钢筋混凝土柱在火灾中的性能模拟英文文献和中文翻译(6)
The calculated tem-perature distributions over the cross-section (see Fig. 10) well agree with those presented in Eurocode 2(2002). Fig. 10 also shows the changing of the temperature with time in the most and the least exposed steelbars. We can observe that the temperature in the bars is much smaller than the temperature of the sur-rounding air (dashed line in the figure), which is the consequence of the relatively thick concrete cover.In the mechanical analysis, each column was modelled by six equally long beam finite elements of thefourth-order. We used Lobatto s integration for the line integrals and Gaussian integration for the cross-sectional integrals. Concrete with siliceous aggregate and the cold formed reinforcing steel were assumedto define the dependence of concrete and reinforcing steel material parameters on temperature. We assumethat steel is only lightly sensitive to creeping. Constant k2, characterizing the transient strain increment, wastaken to be equal 1.8.When disregarding creep of concrete and steel, and also transient strain in concrete, our numerical modelyields the fact that the resistance time of column C1 is practically insensitive to the eccentricity of the axialforce, the resistance times of the three columns being 121.5, 117.3 and 121.3 min for e = 0; 0.015; and 0.04[m]. These values are only slightly smaller than the value suggested by Eurocode 2, i.e. RC1¼ 123.8 min. Thetime developments of the axial and lateral displacements are displayed in Fig. 11 for all three cases. Notethat column C1 collapsed after an excessive growth of the lateral deflection.In Section 4.1.1 we observed that the effects of creep of concrete and steel as well as the transient strainsin concrete onto the fire resistance time were small. This is not true in the present case of column C1. When these strains are considered, the numerically found resistance times are about 10% smaller, i.e. 110.8, 106.0and 111.7 min, respectively. Fig. 12 shows the related time graphs of the displacements. Observe that theaxial displacement, u*, increases during the first 80–90 min (reaching the value 1.15 cm for the eccentricity4 cm), and decreases afterwards. It takes the value about 0 cm at the collapse of the column. By contrast,the lateral displacement increases all the time. A rather substantial lateral displacement may be observed atthe collapse (w cr¼ 11.02 cm for e = 4 cm).The fire resistance time for column C2, calculated by the non-linear analysis, is 102.3 min, when we dis-regard the creep strains in concrete and steel and the transient strains in concrete, and 92.1 min otherwise.This is 16.6 min less than the Eurocode 2 fire resistance time. Like column C1, column C2 failed after anexcessive growth of the lateral deflection. Fig. 13 shows the variation of the axial and lateral displacementswith time. When all kinds of strain contributions are considered, the axial and the lateral displacements atthe collapse time are 0.78 cm and 9.48 cm, respectively.Fig. 14 shows the deformed shapes and the distribution of bending moments M for the two columns att = 60 min and at the time of collapse, tcr. The contribution of all kinds of strain is considered. The dis- 5. ConclusionsWe described a two-step finite element formulation for the thermo-mechanical, transient, non-linearanalysis of the behaviour of the reinforced concrete columns in fire. In the first step, we determine the tem-perature distributions over the cross-sections of the column in fire as a function of time. These constitutethe time and space dependent temperature load of the column. In the mechanical analysis, we employ ournew strain-based 2D geometrically exact and materially non-linear beam finite elements to model the col-umn. These finite elements are special because they satisfy the conditions that the equilibrium and consti-tutive axial forces and bending moments coincide at the integration points.Because the thermo-mechanical processes in the column during fire are really complicated, such a beam-based analysis might seem to be too simple to predict realistic behaviour. Yet the comparison with theexperiments performed by Lin et al. (1992) on clamped reinforced concrete columns, made in our firstnumerical example, show a very good agreement in the resistance time. The agreement in the variationof the axial displacement with time was only qualitative.