DOI:10.14311/APP.2021.30.0001 Acta Polytechnica CTU Proceedings 30:1–6, 2021 © Czech Technical University in Prague, 2021 available online at http://ojs.cvut.cz/ojs/index.php/app SIZE EFFECT ON THE ULTIMATE DRYING SHRINKAGE OF CEMENT MORTAR: 1-YEAR EXPERIMENT AND NUMERICAL MODELING Lenka Dohnalová∗, Petr Havlásek, Vít Šmilauer, Pavel Reiterman, Vendula Davidová Czech Technical University in Prague, Faculty of Civil Engineering, Department of Mechanics, Thákurova 7, 166 29 Prague 6, Czech Republic ∗ corresponding author: lenka.dohnalova@fsv.cvut.cz Abstract. The magnitude and time evolution of shrinkage are influenced by numerous factors which are implemented in the design codes often in a different way. The time-dependent behavior of concrete in structures sensitive to creep and shrinkage should be verified by means of short-term laboratory measurements. Extrapolation of drying shrinkage from short-term measurements is an ill- posed problem. The process is extremely slow but can be accelerated by reducing the specimen size. The knowledge of the size-effect on drying shrinkage is a necessity to establish the transition from the laboratory to the structural size. In the literature, the experimental data on such size-effect are insufficient. For this reason a new experiment was developed to study this phenomenon on small-scale specimens made of cement mortar and the results from the first year are summarized in this paper. The measured data are validated by coupled FEM hygro-mechanical simulations. Keywords: Concrete, drying, modeling, shrinkage, size effect. 1. Introduction Creep and shrinkage of concrete are complex time- dependent processes whose physical origin remains still fully unresolved. The importance of both phe- nomena and the need for their correct prediction grew with the boom of pre-stressed concrete technology which allowed to design slender structures with larger spans and which allowed to utilize concrete poten- tial economically. At the same time, such challenging structures are generally more sensitive to the time- dependent deformations in concrete. The magnitude and time evolution of shrinkage is influenced by numerous factors (ranging from con- crete composition to ambient conditions) which are not approached consistently among current predic- tion models and design codes for concrete. To name a few, the subtle interplay between the development of drying shrinkage and drying creep (investigated e.g. in [1]) can determine whether the durability of con- crete structure will be jeopardized by tensile cracking or the shrinkage-induced tensile stresses are relieved sufficiently fast by increased compliance. A disagree- ment among the different models prevails also in the question of the size-effect on the ultimate value of drying shrinkage [2] which is addressed in this paper. The resulting discrepancies between the design codes give rise to many uncertainties which can drastically affect the requirements on the design of shrinkage- and creep-sensitive structures. The pre- diction of the behavior of such structures can be improved by laboratory measurements on specimens prepared from the same concrete composition. This approach can be successfully utilized to specify the estimated concrete creep based on short-time mea- surements. However, the direct applicability to drying shrink- age is rather limited because the extrapolation presents an ill-posed problem. The characteristic time and the ultimate shrinkage cannot be deter- mined uniquely until the rate plotted in log-scale significantly decreases [3]. In the case of standard laboratory specimens 70 × 70 × 285 mm or 100 × 100 × 500 mm, a subtle decrease in shrinkage rate (in log scale) occurs after approx. 