Determination of temperature fields and stresses during the construction of a massive monolithic foundation slab of a wind turbine tower

. The article proposes a method for determining temperature fields and stresses during the construction of massive monolithic structures in a two-dimensional axisymmetric formulation. The solution is performed using the finite element method. The calculation takes into account the shrinkage of concrete, as well as the change in its physical and mechanical characteristics over time. The problem of calculating a massive monolithic foundation of a wind turbine is presented. Recommendations are given to reduce the risk of early cracking.


Introduction
When concreting massive reinforced concrete structures, exothermic reactions of concrete hardening can occur, which can lead to an uneven distribution of temperatures in the core of the reinforced concrete structure and on its surface, which in turn can lead to the formation of cracks and a decrease in the performance of the structure [1].
There is a large amount of literature on the regulation of the temperature regime of hardening of massive reinforced concrete structures [2][3][4][5][6][7]. At the same time, there are relatively few publications on the determination of thermal stresses caused by the internal heat release of concrete in massive structures [8][9][10]. In these publications, very simplified approaches are used for the calculation or the ready-made software products, which do not allow taking into account the change in the physical and mechanical characteristics of the material over time.
The purpose of this work is to develop a methodology for calculating the temperature field and the stress-strain state in the axisymmetric foundations of wind turbines.

Materials and methods
A massive monolithic foundation slab for a wind turbine is considered (Fig. 1). The structure is circular in plan, with a radius of 9 m, and has a variable height, which varies from 1.5 m at the edge to 3 m in the middle. The calculation of the temperature field is carried out together with the soil mass, which is assumed to be cylindrical in shape with a radius of 15 m. The temperature field is calculated in a two-dimensional axisymmetric formulation using the differential equation: where is the coefficient of thermal conductivity, is the temperature, 0 is the material density, is the specific heat, is the specific power of internal heat sources. On surfaces where convective heat transfer occurs, the boundary conditions have the form: where n is the normal to the surface, h is the heat transfer coefficient, ∞ is the medium temperature. At boundaries very remote from the region where internal heat release occurs, the temperature can be assumed to be known.
When applying the finite element method, the solution of the heat conduction problem is reduced to a system of differential equations [11]: where [C] is the damping matrix, {T} is the vector of temperature nodal values, [K] is the thermal conductivity matrix, {F} is the load vector containing the contribution of internal heat release sources and environmental conditions. When approximating the time derivative of temperature using the Euler method, the system of differential equations (3) at each time step reduces to a system of linear algebraic equations: The matrices included in (3) for a separate finite element are determined by the formulas: where V is the volume of the element, S is the surface area over which convective heat transfer occurs, [N] is the matrix of shape functions, [B] is the matrix of gradients of shape functions.
To solve the problem of heat conduction and problem of calculating the stress-strain state, we use plane triangular FE of the axisymmetric problem with linear shape functions.
In the presence of forced deformations, the system of FEM equations for determining the stress-strain state has the form: where [K] is the stiffness matrix, {U} is the vector of nodal displacements, {F} is the vector of external nodal loads, {Ff} is the contribution of forced deformations to the load vector.
In the problem of calculating stresses caused by internal heat release and shrinkage of concrete, the vector {F} is equal to zero. As applied to our problem, we write system (6) in increments: The stiffness matrix and the contribution of forced deformations to the load vector are determined by the formulas: where [ ] is the matrix of elastic constants, {Δ } forced strain increment vector.
where ℎ is the shrinkage deformation, is the coefficient of linear thermal expansion of concrete.
The modulus of elasticity of concrete is determined as a function of compressive strength R according to the empirical formula given in [11]: The compressive strength of concrete is calculated as a function of its degree of maturity according to the formula presented in [11]: where R28 is the strength of concrete at the age of 28 days, ̅ = / , t is the age of concrete in hours, = ∫ ( ) 0 is the degree of maturity of concrete. The shrinkage strain is determined by the empirical formula [11]: where B = (R -12) MPa is the class of concrete, a and b are empirical coefficients.

Results and discussion
When calculating on the upper surface of the foundation and the upper surface of the soil, the conditions of convective heat transfer were taken with the heat transfer coefficient h = 20 W/(m 2 ⋅K) (free surface open for heat transfer). On the side and inclined surfaces of the foundation, the heat transfer coefficient was taken equal to 10 W /(m 2 • K) (surface covered with formwork). On the lower and side surfaces of the soil, the temperature was assumed to be set and equal to 20 °C. The air temperature was also taken equal to 20 °C. Thermal and physical characteristics of soil: λ = 1.5 W/(m⋅K), ρ = 1600 kg/m 3 , c = 1875 J/(kg⋅K). Soil elasticity modulus Eg = 300 kPa. Thermal and physical characteristics of concrete: λ = 2.67 W/(m⋅K), ρ = 2500 kg/m 3 , c = 1000 J/(kg⋅K).The coefficient of linear thermal expansion of the soil was taken equal to the coefficient of linear thermal expansion of concrete (α = 10 -5 1/K). The lower nodes of the soil mass were assumed to be fixed in z, and the line r = 0 was assumed to be fixed from radial displacements.
The calculation was carried out for fast-hardening B25 class concrete. The heat release function for 1 m 3 of concrete was taken as: where t is time in days, Q28 = 130 MJ⁄m 3 , k = 0.13, x = 0.42. In addition to the analysis of the stress strain state in a two-dimensional formulation, the calculation was carried out according to the simplified method given in the work [11]. Fig. 2 shows a graph of the change in time of the maximum temperature in the foundation, as well as the temperatures at its upper and lower surfaces at r = 0. There was no significant difference in temperatures when calculating in one-dimensional and two-dimensional formulations.
The dashed line in Fig. 3 corresponds to the maximum values of stresses , obtained by a simplified method. It can be seen from this graph that the results of the calculation according to the simplified method and in the two-dimensional formulation are quite close, however, this coincidence is rather accidental. The simplified technique should predict the stresses at the center of the foundation, but the maximum in the considered problem is not observed at the center. At r = 0, the stresses are significantly lower (red line in Fig. 3) than in the calculation using the simplified method. The stress distribution depending on r and z at t = 200 h is shown in Fig. 4.  Fig. 3 shows that for the considered structure there is a risk of early cracking at the age of about 50 hours. A layer of insulation made of extruded polystyrene foam PENOPLEX 2 cm thick with thermophysical characteristics λ = 0.034 W/(m•K), ρ = 30 kg⁄m 3 , c =1650 J/(kg⋅K) was installed between the base of the foundation and the ground. The calculation showed that in this case the risk of early crack formation is excluded.

Conclusion
A technique for calculating the temperature field and the stress-strain state during the construction of axisymmetric structures from monolithic reinforced concrete in a twodimensional formulation based on the finite element method has been developed. The practical problem of determining the temperature field and stress strain state in a massive monolithic foundation of a wind generator has been solved. A comparison with the calculation by the simplified method presented in the paper [11] is made. Recommendations are proposed to reduce the risk of early cracking in the considered structure.