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PROCEEDINGS OF THE ELEVENTH INTERNATIONAL CONFERENCE ON CIVIL, STRUCTURAL AND ENVIRONMENTAL ENGINEERING COMPUTING
Edited by: B.H.V. Topping
The Modular Group Supermatrix Procedure in Structural Analysis
Serbian Academy of Sciences and Arts, Beograd, Serbia
G.M. Zlokovic, "The Modular Group Supermatrix Procedure in Structural Analysis", in B.H.V. Topping, (Editor), "Proceedings of the Eleventh International Conference on Civil, Structural and Environmental Engineering Computing", Civil-Comp Press, Stirlingshire, UK, Paper 135, 2007. doi:10.4203/ccp.86.135
Keywords: structural analysis, computing methods, group supermatrix procedure, theory of groups, G-invariant subspaces, finite element analysis.
Modular structuring is a novel contribution to the group supermatrix procedure developed by the author in his books and papers on structural analysis (vibration, stability and statics) and finite element analysis. The procedure applies symmetry groups and G-invariant subspaces for systems with symmetry properties and also for nonsymmetrical systems. In comparison with conventional procedures, the main advantage of the group supermatrix procedure lies in formulations of much smaller shape functions and matrices, which provides large reductions in the amount of computation.
The new enhancement of the group supermatrix procedure is accomplished by developing a group supermatrix chain of symmetry groups C2, D2, D2h, which is utilized for the modular structuring of the procedure. This chain displays the common pattern of cumulative relationships of group character tables, elements, shape functions and boundary conditions, as well as of group supermatrices in normal and diagonal forms.
In the conventional procedure of the finite element analysis, as shown in the example of a square frame, a fourth part of the frame was modeled and the system with three stiffness equations obtained, which was the greatest possible utilization of symmetry in the conventional way. However, the maximum possible utilization of symmetry is achieved by formulating the problem in G-invariant subspaces.
Solution by the group supermatrix procedure is accomplished by superposing solutions for symmetry adapted loads in particular G-invariant subspaces, where their independent sets of equations are obtained after assembling the stiffness matrices. In examples of square and cubic frames, the alternate method of this procedure provides the solution by basis transformation TDT-1 of the diagonal matrix D of values of load members from the subspaces, where T is the group transformation matrix. The entire load of the system is composed of sets of unit forces that pertain to particular symmetry types of the subspaces. In the diagrams in the subspaces loaded bars with fixed and not fixed ends are located and their displacements and rotations taken from tables of load members, each bar having its ends fixed or not fixed in dependence of symmetry properties of the particular subspace.
The group supermatrix modular structuring defines positive directions of axes parallel to global coordinate axes by rules that produce the same directions or contrary to those of the global coordinate axes. Each node of a structure in a subspace possesses its own set of positive directions of displacements and rotations. This is different in comparison with the conventional analysis, where positive directions of axes in local coordinate systems, which are parallel to global coordinate axes, have the same directions as respective global coordinate axes.
In comparison with the conventional procedure, in referenced books and papers of the author, analyses of complex structures exhibit large reductions in many aspects (formulation, input of data, necessary memory of the computer, amount of computation and duration of working time of the computer).
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