A simple four-node solid shell element for geometric non-linear static and dynamic analysis

A simple four-node solid shell finite element is developed for geometric non-linear static and dynamic analysis. The solid shell formulation differs from the more common degenerate solid shell formulation. The degenerate solid shell formulation uses rotational angles to describe the kinematics of deformation and requires a constant element thickness during deformation. The solid shell formulation replaces both limitations by using a six-component vector field to describe the locations of the shell mid-surface and outer surfaces (relative to the mid-surface). The vector approach allows thickness deformations. Vector components replace the angular terms thus removing small angle restrictions incurred in the degenerate shell formulation when large rotations are considered.; To accommodate larger displacements and rotations, a geometric non-linear element is developed based on the Hellinger-Reissner principle with an independent strain field. This method requires both geometry-based displacements and assumed strains. The geometry-based displacement field includes internal degrees of freedom so-called bubble functions, which make the element less sensitive to distortion. The choice of assumed strain fields accommodates the deformation modes present in the displacement field, precluding element locking and excluding undesirable spurious kinematic modes.; Following a geometric non-linear static formulation, the capabilities are extended to include dynamic analysis, by conserving energy between successive time steps intervals. An algorithm to obtain incremental time derivatives of the vector components is developed. Time variations of the derivatives are integrated using an averaged acceleration model.; Numerical examples of the finite elements developed herein for both static and dynamic analyses are presented. A series of classic benchmark test cases for both linear and non-linear analyses are analyzed. All elements developed typically show good convergence to either exact or referenced solutions. Extreme skew angles caused some difficulties when converging on a singularity, though.; The vector-based displacement field used in the solid shell element allows larger displacement increments than the degenerate shell approach. Extremely large increments are possible in a single step when performing geometric non-linear static analyses. This research also demonstrates that large increments are allowed when considering dynamic analyses. The solution appears stable at relatively large time steps. As anticipated with larger time steps, though, there is some high-frequency truncation.