Benchmark example problems for beams at elevated temperatures

The paper presents development of a series of solutions for beams at elevated temperatures which are supposed to serve as benchmark problems for applications of computational models in fire structural engineering. Three cases of loading i.e. pure bending, central force, and uniformly distributed loading, are considered for a simply supported, and fixed on both ends beams at uniformly distributed elevated temperature varying in time. The results are provided in terms of the midspan deflection for specified loading levels and temperatures. The results mainly obtained using finite element (FE) models and two commercial codes, are verified through comparison with analytical solutions for simplified cases and through parametric study aimed to examine the effect of modelling parameters. The numerical results are subjected to mesh density study using the grid convergence index (GCI) concept

Invariant subspaces of X** under the action of biconjugates

summary:We study conditions on an infinite dimensional separable Banach space $X$ implying that $X$ is the only non-trivial invariant subspace of $X^{**}$ under the action of the algebra $\mathbb{A}(X)$ of biconjugates of bounded operators on $X$: $\mathbb{A}(X)=\lbrace T^{**}\: T \in \mathcal {B}(X)\rbrace$. Such a space is called simple. We characterize simple spaces among spaces which contain an isomorphic copy of $c_{0}$, and show in particular that any space which does not contain $\ell _1$ and has property (u) of Pelczynski is simple