Mathematical Analysis of the Euler-Bernoulli Beam Subject to Swelling Pressure

Abstract

instability. Due to the complexity of interactions between expansive solid and solid-liquid equilibrium,
the forces exerting on retaining structures from swelling are highly nonlinear. In this paper,
we consider the initial/boundary value problem of an Euler-Bernoulli elastic beam subject to the
swelling pressure with one end clamped and another end free. We are interested in establishing
and validating a mathematical model for dynamic deflections of an elastic Euler-Bernoulli beam
with constant cross-sectional area subject to swelling pressure and some initial and boundary conditions.
We built a sequence of functions by using the Galerkin approximation method and the
eigenfunctions of the corresponding 4th order eigenvalue problem. It has been showed that the
sequence of solutions to the ODE systems converges to the unique solution and that the weak solution
is also a classical solution. This work is our initial attempt to study a semi-linear hyperbolic
problem based on the Euler-elastic beam theory and some simplistic swelling pressure model in
soil and rock mechanics.