# Shell Formulations

QPH shell is the Belytschko Leviathan ^{1} shell for linear models or quasi-static
analysis is identical to a QPPS shell analysis, only one difference being explained in Fully-integrated Shell Element QBAT.

The QPPS shell is a
new One-point Quadrature, General Nonlinear Quadrilateral Shell Element with Physical
Stabilization. This shell is a Belytschko Leviathan ^{1} shell modified by Zeng and Combescure. ^{2}

The physical stabilization is applied which enables the explicit evaluation of the stabilizing forces based on the general degenerated shell formulation and which does not require any input parameters. An optimized choice of the moduli is made in order to compute the stabilized forces for nonlinear material so that element's behavior is improved with respect to similar physical stabilization elements. The cost efficiency of the element is demonstrated by numerical examples, as compared with a fully-integrated 4-node element.

The QEPH shell is a new improved element with respect to QPH,QPPS. The improvements will be explained in Fully-integrated Shell Element QBAT. As the QEPH is very efficient, it replaces QPH,QPPS in the applications.

The QBAT shell is a new
fully-integrated 4-node element based on Q4
$\gamma $
24 shell of Batoz and Dhatt ^{2} as discussed in Fully Integrated Formulation.

The general formulation of the degenerated continuum quadrilateral shell (for which all these elements used) is given in General Degenerated 4-Node Shell Formulations. The difficulties in evaluating the stabilized stiffness are also described. Fully-integrated Shell Element QBAT presents the detailed formulations for the one-point quadrature shell element based on the general formulation, and compares it with that of Belytschko and Leviathan.

^{1}Belytschko T. and Leviathan I., “Physical stabilization of the 4-node shell element with one-point quadrature”, Computer Methods in Applied Mechanics and Engineering, 113:321-350, 1992.

^{2}Zeng Q. and Combescure A., “A New One-point Quadrature, General Nonlinear Quadrilateral Shell Element with Physical Stabilization”, Int. Journal Num. Methods in Engineering 42, 1307-1338, 1998.