The principles of the matrix theory of linear structural analysis are based on a few simple hypotheses. The techniques of using these, and their adaption to the solution of specific problems, with the aid of an electronic digital computer, are varied, and it has been the purpose of this thesis to examine one of these techniques, namely that expounded by Professor J.H. Argyris of Imperial College, London. Recognizing the force and the displacement methods as a fundamental subdivision of the whole, the thesis is similarly partitioned into discussions of these topics. The thesis begins in Chapter 1 with a brief recapitulation of some of the concepts of continuum mechanics as applied to a stressed, Hookian media. Although this approach is useful in illustrating that the infinitesimal element is, in effect, the structural element of the continuous media comparable with the member of the structure as defined in Chapter 2, the summation sign being replaced by the integral sign, it has not been pursued further here. Rather, it is used as that foundation on which the remainder of the work can be justified should the need arise. The originality of the thesis must thus rest on the chapters 2, 3 and 4. Much of this work has been published and the relevant publications are given on page 315. Copies of those publications available are included with this submission.
The main topics of Chapter 2 are the treatment of elastic strain energy and its adaption for the calculation of generalized flexibility matrices; the application of these generalized flexibility matrices to specific problems, and the matrix force method for the calculation of Euler Loads of slender elastic columns. An essential feature of the work is that methods are carried to that degree of refinement necessary for practical application. Thus, for example, the programme for calculating bTfb is written (a) to accommodate the maximum size of the resulting flexibility matrix possible in the computer available, (b) to minimize the data input of the b and f matrices by suitable partitioning. This programme is then adapted as a standard part of the indeterminate analysis programme (Section 2.10) and of the displacement analysis programme (Section 3.5). It is gratifying to observe that these programmes have found a ready acceptance by the engineering profession. A study of generalized forces offers many avenues for research and here they have been used in the calculation of influence lines for redundant moments of continuous beams and in the stress analysis of folded plate structures. Special computer programmes have been developed for both of these topics, though for brevity they are omitted from this thesis. Although the thesis is devoted entirely to expounding the facility of the matrix methods of analysis, it is well to observe that for the calculation of beam deflections Simpson's Rule has been used to integrate the pertinent moment-inertia functions.
The result of the folded plate analysis is pleasing in that it leads immediately to an approximate analysis of barrel vault shells and to the extension of the analysis to continuous structures of this type.
The calculation of Euler Loads appears to be a useful innovation. Non-uniform members are handled with no effort, as illustrated in Section 2.13, and the analysis is extended to statically indeterminate structures. Its disadvantages are that large memory storage is required and that there is no indication given as to the magnitude of the second Euler Load. By using two different intervals of subdivision of those members in compression and Richardson's Extrapolation procedure on the two solutions so obtained, values very close to the theoretically exact Euler Loads are obtained.
Chapter 3 is devoted to a discussion of the displacement method of analysis. Apart from elementary topics parallel to those of the beginning of Chapter 2, the emphasis is on the tensorial nature of coordinate transformations and on the analysis of multi-storey building frames. An automatic method of analysis is readily built up if one recognises that distortions of a member due to a displacement of one of its ends due to a displacement at another point can be obtained by three matrix multiplications. These perform the transformations (a) from the given point to the member end; (b) from frame coordinates at the member end to member coordinates; (c) from displacements in member coordinates to member distortions. This method is exploited fully in the analysis of the planar pile headstock, Section 3.11. (The programme is omitted for brevity.) The extension to the three dimensional pile headstock and to the general frame are outlined. Finally, a method of partitioning of multi-storey building frames is devised and the results given from the ensuing approximate method of analysis.
The first topic of Chapter 4 is the application of matrix theory for an analytic explanation of the iterative process of moment distribution as devised by Hardy Cross. Briefly, this entails two separate processes; (a) an iterative procedure for calculating rotations of a structure with non-translatory joints; (b) a procedure for calculating joint translations of the structure by matrix partitioning. The remaining Section 4.2 gives the matrix structure of the finite difference equations developed in the approximate analysis of beams on elastic foundations. ..................................