Flexibility-based finite element analysis for structures with random material properties

In this paper, we propose a flexibility-based finite element formulation for beams with stochastic bending stiffness. We formulate the element-level finite element equilibrium equation in terms of the element flexibility, so that the stochasticity appears on the right side of the equilibrium equation as the random flexibility. As a result, the unknown internal forces, which are dependent on the stochastic bending stiffness for statically indeterminate beams, appear explicitly in the global finite element equilibrium equations. The internal forces associated with the mean stiffness are used to approximate the unknown internal forces in the process of computation. With the new formulation, the mean and covariance of the displacement for stochastic beams can be directly calculated in terms of mean and covariance function of the flexibility, which can be evaluated from one-dimensional probability density function and the correlation function of the stiffness by the Monte Carlo simulation. The numerical example is given to illustrate the advantages of the new formulation over the conventional perturbation solution.