Keywords: buckling, critical temperature, column, fire, stability, steel.

The present article presents a new analytical procedure for the determination of the critical temperature of a straight, geometrically perfect and axially loaded steel column exposed to fire. A series of standard simplifications and assumptions need to be introduced, however, to enable the analytical solution to be derived. In particular, we assume that a steel column can be realistically modelled by a kinematically exact planar beam model of Reissner [1] neglecting the effect of shear strain. Next, we assume a non-linear, temperature dependent material law, which accounts for both viscous and plastic strains. The mathematical expressions for the stress-strain law of steel at high temperatures are taken from EC3 [2] along with the explicit expressions for temperature-dependent material parameters. As the walls of the steel sections are thin, we further assume that the temperature field in the column is uniform, but somewhat delayed with regard to temperature of the surrounding gas. After the thermo-mechanical equations are set up, the fundamental equilibrium solution of the column is obtained and the set of linearized equations at the fundamental equilibrium state are derived. The condition for the existence of the non-trivial solution of the linearized equations supplies us with the value of the critical temperature. Considering that steel at high temperature behaves in accordance with the material model proposed by European standard EC3 [2], the critical temperature is determined exactly.

Numerical examples show that the critical temperature highly depends on both the slenderness of a column and the material model of steel at elevated temperatures. While the effect on the critical temperature is substantial, the shape of the buckling mode remains practically unchanged. We can also see that the simplified method proposed by EC3 can be unsafe for moderate slendernesses. This may be due to the fact that method EC3 ignores the fact that the tangent modulus depends on deformation and hence remains constant if the temperature is constant. Namely, once deformation is in the non-elastic range, the tangent modulus significantly decreases. Because the critical temperature depends on the tangent modulus, it also decreases when the tangent modulus decreases.

1

E. Reissner, "On one-dimensional finite-strain beam theory: The plane problem", Journal of Applied Mathematics and Physics (ZAMP), 23, 795-804, 1972. doi:10.1007/BF01602645

2

Eurocode 3, "Design of Steel Structures, Part 1.2: Structural fire design (draft)", European Committee for Standardization, 2003.

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