Proposed methodology
The proposed methodology for the prediction of in situ dam concrete creep strains in compression is based on a composite model in which the equivalent matrix is the wet-screened concrete and the inclusions are the removed aggregate in the wet-screening procedure. The study is divided into three main parts: (i) an experimental study for the evaluation of the compressive creep strains obtained in situ of the dam concrete and the two wet-screened concretes (#76 and #38 concrete); (ii) the fit of the experimental results to model B3 considering the measured temperature for each type of concrete to obtain a reference temperature state; and, (iii) the prediction of the dam concrete compressive creep strains based on the compressive creep strains of the wet-screened concrete using an equivalent two-phase composite model (Composite #76-Dam and Composite #38-Dam). Figure 1 presents the schematic view of the proposed procedure for the validation of the equivalent composite model for the prediction of dam concrete compressive creep strains.
The experimental programme was defined in order to measure the development of the in situ compressive creep strains of dam concrete and of two wet-screened concretes, the concrete with MSA=76 mm (#76 concrete) and the concrete with MSA = 38 mm (#38 concrete). The #76 and #38 concretes were obtained by removing the aggregates larger than 76 and 38 mm, respectively. The compressive creep strains were measured in creep cells, concrete specimens embedded in the dam’s core (Sect. 2.2).
The comparison between the different test conditions (such as the temperature variations) was achieved by fitting the test results to the model B3 at constant elevated temperatures considering the equivalent age method (Sect. 2.3.2). An optimization procedure considering the equivalent age method allowed for evaluation of the model parameters at reference temperature state, since the temperature variations are taken into account.
The prediction of dam concrete delayed behaviour was obtained using an equivalent two-phase composite model (Sect. 2.3.3). The proposed innovation of this paper is to consider the wet-screened concrete, obtained from the dam concrete as an equivalent aging viscoelastic matrix and the removed aggregates as the elastic inclusions.
A simple parallel and series composite model and its adaptation to aging materials considering the age-adjusted effective modulus method (Sect. 2.3.1) was used [47]. The input parameters are the volume fractions of equivalent matrix (wet-screened concrete) and the inclusions (removed aggregates) and the creep compliance of each wet-screened concrete. Since two type of wet-screened concretes were tested, two composite models were developed: Composite #76-Dam, using the test results of the concrete obtained from sieving the aggregates larger than 76 mm and Composite #38-Dam, using the #38 mm wet-screened concrete test results. Each composite prediction was compared with the model B3 fit of the experimental results of dam concrete.
In the following sections the experimental setup, the general concepts of delayed behaviour of concrete, the methods of analysis and the used models are described. The methods of analysis include the age-adjusted effective modulus method [49], the model B3, its extension to basic creep at constant elevated temperature [29] and the two-phase composite model proposed by Granger and Bažant [47]) using the age-adjusted effective modulus method for the evaluation of the delayed strains of the composite.
Experimental setup for in situ testing using creep cells
The in situ characterization of dam concrete relies on a specific experimental setup based on creep cells. Creep cells (CC) are cylindrical specimens placed inside the dam, cast into expanded polystyrene (EPS) hollow cylinders (Fig. 2). The creep cells are subjected to the same thermohygrometric conditions as the structural concrete since its top face is connected to the dam concrete and are cast and completely covered with the surrounding lift.
Figure 2 shows a general view of a creep cell setup. To separate the effect of applied stress and evaluate the creep strains development, two identical creep cells are installed, an active cell and a free or non-stress cell, placed next to each other. The loading system is connected to the active creep cell, therefore, both the shrinkage strains, the thermal strains and the stress derived strains can be measured (total strains). The non-stress creep cell, installed without a loading system, measures only the shrinkage and thermal strains (non-stress strains).
