Extending the analysis of the Euler-Bernoulli model for a stochastic static cantilever beam: Theory and simulations

In this paper, we present a comprehensive probabilistic analysis of the deflection of a static cantilever beam based on Euler-Bernoulli's theory. For the sake of generality in our stochastic study, we assume that all model parameters (Young's modulus and the beam moment of inertia) are random variables with arbitrary probability densities, while the loads applied on the beam are described via a delta-correlated process. The probabilistic study is based on the calculation of the first probability density function of the solution and the probability density of other key quantities of interest, such as the shear force, and the bending moment, which are treated as random variables too. To conduct our study, we will first calculate the first moments of the solution, which is a stochastic process, and we then will take advantage of the Principle of Maximum Entropy. Furthermore, we will present an algorithm, based on Monte Carlo simulations, that allows us to simulate our analytical development computationally. The theoretical findings will be illustrated with numerical examples where different realistic probability distributions are assumed for each model random parameter.