Fourier Analysis of the Theory of Structures of Certain Types

The primary objective of this thesis is to apply the use of Fourier series expansion to the solution of various problems arising in the study of Beam Structures. Following a statement of the theorem itself, a few of the special cases of the series expansion are demonstrated. Included in these sections are the expansions of the Sin-Harmonic, Cos-Harmonic, Odd-Harmonic and the Bi-Symmetric Functions. Sections seven through ten give examples of the Fourier series expansion of an even function, odd function and the expansion of a function, having finite discontinuities, as a special case in section ten. Sections twelve and thirteen give a discussion of the differentiation and integration of Fourier series term by term. The next five sections deal with the methods of determining the Fourier coefficients. In addition to the method of integration, are included the methods of the Numerical solution, the Graphical solution, the combination Graphical and Mechanical method of solution, and the energy method of solution. Section nineteen, which deals with the general theory of beams, was included to introduce the reader to some of the engineering terms and formulas that the author felt were necessary for an understanding of the sections to follow. Section twenty deals wit the solution of an elastically supported beam. Included is the general solution of a beam of this type, the symbols used for the various terms such as deflection, restoring force and loading conditions. The problem is carried on, in section twenty one, to the case of a beam without elastic support. The problem of a beam with a concentrated load acting at its center is set up and expanded in Fourier series in section twenty two. In section twenty three is a discussion of a simple beam with a concentrated load acting at the center. The solution is an application of the Energy method given in section eighteen. Section twenty four is introduced next to extend the previous Fourier series expansion to the case where the limits of integration extend to infinity. The relationship that is found is known as the Fourier Integral Theorem. This theorem is then applied to the solution of an elastically supported beam of infinite span as shown in section twenty five and a uniformly loaded beam of infinite span as shown in section twenty six