The creep cells are placed one or two lifts above a visiting gallery of the dam, where the external loading system is placed. A flat-jack, placed in the basis of the active cells, filled with oil and controlled by the loading system, applies a distributed load to the specimen. The oil pressure is controlled by a closed hydraulic system which allows fast pressure variations (for modulus of elasticity tests) and can maintain a constant oil pressure (for compression creep tests). Each active creep cell has its individual loading system. The pressure is kept constant over time with the aid of a pressure storage device where nitrogen gas compensates the small pressure decay due to creep deformation of the specimen [21] (Fig. 2). The strain variations are measured with Carlson strainmeters placed inside each creep cell connected to a reading unit located at the gallery of the dam.
The experimental creep compliance, \(J_\mathrm{exp}(t,t')\), at time t for a loading age of \(t'\), considered to be a measurement of creep if the stress is kept constant, are obtained by subtracting the free strains, measured in the non-stress cell, from the total strains, measured in the active cell, and dividing by the applied stress (Eq. 1).
$$\begin{aligned} J_\mathrm{exp}(t,t')=\frac{\varepsilon _\mathrm{active}(t,t')-\varepsilon _{{\rm non}{\text{-}}{\rm stress}}(t,t')}{\sigma (t')} \end{aligned}$$
(1)
If stress is maintained constant since the first load and the temperature is kept constant, one can compare experimental creep compliance, \(J_\mathrm{exp}(t,t')\), with theoretical creep compliance, \(J(t,t')\).
The advantage of creep cells is the possibility to characterize the deformability properties over time in the environmental conditions of the dam’s core. Also allows the testing of large specimens, suited for the dam concrete and its large aggregates, since the surrounding mass concrete is used as the reaction frame. This type of experimental installation was developed in the past by the National Laboratory for Civil Engineering (LNEC) [77, 78] and, for this particular study, several technical improvements were implemented. The main new features are: (i) the use of a standard flat-jack (with rigid interface platens), larger in diameter (\(\phi\)) and increasing the \(\phi\)-MSA ratio of dam concrete specimen (Fig. 5); (ii) the use of three measurement devices in the dam concrete specimen for a more reliable reading; and, (iii) the use of expanded polystyrene (EPS) hollow cylinder as the mould to separate the concrete specimen from the dam’s body (instead of metallic moulds use in the past, with higher rigidity). The procedure for the experimental installation was: (i) placement of the loading system (the storage device in the visiting gallery and the flat-jack of each creep cells) and purge the hydraulic circuit; (ii) placement of the measuring devices inside the active and non-active cells and the electric cables; (iii) wet-screening of the dam concrete; (iv) cast of each creep cell with dam concrete and wet-screened concrete; (v) setting the initial reading values in each creep cell; (vi) cast the surrounding lift.
Methods of analysis and analytical models
Creep strains
Generally, the delayed behaviour of concrete is described as a total strain, \(\varepsilon (t,t')\), resultant of a stress, \(\sigma (t')\), applied at the age of \(t'\) and kept constant until t, and an hygrothermal strain, \(\varepsilon ^0(t)\), such as drying shrinkage, thermal or chemical strains.
$$\begin{aligned} \varepsilon (t,t')=\varepsilon ^i(t')+\varepsilon ^c(t,t')+\varepsilon ^0(t) \end{aligned}$$
(2)
Considering that no cracking occurs, the stress-dependent strain can be expressed as the sum of an instantaneous strain, \(\varepsilon ^i(t')\) and of a creep strain, \(\varepsilon ^c(t,t')\). The instantaneous and creep strains can be expressed as a function of stress, obtaining the creep compliance, \(J(t,t')\).
$$\begin{aligned} \varepsilon (t,t')=J(t,t')\sigma (t')=\varepsilon ^i(t')+\varepsilon ^c(t,t') \end{aligned}$$
(3)
$$\begin{aligned} J(t,t')= \frac{1}{E(t')}+\frac{\varepsilon ^c(t,t')}{\sigma (t')} \end{aligned}$$
(4)
The stress, \(\sigma (t,t')\), obtained from a given strain, \(\varepsilon (t')\) is related to the relaxation function.
