p-version finite elements in structural dynamics and stability

Abstract

The performance of the finite element method (FEM) may be improved in three ways. The first is by the h-version to refine the finite element mesh and the second is by the p-version to increase the order of polynomial shape functions for a fixed mesh. The third is simultaneously to refine the mesh and increase the degrees of elements uniformly or selectively, called the h-p version, which is a combination of the first two methods. The advantages of the p-version elements over the h-version are: (i) they have better conditioned matrices; (ii) they do not require a change in the mesh and can be easily used in the adaptive analysis; (iii) just one element can predict accurate solutions for a simple structure; (iv) they tend to give the same accurate results with far fewer degrees of freedom (DOF); and (v) they can overcome some locking problems. The main objectives of present study are to give a wide range of application of the Fourier p-elements and polynomial p-elements to investigate the vibration problems, buckling problems and dynamic stability problems of various conservative linear structures including beam-columns, straight Timoshenko beams, pre-twisted straight beams, Mindlin plates and open thin shell panels. The natural frequencies, buckling loads and the relation of frequencies and various buckling loads are considered. Good agreement is achieved with the available results. The main work can be divided into five parts. Firstly, the numerical examples of uniform or tapered beam-columns with or without end mass are considered and compared with results of dynamic stiffness method. New results of straight beamcolumns subjected to uniformly distributed follower tension are originally reported. Secondly, the axial-torsional buckling of space straight beams based on unequally shared end torque theory is studied. Then flexural-torsional buckling problems of space straight beams subject to end moments, end shear loads and distributed shear loads are investigated. Thirdly, the effects of pre-twist rate and rigidity ratio on dynamic stability of pre-twisted straight beams are given. The natural frequencies and buckling loads of pre-twisted beams subject to axial loads, torque, moments and shears are discussed in detail. Fourthly, dynamic stability problems of Mindlin plates with rectangular, skew, trapezoidal, triangular, polygonal shapes are analyzed. Problems of plate systems composed of rectangular and/or trapezoidal elements with different thicknesses are discussed. Finally, the influence of aspect ratio, circumferential angles on the natural frequencies and vibration mode shapes of open thin cylindrical, conical and spherical shell panels are studied. The buckling problems of cylindrical shell panels under axially compressed loads are finally investigated.