Application of the finite element method for thermal calculations of unprotected steel building structures within the framework of development of design documentation for their fire protection

Introduction. The application of modern computer technologies made it possible to achieve high practical and economic results in the construction of modern buildings and structures. This is due to the fact that modern software complexes allow to predict with sufficient accuracy the behaviour of building structures under the influence of various factors, including high-temperature influence. It should be noted that in the overwhelming majority of modern software systems, the most widespread are grid methods for solving the Fourier differential equation of heat conduction, namely the finite element method. The calculation result obtained using the finite element method depends on various factors that may not always be obvious when solving a particular problem, but in order to obtain the necessary accuracy of the solution, they must be taken into account in the modelling process.

Aims and Objectives. The aim of the work is to assess the convergence of the numerical solution of the Fourier differential heat conduction equation by the finite element method when performing the thermal calculation of heating of unprotected steel building structures within the framework of the development of design documentation for their fire protection, as well as validation of the obtained results of mathematical modelling with the known results of calculation and analytical solutions.

Methods. A steel column of I-beam section of profile No. 20 according to GOST 8239 was taken as a modelling object. Modelling of heating of the investigated structure by the finite element method was carried out without the use of fire protection when simplifying the dimensionality of the problem from three-dimensional to two-dimensional. Verification of the obtained modelling results was performed by the criterion of convergence of the numerical solution of the problem at the modelled time interval (60 min) at the sequence of meshes (three meshes with the number of degrees of freedom (DOFs): 200, 2,084, 7,102) and time steps (ten time steps: 0.05, 0.1, 0.5, 1, 2, 3, 5, 10, 15, 30 s). Validation of the obtained results was performed by comparison with the results of thermal calculations of steel structures, set out in the book “Calculation of fire resistance of building structures” by A.I. Yakovlev.

Results and Discussion. As a result of carrying out a series of thermal calculations of heating of the studied steel building structure with different grid steps and time steps, it was found that the modelling time step had a greater influence on the convergence of the obtained results than the grid step. At the same time, the change in the convergence of the obtained results in the studied time interval for all modelling variants occurred unevenly, namely: at the beginning of the modelled time interval, the difference of the obtained temperatures in the cross-section of the structure first increased and then decreased. Comparison of the obtained results with the results of thermo­technical calculations stated in the book by A.I. Yakovlev showed that the obtained average temperature in the cross-section of the structure was lower than the temperature stated in the book by A.I. Yakovlev, while the difference between the obtained times of reaching the critical temperature (450–750 °C) increases both with increasing the value of the critical temperature and with increasing the reduced thickness of the metal.

Conclusion. The assessment of convergence of the obtained results of modelling the heating of steel building structure by the finite element method and their validation with the known design and analytical solutions have shown that the application of the finite element method in the performance of thermal calculations within the development of design documentation for fire protection of steel building structures has its own features, which must be taken into account to obtain the required accuracy of the solution.

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