$$\begin{aligned} \sigma (t,t')=R(t,t')\varepsilon (t') \end{aligned}$$
(5)
which can be approximated, as proposed by Bažant and Kim [50], by,
$$\begin{aligned} R(t,t')=\frac{0.992}{J(t,t')}-\frac{0.15}{J(t,t-1)}\left[ \frac{J(t-\Delta t,t')}{J(t,t'+\Delta t)}-1\right] , \quad \Delta t=\frac{t-t'}{2} \end{aligned}$$
(6)
The age-adjusted effective modulus method, AAEM method [49], based on the linear principle of superposition, is used for obtaining an approximate solution of the creep strains development by assuming that,
$$\begin{aligned}&\varepsilon (t,t')-\varepsilon ^0(t)=\varepsilon _0+\varepsilon _1 \phi (t,t'), \quad t>t' \nonumber \\&\sigma (t')=0, \quad t<t' \end{aligned}$$
(7)
where \(\varepsilon _0\) and \(\varepsilon _1\) are given constants. The advantage of this method is that the creep analysis converts into an elastic analysis considering an incremental form,
$$\begin{aligned} \Delta \sigma (t) = E''(t,t')\left( \Delta \varepsilon (t,t')-\Delta \varepsilon ''(t,t')\right) \end{aligned}$$
(8)
in which,
$$\begin{aligned} \Delta \varepsilon (t,t')=\varepsilon (t)-\varepsilon (t'), \; \Delta \sigma (t,t')=\sigma (t)-\sigma (t') \end{aligned}$$
(9)
$$\begin{aligned} \Delta \varepsilon ''(t,t')=\frac{\sigma (t')}{E(t')}\phi (t,t')+\varepsilon ^0(t)-\varepsilon ^0(t') \end{aligned}$$
(10)
$$\begin{aligned} E''(t,t')=\frac{E(t')}{1+\chi (t,t') \phi (t,t')} \end{aligned}$$
(11)
$$\begin{aligned} \phi (t,t')=J(t,t')E(t')-1 \end{aligned}$$
(12)
$$\begin{aligned} \chi (t,t')=\left( 1-\frac{R(t,t')}{E(t')}\right) ^{-1}-\frac{1}{\phi (t,t')} \end{aligned}$$
(13)
where \(\phi (t,t')\) and \(\chi (t,t')\) are, respectively, the creep coefficient and the age coefficient and \(E''(t,t')\) is the age-adjusted effective modulus.
This method was used by [47] to introduce the aging viscoelasticity of concrete into a composite two-phase model (Sect. 2.3.3).
Prediction model for the concrete creep strains
Model B3, proposed by Bažant and Baweja [28, 29], describes creep compliance as the sum of the asymptotic elastic strains due to unit stress, \(q_1\), the basic creep compliance, \(C_0(t,t')\), and the drying creep compliance, \(C_d(t,t_0,t')\) (Eq. 14). Its strong points are related to the fact that the creep compliance rate, \(\dot{C}_0(t,t')\), is derived according to the guidelines of RILEM TC-107 [51], has been fitted from multi-decade laboratory tests [52], is based on the micromechanics of aging considered in the solidification theory [53, 54] and has been shown to have lower coefficients of variation of errors for dam concrete [29].
$$\begin{aligned} J(t,t' )=q_{1} +C_{0} (t,t')+C_{d} (t,t',t_0) \end{aligned}$$
(14)
For the dam body, due to the large thickness of the dam and the slow water diffusion in concrete, only a small layer of the upstream and downstream (during construction) is subjected to cyclic drying and wetting [59] and the moisture exchange with the environment is small. For this reason, in this study, drying creep strains can be considered negligible [7, 60]. The basic creep compliance, \(C_0(t,t')\), can be expressed as a linear combination of material parameters and time-dependent variables.
$$\begin{aligned} C_{0} (t,t')=q_{2} \, Q(t,t')+q_{3} \, \ln \left[ 1+(t-t')^{n} \right] +q_{4} \, \ln \left( \frac{t}{t'} \right) \end{aligned}$$
(15)
where \(Q(t,t')\) is a binomial integral with no analytical expression but can be approximated by Eqs. 16–19, with an error less than 1 % for \(n=0.1\) and \(m=0.5\) for a large range of loading age and time under loading [28, 29].
