The Rayleigh-Ritz Method for Structural Analysis (Mechanical Engineering and Solid Mechanics Series)

A presentation of the theory behind the Rayleigh-Ritz (R-R) method, as well as a discussion of the choice of admissible functions and the use of penalty methods, including recent developments such as using negative inertia and bi-penalty terms. While presenting the mathematical basis of the R-R method, the authors also give simple explanations and analogies to make it easier to understand. Examples include calculation of natural frequencies and critical loads of structures and structural components, such as beams, plates, shells and solids. MATLAB codes for some common problems are also supplied. PREFACE xi INTRODUCTION AND HISTORICAL NOTES xiii CHAPTER 1. PRINCIPLE OF CONSERVATION OF ENERGY AND RAYLEIGH S PRINCIPLE 1 CHAPTER 2. RAYLEIGH S PRINCIPLE AND ITS IMPLICATIONS 11 CHAPTER 3. THE RAYLEIGH RITZ METHOD AND SIMPLE APPLICATIONS 21 CHAPTER 4. LAGRANGIAN MULTIPLIER METHOD 33 CHAPTER 5. COURANT S PENALTY METHOD INCLUDING NEGATIVE STIFFNESS AND MASS TERMS 39 CHAPTER 6. SOME USEFUL MATHEMATICAL DERIVATIONS AND APPLICATIONS 55 CHAPTER 7. THE THEOREM OF SEPARATION AND ASYMPTOTIC MODELING THEOREMS 67 CHAPTER 8. ADMISSIBLE FUNCTIONS 81 CHAPTER 9. NATURAL FREQUENCIES AND MODES OF BEAMS 89 CHAPTER 10. NATURAL FREQUENCIES AND MODES OF PLATES OF RECTANGULAR PLANFORM 113 CHAPTER 11. NATURAL FREQUENCIES AND MODES OF SHALLOW SHELLS OF RECTANGULAR PLANFORM 133 CHAPTER 12. NATURAL FREQUENCIES AND MODES OF THREE-DIMENSIONAL BODIES 149 CHAPTER 13. VIBRATION OF AXIALLY LOADED BEAMS AND GEOMETRIC STIFFNESS 161 CHAPTER 14. THE RRM IN FINITE ELEMENTS METHOD 181 BIBLIOGRAPHY 197 APPENDIX 203 INDEX 229