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The only unknown functions in the functional, theextensional strain, e, and the pseudo-curvature, j, are approximated by the Lagrangian interpolationscheme. The remaining unknown functions, i.e. displacements, the rotation and the internal forces and mo-ments, appear in the functional only through their boundary values. Our finite elements differ to the estab-lished ones also because they enforce the constitutive and equilibrium internal forces to be equal at theintegration points (Vratanar and Saje, 1998). We use the fibre-based constitutive equations. We assumethe conformity of extensional strains of concrete and steel at their contact. Note that our formulation assumes exact kinematical equations of the beam and is thus not only materially, but also geometricallynon-linear in an exact sense. The theoretical basis of the formulation and its finite element implementationdetails have already been described to some extent (Bratina et al., 2003a,b, 2004; Planinc et al., 2001)andalarge number of the check-analyses have been performed by now proving high accuracy, economy androbustness of the formulation at room as well as at high temperatures. That is why we here present onlythose details that are relevant to the present discussion. The material models used in the analysis aredescribed in Section 3.2.3.1. The system of discretized equations of the structureThe system of the discretized equations of the structure takes the form G(x,k,T, t)= 0 where G denotesthe non-linear algebraic vector function of the nodal unknowns, x, the mechanical loading factor, k, thetemperature, T, and time, t. It is solved by Newton s incremental–iterative method. The solution time inter-val is pided into time steps [tj, tj+1], j =0,1, ... In each step j + 1, the mechanical loading factor increment,Dk, and the temperature increment, DT, are prescribed, and the iterative (i =0,1,2, ... ,n) corrections of theunknowns of the problem, dxjþ1iþ1 , are determined by Newton s method from the system of the linearized dis-cretized equationsrxGðxjþ Dxjþ1i; kjþ1; T jþ1; tjþ1Þdxjþ1iþ1 ¼ Gðxjþ Dxjþ1i; kjþ1; T jþ1; tjþ1Þð5Þwhere rxG Kjþ1Tiis the current tangent stiffness matrix of the structure. The values of the unknowns atthe end of time step [tj, tj+1] are determined by the equationxjþ1¼ xjþX ni¼0dxjþ1iþ1 .When the tangent stiffness matrix becomes singular (detKTi = 0), the structure reaches its critical statewhich is either the limit or the bifurcation point. This state is described by the quadruple xcr, kcr, Tcr, andtcr, and is assumed to represent the bearing capacity of the structure (the collapse). The related time andthe related temperature are termed the fire resistance time and the critical temperature . During a typicalfire analysis, the mechanical loading factor is kept constant (k = const.), while the temperature rises untilthe collapse takes place at the critical temperature. The singularity of the tangent stiffness matrix indi-cates the global instability of the structure. In addition to the global instability, the local instabilitymay take place during fire. This is defined as the state at which the determinant of the tangent consti-tutive matrix of a cross-section becomes zero (Bratina et al., 2003a, 2004). Once the tangent constitutivematrix becomes singular, the cross-section experiences the strain-softening in the subsequent progressivedeformation while its neighbouring cross-sections have to start the unloading. If strained in the softeningregime, the cross-section is unstable. When the structure is statically determinant and if the geometricallynon-linear effect is not dominant, the local instability of the cross-section instantaneously causes the glo-bal instability of the structure. Notice that concrete has a relatively large strain-softening capability, inparticular at high temperatures, so both mechanisms of the instability have been considered in ourformulation.Remark. The critical temperature (or the collapse) may be defined in various ways. One of the morepopular conditions is that the maximum displacement in the structure reaches a prescribed large value.Note that this condition and the detKT = 0 condition employed here do not yield the same result for thecritical temperature. 3.2. Mechanical properties of concrete and steel. Strain and stress incrementsThe geometric, i.e. total extensional strain increment DD of a generic material fibre is assumed to be thesum of increments of elastic, DDe, plastic, DDp, thermal, DDth, creep, DDcr, and transient strain increment,DDtr, the latter being non-zero in concrete and vanishing in steel:DD ¼ DDe þ DDp þ DDth þ DDcr þ DDtr. ð6ÞThe sum of elastic and plastic parts of the strain will be termed the mechanical strain, Dr = De + Dp.Weassume that the relationship between the mechanical part and the longitudinal normal stress, r, is given bythe constitutive law r ¼ FðDr; T Þ, where F is a functional pertinent to the chosen material. In the presentfire analysis, we use the constitutive laws of concrete and reinforcing steel as suggested by Eurocode 2(2002). The graphs of the relationships are depicted in Fig. 1. The figure puts it clear that the increase intemperature decreases the strength of material and increases its ductility. Both materials, concrete andsteel, exhibit the extensive strain-softening in the post-strength regime. Once functional F is given, thestress increment Drj+1in the time step [tj, tj+1] is given by the relation Drjþ1¼ rjþ1 rj¼ FðDjþ1r ;T jþ1Þ FðDjr; T jÞ. The unloading is assumed as being elastic with an elastic modulus taken at the currenttemperature. An isotropic model of the strain-hardening is assumed in the loading–unloading cycles.The thermal strain in concrete, Dth,c, is assumed to be a function of the current temperature and is givenby the relation Dth,c = s(T). The approximation of functional s is defined in Eurocode 2 (2002) and is alsoadopted here. The thermal strain increment in time step [tj, tj+1] is thus determined by the equationDDjþ1th;c ¼ sðT jþ1Þ sðT jÞ¼ Djþ1th;c Djth;c.The concrete creep strain, Dcr,c, is assumed to be a function of the current stress, time and temperature.Here we employ the model proposed by Harmathy (1993)Djþ1cr;c ¼ b1rjþ1cfjþ1cTðtjþ1Þ1=2edðT jþ1 293Þ; ð7Þwhere fjþ1cT is strength of concrete at temperature Tj+1[K], tj+1[s] is time and b1 [s 1/2] and d [K 1] areempirical constants of material. The least square method analysis of the data of creep tests, performedby Cruz (1968), gives b1 = 6.28 · 10 6s 1/2and d = 2.658 · 10 3K 1. Fig. 2 shows the comparisons be-tween the experimental and analytical results using (7) for creep strains at various temperatures in the rangefrom 24 C to 649 C. Note that the agreement is satisfactory.The creep strain increment DDcr,cj+1in time step [tj, tj+1] is given by the equation DDjþ1cr;c ¼ Djþ1cr;c Djcr;c. The transient strain in concrete (Dtr,c) has been found to have an important influence on mechanicalbehaviour of concrete during the first heating of concrete (see Anderberg and Thelandersson, 1976). It isirrecoverable and is the result of the physico-chemical changes that take place only under the first heating.Formally, it may be defined as the part of the total strain obtained in stressed concrete under heating thatcannot be accounted for otherwise. In our formulation we use the transient strain model of Anderberg andThelandersson (1976)DDjþ1tr;c ¼ k2rjþ1cfc0DDjþ1th;c ; ð8Þwhere fc0 is strength of concrete at the room temperature; k2 is a dimensionless constant whose value rangesfrom 1.8 to 2.35. This model assumes that the change of the transient strain is linearly dependent on thechange of the current thermal strain.The thermal strain of steel,
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