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This is also the approach followed in thepresent paper.We are particularly interested in the behaviour of a reinforced concrete column in fire, because col-umns are fundamental for the bearing capacity of skeletal structures. The deformation of the columnexhibits geometrical effects such as buckling, and material effects such as progressive material softeningwith the temperature rise, spread of the yield through the cross-section and the redistribution of stressesin axial and cross-sectional directions (Najjar and Burgess, 1996). Thus, the fire analysis of columns rep-resents a severe test of the accuracy of a numerical formulation. Our goal is to show that a beam-basedmodel, when appropriately deduced, gives sufficiently reliable results for the resistance time when appliedin the fire analysis of reinforced concrete columns. We prove this by comparing experimental, numericaland building code (Eurocode 2, 2002) results. We also show that the Eurocode 2 results for the fire resistance time of reinforced concrete columns could be non-conservative. But let us first describe howthe fire resistance of the reinforced concrete column is estimated by the European building code (Euro-code 2, 2002)!2. The method for assessing the fire resistance time of reinforced concrete columns according to theEuropean standard (Eurocode 2, 2002)The standard fire resistance is defined as the ability of a structure or its part to keep the bearingcapacity during a standard fire exposure, for a specified standard period of time such as 30, 60 or90 min. The standard fire exposure is described by an increasing temperature–time curve of the surround-ing air as experienced in typical hydrocarbon fires. The temperature–time curve given in Eurocode 1(1995) readsT ðtÞ¼ 20 þ 345 log10ð8t þ 1Þ; ð1Þwhere T [ C] is the gas temperature in the fire compartment at time t [min] (see Fig. 10(a)). Eurocode 2(2002) gives a simple formula for the fire resistance time of the reinforced concrete column subjected mainlyto compression and being a member of a non-sway structure. This formula readsR ¼ 120Rgfi þ Ra þ Rl þ Rb þ Rn120 1.8½min ; ð2ÞwhereRgfi ¼ 83 1 lfix þ 1x þ 0.85acc"#;Ra ¼ 1.6ða 30Þ;Rl ¼ 9.6ð5 l0;fiÞ;Rb ¼ 0.09b0; ð3ÞRn ¼ 0 for n ¼ 4 ðcorner bars onlyÞ;12 for n > 4. The reduction factor for the design load level in fire, lfi, is determined by the equationlfi¼ NEd;fiNRd; ð4Þwhere NEd,fi is the design axial load in fire, and NRd is the design resistance of the column at room temper-ature. NRd is calculated by the use of Eurocode 2 (1991) with the partial safety factor cm for the roomtemperature design including the second-order geometrical effects with the initial eccentricity being equalto the eccentricity of NEd,fi.The remaining parameters in Eq. (3) are: a [mm] is the axis distance to the longitudinal steel bars; l0,fi [m]is the effective (buckling) length of the column exposed to fire; b0=2Ac/(b + h) [mm] for the rectangular cross-section whose sides are b and h and Ac is its area; x = As fyd/Acfcd denotes the reinforcement ratio atroom temperature, with As, fyd and fcd being the area of the steel bars, the yield strength of steel and thecompression strength of concrete, respectively; and acc is the coefficient which accounts for the reductionof compressive strength of concrete (acc = 0.85) (Eurocode 2, 1991).3. The thermo-mechanical response and the fire resistance time of a reinforced concrete structure in fire:The description of the computational modelThe present thermo-mechanical analysis of reinforced concrete planar beam structures consists ofconsecutive separate thermal and mechanical time-dependent analyses. In the thermal analysis, we as-sume that the temperature of the surrounding air is a prescribed function of time. We further assumethat the air temperature is spatially homogeneous. Then the heat transfer in the longitudinal directionof the beam can be neglected, and the 2D thermal transient analysis over only a typical cross-section ofthe beam is sufficient to determine the temperature distribution in the whole beam during fire. Note thatthe temperature determined in such a way changes across the section in a very general way. Because wedeal with the planar deformations only, we here assume that the cross-section is symmetrical with re-spect to the plane of deformation, so that temperature is also distributed over the section in a symmet-rical way. Both the temperature and its gradient are important in the mechanical analysis; the formerdictates the values of material moduli, while the latter implies the temperature–driven stresses. In orderto obtain the temperature and its gradients with the sufficient accuracy, the cross-section has to be mod-elled in the thermal analysis with a relatively dense finite-element mesh. We note in passing that thelinear variation of the temperature across the section is often sufficient in the fire analysis of steel struc-tures, see, e.g. Zhao (2000). The steel bars occupy only a small portion of the section and were thereforedisregarded in the thermal analysis. Lie and Irwin (1993) show that the differences in temperature in theconcrete and in the embedded steel bar at the contact are small. The temperature in the steel bar istherefore assumed as being that of concrete at its location. We also disregard the moisture transportand its evaporation and condensation in concrete, although the effect of moisture distribution historiesmay be significant, particularly for a higher initial moisture content. Here we assume that this is not thecase. Then the flame emissivity of fire, er, may have more significant effect than the moisture, which canthus safely be ignored (see Figs. 8–10 in Huang et al., 1996). The effect of moisture, on the other hand,can be very important in high strength concretes where the build-up of the high pore pressure duringheating may cause the spalling of concrete, which results in the loss of strength of concrete, and in theincrease of the transfer of heat to the inner layers of the section (Kodur et al., 2004). The spalling, how-ever, does not seem to be influential enough for the normal strength concrete studied here to beconsidered.Once the time-changing distribution of the temperature over the cross-section has been determined, it isimposed on a planar beam as a thermal load. Along with the self weight of the structure and additionalforces required by the design rules, these loads constitute the time-dependent loading set of the structure.We use our novel finite element formulation to determine the mechanical response of the planar frame (Bra-tina et al., 2003a,b, 2004). The formulation is based on the modified Hu–Washizu functional of the kine-matically exact planar beam theory of Reissner (1972).
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