The spectral and dynamic stiffness methods in solid and structural mechanics

Abstract

There are two different concepts of spectral methods. One replaces time t by frequency ω so that the parameter ω is built-in as a parameter in the formulation. This is called the dynamic stiffness (Leung 1993) or the spectral element method (Lee 2009). The finite strip method (Cheung and Tham 1996) is one of the variants. The other refers to methods that achieve spectral accuracy, i.e., for sufficient number N of terms or grid points, the error approaches N-∞, while the error of the second order methods approaches N-2 and the fourth order approached N-4 etc. Therefore, the thesis is separated mainly in these two parts. In order not to mix up these two parts, we shall use the dynamic stiffness method for the first one and the spectral method for the second category. The main objectives of present study are to give a wide range of application of the Dynamic Stiffness Method and to develop an analytical technique for the solutions of eigenfrequency and buckling problems for various kinds of structures with different boundary conditions, including uniform and non-uniform members, one- and two- dimensional problems. In this thesis, the analytical formulation of the dynamic stiffness method is introduced. Some useful computational techniques are discussed. These include the Hamilton principle, Leung's theorem, Sturm's theorem, and the Wittrick-Williams algorithm. Solid examples using Timoshenko columns are also discussed. A more complicated application of active multi-layer beam is also investigated. The thesis devoted to the study of non-uniform member, tapered Timoshenko column and spinning beam. Governing equations are derived and solution by power series is given. Finally, several beams are joined together as step beams. Two-dimensional problems represented by laminated composite plates are solved and the numerical results are compared with those of existing literature. Some considerations concerning the use of dynamic stiffness method will be discussed. Also, certain techniques that are incorporated with the dynamic stiffness method will be presented in detail. The spectral method is formed and one-dimensional problems using axial deformation of a beam is introduced as an illustrative example. Very general straight beam problems with initial secondary forces and moments are covered and two-dimensional elasticity is discussed in details. The elasticity equations are reformulated in a convenient form for easy programming. Three dimensional problems and Mindlin plate problems are investigated. Numerical examples for static and natural vibration analyses for simple elements as well as composite structures are given. The main contributions of the thesis include the general formulation of the dynamic stiffness matrix of any one-and two-dimensional members and the general formulation of the spectral methods for one-, two- and three dimensional problems.

Date of Award	2 Oct 2015
Original language	English
Awarding Institution	City University of Hong Kong
Supervisor	Yee Tak Andrew LEUNG (Supervisor)