100 days while the almost ultimate shrinkage strain is reached after ap- prox. 1000 days of drying [4]. The size-effect on drying shrinkage is in a cer- tain way introduced in majority of design codes or models for time-dependent behavior of concrete. The origin of the size-effect can be sought in non- uniform distribution of relative humidity over the cross-section which gives rise to non-uniform shrink- age strains. The shrinkage strains are internally re- strained which produces highly nonlinear distribution of self-equilibrated stresses. Owing to the presence of these stresses in the early stages of drying, con- crete is extremely prone to surface cracking. The generated stresses become redistributed as the con- crete dries and their magnitude is gradually relieved by concrete creep. This complicated interplay causes that the value of ultimate drying shrinkage decreases with increasing size of the concrete member. If the size-effect on the ultimate shrinkage is known, the time-demanding experiments can be sig- nificantly accelerated when the dimensions of a spec- 1 http://dx.doi.org/10.14311/APP.2021.30.0001 http://ojs.cvut.cz/ojs/index.php/app L. Dohnalová, P. Havlásek, V. Šmilauer et al. Acta Polytechnica CTU Proceedings Figure 1. Images of the experimental setup: frames for automated (left) and manual (right) readings. imen become reduced. The rate of drying and thus (drying) shrinkage scales with the square of the ef- fective thickness D defined in B3 model [5] as 2V /S where V is the volume of the specimen and S is the surface exposed to drying. For example in the case of an infinite prism with square cross-section, D co- incides with half of its thickness. In this paper, pa- rameter D is used to express the specimen size. The comprehensive creep and shrinkage database assembled at the Northwestern University [6] com- prises a remarkable number of experimental results gathered from the entire world. However, the promis- ing content of this database becomes rather limited when only a specific phenomenon is to be identi- fied. Only a small number of experimental studies ex- amined drying shrinkage of specimens with different sizes prepared from the same concrete mixture and subject to identical conditions. This selection needs to be reduced even further because in many cases the experiment was terminated prematurely before the ultimate shrinkage was reached. A recent paper by the present authors [2] at- tempted to evaluate the aforementioned size-effect based on the data available in the literature and the aforementioned database. The experiments unani- mously indicate that the ultimate drying shrinkage decreases with specimen size, but this relationship cannot be evaluated more accurately owing to insuf- ficient relevant experimental data. The lacking data stimulated the present authors to design and execute a new narrow-oriented exper- iment focused directly on the size-effect on drying shrinkage. This paper presents and summarizes the measured data collected within the first year of the experiment and compares the strain development in time with the results obtained from coupled hygro- mechanical finite element simulations. 2. Materials and methods 2.1. Experimental setup The proposed setup for the shrinkage and moisture loss measurement comprised a large variety of pris- matic specimens with uniform length 400 mm but with cross-section ranging from 20 × 20 mm (D = 10 mm) to 100 × 100 mm (D = 50 mm). In or- der to guarantee material homogeneity and to obtain representative data even on the smallest specimens, cement mortar with 2 mm maximum aggregate size was preferred at the expense of concrete. The composition of the mortar mix had stan- dard sand-to-cement ratio 3:1 by mass. Stan- dard CEN siliceous sand and the blended binder CEM II 32.5 B-S with slag content 29% were used. The typical water-to-cement ratio was reduced from w/c = 0.5 to 0.45 to match the behavior of mortar in a typical structural concrete. Compressive strength and Young’s modulus measured at the age of 33 days on standard cylinders 100 × 200 mm were 33.9 MPa and 40.9 GPa, respectively. The slender prisms were cast into a custom-built steel formwork with plastic bottom. The goal was to resemble curing in sealed conditions. For this reason the top was covered with 1 mm thick PE sheet fol- lowed by a damp cloth. The following day the spec- imens were transported to a sealed container with henv = 96%. The specimens were demolded at the age of 31 days when the drying began. The ends of the prisms were equipped by metal sheets with a central dent to pro- vide suitable surface for the measurement as well as moisture sealing. The reference measurement was done once the glue had hardened. Altogether 11 mortar prisms with one companion steel specimen for temperature compensation were placed in a special measuring frame, see Fig. 1 left. Each specimen rested on a bolt with round