Preface

Chapter 1 Introduction

Chapter 2 Vibration Analysis of Tapered Euler-Bernoulli Beams

2.1 Introduction

2.2 The Rayleigh-Ritz Method for the Tapered Beams

2.3 A New Set of Admissible Functions

2.3.1 The coefficients for a truncated beam

2.3.2 The coefficients for a sharply ended beam

2.3.3 The tapered beam with rigid body motion

2.4 Convergency and Comparison Studies

2.4.1 Convergency study

2.4.2 Optimum expanding point of Taylor series

2.5 Numerical Results

2.6 Concluding Remarks

Chapter 3 Vibration Analysis of Tapered Euler-Bernoulli Beams with Intermediate Supports

3.1 Introduction

3.2 The Rayleigh-Ritz Method for Tapered Beams with Intermediate Supports

3.3 A Set of Static Tapered Beam Functions

3.3.1 The truncated beam

3.3.2 The sharply ended beam

3.3.3 The tapered beam with motions of rigid body

3.4 Numerical Examples

3.5 Concluding Remarks

Chapter 4 Vibration Analysis of Multi-span Timoshenke Beams

4.1 Introduction

4.2 Eigenfrequency Equation

4.3 Static Timoshenko Beam Functions

4.4 Convergence and Comparison Studies

4.5 Numerical Examples

4.6 Concluding Remarks

Chapter 5 Vibration Analysis of Tapered Timoshenke Beams

5.1 Introduction

5.2 Eigenfrequency Equation of Tapered Beam

5.3 The Static Timoshenko Beam Functions (STBF)

5.3.1 Truncated beam

5.3.2 Sharply ended beam

5.4 Convergence and Comparison Study

5.5 Numerical Results

5.6 Conclusions

Chapter 6 Estimation of Dynamic Characteristics of a Spring-Mass-Beam System

6.1 Introduction

6.2 Governing Differential Equations

6.3 Galerkin Solutions

6.4 Basic Characteristics of Solutions

6.5 Static Beam Functions

6.6 Determination of Factors

6.7 An Example

6.8 Characteristics of Solutions

6.9 Conclusions

Chapter 7 Vibration Analysis of Kirchhoff Rectangular Plates

Part Ⅰ Using Static Beam Functions under Point Loads

7.1 Introduction

7.2 Sets of Static Beam Functions under Point Loads

7.3 Rayleigh-Ritz Solution for Rectangular Plates

7.4 Numerical Results

7.5 Concluding Remarks

Part Ⅱ Using Static Beam Functions under Sinusoidal Loads

7.1 Introduction

7.2 The Set of Static Beam Functions

7.3 The RayleighoRitz Approach

7.4 Numerical Results

7.5 Concluding Remarks

Chapter 8 Vibration Analysis of Kirchhoff Rectangular Plates with Elastic Edge Constraints

8.1 Introduction

8.2 The Set of Static Beam Functions

8.3 The Rayleigh-Ritz Solution

8.4 Numerical Examples

8.5 Discussion and Conclusions

Chapter 9 Vibration Analysis of Kirchhoff Rectangular Plates with Intermediate Line-supports

Part Ⅰ Using a Combination of Vibrating Beam Functions and Polynomials

9.1 Introduction

9.2 Mathematical Model

9.3 Numerical Examples

9.4 Concluding Remarks

Part Ⅱ Using the Static Beam Functions for Beam with Point-supports

9.1 Introduction

9.2 A New Set of Admissible Functions

9.3 Eigenfrequency Equation

9.4 Some Numerical Results

9.5 Conclusions

Chapter 10 Vibration Analysis of Kirchhon Rectangular Plates with Elastic Intermediate Line-supports and Edge Constraints

10.1 Introduction

10.2 A Set of Static Beam Functions

10.3 Formulation of Eigenvalue Equation

10.4 Numerical Examples

10.5 Conclusions

Chapter 11 Vibration Analysis of Kirchhoff Rectangular Plates with Elastic Point-supports

11.1 Introduction

11.2 Sets of Static Beam Functions under Point Loads

11.3 Eigenvalue Problem with Rayleigh-Ritz Method

11.4 Numerical Results

11.5 Conclusion

Chapter 12 Vibration Analysis of Symmetrically Laminated Rectangular Plates with Intermediate Line-supports

