Closed-form solutions and uncertainty quantification for gravity-loaded beams

Typically, the cantilever non-uniform gravity-loaded Euler–Bernoulli beams are numerically modeled as the governing equation for free vibration analysis does not yield an exact solution. We show that, for certain polynomial variations of the mass and stiffness, there exists a fundamental closed form solution to the fourth order governing differential equation for gravity-loaded beams. An inverse problem approach is used to find an infinite number of such beams, with various mass and stiffness distributions, which share the same fundamental frequency. The derived distributions are demonstrated as test functions for a p-version finite element method. The functions can also be used to design gravity-loaded cantilever beams having a pre-specified fundamental natural frequency. Examples of such beams with rectangular cross section are presented. The bounds for the pre-specified fundamental frequency and its variation for beams of different lengths are also studied. In presence of uncertainty, this flexural stiffness is treated as a spatial random field. For known probability distributions of the natural frequencies, the corresponding distribution of this field is found analytically. This analytical solution can serve as a benchmark solution for different statistical simulation tools to find the probabilistic nature of the stiffness distribution for known probability distributions of the frequencies.