The MITC9 shell element in plate bending: mathematical analysis of a simplified case

We consider the 9-node shell element referred to as the MITC9 shell element in plate bending solutions and present a simplified mathematical analysis. The element uses bi-quadratic interpolations of the rotations and transverse displacement, and the “rotated Raviart-Thomas” interpolations for the transverse shear stresses. A rigorous mathematical analysis of the element is still lacking, even for the simplified case of plate solutions (that is, flat shells), although the numerical evidence suggests a good and reliable behavior. Here we start such an analysis by considering a very simple particular case; namely, a rectangular plate, clamped all around the boundary, and solved with a uniform decomposition. Moreover, we consider only the so-called limit case, corresponding to the limit equations that are obtained for the thickness t going to zero. While the mathematical analysis of the limit case is simpler, such analysis, in general, gives an excellent indication of whether shear locking is present in K. J. Bathe (B) MIT, Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA e-mail: [email protected] F. Brezzi · L. D. Marini Istituto di Matematica Applicata e Tecnologie Informatiche (IMATI), Consiglio nazionale delle Ricerche (CNR), Via Ferrata 5/A, 27100 Pavia, Italy F. Brezzi Centro di Simulazione Numerica Avanzata (CeSNA), Instituto Universitario di Studi Superiori (IUSS), Lungoticino Sforza 56, 27100 Pavia, Italy L. D. Marini Dipartimento di Matematica, Università di Pavia, Via Ferrata 5/A, 27100 Pavia, Italy the real case t > 0. We detail that the element in the setting considered shows indeed optimal behavior.