12.1 Introduction

12.2 A Set of Static Beam Functions

12.3 Eigenfrequency Equation

12.4 Numerical Results

12.4.1 Accuracy and convergency study

12.4.2 Numerical examples

12.5 Concluding remarks

Chapter 13 Vibration Analysis of Asymmetrically Laminated Rectangular Plates with Internal Line-supports

13.1 Introduction

13.2 Energy Functional

13.3 Rayleigh-Ritz Solution

13.4 Trial Functions

13.5 Convergence and Comparison Study

13.6 Numerical Results

13.7 Conclusion

Chapter 14 Vibration Analysis of Composite Rectangular Plates with Point-supports

14.1 Introduction

14.2 Static Beam Functions

14.2.1 The static beam functions under sine series loads

14.2.2 The static beam functions under a point-load

14.3 Eigenfrequency Equation

14.4 Admissible Functions

14.5 Comparison and Convergence

14.5.1 Isotropic square plates with point-supports

14.5.2 Laminated square composite plates

14.6 Numerical Results

14.7 Conclusions

Chapter 15 Vibration Analysis of Tapered Kirchhoff Rectangular Plates

15.1 Introduction

15.2 The development of a set of tapered beam functions

15.3 The Rayleigh-Ritz method

15.4 Numerical examples

15.5 Concluding remarks

Appendix

Chapter 16 Vibration Analysis of Tapered Kirchhoff Rectangular Plates with Intermediate Line-supports

16.1 Introduction

16.2 The Rayleigh-Ritz Method for Tapered Rectangular Plates

16.3 A Set of Static Beam Functions

16.3.1 The truncated beam

16.3.2 The sharp ended beam

16.3.3 The tapered beam with rigid body motions

16.4 Numerical Examples

16.5 Conclusions

Chapter 17 Vibration Analysis of Mindlin Rectangular Plates

17.1 Introduction

17.2 A Set of Static Timoshenko Beam Functions

17.3 Eigenfrequency Equation of Mindlin Plate

17.4 Comparison and Convergency Studies

17.5 The Parametric Study

17.6 Conclusions

Chapter 18 Vibration Analysis of Mindlin Rectangular Plates with Elastically Restrained Edges

18.1 Introduction

18.2 Rayleigh-Ritz Formulae for Mindlin Rectangular Plates

18.3 A Set of Static Timoshenko Beam Functions

18.4 Comparison and Convergency Studies

18.5 Numerical Results

18.6 ConclusionsChapter 19 Vibration Analysis of Mindlin Rectangular Plates

with Intermediate Line-supports

19.1 Introduction

19.2 Rayleigh-Ritz Solution of Mindlin Plate

19.3 Static Timoshenko Beam Functions

19.4 Convergence and Comparison Study

19.5 Numerical Results

19.6 Conclusions

Chapter 20 Vibrations Analysis of Tapered Mindlin Plates

20.1 Introduction

20.2 The Eigenfrequency Equation of Tapered Plates

20.3 Two Sets of Static Timoshenko Beam Functions (STBF)

20.3.1 Truncated beam

20.3.2 Sharp-ended beam

20.4 Convergence and Comparison Studies

20.5 Numerical Results

20.6 Concluding Remarks

Chapter 21 Vibration Analysis of Thick Rectangular Plates with Internal Line-supports

21.1 Introduction

21.2 Trial Functions

21.3 Numerical Examples

21.3.1 Preliminary assessment: simply supported laminated plates

21.3.2 Continuous rectangular plates

21.4 Conclusions

Chapter 22 Vibration Analysis of Layered Thick Rectangular Plates with Internal Point-supports

22.1 Introduction

22.2 Two Sets of Static Beam Functions

22.2.1 Static beam functions under a series of sinusoidal loads

22.2.2 Static beam functions under a series of point-loads

22.3 Finite Layer Formulation

22.4 Basic Functions

22.5 Numerical Studies

22.5.1 Convergence and comparison

22.5.2 Numerical examples

22.6 Concluding Remarks

Appendix A

Appendix B

Chapter 23 Vibration Analysis of Rectangular Tanks Partially Filled with Liquid

23.1 Introduction

23.2 Basic Equations

23.3 Solution of Velocity Potential

23.4 Rayleigh-Ritz-Galerkin Method

23.4.1 Rayleigh quotient

23.4.2 Eigenfrequency equation

23.5 Admissible Functions

23.6 Numerical Results

23.6.1 Convergence and comparison study

23.6.2 Parametric effect study

23.7 Conclusions

References