$$\begin{aligned} Q(t,t')=Q_{f} (t')\, \left[ 1+\left( \frac{Q_{f} \left( t' \right) }{Z\left( t,t' \right) } \right) ^{r(t' )} \right] ^{-{1 /r(t' )}} \end{aligned}$$
(16)
$$\begin{aligned} r\left( t' \right) =1.7\, \left( t' \right) ^{0.12} +8 \end{aligned}$$
(17)
$$\begin{aligned} Z\left( t,t' \right) =\left( t' \right) ^{-m} \, \ln \left[ 1+\left( t-t' \right) ^{n} \right] \end{aligned}$$
(18)
$$\begin{aligned} Q_{f} \left( t' \right) =\left[ 0.086\, \left( t' \right) ^{{2 /9} } +1.21\, \left( t' \right) ^{{4/9} } \right] ^{-1} \end{aligned}$$
(19)
Considering a load duration, \(\Delta t\), usually taken to be 0.01 days, the static modulus of elasticity yields from the creep compliance (Eq. 14),
$$\begin{aligned} E(t')=\frac{1}{A_0+\frac{A_1}{\sqrt{t'}}} \end{aligned}$$
(20)
where \(A_0=q_1+q_3 ln(1+\Delta t^n)\) and \(A_1=q_2 ln(1+\Delta t^n)\).
Each term of the sum has a physical meaning: \(q_1\) is the asymptotic elastic part, \(q_2\) refers to aging viscoelasticity, \(q_3\) refers to non-aging viscoelasticity and \(q_4\) refers to aging flow. Since it is a linear combination of time-dependent variables, the fit to experimental data is easier than other creep models.
The temperature and moisture conditions have an important role in the development of the mechanical properties, especially on creep strains. The influence of temperature on the properties development is mainly ruled by the composition of the binder due to changes of cement hydration rate. The replacementof cement by fly ash is known to decrease the rate of property development [61–63] due to the late chemical reactions with the calcium silicates.
To model the effect of temperature variations in the hardening of concrete, several authors use the equivalent age method (or Arhenius maturity) [35] with equivalent activation energies, calibrated for tests at different temperatures, for different mechanical properties and different types of concrete [36, 64–66]. Particularly for the investigation of temperature effect on the creep of concrete, some experimental studies have been done [37, 67–69].
According to this method the original compliance (Eq. 15) yields a new expression (Eq. 21) to take into account constant elevated temperatures, T(t), in degrees Celsius [29].
$$\begin{aligned} C_{0} \left( t,t',T\right) =R_{T} \left[ q_{2} \, Q\left( t_{T} ,t'_{e} \right) +q_{3} \, \ln \left[ 1+\left( t_{T} -t'_{e} \right) ^{n} \right] +q_{4} \, \ln \left( \frac{t_{T} }{t'_{e} } \right) \right] \end{aligned}$$
(21)
where \(t'_{e}\) and \(t_{T}-t'_{e}\) are the equivalent age and the equivalent loading time both with the respective activation energy, \(U_h\), for the cement hydration reactions and \(U_c\), for describing the acceleration of creep rate. \(U'_c\) refers to magnification of creep due to temperature increase, defined by the Eqs. (27–29).