head. Axial shortening was measured at the top using lin- ear transducer with internal spring return MMR10- 12 (effective stroke 12.7 mm, total resistance 10 kΩ). Data logging occurred automatically, the setup did not allow to measure the moisture loss due to drying. For this reason, a second group consisting of 15 specimens was measured in a more conventional ap- proach using stiff measuring frame equipped with dig- ital indicator Sylvac with 1 µm precision, see Fig. 1 right. This setup allowed to determine also the evap- orated water which was recorded using precision bal- ance scales; KERN 572-39 (range 4.2 kg, readout 0.01 g) and T-scale (range 15 kg, readout 0.1 g) for sample with cross-section 100 × 100 mm. Minimum difference between the coefficient of thermal expan- sion of steel and mortar allowed relative reading, cir- cumventing temperature compensation. The experiment was executed in a laboratory with- out humidity and temperature control. In order to take the inevitable fluctuations into considera- tion, both temperature and relative humidity were recorded. The frequency corresponded to the auto- matic and manual measurements. 2.2. Numerical modeling The consistency of the measured data was verified using one-way coupled hygro-mechanical simulations 2 vol. 30/2021 Size effect on the drying shrinkage 0 200 400 600 800 1000 1200 1400 400 1 10 100 0.3 0.4 0.5 0.6 0.7 0.8 0.9 humidity Sh ri nk ag e, ε sh [ × 1 0- 6 ] A m bi en t h um id ity , h en v Drying duration, t - t0 [day] D = 10 mm D = 13 mm* D = 15 mm D = 20 mm* D = 20 mm D = 25 mm D = 30 mm D = 50 mm 0 200 400 600 800 1000 1200 400 1 10 100 0.3 0.4 0.5 0.6 0.7 0.8 0.9 humidity Sh ri nk ag e, ε sh [ × 1 0- 6 ] A m bi en t h um id ity , h en v Drying duration, t - t0 [day] D = 10 mm D = 13 mm* D = 15 mm D = 20 mm* D = 20 mm D = 25 mm D = 30 mm D = 50 mm Figure 2. Evolution of shrinkage measured automatically (left) and manually (right). The shaded area corresponds to the experimental range in the case more identical specimens. Asterisk denotes specimens with rectangular cross- section (square cross-section otherwise). 0 200 400 600 800 1000 1200 1400 400 1 10 100 0.3 0.4 0.5 0.6 0.7 0.8 0.9 humidity Sh ri nk ag e, ε sh [ × 1 0- 6 ] A m bi en t h um id ity , h en v Drying duration, t - t0 [day] D = 10 mm D = 15 mm D = 20 mm D = 25 mm D = 30 mm D = 50 mm Figure 3. Comparison of the average shrinkage strain measured automatically (solid lines) and man- ually (symbols). The two techniques gave good agree- ment during the first 6 months but afterwards started to diverge which was attributed to the deformation of the plywood back of the shrinkage frame for auto- mated measurements. in open-source finite element package OOFEM [7]. Concrete drying is, in the present study, described by a widely recognized model proposed by Bažant and Najjar [8]. Under certain simplified assumptions, the governing equation for the diffusion of water vapor 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 describe the relative humidity threshold and the rate of the tran- sition. The ambient relative humidity is prescribed using mixed boundary condition which relates the humidity flux Jh with the humidity difference at the boundary via surface factor f . In each time step, the moisture transport is fol- lowed by the structural sub-problem which utilizes the computed field of relative humidity. The mechan- ical problem uses constitutive model based on the Microprestress-solidification (MPS) theory [9]. Un- der constant temperature and sealed conditions, the MPS model reduces to the B3 model [5]. However, changes in relative humidity give rise not only to volume changes (shrinkage or swelling), but also to additional creep (Pickett effect). The model has been modified [10] 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 (3) where ksh is a material parameter usually treated as a humidity- and age-independent constant. The MPS model is extended with the cohesive crack model and fixed orientation of cracks to consider material soft- ening if the tensile strength is reached. However, the formed cracks do not influence the humidity trans- port, allowing one-way coupling scheme. The computational model was calibrated in sev- eral subsequent steps. The basic creep compliance, which is in the case of the B3 model described by 4 