diffusivity, the lower the curvature and vice versa (compare e.g. the dashed blue and black lines #8 and #9). The higher this diffusivity the lower is the gradient of relative hu- midity between the sealed and drying surface of the cross-section and thus the lower is the resulting cur- vature. The second peak appears when the diffusivity function has a steep transition between high and low relative humidity which is controlled by parameter n in (2). The preceding observations are confirmed in Fig- 0 200 400 600 800 1000 1200 1400 0.1 1 10 100 1000 10000 t0 = 28 d Calibration: simulation henv = 50% henv = 30% henv = 70% henv = 30, 50, 70% A xi al s hr in ka ge [1 0- 6 ] Duration of drying [day] ref 2 3 4 5 6 7 8 9 10 Figure 6. Best fits of (reference) axial shrinkage caused by symmetric drying at three levels of ambient relative humidity henv= 30, 50, 70% (D = 150 mm, t0 = 28 days). 0 10 20 30 40 50 60 70 80 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 t0 = 28 d Calibration: simulation henv = 30, 50, 70% D i� us iv ity [m m 2 / da y] Relative humidity [-] ref, f = 1.5 2, f = 1.5 3, f = 1.7 4, f = 1.3 5, f = 1.3 6, f = 1.7 7, f = 1.5 8, f = 1.3 9, f = 1.5 10, f = 1.7 Figure 7. Diffusivity functions giving the best agree- ment with the axial shrinkage from the reference sim- ulations (symmetric drying at three levels of ambient relative humidity henv= 30, 50, 70%, D = 150 mm, t0 = 28 days). ures 4 and 5 where the calibration was done on the data from simulation with reference parameters but at “laboratory conditions” (t0 = 28 days, henv= 50%). The identified diffusivity functions are very close to the reference case in the region from henv to henv+ 10%–15%, while beyond this limit the func- tions are distinctively different which has an impact on the evolution of curvature. Incorporating weight loss does not narrow the scat- ter of the identified diffusivity functions (figures are not presented here). From the findings presented above it is natural to expect that if the separate measurements of axial shrinkage are performed at different levels of rela- tive humidity, then the identified diffusivity functions will be closer to each other in the relative humidity slightly above the prescribed levels. To validate this statement, 2 additional values of relative humidity 27 Pavel Horák, Petr Havlásek Acta Polytechnica CTU Proceedings -1000 0 1000 2000 3000 4000 5000 6000 7000 0.01 0.1 1 10 100 1000 10000 t0 = 28 d, henv = 30, 50, 70% Calibration: Simulation C ur va tu re [1 0- 6 / m ] Duration of drying [day] ref 2 3 4 5 6 7 8 9 10 Figure 8. Prediction of curvature of 75 mm thick slab exposed to drying from one side only at henv= 50% which started at t0 = 28 days. The line types and the material parameters correspond to identification at 3 different levels of henv(Figs. 6 and 7). 0 200 400 600 800 1000 0.1 1 10 100 1000 10000 t0 = 28 d, henv = 50% Calibration: simulation 150 mm 75 mm 300 mm A xi al s hr in ka ge [1 0- 6 ] Duration of drying [day] ref 2 3 4 5 6 7 8 9 10 Figure 9. Best fits of (reference) axial shrinkage of 75 mm, 150 mm, and 300 mm thick slabs exposed to symmetric drying at henv= 50% (t0 = 28 days). 0 10 20 30 40 50 60 70 80 0.4 0.5 0.6 0.7 0.8 0.9 1 t0 = 28 d, henv = 50% Calibration: simulation D i� us iv ity [m m 2 / da y] Relative humidity [-] ref, f = 1.5 5, f = 1.5 6, f = 1.5 8, f = 1.5 Figure 10. Diffusivity functions giving the best agreement with the reference simulations of axial shrinkage of symmetrically drying slabs with various thicknesses (henv= 50%, t0 = 28 days, Fig. 9). 0 10 20 30 40 50 60 70 80 0.4 0.5 0.6 0.7 0.8 0.9 1 t0 = 28 d, henv = 50% Calibration: reverse D i� us iv ity [m m 2 / da y] Relative humidity [-] ref, f = 1.5 2, f = 1.5 3, f = 1.5 4, f = 1.5 5, f = 1.5 6, f = 1.5 7, f = 1.7 8, f = 1.5 9, f = 1.5 10, f = 1.5 Figure 11. Diffusivity functions calibrated in- versely on the evolution of curvature of 75 mm thick slab drying from one side (henv= 50, t0 = 28 days) obtained from the simulation with the reference set of material parameters. were added and the analysis was done at 30%, 50% and 70%. The 10 best fits are almost indistinguish- able from the reference solution of axial shrinkage (see Fig. 6) and indeed, the identified diffusivity functions form a narrow band which widens with increasing rel- ative humidity (Fig. 7). This has a direct impact on the computed curva- ture shown in Fig. 8 where the scatter in the first peak persists (caused by high scatter of the maximum diffusivity), but beyond 100 days of drying the agree- ment between the solutions becomes significantly bet- ter. Next, it was attempted to identify the diffusivity function based on the shrinkage evolution of speci- mens (i.e. models) with different effective sizes. The reference thickness D = 150 mm was doubled to 300 mm and reduced to 75 mm. The 10 best fits match the solution with reference parameters per- fectly (Fig. 9), yet the diffusivity functions show con- siderable scatter, see Fig. 10. Also it needs to be emphasized that such approach can turn out to be very demanding on the experimental duration since the evolution scales with D2. To summarize, and as illustrated in Figs. 12 and 13 it turned out that the shrinkage of symmetrically drying specimen is significantly less sensitive to the diffusivity function compared to the curvature pro- duced by non-symmetric drying. For this reason the diffusivity identified from axial shrinkage should be used with caution and to members with similar size and boundary conditions. On the other hand the dif- fusivity identified from the evolution of curvature (see Fig. 11) exhibits significantly less scatter and so does the back-calculated axial shrinkage which matches the reference case perfectly (not presented here). 