$$\begin{aligned} t'_{e} =\int _{0}^{t' }\beta _{T} \left( \tau \right) {} d\tau \end{aligned}$$
(22)
$$\begin{aligned} t_{T} -t'_{e} =\int _{t'}^{t}\beta '_{T} \left( \tau '\right) {} d\tau ' \end{aligned}$$
(23)
$$\begin{aligned} \beta _{T} =\exp \left[ \frac{U_{h} }{R} \left( \frac{1}{T_{ref}+273 } -\frac{1}{T+273} \right) \right] \end{aligned}$$
(24)
$$\begin{aligned} \beta '_{T} =\exp \left[ \frac{U_{c} }{R} \left( \frac{1}{T_{ref}+273} -\frac{1}{T+273} \right) \right] \end{aligned}$$
(25)
$$\begin{aligned} R_{T} =\exp \left[ \frac{U'_{c} }{R} \left( \frac{1}{T_{ref}+273 } -\frac{1}{T+273} \right) \right] \end{aligned}$$
(26)
where \(T_{ref}\) is the reference temperature in degrees Celsius, T is the measured temperature in degrees Celsius and R is the gas constant (8.31 J K\(^{-1}\) mol\(^{-1}\)) and, according to experimental fit to laboratory tests [29], the activation energies, \(U_h\), \(U'_c\) and \(U_c\), can be predicted by the following expressions:
$$\begin{aligned} \frac{U_{h}}{R} =5000~^{\circ} \,K \end{aligned}$$
(27)
$$\begin{aligned} \frac{U_{c}}{R} =3418\left[ \left( {w/c} \right) \, \left( c\right) \right] ^{-0.27} \left( f_{c,28} \right) ^{0.54} \end{aligned}$$
(28)
$$\begin{aligned} \frac{U'_{c}}{R} =0.18\frac{U_{c}}{R} \end{aligned}$$
(29)
where the w, c and \(f_{c,28}\) are the water content, the cement content and the compressive strength at the age of 28 days. In order to take into account the temperature effect of a specific concrete composition on the creep development, the activation energies can be adjusted to the obtained experimental results [35, 36].
Composite model for the characterization of the deformability properties
The second part of the study concerns the use of composite models to predict the delayed behaviour of concrete with different coarse aggregate contents.
The heterogeneity of concrete can be studied using models where the meso-structure is taken into account. The first composite models applied to concrete concerned the elastic behaviour using approaches based on uniaxial rheological models [41–43] and on homogenization models, such as the variational approach considering spherical inclusions (Hashin–Shtrickman bounds) [70], the self-consistent model considering ellipsoidal inclusions [71] and the Mori–Tanaka method [72]. The prediction of the aging viscoelastic behaviour of the materials using composite model was developed with the work of Counto and Popovics [43, 48] and later with Granger, Bažant and Baweja [47, 73], based on the uniaxial rheological models, and, more recently, Sanahuja and Lavergne, using homogenization concepts [74, 75].
The chosen model is the two-phase coupled series and parallel composite model, described by Granger and Bažant [47], which considers the mortar as the aging viscoelastic material and the coarse aggregates as the elastic inclusions. The model is based in a simple uniaxial rheological model which is strongly related to the physical behaviour of the material. The extension to triaxial behaviour was developed later by Baweja et al. [73]. It is considered that a part of the mortar is placed in series with the aggregates (related with parameter \(\alpha\)) and another part is placed in parallel (related with parameter \(\beta\)) (Fig. 3b). The series portion can be perceived as the amount of mortar that separates the coarse aggregates avoiding their direct contact and the parallel portion corresponds to the remaining volume between the aggregates.
The composite model estimates the modulus of elasticity of the composite material based on the modulus of elasticity of the mortar, \(E_m\), the modulus of elasticity of the aggregate, \(E_{a}\), its respective unit volume, \(V_{a}\) and the proportion of mortar placed in series and in parallel defined by \(\beta\) (Eq. 30).
$$\begin{aligned} \frac{1}{E_c}=\frac{1-\beta }{E_m}+\frac{\beta }{\alpha E_{a}+\left( 1-\alpha \right) E_m} \end{aligned}$$
(30)
where \(\alpha \beta =V_{a}\).
