CHAPTER 2 REVISION OF CONCRETE CONTRIBUTION
3.3 Inclusion of shrinkage effects into 2D-PARC model
3.3.2 Model B
As better discussed in Chaper 2, matrix [Dc] is expressed as a function of the secant values of concrete Young modulus Ec and Poisson coefficient νc, which are properly updated during the analysis to account for material nonlinearity. These two parameters are evaluated through an iterative procedure based on the bisection method, by progressively narrowing a predefined stress interval until the achievement of a convergence check on stresses. In presence of shrinkage, the expression of this convergence check is properly modified by relating the concrete stresses evaluated in the principal directions 1-2 only to the corresponding concrete net strains, so obtaining:
tolE
shi ic shj c jc
ic
2
1 . (3.13)
where subscript i and j are respectively equal to 1 and 2 in case of biaxial tension and conversely to 2 and 1 in case of tension-compression or biaxial compression and tol represents the chosen tolerance value, adequately small. Actually the convergence check can be always performed indistinctly on both principal stresses, owing to the isotropic nature of the adopted constitutive model for concrete (see §2.3). When the condition reported in Equation (3.13) is satisfied, convergence is assumed to be reached on stresses. Consequently, the corresponding elastic parameters Ec and represent the updated values that can be used for the construction of concrete stiffness matrix [Dc] of the analyzed integration point.
As regards tension stiffening contribution, it should be reminded that the non-uniform distribution of the strains in the reinforcement is locally evaluated in 2D-PARC model by solving – at each integration point – the governing equations of the problem (i.e. equilibrium conditions for the whole section, for concrete and for steel, as well as compatibility condition) through the finite difference method, [60]. Shrinkage effects should be then properly included in the evaluation of stress and strain distribution in concrete. Moreover, it is necessary to ensure global compatibility by imposing that the mean value of steel strains computed between two adjacent cracks from the tension stiffening formulation is equal to the total strain evaluated in the global procedure, which is in turned modified in presence of shrinkage according to the flow chart reported in Figure 3.4.
Model B evaluates the actual value of the shrinkage strain associated to each integration point as a function of the moisture field in the element. As explained in the following, this latter is in turn obtained by performing an
“equivalent” thermal analysis by exploiting the analogy between the equations governing the moisture and the thermal problem, with a simple substitution of the corresponding parameters. Finally, moisture field is related to shrinkage strains and the stress-strain field in the element is computed by performing the structural analysis. Following this procedure, the differential shrinkage strains (and the related eigen-stresses) between the faces exposed to drying and the core are taken into account in the analysis, so leading to more realistic results.
Simulations are performed by considering two separate mechanisms, namely drying shrinkage and autogenous shrinkage, both related to the decrease of internal relative humidity.
As already explained in §3.2.1, concrete elements subjected to ambient conditions loss indeed capillary water through diffusion and evaporation until the internal relative humidity of concrete is in balance with that of the surrounding environment. A volumetric change of the element takes place due to drying, so leading to the called drying shrinkage. This phenomenon is the most important and often is the only one considered in simulations (for further details, e.g. [149–
152]). In this thesis, also autogenous shrinkage is considered, even if this latter effect gains relevance especially at early-age and for concrete characterized by low water-cement ration (tipically w/c < 0.39) - among others, [153,154]. This phenomenon is related to the process of hydration of the cement paste that causes an additional decrease in moisture content, the so called self-desiccation.
Hence, concrete shrinkage can be successfully predicted only by relating it to the moisture state and its variation in space and time. In more detail, the moisture distribution in a concrete element is non-uniform. As a matter of fact, as already mentioned in §3.2.1, neither autogenous nor drying shrinkage occur uniformly: the rate of hydration is indeed higher in the core because of the presence of stronger chemical reactions and also the drying process proceeds with different speed depending on the distance of each point of the element to the drying surface, leading to a non-uniform field as well. An uneven tendency to shrink occurs: while the exposed surfaces tend to shrink a lot the inner fibers have no tendency to shrink. Differential shrinkage occurs, leading in turn to self-equilibrated stresses: to restore compatibility, the external fibers are subjected to tension while the core is compressed.
Since shrinkage at each point of the element is related to the moisture distribution within the member and its evolution with time, the first step of the numerical analysis is related to the determination of the moisture field in concrete.
There is a general agreement in recognizing that moisture flow in concrete subjected to drying is governed by diffusion principles (among others, [134,155–
158]) especially when the average moisture content decreases below 70 to 80%
of the initial saturation, [159].
Mass transfer equation for an isotropic material subjected to isothermal condition can be written as ([124,158,160]):
t W W grad D t div
W n
e e
. (3.14)
where D is the diffusion coefficient and We and Wn stands respectively for evaporable and non-evaporable water. Thus, the last term of Equation (3.14) is related to the consumption of water due to chemical reactions.
