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This book presents simplified analytical methodologies for static and dynamic problems concerning various elastic thin plates in the bending state and the potential effects of dead loads on static and dynamic behaviors. The plates considered vary in terms of the plane (e.g. rectangular or circular plane), stiffness of bending, transverse shear and mass. The representative examples include void slabs, plates stiffened with beams, stepped thickness plates, cellular plates and floating plates, in addition to normal plates. The closed-form approximate solutions are presented in connection with a groundbreaking methodology that can easily accommodate discontinuous variations in stiffness and mass with continuous function as for a distribution. The closed-form solutions can be used to determine the size of structural members in the preliminary design stages, and to predict potential problems with building slabs intended for human beings' practical use.

Simplified Analytical Methods of Elastic Plates

Part  I  Static and Dynamic Analyses of Normal Plates

1   Static and Dynamic Analyses of Rectangular Normal Plates

1.1   Introduction 

1.2   Equilibrium Equations of the Plate Element

1.3   Relationships Among Stress, Strain, and Displacements

1.4   Stress Resultants and Stress Couples Expressed in Term of w

1.5   Boundary Conditions of the Bending Theory

1.6   Analytical Method of Static Rectangular Plates Used the Galerkin Method

1.7   Selection of Shape Functions for Static Problems

1.8   Free Transverse Vibrations of Plates without Damping

1.9   Forced Vibrations of Rectangular Plates

1.10 Dynamic Response of Sinusoidal Dynamic Loads

1.11 Conclusions

References

2   Static and Dynamic Analyses of Circular Normal Plates

2.1   Introduction

2.2   Governing Equations of Uniform Circular Plates

2.3   Governing Equations of Circular Plates Subjected to Rotationally Symmetric Loading

2.4   Conclusions

References

3   Static and Dynamic Analyses of Rectangular Normal Plates with Edge Beams

3.1   Introduction

3.2   Governing Equations of a Normal Plate with Edge Beams

3.3   Static Analysis Used the Galerkin Method

3.4   Numerical Results for Static Solution

3.5   Free Transverse Vibrations of a Plate with Edge Beams

3.6   Numerical Results for Natural Frequencies

3.7   Forced Vibrations of a Plate with Edge Beams

3.8   Approximate Solutions for Forced Vibrations

3.9   Numerical Results for Dynamic Responses

3.10 Conclusions

Appendix A3.1

Appendix A3.2

References

Part II    Static and Dynamic Analyses of Various Plates

4   Static and Dynamic Analyses of Rectangular Plates with Voids

4.1   Introduction

4.2   Governing Equations of Plates with Voids

4.3   Static Analyses to Rectangular Plates with Voids

4.4   Numerical Results

4.5   Relationships between Theoretical and Experimental Results

4.6   Conclusions for the Static Problems

4.7   Free Transverse Vibrations of a Plate with Voids

4.8   Numerical Results for Natural Frequencies

4.9   Relationships between Theoretical Results and Experimental Results for Natural Frequencies

4.10 Forced Vibrations of Plates with Voids

4.11 Dynamic Analyses Based on the Linear Acceleration Method 

4.12 Closed-form Approximate Solutions for Forced Vibrations

4.13 Numerical Results for Dynamical Responses; Discussions 

4.14 Conclusions for Free and Forced Vibrations

References

5   Static and Dynamic Analyses of Circular Plates with Voids

5.1   Introduction

5.2   Governing Equations of a Circular Plate with Voids

5.3   Static Analysis

5.4   Numerical Results for Static Problems

5.5   Free Transverse Vibrations of Plate with Voids

5.6   Numerical Results for Natural Frequencies

5.7   Forced Vibrations of Plates with Voids

5.8   Closed-form Approximate Solutions for Forced Vibrations 

5.9   Numerical Results for Dynamic Responses: Discussions 

5.10 Conclusions

References

6   Static and Dynamic Analyses of Rectangular Cellular Plates

6.1   Introduction

6.2   Governing Equations of a Cellular Plate with Transverse Shear Deformations along with Frame Deformation

6.3   Transverse Shear Stiffness of Cellular Plates

6.4   Stress Resultants and Stress Couples of Platelets and Partition

6.5   Static Analysis

6.6   Numerical Results for Static Calculation

6.7   Free Transverse Vibrations of Cellular Plates

6.8   Numerical Results for Natural Frequencies 

6.9   Forced Vibration of Cellular Plates

6.10 Approximate Solutions for Forced Vibrations

6.11 Numerical Results for Dynamic Responses 

6.12 Conclusions 

Appendix A6.1

Appendix A6.2

Appendix A6.3

References

7   Static and Dynamic Analyses of Circular Cellular Plates

7.1   Introduction

