Superconvergence and error estimation of finite element solutions to fire-exposed frame problems

Abstract

When a fire reaches the point of flashover the hot gases inside the burning room ignite resulting in furnace-like conditions. Thereafter, the building frame experiences temperatures sufficient to compromise its structural integrity. Physical and mathematical models help to predict when this will happen. This thesis looks at both the thermal and structural aspects of modelling a frame exposed to a post-flashover fire. The temperatures in the frame are calculated by solving a 2D heat conduction equation over the cross-section of each beam. The solution procedure uses the finite element method with automatic mesh generation/adaption based on the Delaunay triangulation process and the recovered heat flux. With the Euler-Bernoulli assumption that the cross-section of a beam remains plane and perpendicular to the neutral line and that strains are small, an error estimator, based on the work of Bank and Weiser [9], has been derived for finite element solutions to small-deformation, thermoelastic and thermoplastic frame problems. The estimator has been shown to be consistent for all finite element solutions and asymptotically exact when the solution involves appropriate higher degree polynomials. The asymptotic exactness is shown to be closely related to superconvergence properties of the approximate solution in these cases. Specifically, with coupled bending and compression, it is necessary to use quadratic approximations, instead of linear, for the compression and twisting terms to get a global O(h2) rate of convergence in the energy norm, some superconvergence properties and asymptotic exactness with the error estimator.