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This suggests that the specimen was probably almost dry whenthe test started. Consequently, the transient strain can be neglected. The creep strain in steel (Dcr,s) doesnot effect the axial displacement considerably (see Fig. 5), because the highest temperature in reinforcingsteel is not greater than 400 C and because steel Au 50 which is the least sensitive to creep was used.Surprisingly, the calculated fire resistance time does not depend much on which strain parts are consid-ered or neglected and rather well equals to the resistance time measured in experiment. Fig. 6 also shows thevariation with time of the calculated lateral displacement w* at the mid-point of the column. A sudden in-crease in w* well indicates the onset of the buckling of the column at time 205 min and the subsequent crit-ical state at about 215 min, which is close to 208 min reported in Lie and Irwin (1993). Fig. 6 further showsthe results of our model when the mechanical analysis using thermal parameters according to Eurocode 1(1995) is made (Fig. 5), and the thermal insulation at the bottom and at the top of the column is disre-garded. The results for the displacements and the resistance time (tcr = 211.7 min) do not differ essentiallyin comparison with the results that employed the more realistic temperature distributions. Hence, in theparticular column in fire considered here, the accuracy of the temperature field was not the essential factor.Fig. 7 shows the distribution of strains (e,j) along the centroidal axis of the column at t = 90 min and atthe instant of the numerical collapse, tcr = 214.4 min. All kinds of the strain parts were taken into account.As expected, the least values of strains appear at the top and at the bottom of the column in the insulatedareas. Until the buckling starts at 205 min, the column remains straight and the pseudocurvature zero. Atthe critical time, tcr = 214.4 min, when the column collapses, the longitudinal strain is still dominant, i.e.about 30-times larger than the corresponding maximal pseudocurvature. Note that the elongation at theexposed part of the column at 214.4 min is roughly 10-times bigger than at 90 min. By contrast, the elon-gation at the insulated parts remains practically constant.The results of the numerical analysis make it possible to assess the contribution of inpidual strain partsto the integral response of the column. Fig. 8 shows the isolines of thermal, geometric, mechanical, creepand transient strains and stress in concrete at the mid-point cross-section at 180 min. At this instant thecolumn is still unbuckled; that is why the geometrical strain is homogeneous across the section. The remain-ing strains vary over the cross-section. With the exception of the mechanical strain, the strains attain their maximal values at the surface of the cross-section. Note that the concrete transient strains are comparablein size with thermal or mechanical strains. They are, however, compressive, in contrast to thermal strainswhich are tensile. Mechanical strains are also compressive; they are maximal at regions positioned a fewcentimetres away from the surface of the section. The stresses in concrete are also not homogeneous overthe section. They are maximal at the centre region of the section. Fig. 8(d) shows that about one third of thesection is practically not stressed at this instant.4.1.2. Comparisons with Eurocode 2 (2002)According to Eurocode 2, the fire resistance time of a reinforced concrete column is given by formulae(2)–(4). First, we have to calculate the design resistance of the column at room temperature. The relatedprocedure which is well described in Eurocode 2 (1991) requires the second order effects to be includedvia the geometrical imperfection and prescribes different partial safety factors for action (c = 1.35) andfor material properties (cc = 1.5, cs = 1.15). Assuming the geometrical imperfection to be ea = 0.49 cm,we obtain NRd = 2394 kN; the design resistance calculated by our numerical model is 2556 kN, which isabout 7% larger value. The effective length of the clamped beam is l0,fi = 1.91 m. The selected reduction fac-tor for the design load level in fire was lfi = 0.446 and lfi = 0.418, respectively. Inserting these values intoEq. (2) gives REC2 = 180.8 min (at Tcr = 1110 C) for NRd = 2394 kN and REC2 = 186.7 min (atTcr = 1115 C) for NRd = 2556 kN. Eurocode 2 assumes that the column is not insulated, in contrast to the present column which is partlythermally insulated. In order to make a fair comparison, in our numerical calculations we also assumed thatthe column is not insulated. The results of the analysis are shown in Fig. 6 for displacements u* and w*asfunctions of time. The calculated resistance time is 211.7 min. This is remarkably more than the resistancetime 180.8 min predicted by Eurocode 2. A similar conservativeness of Eurocode 2 with regard to the resis-tance time of circular reinforced concrete columns was reported by Franssen and Dotreppe (2003).4.2. Simply supported reinforced concrete columns: Comparisons with Eurocode 2 (2002)We study the behaviour of two simply supported perfectly straight columns, here denoted by C1 and C2.Both columns are subjected to a compressive axial load and bending moments at the supports. Column C1(height 4 m) is subjected to an eccentric axial force. We consider three different eccentricities, e = 0; 0.015;and 0.04 [m], and study their effect. Column C2 (height 4.5 m) is subjected to a smaller axial force and large,but unequal boundary moments (see Fig. 9). The self weight of the columns is modelled as an axial traction.The variation of the surrounding air temperature with time is assumed after Eurocode 1 (1995) (Eq. (1)).Geometrical and mechanical data along with the load disposition are presented in Fig. 9.The amount of the reinforcement and the design resistance of the columns were calculated according toEurocode 2 (1991). The partial safety factors for actions and for material properties were taken to bec = 1.35 and cc = 1.5, cs = 1.15, respectively. The resistance of column C1 at room temperature was foundto be NRd,C1= 1350 kN for all three values of eccentricities. The related areas of the steel reinforcement are:As ¼ A0s¼ 7.56 cm2for e =0; As ¼ A0s¼ 10.68 cm2for e = 0.015; and As ¼ A0s¼ 16.04 cm2for e = 0.04. Theresistances of these columns, obtained by the present numerical method, are 1527 kN, 1451 kN and1399 kN. These values are from 4–13% bigger than the one obtained from the code. The resistance of col-umn C2 at room temperature is NRd,C2= 945 kN, MRd,1,C2= 47.3 kN m or MRd,2,C2= 75.6 kN m. Fromthis As ¼ A0s¼ 6.32 cm2follows. The selected reduction factors for the design load level in fire were lfi = 0.5and lfi = 0.4 for columns C1 and C2, respectively (see Fig. 9 and Eq. (4)). The corresponding fire resistancetimes by Eurocode 2 (2002) are RC1¼ 123.8 min (at Tcr = 1054 C) and RC2¼ 108.7 min (Tcr being1034 C). Note that the resistance time in Eurocode 2 does not depend on the eccentricity of the loadand/or the area of the reinforcement; that is why all of the three cases of column C1 have the sameresistance time. Thermal parameters of concrete, i.e. conductivity kc, specific heat cc, convection heat transfer coefficienthc, emissivity er and density qc, were chosen on the basis of the data from Eurocode 2 (2002) for concretewith the siliceous aggregate and for the moisture content 1.5% of the concrete weight.
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