Dth,s, is assumed to be a function of the current temperature, see Eurocode 2(2002). Accordingly, the thermal strain increment in steel is determined by the equationDDjþ1th;s¼ Djþ1th;s  Djth;s.The creep strain of steel (Dcr,s) becomes considerable when the temperature of steel exceeds 400  C.There is a number of creep models available for steel at high temperature. In our research we employthe model proposed by Williams-Leir (1983). This model assumes that the creep strain is a function ofthe current stress and temperature in steel, and that its evolution in time is governed by the differentialequation_ Dcr;s ¼ sgnðrsÞb1coth2ðb2jDcr;sjÞ. ð9ÞMaterial parameters b1 and b2 are the functions of stress and temperature in steel, see Williams-Leir (1983).Even if we assume that rs and T are given functions of time, Eq. (9) is a complicated differential equationthat needs to be integrated numerically. Here we employ the iterative solution method in which the timederivative is substituted by the implicit differential quotient. The creep strain increment of steel in time step[tj, tj+1] is thus determined by the equationDDjþ1cr;s¼ sgnðrjþ1sÞb1coth2ðb2jDjþ1cr;sjÞDtjþ1. ð10ÞThe flowchart of the incremental-iterative solution procedure of the mechanical response of the structureis given in Box 1. Remark. The majority of the numerical formulations of the fire analysis of the reinforced concretestructures do not differentiate between the plastic and creep strains in concrete. They employ the combinedplastic strain which includes both the plastic and the creep strain parts. The stress in concrete is then takento be a function of this combined strain (Lie and Irwin, 1993; Zha, 2003). Such a material model cannotaccount for the rates of temperature and creep strain properly, neither is able to pide the resultingcombined plastic strain into the physical plastic and creep parts. The transient creep in concrete is usuallyignored (Zha, 2003).By contrast, the present formulation considers each of the physical strain parts separately, thus enablingan engineer to follow the time variation of each particular strain and to assess its contribution to the totalstrain. This holds true for both concrete and steel materials.4. Numerical examples4.1. Clamped reinforced concrete column4.1.1. Comparisons with experiment (Lin et al., 1992)In this example we compare the results of our numerical model with the experimental results of full-scalelaboratory fire tests on the centrically loaded reinforced concrete column, performed by Lin et al. (1992)and reported on by Lie and Irwin (1993). The geometric and loading data are given in Fig. 3. The self-weight of the column is modelled as an axial traction. In order to simulate a fire situation, the columnwas exposed to hot surrounding air in such a way that the air temperature (generated by the furnace)was changing according to the ASTM fire curve (1976). The yield stress of the reinforcing bars wasfy0 = 42 kN/cm2, while the strength of siliceous aggregate concrete at room temperature wasfc0 = 3.61 kN/cm2. The measured fire resistance time of the column was 208 min (Lie and Irwin, 1993).The related critical temperature was 1087  C.The remaining material parameters and their temperature dependence, needed in the numerical analysisof the mechanical response, were estimated using the given strengths and the data from Eurocode 2 (2002).In the first step, we have to determine the temperature distribution over the cross-section at discretetimes tjduring fire. The 2D transient heat conduction problem was solved by our computer programme (Saje and Turk, 1987). The transfer of heat due to the radiation and the air convection at the boundarieswere taken into account in a standard way. Thermal parameters, such as the conductivity kc, the convectionheat transfer coefficient hc and the emissivity er, were not presented in the report by Lie and Irwin (1993)and were hence selected in such a manner that the calculated and the measured temperatures in concreteagreed as much as possible (er = 0.3, hc = 20 W/m K, the graph of kc is depicted in Fig. 4). The remainingparameters, needed in the analysis of the temperature field, were estimated on the basis of Eurocode 2(2002). End parts of the column were thermally insulated, so that the actual length of the column exposedto fire was 310 cm. Fig. 5 shows the calculated and the measured temperature distributions in concretecross-section along its centerline at various times, as well as the calculated isothermals. We see that the cal-culated and the measured temperatures agree well. The figure also shows the temperature distributions sug-gested by Eurocode 1 (1995); these distributions assume that the column is not insulated. The disagreementbetween the measured and the Eurocode 1 distributions is rather big. The Eurocode curves exhibit largertemperatures than the experiment at the surfaces of the cross-section, and substantially smaller tempera-tures at the central region of the cross-section. The calculated time-dependent temperature field varyingover the cross-section was used as the thermal load of the column in the subsequent mechanical analysis.The mechanical response of the column subjected to thermal and mechanical loads was obtained by ourcomputer programme, made in Matlab (MathWorks, 1999). The column was modelled by six beam finiteelements with e and j being interpolated with the Lagrangian polynomials of the fourth order. In orderto initiate the buckling, the axis of the column was made imperfect with small eccentricity 0.01 cm. Thecross-sectional integration needed to determine the constitutive axial force and constitutive bending mo-ment and the cross-sectional constitutive tangent stiffness matrix was performed numerically. We usedthe 3 · 3-point Gaussian integration with the total of 180 integration points over one half of the cross-section.Fig. 6 shows how the measured axial displacement was changing with time. We can see that the axialdisplacement was increasing during the first 120 min. This corresponds to the elongation of the columnwhich is due to the growing thermal strain. Subsequently, the axial displacement started decreasing. Thiscaused the shortening of the column which was due to the rapid increase of creep and transient strains. Thiskind of the behaviour is typical for the reinforced concrete columns.We find it interesting to study the effect of inpidual strain parts in concrete and steel on the mechanicalbehaviour of the column. If only the thermal strain (Dth) is considered (while Dcr and Dtr neglected), theaxial displacement u* of the column becomes maximal at about 90 minutes and is greater than the mea-sured one (see Fig. 6). This remains true if the creep strain of concrete (Dcr,c) and, particularly, the transientstrain (Dtr) are also considered, yet the maxima are now lower and the final axial displacement at the collapse at about 210 minutes is considerably bigger. The calculated displacement at the collapse is this timegreater than the measured one, although we used the least recommended value (1.8) for the constant k2 inthe transient strain increment expression. 钢筋混凝土柱在火灾中的性能模拟英文文献和中文翻译(4):http://www.751com.cn/fanyi/lunwen_56542.html