*7.4. Limitations of One-Dimensional Approach*

The proposed one-dimensional model was able to predict well both the heating and cooling phase, which is important in relation to the occurrence of undesirable tensile stresses. However, the one-dimensional model was not recommended to describe the temperature evolution of concrete plates with undefined strictly boundary conditions, as shown in the example discussed below. It concerns the bottom slab of stage I, for which the A-A measurement cross-section was located just next to the protruding web reinforcement and could not be insulated well with the styrofoam layer. The second example with irregular boundary conditions was the top and bottom plate monitored in stage No. III, where some disorder was the temperature coming from the hardening web. For these elements, the selected model parameters were modified. The obtained solutions were satisfactory, but for special cases, the calculations should be made with a two-dimensional model.

For the bottom slab, 80 cm thick (stage I) simulations in two variants were carried out. The first one for thermophysical coefficients specified for the 93 cm slab (Table 5) with modification of *n*parameter according to Figure 19b. The second approach with the change of heat transfer coefficients α*styr <sup>s</sup>* and α*f c* (Table 9). The results of the calculations are shown in Figure 30. Considering the temperature distribution at the points located near the upper surface of the plate (p1, p2—Figure 30a) it could be concluded that there was no full insulation with styrofoam and the influence of protruding nearby reinforcement was substantial, hence the adopted coefficients of heat transfer α*styr <sup>s</sup>* and α*f c* as in the case of a regular slab (93 cm, thick) was a wrong idea. Modification of these parameters improved the solution (Figure 30b).

For the A-A cross-section (80 cm thick), after 24 h, the temperature difference between points p1 and p2 was 19.3 ◦C (Figure 30a). Such information prompted the contractor to insulate the upper surface of the slab, which in 25–94 h was covered with a styrofoam layer, 5 cm thick. This was particularly visible in the thermal history of points located in the upper zone of the plate, i.e., p1. At this time, there was no cooling of the element, and a constant temperature was maintained.

**Table 9.** Thermophysical parameters—bottom slab, 80 cm thick, stage I.

**Figure 30.** The concrete temperature of bottom slab, 80 cm thick (stage I): (**a**) Variant No. 1 and (**b**) variant No. 2.

In stage III, the box section of the bridge was concreted in one step. Thus, for the upper surface of the bottom slab and the bottom surface of the top slab, the influence of the temperature from the embedded web was significant. Therefore, a replacement heat transfer coefficient was adopted, indicating it with a symbol α*conc <sup>s</sup>* (Table 10, Table 11). The others parameters, except *A*0/κ for the top plate and *n* for the bottom plate (modification *n* according to Figure 19b) remained unchanged relatively to the regular element for stage III, i.e., web (Table 10). For the bottom surface of the top plate the 10 days web temperature from point p6 (*T*<sup>10</sup> *days env*\_*<sup>t</sup>* ) and for the upper surface of the bottom plate an average temperature from point p8 (*T*<sup>10</sup> *days env*\_*<sup>b</sup>* ) was considered. The results (Figures 31 and 32) were satisfying, although they required additional assumptions.


**Table 10.** Thermophysical parameters—bottom slab, 35 cm thick, stage III.


**Table 11.** Thermophysical parameters—top slab, 57.2 cm thick, stage III.

**Figure 31.** The concrete temperature of bottom slab 35 cm thick (stage III): (**a**) Constant ambient temperature and (**b**) variable, measured ambient temperature.

**Figure 32.** The concrete temperature of top slab 57.2 cm thick (stage III): (**a**) Constant ambient temperature and (**b**) variable, measured ambient temperature.

The thermal changes in the top plate differed the most (Figure 32). The max. concrete temperature occurred at point p4 was 42.8 ◦C. This temperature was lower by 16.8 ◦C than the maximum temperature recorded for the same element during the second stage of investigations. The various boundary conditions (formwork and free surface) and low ambient temperature generate difference (15.7 ◦C) between the center (p4) and the surface (p1) of the element. In general, guidelines state that the temperature difference must remain smaller than 15–20 ◦C depending on the element thickness.

The one-dimensional approach has several limitations. Early-age concrete is difficult for modeling because it is a complex material, which is additionally subjected to transformations as a result of cement hydration. The contractors use different types of formwork, insulation and the weather is changing all the time during construction process. Due to this fact, the 1–D way is not universal. Nevertheless, this fundamental method gives a possibility to control the early age behavior of concrete with quite good agreement. In the case of 1–D problem we could solve the partial differential equation using the own code of the finite difference method. There are no hardware restrictions and calculation time is short. A numerical, one-dimensional model is always a good complement to the measuring system. Many researchers [8,9,12] use this method to predict temperature evolution of concrete.
