Probabilistic analysis of a cantilever beam subjected to random loads via probability density functions

Abstract

This paper addresses the probabilistic analysis of the deflection of a cantilever beam by means of a randomization of the classical governing fourth-order differential equation with null boundary conditions. The probabilistic study is based on the calculation of the first probability density function of the solution, which is a stochastic process, as well as the density function of further quantities of interest associated with this engineering problem such as the maximum slope and deflection at the free end of the cantilever beam, that are treated as random variables. In addition, the probability density function of the bending moment and the shear force will also be computed. The study takes extensive advantage of the so called Random Variable Transformation method, also known as Probability Transformation Method, that allows us to fully unify the probabilistic analysis in three relevant cases commonly studied in the deterministic setting. All the theoretical findings are illustrated via detailed numerical examples corresponding to each one of the three scenarios.

Branching Processes for Epidemics Study

“…The mass function allows to represent the degree of knowledge of the possible underlying values of a random entity. The latter entails all the probabilistic information and hence finding it is the better option to characterize the problem of the random object regardless the context [17]. The pmfs in this context allow us also to understand the stochastic evolution of the disease over time, evaluate statistics, check the influence of the contagion's random variable and obtain the extinction probability of the disease.…”

“…ds (1) s=1 . (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20) This numerical approach is a time-dependent and not a local methodology. Once the number of experiments n r is determined, each realization of the branching process gives values of infected members per generation.…”

“…Its expectation value is E (X 4 ) = 19.448 according to Eq. (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20). For a tolerance of ξ = 0.001, the sample statistic ξ4 from 48518 realizations attempts is acceptable.…”

“…The likelihood functions were built according to Eq. (4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18). Before discussing these results, the mode of P paths are presented in Figure 4.15.…”