7.2   Governing Equations of a Circular Cellular Plate with Transverse Shear Deformations along with Frame Deformation

7.3   Transverse Shear Stiffness of Cellular Plates

7.4   Stress Resultants and Stress Couples of Platelets and Partition

7.5   Static Analysis

7.6   Numerical Results for Static Problem

7.7   Free Transverse Vibrations of Cellular Plates

7.8   Numerical Results for Natural Frequencies

7.9   Forced Vibration of Cellular Plates

7.10 Numerical Results for Dynamic Responses

7.11 Conclusions

Appendix A7.1

Appendix A7.2

Appendix A7.3

Appendix A7.4

References

8   Static and Dynamic Analyses of Rectangular Plates with Stepped Thickness

8.1   Introduction

8.2   Governing Equations of Rectangular Plates with Stepped Thickness

8.3   Static Analysis

8.4   Numerical Results for Static Solution

8.5   Free Transverse Vibrations of Plate with Stepped Thickness

8.6   Numerical Results for Natural Frequencies

8.7   Forced Vibrations of Plate with Stepped Thickness

8.8   Approximate Solutions for Forced Vibrations

8.9   Numerical Results for Dynamic Responses

8.10 Conclusions

Appendix A8.1

References

Part  III   Static and Dynamic Analysis of Special Plates

9   Static and Dynamic Analyses of Rectangular Plates with Stepped Thickness Subjected to Moving Loads

9.1   Introduction

9.2   Governing Equations of Plate with Stepped Thickness Including the Effect of Moving Additional Mass

9.3   Forced Vibration of a Plate with Stepped Thickness

9.4   Approximate Solution Excluding the Effect of Additional Mass due to Moving Loads

9.5   Numerical Results

9.6   Conclusions 

References

10 Static and Dynamic Analyses of Rectangular Floating Plates Subjected to Moving Loads

10.1 Introduction

10.2 Governing Equations of a Rectangular Plate on an Elastic Foundation

10.3 Free Transverse Vibrations

10.4 Forced Transverse Vibrations

10.5 Approximate Solutions for Forced Transverse Vibration

10.6 Numerical Results

10.7 Conclusions

Appendix A10.1

References

Part IV   Effects of Dead Loads on Elastic Plates

11 Effects of Dead Loads on Static and Dynamic Analyses of Rectangular Plates

11.1 Introduction

11.2 Governing Equations Including the Effect of Dead Loads for Plates

11.3 Formulation of Static Problem Including the Effect of Dead Loads

11.4 Numerical Results

11.5 Approximate Solution

11.6 Example

11.7 Transverse Free Vibration Based on the Galerkin Method

11.8 Closed-form Solution for Transverse Free Vibrations

11.9 Dynamic Analyses Based on the Galerkin Method

11.10 Dynamic Analyses Based on the Approximate Closed-form Solution

11.11 Numerical Results to Dynamic Live Loads

11.12 Method Reflected the Effect of Dead Loads in Dynamic Problems

11.13 Conclusions

Appendix A11.1

References

Part V   Effects of Dead Loads on Elastic Beams

12 Effects of Dead Loads on Static and Free Vibration Problems of Beams

12.1 Introduction 

12.2 Advanced Governing Equations of Beams Including Effect of Dead Loads

12.3 Numerical Results Using Galerkin Method for Static Problems

12.4 Closed-form Solutions Including Effect of Dead Loads in Static Problems

12.5 Proposal How to Reflect the Effect of Dead Load on Static Beams

12.6 Free Transverse Vibrations of Uniform Beams

12.7 Numerical Results for Free Transverse Vibrations of Beams Using Galerkin Method

12.8 Closed-form Approximate Solutions for Natural Frequencies

12.9 Conclusions

Appendix A12.1

References

13 Effects of Dead Loads on Dynamic Problems of Beams

13.1 Introduction

13.2 Dynamic Analyses of Beams Subject to Unmoving Dynamic Live Loads

13.3 Numerical Results for Beams Subject to Unmoving Dynamic Live Loads

13.4 Approximate Solutions for Simply Supported Beams Subject to Unmoving Dynamic Live Loads

13.5 How to Import the Effect of Dead Loads for Dynamic Beams Subject to Unmoving Dynamic Live Loads

13.6 Dynamic Analyses Using the Galerkin Method on Dynamic Beams Subject to Moving Live Loads

13.7 Various Moving Loads

13.8 Additional Mass due to Moving Loads

13.9 Approximate Solutions of Beams Subject to Moving Live Loads

13.10 Numerical Results for Beams Subject to Moving Live Loads

13.11 Conclusions

References

Part VI   Recent Topics of Plate Analysis

14 Refined Plate Theory in Bending Problem of Uniform Rectangular Plates

14.1 Introduction

14.2 Various Plate Theories

14.3 Analysis of Isotropic Plates Using Refined Plate Theory

14.4 The Governing Equation in RPT

14.5 Simplified RPT

14.6 Static Analysis Used Simplified RPT

14.7 Selection of Shape Functions for Static Problems

14.8 Free Transverse Vibrations of Plates without Damping

14.9 Forced Vibration of Plates in Simplified RPT

14.10 Advanced Transformation of Uncoupled Form in Simplified RPT

14.11 Advanced RPT

14.12 Conclusions

References