By using the U-transformation method, it is possible to uncouple linear simultaneous equations, either algebraic or differential, with cyclic periodicity. This book presents a procedure for applying the U-transformation technique twice to uncouple the two sets of unknown variables in a doubly periodic structure to achieve an analytical exact solution.Explicit exact solutions for the static and dynamic analyses for certain engineering structures with doubly periodic properties — such as a continuous truss with any number of spans, cable network and grillwork on supports with periodicity, and grillwork with periodic stiffening members or equidistant line supports — can be found in the book. The availability of these exact solutions not only helps with the checking of the convergence and accuracy of numerical solutions, but also provides a basis for optimization design for these types of structures.The study of the force vibration and mode shape of periodic systems with nonlinear disorder is yet another research area which has attained considerable success by the U-transformation method. This book illustrates the analytical approach and procedure for the problems of localization of the mode shape of nearly periodic systems together with the results.