parameters q1 − q4, was determined from empir- ical prediction formulae using compressive strength and concrete composition as input parameters. Af- terwards, parameters controlling short-term creep q1 and q2 were adjusted to match the results of the 1-hour creep experiment. The value of the drying creep parameter, k3 = 26 was identified in the previ- ous analysis of Bryant and Vadhanavikkit experiment 3 L. Dohnalová, P. Havlásek, V. Šmilauer et al. Acta Polytechnica CTU Proceedings B3 model MPS model Moisture diffusivity + mixed b.c. q1 q2 q3 q4 k3 ft Gf ksh C1 α0 hc n f [×10−6/MPa] [ - ] [MPa] [N/m] [ - ] [mm2/day] [ - ] [ - ] [ - ] [mm/day] 15.5 150 2.08 9.41 26 3.0 100 0.0016 30 0.15 0.7 12 0.3 Table 1. Summary of material parameters [11]. The response is only very slightly sensitive to the value of tensile strength ft and fracture energy Gf , which were set to typical values for concrete (see Tab. 1 which summarizes the values of all basic pa- rameters used in simulations). The remaining parameter of MPS model ksh, and the parameters of the transport model were cal- ibrated to achieve reasonable agreement with the shrinkage results of the performed experiment. The aim was to achieve the best possible match for all prism sizes. The main emphasis was placed on the ini- tial development of shrinkage (approx. first 6 months of drying) when both groups of the specimens show similar development of shrinkage. Later, when there is no longer a match between the specimen groups with a different measurement method, the aim was to achieve the best possible match with the results of the manually measured prisms. For the purpose of identification of material pa- rameters, the evolution of ambient relative humidity was defined by a piecewise-linear function according to the measured values. In addition, the analysis was repeated for a partially adjusted development of am- bient humidity (beginning according to the experi- ment, after 91 days of drying assumed humidity fixed at henv = 60%) achieve final values of shrinkage and to demonstrate more clearly the size-effect on drying shrinkage predicted by the model. 3. Results and discussion The development of shrinkage in time is for both au- tomated and manual measurement techniques shown in Fig. 2. In the case of more identical specimens, the mean response is shown in solid line while the shaded area corresponds to the measured experimen- tal range. The obvious fluctuations in shrinkage strain are caused by oscillating value of ambient rel- ative humidity which is shown in dashed black line. The average responses obtained with the two tech- niques are compared in Fig. 3. Except of the largest specimen, the measurements demonstrate very good agreement during the first 6 months of drying. Sud- den decrease in relative humidity from 60% to 40%, which occurred afterwards, caused rapid acceleration of drying accompanied by increase in shrinkage rate. This decrease also led to separation of the experi- mental trends determined by the two methods; in all cases the shrinkage strain measured automatically ex- ceeded the manual readings. The reason for this was not clarified even by thorough analysis of the mea- sured data. Still, the most probable cause seems to 0 200 400 600 800 1000 400 1 10 100 Sh ri nk ag e, ε sh [ × 1 0- 6 ] Drying duration, t - t0 [day] D = 10 mm D = 15 mm D = 20 mm D = 25 mm D = 30 mm D = 50 mm Figure 4. Comparison of the experimental measure- ments (symbols) with the results obtained from FEM simulations (solid lines). be connected to the frame for automated measure- ment. The measuring frame was attempted to be sufficiently stiff for the current measurement. Since the frame serves only as a support for the specimens and is loaded merely by their self weight, the se- lected material was 26 mm thick plywood for con- crete molds. However, this material turned out to be insufficiently stable under variable relative humidity of the ambient environment since non-negligible vol- ume changes were observed. These changes could not be completely eliminated by incorporating the mea- surement of the steel standard primarily intended for temperature compensation. The difference between the shrinkage of geometrically identical specimens de- termined by the automated and manual measurement technique was not uniform and was