Methods of analysis of periodic structures

A number of existing methods, suitable for the analysis of periodic structures are reviewed. Most of these methods are related to the well known Transfer Matrix Method (TMM). It is seen that TMM, though efficient, suffers from a number of numerical stability problems which will prevent a solution being obtained in certain circumstances. Two lines of approach are selected for further development which make both numerical stability and solution efficiency possible. The first is an algebraic solution of the governing difference equations known as the Matrix Difference Method (MDM). Existing closed form solutions by MDM are given for box girders and for beams on elastic supports with axial and transverse loading. The latter is used for a parametric study and in the analysis of U-frame bridges. MDM is also used to investigate the characteristics of vierendeel girders, tall building frames and a counterbraced truss. An upper limit of problem size for MDM analysis is suggested. The second approach to periodic structure analysis is a numerical one in which the structure is modelled using finite elements. A new general method for the analysis of infinitely long periodic structures, the Deflection Transfer Matrix Method (DTMM), is developed. The method involves an iterative procedure, similar to Gaussian elimination, to find the reduced structure stiffness matrix. DTMM is programmed in FORTRAN and is interfaced with a comprehensive finite element program (LUSAS). Results for the analysis of a closed cylindrical shell and an infinite plate on elastic subgrade are compared with F.E. and published solutions. The method is appraised and suggestions for further developments are given in detail. Extending the numerical approach to finite length periodic structures, an eigenvalue method, used previously for the static analysis of deep beams, is developed. The theory is extended to make it general and the method is programmed in FORTRAN and interfaced with LUSAS. Results are obtained for deep beams and a large coupled shear wall, comparison being made with a normal F.E. solution and existing experimental and TMM solutions. The method (EST) is found to be versatile, stable and efficient. Suggestions for further development are made.