Alternatively, the nonlinear diffusion equation can be written as a function of the internal pore humidity h instead of the water concentration W as the driving potential, so obtaining (see e.g. [57,126,155,158]):
t h h grad D t div
W t
h s
h
1
. (3.15)
being Dh the moisture diffusion coefficient and hs the self-desiccation humidity.
Thus, the first term at the right-hand side accounts for the moisture change due to drying, while the latter for humidity variation caused by water consumed in chemical reactions.
This transformation allows to easily consider the boundary conditions ([161,162]) and to better link moisture field with a subsequent shrinkage analysis ([124,155]). Moreover, internal profiling of moisture distribution becomes more feasible utilizing h instead of W ([126,158,163–166]). Thus, in this work the formulation in terms of internal pore humidity h is applied.
To pass from Equation (3.14) to Equation (3.15) the factor (∂W/∂t)-1 that represents the reciprocal of the slope of the moisture sorption isotherm curve (also called moisture capacity) must be added. However, considering that the moisture capacity of concrete at usual environmental conditions (h>50%) can be assumed reasonably constant ([162]), Equation (3.15) can be rewritten as:
D gradh
htt div
h * s
. (3.16)
by assembling terms (∂W/∂t)-1 and Dh to obtain a single coefficient D*.
This latter, for isothermal conditions, can be expressed as a function of the pore relative humidity (0 < h < 1) as:
p
c
*
h D h
h
D 1 1 1
1
1
. (3.17)
where D1 is the maximum of D*(h) for h = 1, hc is the pore relative humidity at D*(h) = 0.5 D1, α is the ratio between D0 and D1, being D0 the minimum of D*(h) for h = 0, while p is an exponent. According to Model Code 2010 [57], α = 0.05,
hc = 0.80, p = 15 and D1 = D1,o / (fcm - 8) are commonly assumed, D1,o being equal to 1·10-8 m2/s and fcm being the mean concrete compressive strength.
As regards the boundary conditions to be applied to the moisture field, two main different approaches can be used. A possible solution is to impose the value of environmental humidity to the concrete surfaces, so-called Dirichlet boundary condition (see e.g. [167]). Alternatively, Neumann boundary condition can be set (see e.g. [124,127,158]). This latter better represents reality and is the most widespread strategy, consequently it is also adopted in this work. It consists in the introduction of a proportionality factor fboundary between the exposed surface flux and the difference between the environmental humidity hen and the relative humidity of the exposed surface hsurf, so obtaining:
en surf
boundary surf
* f h h
n
D h
. (3.18)
where n is the normal axis of the boundary surface.
3.3.2.1 Adopted modeling strategy
Moisture distribution within plain and reinforced concrete elements is herein evaluated by exploiting the formal analogy between the moisture and thermal problems, through a simple substitution of the parameters governing the corresponding diffusion equations. There are indeed striking resemblances between Equation (3.16) with its boundary condition – Equation (3.18) – describing the diffusion problem and the well-known heat equations.
This choice is related to the fact that the adopted commercial FE Code (ABAQUS) has only the heat transfer option and not the moisture diffusion one.
Moreover, temperature belongs to those field variables which are passed from ABAQUS to the user-material subroutine, where 2D-PARC is implemented, that describes the constitutive behavior of the material.
As a consequence, the mutual correspondence between thermal and moisture quantities is exploited. In more detail, temperature is considered in spite of moisture content and thermal conductivity in spite of moisture diffusivity.
Furthermore, density and specific heat are set equal to unity in thermal analyses, so as to avoid the presence of extra-multipliers that are necessary in the heat transfer equation, but not in the moisture diffusion one. As regards boundary conditions, the moisture emissivity coefficient is substituted by convection heat transfer one. Moreover, the self-desiccation is taken into account by considering the internal heat generation in place.
A sequentially-coupled thermo-mechanical analysis is then performed; in this way, temperature (i.e. moisture) values in each node of the FE mesh are read by 2D-PARC subroutine and used for the evaluation of the free shrinkage strain vector {sh}. The assumption that micro-cracking due to shrinkage does not influence the moisture transport has been highlighted by several research ([168–
170]), since the usual width of these cracks is remarkably lower than the limit value, according to [171], which begins to influence the drying rates.
Several numerical formulations relate the moisture potential to the local unrestrained shrinkage, among them the well-known relation suggested in [148]
is applied in this work. In more detail, the free shrinkage strain is calculated as a function of pore relative humidity h through the following relation:
h ksh
sh
. (3.19)
where the shrinkage coefficient ksh is defined as the product between the ultimate drying shrinkage at complete drying (s0) and the ratio of elastic modulus with time (Ec(t0) / Ec(t)). The same approach has been applied also in [127,172–176]
Since shrinkage-induced stresses develop gradually with time, creep effects should be also considered in the analyses. To this aim, the compliance function is evaluated in 2D-PARC model by following the procedure described in [177], to which reference is made. The same approach is also adopted for model A; its description is not discussed herein for brevity, being out of the scope of this work.
Once the shrinkage strain vector is evaluated, strain and stress fields in the member are computed by applying the same formulation presented in §3.3.1.