found to be, in a certain way, influenced by their position, namely by the distance from the largest specimen placed at one end (see Fig. 1, the weight of the prism labeled “1” reached almost 10 kg). The joint effect of these two factors seems to be the most probable explanation. Therefore, the data from the manual measurements are preferred in the later stage of the experiment (solid steel structure of the measuring frame insensi- tive to level of relative humidity) despite the intervals between the individual measurements were gradually increasing. As was stated above, the manual measurements al- lowed to determine not only the axial shrinkage but also the evolution of weight loss due to drying. To the best knowledge of the authors, such data are not available in this extent in any of the preceding exper- iments related to the size effect on drying shrinkage. 4 vol. 30/2021 Size effect on the drying shrinkage 0 10 20 30 40 50 60 70 80 400 1 10 100 0.3 0.4 0.5 0.6 0.7 0.8 0.9 humidity W ei gh t l os s [k g/ m 3 ] A m bi en t h um id ity , h en v Duration of drying, t - t0 [day] D = 10 mm D = 15 mm D = 20 mm D = 25 mm D = 30 mm D = 50 mm 0 10 20 30 40 50 60 70 80 400 1 10 100 W ei gh t l os s [k g/ m 3 ] Drying duration, t - t0 [day] D = 10 mm D = 15 mm D = 20 mm D = 25 mm D = 30 mm D = 50 mm Figure 5. Experimentally measured moisture loss (left) and its comparison to FEM analysis (right). After 100 days of drying the model exhibits excessive sensitivity to changes in henv. 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 0 50 100 150 200 250 300 350 400 reference size, D = 10 mm Si ze e ff ec t Drying duration t - t0 [day] D = 13 mm* D = 15 mm D = 20 mm* D = 20 mm D = 25 mm D = 30 mm 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 0 50 100 150 200 250 300 350 400 0.3 0.4 0.5 0.6 0.7 0.8 reference size, D = 10 mm Si ze e ff ec t A m bi en t h um id ity , h en v Drying duration t - t0 [day] D = 15 mm D = 20 mm D = 25 mm D = 30 mm humidity Figure 6. Time development of the experimentally measured (left) and computed (right) size-effect on drying shrinkage. The data points correspond to the average values obtained from manual measurement. The ambient relative humidity recorded in the experiment resulted into the FEM response shown in solid lines while the modified history of humidity (henv = 60% since 91 days of drying) led to the results shown in dashed lines. Yet such data provide better understanding of the time development of the drying process. The mea- sured evolution of moisture loss normalized per vol- ume of the specimen is shown in the left part of Fig. 5. The sensitivity to changes in ambient relative humid- ity which occurred in the later stage of drying is in the case of moisture loss not so pronounced as in the case of drying shrinkage. In the finite element simulations the driving force for the moisture transport is the relative humid- ity, therefore the moisture loss needs to be back- calculated. The value of moisture capacity (slope of the desorption isotherm, for simplicity assumed as linear) which gave the best agreement with the experimental data as shown in right part of Fig. 5 was k = 120 kg/m3. In the simulations, the de- velopment of ambient relative humidity corresponded with the experimental measurements. The agreement with the experimental data is satisfactory only in the first 100 days of drying; afterwards, the model over- estimates the rate of the weight loss or gain caused by changes in henv. The discrepancy in the initial value of weight loss stems from the linear desorption isotherm which is not realistic in the case of high rel- ative humidity. The ongoing changes in henv did not allow the spec- imens to reach their ultimate value of drying shrink- age. For this reason the size effect on drying shrink- age is here evaluated within the entire duration of the experiment and is related to the thinnest speci- men with cross-section 20 × 20 mm (D = 10 mm). Naturally, the strongest size-effect can be observed shortly after the onset of drying when the difference between the moisture loss of specimens with different size is most remarkable. On the other hand with the increasing duration the size-effect is slowly diminish- ing until the equilibrium with the ambient conditions is reached. The expected evolution of the size-effect on