Thin structural theories such as the shear-deformable Timoshenko beam and Reissner-Mindlin
plate theories have seen wide use throughout engineering practice to simulate the response of
structures with planar dimensions far larger than their thickness dimension. Meshless methods
have been applied to construct numerical methods to solve the shear deformable theories.
Similarly to the finite element method, meshless methods must be carefully designed to overcome
the well-known shear-locking problem. Many successful treatments of shear-locking in
the finite element literature are constructed through the application of a mixed weak form. In
the mixed weak form the shear stresses are treated as an independent variational quantity in
addition to the usual displacement variables.
We introduce a novel hybrid meshless-finite element formulation for the Timoshenko beam
problem that converges to the stable first-order/zero-order finite element method in the local
limit when using maximum entropy meshless basis functions. The resulting formulation is free
from the effects shear-locking.
We then consider the Reissner-Mindlin plate problem. The shear stresses can be identified as
a vector field belonging to the Sobelov space with square integrable rotation, suggesting the use
of rotated Raviart-Thomas-Nedelec elements of lowest-order for discretising the shear stress field. This novel formulation is again free from the effects of shear-locking.
Finally we consider the construction of a generalised displacement method where the shear
stresses are eliminated prior to the solution of the final linear system of equations. We implement
an existing technique in the literature for the Stokes problem called the nodal volume
averaging technique. To ensure stability we split the shear energy between a part calculated
using the displacement variables and the mixed variables resulting in a stabilised weak form. The method then satisfies the stability conditions resulting in a formulation that is free from
the effects of shear-locking.Open Acces
