Reduced minimization of product and non-product type in Mindlin plate elements

This paper presents concepts of two-dimensional reduced minimization of product type and non-product type: the analytical reduced minimization theory in one dimension when extended to that of Mindlin plate elements reveals that the two-dimensional reduced minimization for Lagrangian plate elements can be obtained by the successive use of the one-dimensional reduced minimization. But this relation does not hold for serendipity elements due to an incompleteness of displacement functions. This paper examines e¤ects of various integrations of shear strain energy by applying the one-dimensional successive reduced minimization to Lagrangian and serendipity plate elements. The results show why the selective 2]3 and 3]2 rule for the transverse shear strain energies of an 8-noded plate element cannot eliminate locking and why a conventional 12-noded plate element of serendipity family does not have one-order-lower, optimal Gauss strain locations