28 vol. 30/2021 Modeling of Concrete Subjected to Symmetric and Asymmetric Drying 0 10 20 30 40 50 60 70 80 0.4 0.5 0.6 0.7 0.8 0.9 1 t0 = 28 d, henv = 50% C1 60, alpha0 0.05, hc 0.74, n 6, f 1.5 C1 75, alpha0 0.10, hc 0.80, n 16, f 1.1 D i� us iv ity [m m 2 / da y] Relative humidity [-] Reference, f = 1.5 Modi�ed, f = 1.1 henv Figure 12. Diffusivity function with the “reference” and “modified” set of material parameters. 4. Conclusions This paper investigated drying shrinkage of concrete slabs exposed to symmetric and asymmetric drying at constant ambient relative humidity. The problem was approached by means one-way coupled finite element simulations in program OOFEM. The moisture trans- port was described using the the Bažant-Najjar model while the time-dependent behavior of concrete by the modified MPS model. The emphasis was on put on identification of material parameters of the model for moisture transport and their uniqueness. The follow- ing conclusions can be drawn from the analysis: • Identification of parameters based the evolution of axial shrinkage of symmetrically drying concrete does not yield unique set of material parameters. The reference data could be reproduced with var- ious sets of parameters with accuracy similar or better than the typical experimental scatter. • Incorporating the shrinkage data measured at dif- ferent levels of relative humidity helps to narrow the identified diffusivity functions. This is not the case of the data on moisture loss or shrinkage of specimens with different size. • The behavior of concrete drying non-symmetrically is considerably more sensitive to the values of ma- terial parameters. The parameters identified from the evolution of curvature give more consistent dif- fusivity functions and result into almost identical axial shrinkage. • The presence and the character of the second maxi- mum in the evolution of curvature is most sensitive to the exponent n which defines the steepness of the transition between small and high diffusivity. • The numerical simulations of concrete drying from one side only exhibit very little sensitivity to con- crete creep (Fig. 13). -200 0 200 400 600 800 1000 1200 0.01 0.1 1 10 100 1000 10000 -1000 0 1000 2000 3000 4000 5000 6000 t0 = 28 d, henv = 50% A xi al s hr in ka ge [1 0- 6 ] C ur va tu re [1 0- 6 / m ] Duration of drying [day] Reference Modi�ed Figure 13. Evolution of axial shrinkage and cur- vature obtained with the “reference” and “modified” set of material parameters of the diffusivity function (Fig. 12) and D = 150 mm, t0 = 28 days, henv= 50%. Thin lines correspond to the behavior without vis- coelastic effects. Acknowledgements The authors gratefully acknowledge financial support from the Grant Agency of the Czech Technical University in Prague, project number SGS20/038/OHK1/1T/11, and from the Czech Science Foundation (GA ČR), project number 19-20666S. References [1] V. Šmilauer, P. Havlásek, T. Gasch, et al. Hygro-mechanical modeling of restrained ring test: COST TU1404 benchmark. Construction and Building Materials 229:116543, 2019. doi:https://doi.org/10.1016/j.conbuildmat.2019.07.269. [2] B. Patzák. OOFEM home page, 2000. http://www.oofem.org. [3] A. H. Bryant, C. Vadhanavikkit. Creep, shrinkage-size, and age at loading effects. ACI Materials Journal 84:117–123, 1987. [4] Z. P. Bažant, L. J. Najjar. Nonlinear water diffusion in nonsaturated concrete. Materials and Structures 5:3–20, 1972. doi:10.1007/BF02479073. [5] Fédération Internationale du Béton. Model Code 2010. No. vol. 65 in fib Bulletin. International Federation for Structural Concrete (fib), 2012. [6] Z. Bažant, P. Havlásek, M. Jirásek. Microprestress-solidification theory: Modeling of size effect on drying creep. In N. Bicanic, H. Mang, G. Meschke, R. de Borst (eds.), Computational Modelling of Concrete Structures, pp. 749–758. CRC Press/Balkema, EH Leiden, The Netherlands, 2014. [7] Z. P. Bažant, A. P. Hauggaard, S. Baweja, F. J. Ulm. Microprestress solidification theory for concrete creep. I: Aging and drying effects. Journal of Engineering Mechanics 123:1188–1194, 1997. 29 https://doi.org/https://doi.org/10.1016/j.conbuildmat.2019.07.269 http://www.oofem.org https://doi.org/10.1007/BF02479073 Pavel Horák, Petr Havlásek Acta Polytechnica CTU Proceedings [8] Z. Bažant, S. Baweja. Creep and shrinkage prediction model for analysis and design of concrete structures: Model B3. Adam Neville Symposium: Creep and Shrinkage - Structural Design Effects 2000. [9] P. Havlásek. Creep and Shrinkage of Concrete Subjected to Variable Environmental Conditions, PhD. Thesis. Czech Technical University in Prague, 2014. 30