Unique decomposition of element stiffness numeric matrices for three-noded plate bending triangles
✍ Scribed by L. S. D. Morley
- Publisher
- John Wiley and Sons
- Year
- 1998
- Tongue
- English
- Weight
- 246 KB
- Volume
- 43
- Category
- Article
- ISSN
- 0029-5981
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✦ Synopsis
This work is concerned with the numeric sti ness matrices of three-noded triangular plate bending ÿnite elements; in particular with those numeric sti ness matrices, which are freedom-deÿcient 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 K 6 ∈ R 6×6 of the corresponding Kirchho sixnoded 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 modiÿcation or design, are derived explicitly and are succinct in application. Decomposition of the numeric sti ness matrix leads to exceptional versatility in objective modiÿcation, e.g. design of the matrix by adaptive process. Attention is conÿned here to the sti ness matrices K 9 ∈ R 9×9 of nine-degree-of-freedom three-noded Kirchho plate bending triangles with their single-degree-of-freedom deÿciency.
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-deÿnite matrix M3 ∈ R 3×3 which is designated the 'mechanism restraint' matrix. It is necessary only that the designed coe cients are such that the matrix M3 remains symmetric and positive deÿnite so as to ensure retention of patch test satisfaction on restoration to the newly designed K9.
The illustrative examples provide a ÿrst 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 ÿeld, can become a practical reality.
An appendix lists the Fortran computer codings which are used in the examples to calculate the sti ness matrix K 6 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. ?