The Morley plate element in the frame of a generalized, frequency-dependent hybrid finite element formulation

Resumo

The Morley plate element is the simplest triangular finite element for homogeneous, isotropic material,
and represents constant bending curvature/moment exactly, as the flexural counterpart of the membrane, constant
strain/stress finite element. It has six degrees of freedom: three corner-node transversal displacements and three
edge rotations. We propose a slightly modified, improved Morley element based on a frequency-dependent hybrid
finite element formulation to be used in the frame of a generalized modal analysis for stiffness and mass matrices
given as frequency power series. The domain stress solution satisfies the homogeneous elastodynamic equilibrium
equations for moderately thick plates, as we resort to the concept of mean transversal shear distortion proposed
in a previous conference contribution (PANACM/CILAMCE 2021). We show that the formulation for just one
mass matrix corresponds to a plain displacement formulation, as proposed in the literature for the thin-plate, static
problem (although introducing some due corrections). Some numerical tests with one and two mass matrices show
that the model can be seamlessly applied to both moderately-thick and thin plate problems – thus without the
shear-locking inconvenience – and in spite of its shape-function simplicity ensures good, asymptotic convergence
for natural frequencies. As we have a similar generalized modal development for the membrane triangle, this leads
to the simplest – and consistently – conceivable shell element.