drying shrinkage is confirmed by the experimental measure- 5 L. Dohnalová, P. Havlásek, V. Šmilauer et al. Acta Polytechnica CTU Proceedings ments as shown in the left part of Fig. 6. Even though the ultimate values of drying shrinkage have not been reach so far, the size-effect has become almost stable and there is no indication that is should vanish com- pletely. The processed data are in correct order, the size-effect increases with size. With respect to the reference specimen (D = 10 mm) the size effect is within 20% of that value. As was shown in Fig. 4, the experimentally mea- sured evolution of shrinkage was captured by the nu- merical model very realistically up to ≈ 180 days of drying. However, the mismatch beyond this time causes that the numerically computed size-effect sig- nificantly deviates from the experiment as demon- strated by the smooth solid lines in the right part of Fig. 6. In order to arrive at equilibrated values of drying shrinkage, the simulation was repeated with a modified history of henv, which was fixed at 60% after 91 days of drying. Under such conditions the predicted size-effect becomes almost negligible within one year of drying. 4. Conclusions This paper analyzed and presented the results from the first year of a pilot experiment developed to in- vestigate the size-effect on drying shrinkage. The primary motivation of this research was to verify whether the shrinkage measurements could be accel- erated using smaller specimens than is the current practice. The findings can be summarized as follows. • The difficulties encountered in the first run of the experiment provided the research team with a valu- able feedback for its future improvement. The ex- perimental technique itself is feasible, cheap and capable to deliver good-quality data. A stiff steel frame, automatic monitoring of ambient conditions are a necessity, while the currently used monitoring of moisture loss is sufficient. • The measured data were in expected order and ex- hibited very little scatter among specimens with the same dimensions. This indicates good homo- geneity of the material in the specimens despite their small dimensions. Moreover, the consistency of the data was verified by a computational model and the agreement was promising. • The size-effect on drying shrinkage cannot be prop- erly evaluated unless the experiment is conducted at constant relative humidity, which was not the present case. Despite this deficiency, in the cur- rently examined size range a non-negligible size- effect on drying shrinkage was registered. • The constitutive model used in structural analysis needs to be improved as it strongly underestimates the size-effect on drying shrinkage. Acknowledgements The authors gratefully acknowledge financial support from the Czech Science Foundation (GA ČR), project number 19-20666S. References [1] L. Dohnalová, P. Havlásek. Comparison of drying shrinkage and drying creep kinetics in concrete. Acta Polytechnica CTU Proceedings 15:12–19, 2018. [2] L. Dohnalová, P. Havlásek. Size effect on the ultimate drying shrinkage of concrete - experimental evidence and engineering practice. Acta Polytechnica CTU Proceedings 26:13–18, 2020. [3] Z. Bažant, M. Jirásek. Creep and Hygrothermal Effects in Concrete Structures. Springer Netherlands, 2018. doi:10.1007/978-94-024-1138-6. [4] C. Aguilar. Estudio del comportamiento y desarrollo de una metodologia de prediccion de la retraccion hidraulica de hormigones, PhD. Thesis (in Spanish). Pontificia Universidad Catolica de Chile. Escuela de Ingenieria, 2005. [5] 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. [6] M. Hubler, R. Wendner, Z. Bažant. Comprehensive database for concrete creep and shrinkage: analysis and recommendations for testing and recording. ACI 112:547–558, 2015. [7] B. Patzák. OOFEM home page, 2000. Http://www.oofem.org. [8] Z. P. Bažant, L. J. Najjar. Nonlinear water diffusion in nonsaturated concrete. Materials and Structures 5:3–20, 1972. doi:10.1007/BF02479073. [9] 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. [10] 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. [11] A. H. Bryant, C. Vadhanavikkit. Creep, shrinkage-size, and age at loading effects. ACI Materials Journal 84:117–123, 1987. 6 https://doi.org/10.1007/978-94-024-1138-6 https://doi.org/10.1007/BF02479073