Vibration of Mindlin Plates: Programming the p-Version Ritz Method

Over the last several years, the four authors have jointly conducted research into the analysis of vibrating Mindlin plates as a collaborative project between Nanyang Technological University, The National University of Singapore, and The University of Queensland. The research was prompted by the fact that there is a dearth of vibration results for Mindlin plates when compared to classical thin plate solutions. To generate the vibration results, the authors have successfully employed the Ritz method for general plate shapes and boundary conditions. The Ritz method, once thought to be awkward for general plate analysis, can be automated through suitable trial functions (for displacements) that satisfy the geometric plate boundary conditions a priori. This work has been well-received by academics and researchers, as indicated by the continual requests for the authors' papers and the Ritz software codes. This monograph is written with the view to share this so-called p-Ritz method for the vibration analysis of Mindlin plates and its software codes with the research community. To the authors' knowledge, the monograph contains the first published Ritz plate software codes of its kind.

Preface. Introduction. Background of vibration. Plate vibration. About this monograph. Mindlin Plate Theory and Ritz Method. Mindlin plate theory. Energy functionals. Governing differential equations. Boundary conditions. Relations between Kirchhoff and Mindlin plates. Reduction of Mindlin theory to Kirchhoff. Frequency relationship for a class of plates. Shear correction factor. Ritz method. Preliminary remarks. Application of Ritz method to Mindlin plates. Formulation in Polar Coordinates. Introduction. Energy functionals. Eigenvalue equation. Circular and annular plates. Sectorial and annular sectorial plates. Computer program. Software code: VPRITZP1. Sample files for VPRITZP1. Software code: VPRITZP2. Sample files for VPRITZP2. Benchmark checks. Annular plates. Sectorial plates. Annular sectorial plates. Formulation in Rectangular Coordinates. Introduction. Energy functionals. Eigenvalue equation. Computer program. Software code: VPRITZRE. Input file. Output file. Benchmark checks. Isosceles triangular plates. Trapezoidal plates. Elliptical plates. Formulation in Skew Coordinates. Introduction. Skew coordinates transformation. Energy functional in skew coordinates. Eigenvalue equation. Computer program. Software code: VPRITZSK. Sample files. Benchmark checks. Plates with Complicating Effects. Introduction. Initial inplane stresses. Elastic foundations. Stiffeners. Nonuniform thickness. Line/curved/loop internal supports. Point supports. Mixed boundary conditions. Reentrant corners. Perforated plates. Sandwich construction. References. Appendix I - Gaussian quadrature subroutines. Appendix II - Subroutines for mathematical operations on polynomials. Subject index.