The physical meaning of the free parameter \(\beta\) is related to the amount of paste coupled in series and the product \(\alpha \beta\) is the volume of aggregate per unit volume of concrete, V\(_a\). When \(\beta\) is equal to 1.0 (Fig. 3a), the model derives into Voigt model (purely parallel model [43]) which can be related the maximum compactness, \(V_{a,max}\). The maximum compactness of aggregate is related to the aggregate size distribution and corresponds to the volume of aggregate in the mix necessary to obtain the most compact packing [76]. If the aggregates are in contact with each other, it is expected that the series portion is null and that the composite model turn into a parallel model (\(\beta =1\), Fig. 3b). An example of this type of concrete is the prepacked or preplaced aggregate concrete in which the aggregates are first placed in the forms and then the empty spaces are filled with a fluid mortar. In this case the aggregates are in direct contact with each other and, since there is no compaction procedure, no mortar is coupled in series with the aggregates.
The aging viscoelastic behaviour and the continuous stress transfer from the mortar to the stiffer aggregates are modelled using the AAEM method [49] (Eqs. 7– 13), described earlier in Sect. 2.3.1, in which the stress-strain behaviour is related the coupled series and parallel two-phase composite model conditions.
The total strain, \(\varepsilon (t,t')\), due to a stress, \(\sigma (t')\), is obtained by the sum of the strain of mortar placed in series, \(\varepsilon _m(t,t')\), and the strain of the parallel coupling of mortar and aggregates, \(\varepsilon _{am}(t,t')\). Considering firstly the strain of the parallel coupling of the composite model, \(\varepsilon _{am}(t,t')\), the stresses variations in both the mortar, \(\Delta \sigma _m\), and the aggregates, \(\Delta \sigma _a\), which yield the stress transfer from the mortar to the aggregates over time, can be obtained by
$$\begin{aligned} \Delta \sigma _a(t,t')=E_a \Delta \varepsilon _{par} (t,t') \end{aligned}$$
(31)
$$\begin{aligned} \Delta \sigma _m(t,t')=E''(t,t')\left[ \Delta \varepsilon _m(t,t')-\frac{\sigma _m(t')}{E_m(t')}\phi (t,t')\right] \end{aligned}$$
(32)
and the total stress in the mortar is ruled by the parallel model stress-strain relations,
$$\begin{aligned} \alpha \Delta \sigma _a(t,t')+\left( 1-\alpha \right) \Delta \sigma _m=0 \end{aligned}$$
(33)
yielding,
$$\begin{aligned} \sigma _m(t')=\sigma (t')\frac{E_m(t')}{\alpha E_a+\left( 1-\alpha \right) E_m(t')} \end{aligned}$$
(34)
Considering an unit stress to obtain the concrete’s creep compliance \(J(t,t')\), it is possible to derive the expression for the proposed composite model, taking into account the creep compliance of the mortar placed in series, \(J_m(t,t')\).
$$\begin{aligned} J(t,t')=\frac{\beta }{\alpha E_a+\left( 1-\alpha \right) E_m(t')}\left[ 1+\left( 1-\alpha \right) \frac{E''_m}{E''_{am}}\phi (t,t')\right] +\left( 1-\beta \right) J_m(t,t') \end{aligned}$$
(35)
in which,
$$\begin{aligned} \phi _m(t,t')=E_m(t')J_m(t,t') -1 \end{aligned}$$
(36)
$$\begin{aligned} E''_m(t,t')=\frac{E_m(t')-R_m(t,t')}{\phi _m(t,t')} \end{aligned}$$
(37)
$$\begin{aligned} E''_{am}(t,t')=\alpha E_a+\left( 1-\alpha \right) E''_m(t,t') \end{aligned}$$
(38)
where \(\phi _m(t,t')\) is the creep coefficient of the mortar, \(E''_m(t,t')\) is the age-adjusted modulus of elasticity of the mortar and \(E''_{am}(t,t')\) is the age-adjusted modulus of elasticity of the parallel portion of aggregate and mortar.
In conclusion, given the mortar’s creep compliance, \(J_m(t,t')\), the modulus of elasticity of the inclusions, the fraction volumes of each component and the appropriate parameter \(\beta\) is possible to predict the creep compliance of the concrete, \(J(t,t')\), using simple analytical expressions.