Unique decomposition of element stiffness numeric matrices for three-noded plate bending triangles

This work is concerned with the numeric sti ness matrices of three-noded triangular plate bending nite elements; in particular with those numeric sti ness matrices, which are freedom-de cient and comply with the conditions of the patch test. Subsequent to initial transformation of the rotation connectors for such matrices it is evident that there must exist an unique decomposition to the sti ness matrix K6 2R6 6 of the corresponding Kirchho six- noded constant bending moment triangle. In K6 all six trial functions are themselves synonymous with those which describe the patch test. The transformation matrices of decomposition, and subsequent restoration upon modi cation or design, are derived explicitly and are succinct in application. Decomposition of the numeric sti ness matrix leads to exceptional versatility in objective modi cation, e.g. design of the matrix by adaptive process. Attention is con ned here to the sti ness matrices K9 2R9 9 of nine-degree-of-freedom three-noded Kirchho plate bending triangles with their single-degree-of-freedom de ciency. The decomposition of the element sti ness matrix immediately reveals those six coe cients which are available for design. They control the e ect of transverse shear and are the constituents of a symmetric positive-de nite matrix M3 2R3 3 which is designated the `mechanism restraintʹ matrix. It is necessary only that the designed coe cients are such that the matrix M3 remains symmetric and positive de nite so as to ensure retention of patch test satisfaction on restoration to the newly designed K9. The illustrative examples provide a rst perception of the leap in expectation which is enabled by design of the numeric K9 when uninhibited by formal method. Thus, the feasibility is illustrated of simple adaptive design of K9 with objective to recover cubically varying w displacements over an equilateral patch of equal triangles. This recovery is readily achieved by ad hoc inverse method but raises the issue of uniqueness in design. In highlighting the characteristics of M3 it is evident that there remains a wealth of opportunity for further research before adaptive design of the element sti ness matrix, within an arbitrary prevailing w displacement eld, can become a practical reality. An appendix lists the Fortran computer codings which are used in the examples to calculate the sti ness matrix K6 of the six-noded constant bending moment Kirchho triangle as well as the explicit transformation matrices for decomposition and restoration of the numeric K9 sti ness matrix