Experimental and Analytical Studies for the Size Effect on the Axial Strength of High-Strength Concrete Walls with Various Fire-Damaged Areas

FE model generation and structural analyses are conducted using commercial software Abaqus version 6.10-3 (Dassault Systems, Velizy-Villacoublay, France), using 3D continuum brick with reduced integration point (C3D8R) and 3D truss elements (T3D2) for concrete and steel reinforcements, respectively. Concrete and steels are assumed perfectly bonded because there were stirrups confining steel reinforcements and spalling of high-strength concrete was prevented in advance due to addition of polypropylene fiber in experiments.

The dimensions and variables are determined to be the same as those in the existing experimental data, as shown in Fig. 1 and Table 1. Temperature information of the inside the wall heated with ISO-834 standard heating curve from previously conducted experiments (Chun, 2019; Chun et al., 2018; Kang et al., 2019). Effect of temperature distributions on strength of the wall was included in the analysis by implementing temperature-dependent stress–strain curves of concrete in the corresponding section of the wall as shown in Fig. 4. The model included perfect bond between the concrete and the reinforcing steel bars during the analysis, because (1) vertical steel bars were confined by the lateral bars with enough anchorage length, and (2) there was no de-bonding of steel bars observed from the experiments. In addition, failure was determined when numerical instability occurs due to material and geometrical nonlinearity. As the concrete develops plasticity and wall deformation increases, numerical instability occurs, which leads to termination of analysis after reasonable number of time increment control for convergence. In the ABAQUS, the time increment was increased by 50% in default if the analyses were converged in the two successive increments and decreased by 25% if the analyses were diverged or not converged within 16 iterations (ABAQUS, 2010). For effective calculation, maximum and minimum time increments were set as 100 s and 1× \({10}^{-5}\) sec, respectively. The bottom surface of the model is fixed in all translational directions, while the top head of the wall is restrained in the horizontal direction only for vertical loading prescription. The analysis considers both material and geometrical nonlinearity. The material properties of the concrete at room temperature (RT) are obtained from the material strength tests as shown in Fig. 2 and Table 2, then the temperature-dependent material properties of the concrete are obtained according to the reduction ratios of elastic modulus and strength which are from 100 to 900 °C shown in Table 3. Both reduction ratios of concrete strength and elastic modulus of concrete were adopted from the existing literatures (Cheng et al., 2004, Poon et al. 2001, Poon et al. 2016). From one of the literature, the authors were able to obtain reduction ratios of strength and elasticity per every 100 °C of temperature increase, while the temperature differences between the sections inside the wall model were prescribed with the temperatures smoothly changing from 100 to 900 °C. Therefore, reduction ratios at temperature levels not provided from the literatures (i.e., 150, 250, 350, 650, and 750 °C) were generated by conducting linear interpolation, i.e., reduction ratios at 150 °C was calculated as averaged ones of 10 and 200 °C. In addition, stress–strain curves of fire-damaged and non-damaged steel bars with diameter of 10 mm are obtained as Fig. 3 from the tensile strength tests, which are assigned to the steel elements in heated and unheated side of the walls, respectively. Damaged steel means reinforcing steel bars located near to the wall surface exposed to fire. Because the steel bars located far away from the fire were exposed to temperatures around 100 °C which was not enough to have influence on mechanical behaviors of steel bars, those were grouped as undamaged steel bars and the material properties measured at room temperature were implemented. Likewise, those located to the wall surface exposed to fire were grouped as damaged steel bars and degraded material properties obtained from the tests were used. Considering that the temperature difference at around damaged steel bars were minimal such as 400–650 °C, single level of material properties shown in Fig. 3 was used. The material properties of the concrete at various temperatures are assigned to the corresponding part of the FE model, considering temperature distributions through the wall thickness obtained from the experiments (Chun, 2019; Chun et al., 2018; Kang et al., 2019) and assigned temperatures are shown in Fig. 4.

In this section, the experimental method and results are briefly introduced to show overall tendencies of axial strength reduction due to fire and to validate the analytical models. More specific details of the experiment can be found in the previously published papers (Chun, 2019; Chun et al., 2018; Kang et al., 2019).

Using the analytical approaches validated in Sect. 2.2, the more analytical studies are conducted for the axial strength of fire-damaged walls in real scale. The walls are of various widths of 600, 1,500, and 3,000 mm, as illustrated in Fig. 9a–c, and the heated areas are varied as half-surface, single-surface and double-surfaces. The wall height is fixed as 2,800 mm, and constant thickness of 200 mm is used for all cases which makes slenderness values are controlled to be same as 45.029. These dimensions are determined based on the sizes of the walls in typical residential buildings in Korea. In addition, vertical reinforcements are included in the model to have ratio and intervals similar to those of the short walls used for the validation. Details of the model names, dimensions, heated areas and steel reinforcements are listed in Table 8. Pin support conditions are prescribed to the upper and lower part of the walls as if the walls are constrained by the slabs having 100 mm of thickness. Thus, nodes on the part from the bottom surface to 100 mm of height are restrained in all translational directions, and the nodes on the part from the top surface to 2700 mm of height are restrained in horizontal directions. Loading is applied to the top surface of the wall models in the form of displacement control. The temperature distributions inside the walls and the material contents are assumed as same to those of small sized wall models generated for the validation. Thus, nonlinear and temperature-dependent material properties are also considered in the analyses for the parametric studies by assigning material properties of different temperatures to the corresponding part of the concrete and steel.

Table 9 lists the predicted results of wall models having different wall widths and heated areas. It is common that the greater the heated area, the more the axial strength decreases, regardless of the wall width. In order to fairly compare the axial strengths of the walls having different widths and heated areas, the strength reduction ratios for the heated wall models are calculated using Eq. (1), and found to be 7–16, 35–41, and 56–57% for the walls heated on half-surface, single-surface and double-surfaces, respectively. It is interesting to note that, for the walls in real scale, the reduction ratios of the walls heated on double-surface are much larger than that of the walls heated on single-surface as well as the short walls heated on double-surfaces. This is not observed from the short walls, maybe because the aspect ratios (ratio of height to thickness) of the walls in real scale become much larger than those of the short walls especially when heated on double-surfaces and cause stability issue. Whereas, the strength reduction ratios of the half-surface heated walls in real scale are much smaller than those of the short walls, because the area of undamaged part is large enough to resist the applied load. Regarding the wall width, it seems that the axial strength reduction ratio slightly decreases with the wall width, especially when half of the wall surface is exposed to fire. However, for the walls heated on a full surface or the larger area, it is observed that the effect of wall width on the axial strength reduction ratio is minimal or even slightly increase. This may be explained from the relationship between wall width and the fire-damaged area. Wall width works for residual strength of the fire-damaged walls when only partial area of the surface is exposed to fire, because the increased undamaged part with the increased wall width resists against buckling. Whereas, the walls heated on single and double-surfaces cannot resist local buckling as much as the wall heated on half-surface does and increase of wall width does not significantly reduce axial force reduction ratios.

Fig. 10a–c are the load–displacement curves of the walls predicted from the analyses which shows estimation of axial strength reduction ratios of fire-damaged walls from the proposed equations are well matched with the simulation and experiments. In addition, the solid lines in the graphs show trends of how axial strength reduction ratios of the fire-damaged walls change depending on wall width and heat exposure condition. As commonly shown, the axial strength increases almost linearly with the wall width. In addition, it is found that the maximum displacement at wall failure decreases with the wall width, which tells that the more brittle behavior can be expected from the fire-damaged walls in real scale.